Quantum Simulations of Abelian Lattice Gauge Theories with Ultracold Atoms

Transcription

1 Ludwig-Maximilians-Universität München Technische Universität München Max-Planck-Institut für Quantenoptik Quantum Simulations of Abelian Lattice Gauge Theories with Ultracold Atoms September 2016 Daniel González Cuadra

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3 Ludwig-Maximilians-Universität München Technische Universität München Max-Planck-Institut für Quantenoptik Quantum Simulations of Abelian Lattice Gauge Theories with Ultracold Atoms First reviewer: Prof. Dr. J. Ignacio Cirac Second reviewer: Dr. Erez Zohar Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science within the programme Theoretical and Mathematical Physics Munich, September 2016 Daniel González Cuadra

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5 Abstract In the present work, a quantum simulation scheme for the lattice version of the Abelian-Higgs theory is proposed, using a system of ultracold bosonic atoms trapped in optical lattices. Furthermore, a new way to simulate the pure-gauge Kogut- Susskind Hamiltonian is described. After reviewing the most important properties of both the simulating and the simulated systems, the lattice Abelian-Higgs Hamiltonian is obtained using the transfer matrix method, starting from the action of the theory. The experimental requirements for simulating the Hamiltonian s gauge-invariant interactions using ultracold bosons are described, and the mapping between the simulated and the simulating degrees of freedom is introduced. Both the complete lattice Abelian-Higgs and pure-gauge Kogut-Susskind Hamiltonians emerge effectively from the atomic Hamiltonian provided that the energy scales of the system fulfill the required conditions including some non-desired but small corrections. These correction terms do not violate the symmetry, since the atomic Hamiltonian is already built such that only gauge-invariant terms appear, and, hence, the effective Hamiltonian is gauge invariant as well. The effect of these corrections, and the overall accuracy of the quantum simulation, is studied numerically for a small version of the system. The implementation of a quantum simulation for these abelian lattice gauge theories opens the door for studying interesting high energy phenomena such as the confinement of dynamical charges, or the Brout-Englert-Higgs mechanism beyond the scope of current theoretical and numerical techniques, using a highly controllable low-energy quantum system. v

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7 Contents 1. Introduction 1 2. Ultracold Atoms in Optical Lattices Optical Potentials Periodic Potentials One-Particle Picture: Band Structure Two-Body Interactions Scattering Theory Pseudopotential Ultracold Limit Second Quantization The Bose-Hubbard Model Ultracold Spinor Atomic Gases Fermions Lattice Gauge Theories Gauge Theories in the Continuum Quantum Electrodynamics Wilson Loop Lattice Formulation of Gauge Theories Motivation: The Quark Confinement Problem Space-Time Discretization Wilson s Action Hamiltonian Formalism Transfer Matrix Method for Gauge Theories Kogut-Susskind Hamiltonian Abelian-Higgs Theory The Brout-Englert-Higgs Mechanism: A Toy Model Spontaneous Breaking of Global Symmetries SSB in the Presence of Local Symmetries Lattice Abelian-Higgs Action Phase Diagram Lattice Abelian-Higgs Hamiltonian Confinement in the Presence of Dynamical Matter Quantum Simulations of Abelian Lattice Gauge Theories Quantum Simulation Techniques Gauge Symmetry Plaquette Interactions vii

8 Contents Static Charges: Initializing the System Simulating the Lattice Abelian-Higgs Theory Experimental Requirements Primitive Hamiltonian Effective Hamiltonian Pure-Gauge Theory Full Abelian-Higgs Theory XY Model Initialization and Measurements Numerical Study Summary and Conclusion 81 A. Theoretical Tools 83 A.1. Effective Hamiltonian A.2. Non-Commuting Operators A.3. Transfer Matrix Method B. Feshbach Resonances 89 C. Calculations 93 Bibliography 99 Acknowledgements 107 Declaration of Authorship 109 viii

9 1. Introduction The study of natural systems governed by the laws of quantum mechanics is a complicated task. Most of the usual theoretical and numerical techniques employed to tackle the behaviour of quantum systems become rapidly overwhelmed once their size starts increasing. In particular, the computational resources required to simulate a quantum mechanical system using a classical computer scale exponentially with the number of its constituents. This problem made Richard Feynman think of the concept of a quantum simulator, back in 1982 [1]. According to his idea, such a device would be governed itself by the laws of quantum mechanics and, exploiting this fact, could be used to study other quantum mechanical systems more efficiently. Feynman s idea was formalized, some years later, as an algorithm that could be processed by an universal quantum computer [2]. This is referred to as a digital quantum simulation, since it is based on approximating the simulated system s dynamics by applying a discrete set of operations (quantum gates) on a highly-controllable quantum device. Quantum computers are very interesting from a theoretical point of view, since, in principle, they could simulate any other physical system using only polynomial resources, provided that certain locality conditions are fulfilled [2]. From the experimental side, however and in spite of the great advances over the last decades regarding the manipulation of microscopic quantum systems a practical implementation of a fully operational quantum computer is still a long-term goal. Notwithstanding, having these controllable devices so well studied and reachable makes quantum simulation a relality, by means of simpler quantum devices that can perform calculations beyond the scope of classical machines. In particular, the socalled analog quantum simulators, although much more limited than an universal simulator, can be used to study a broad range of quantum systems within the reach of today s technological progress [3, 4]. Analog simulators act by mapping the degrees of freedom of the simulated system to those of the simulating one. The latter can be controlled in the laboratory and its dynamics can be tailored (in particular, the corresponding Hamiltonian) to be equivalent to those of the system we are trying to study. This allows us to obtain information about systems that can not be accessed experimentally, by investigating others where state preparation and measurements are much easier tasks. Non-universal simulators, although more limited in scope than a full quantum computer, allow us to investigate a broad range of phenomena appearing in quantum many-body systems, in particular which are very hard to deal with using conventional theoretical or numerical tools. Among the relevant platforms that can serve as quantum simulators, both analog and digital, many atomic and optical systems stand out due to their remarkable experimental controllability. Some examples include ultracold atoms [5 9], trapped ions [10 14], photonic systems [15] and Rydberg atoms [16]. Ultracold atoms in optical lattices, in particular, present the possibility of recreating many different interactions such as nearest-neighbour, long-range forces, on-site interactions, hopping 1

10 1. Introduction processes, etc allowing for the simulation of both condensed matter and high energy models, as we will see. Solid-state systems, such as quantum dots [17 19] or superconducting circuits [20, 21], also show prominent results that make them interesting candidates to perform quantum simulations. Using these ideas, many condensed matter Hamiltonians have been considered for quantum simulations, some of them even realized experimentally. Some examples include spin systems [14, 22 24], such as the Ising or Heisenberg models; the Bose- Hubbard model [5, 25]; the Tonks-Girardeau gas [26], or copper-oxide superconductors [18]. External gauge potentials can be simulated as well, allowing for the study of the fractional quantum Hall effect [27] and other topological phenomena [28 31]. Quantum simulations of high energy theories, although more demanding than their condensed matter counterparts, are also possible. Some examples include the simulation of the Dirac [32 34] and Majorana equations [35], the Casimir force [36], the Schwinger mechanism [37] or the oscillations of neutrinos [38]. Simulations of quantum field theories [39, 40], gravitational theories [41] and black holess [42, 43] have been proposed in the last years, some of them realized experimentally as well [44]. Within high energy physics, gauge theories are particularly relevant, since they lie in the core of the Standard Model of particle physics [45 48]. Dynamical gauge degrees of freedom are introduced in quantum field theories to explain the interaction between the basic constituents of matter, such as quarks or electrons. Many techniques have been developed to study such theories. The most satisfactory ones rely on perturbative expansions around small coupling constants. However, these methods lose their validity if one tries to apply them for the study of the so-called non-perturbative phenomena, where the relevant coupling constant present large values. This is the case for the interaction strength between separated quarks, which grows with the distance between them [45, 46] (running coupling). At short distances, the interaction coupling constant present small values (asymptotic freedom), and the perturbative methods can be applied. For long distances, however, the growth in the interaction strength gives rise to the confinement mechanism, implying that no free quarks can be found in nature, a claim that is supported by the experiments [46]. To deal with this and other non-perturbative phenomena, the lattice formulation of gauge theories was developed [49, 50], obtaining a proper framework to perform numerical simulations on these theories using Monte Carlo methods, in particular [51]. These techniques provided a great advance in the understanding of particle physics during the last decades. However, they present some limitations when they are applied to certain cases. An example is the so-called sign problem, which appears in regimes with a finite chemical potential for fermionic particles [52], and becomes problematic for the studying of different phases of QCD, such as the quark-gluon plasma or the color-superconducting phase [53]. The analysis of real-time dynamics is also lacking, since Monte Carlo simulations only allow to calculate euclidean correlation functions (imaginary time). In this context, quantum simulations of gauge theories, in particular their lattice version, provide the opportunity to study regimes of these theories that can not be explained using the current theoretical or numerical techniques nor be accessed experimentally by overcoming the mentioned difficulties (sign problem, real-time dynamics, etc.) that the latter present. In contrast to the quantum simulation of condensed matter models, lattice gauge theories are more complicated to simulate using low-energy systems, such as ultracold atoms, since the 2

11 gauge and Lorentz symmetries are not naturally present in the latter. In addition, the simulation of both the gauge and matter degrees of freedom usually requires the use of bosonic and fermionic atoms, which increases the experimental complexity. Any attempt to simulate theses theories must take these facts into account. Some example of simulation proposals for gauge theories using ultracold atoms include both the continuous [54] and lattice version of QED, focusing specially on the latter case. Among these, in some works the gauge symmetry is obtained as a low-energy effective symmetry, simulating the Kogut-Susskind Hamiltonian with [55, 56] and without matter [57, 58], whereas in others the gauge symmetry is obtained exactly by mapping it to an internal symmetry of the atomic system [59]. Apart from ultracold atoms, other systems have been proposed to perform the simulations, such as superconducting circuits [60], trapped ions [61] and Rydberg atoms [62]. Discrete symmetry groups have been considered as well, both for analog [59, 62, 63] and digital simulations [64, 65], as well as non-abelian theories [66 68]. Recently, the first experimental realization of a quantum simulation of a lattice gauge theory was performed [69], where the real-time dynamics of the Schwinger model in dimensions were simulated on a few-qubit trapped-ion quantum computer. This experiment opened the door for studying High Energy Physics beyond the capabilities of classical simulations, however, many challenges remain to be solved (simulating higher dimensions, larger systems, etc.). This thesis is devoted to the quantum simulation of one particular lattice gauge theory the Abelian-Higgs theory, which contains both gauge and scalar fields using ultracold bosonic atoms confined in an optical lattice. The simulation scheme will also provide a new way to simulate the pure-gauge Kogut-Susskind Hamiltonian. In the first chapters, the most important physics concerning the simulating and the simulated systems will be reviewed. Chapter 2 deals with systems of ultracold atoms. A theoretical explanation of their main features, as well as the most important experimental techniques that are needed to control and manipulate them, will be explained, focusing on the second-quantized Hamiltonian that describe an atomic system in the ultracold regime. In chapter 3, lattice gauge theories will be introduced, paying special attention to the role that gauge symmetry has in the construction of such theories. We will focus on abelian gauge theories and, in particular, on their Hamiltonian formulation, obtained using the transfer matrix method. This formulation of gauge theories is more useful if one intends to simulate them using another Hamiltonian, like the atomic one in this case. The Abelian-Higgs theory will be presented in chapter 4, along with an explanation of the Brout-Englert-Higgs mechanism. This theory will be first written on a discrete lattice, after which the corresponding Hamiltonian will be obtained. In chapter 5, we will explained how quantum simulations of lattice gauge theories can be performed using ultracold atoms. In particular, how the gauge symmetry, and the corresponding degrees of freedom and interactions, can be mapped onto the atomic system. These concepts will be applied to the particular case of the Abelian-Higgs theory, leading to a quantum simulation proposal for the latter, allowing for its phase diagram to be explored, with controlled precision, in an ultracold atomic experiment. We will see how this simulation scheme also serves, in particular, as an new quantum simulation proposal for the pure-gauge Kogut-Susskind Hamiltonian. The accuracy of the simulation will be studied numerically for the latter case. 3

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13 2. Ultracold Atoms in Optical Lattices Ultracold atomic gases consist of neutral atoms trapped and cooled down to almost absolute zero temperature, where quantum effects play a significant role. This is possible thanks to the development of experimental techniques during the last decades, such as laser cooling methods or magneto-optical traps, that allow for a high controllability over these atomic systems [70, 71]. The seminal experiments in which Bose-Einstein condensates were obtained using ultracold and dilute systems of bosonic atoms [72, 73] opened the door to the study of collective quantum phenomena in regimes that were not possible before. In particular, the so-called optical lattices structures made of counter-propagating lasers can be prepared in a way that emulates a periodic lattice structure with atoms sitting on the vertices [5, 7, 9]. Within these lattices, interactions between the atoms can be tuned in order to recreate some of the most important Hamiltonian models describing many-body systems, both in condensed matter and in high energy physics. This approach was first considered for the Bose-Hubbard model [5], followed by an experimental realization where the phase transition between the superfluid and the Mott insulator phases that this model present was observed [25], leading to a revolution in the way strongly correlated quantum systems could be studied. Ultracold atoms in optical lattices have therefore become one of the main platforms where quantum simulators can be designed to study many-body systems, from the Ising model to lattice gauge theories. This chapter is devoted to their description. In the following, we will review the main physical mechanisms that are involved in creating optical lattices and how the atomic interactions needed to realize the desired many-body Hamiltonians can be obtained. Once we have all the ingredients, the Bose-Hubbard model will be derived for a system of ultracold bosonic atoms in second quantization under certain conditions, where the ultracold regime will be specified. Finally, the Bose-Hubbard model will be generalized to include fermions and different atomic internal spin states, providing us with a sufficiently rich toolbox for the quantum simulation of many interesting quantum systems Optical Potentials In this section it will be explained how neutral atoms are trapped using off-resonant lasers and the way periodic optical lattices are generated [5]. Let us begin by introducing a semi-classical approach to explain the interaction between atoms and laser fields. The latter will be consider classical, whereas for the former we will use the quantum formalism. By doing so we are neglecting the atom s influence on the the field, which is not important in this context. The Hamiltonian for a single atom of mass m has the form H A = p2 2m + ω j e j e j (2.1) j 5

14 2. Ultracold Atoms in Optical Lattices Figure 2.1.: Two level atom with an energy difference of ω e between the ground g and the excited state e, subject to a laser field with frequency ω. The first term corresponds to the kinetic energy of the atom with momentum p and the second to its internal energy levels e j with energies ω j. Consider now the classical electric field generated by a laser, E (x, t) = E (x) e iωtˆɛ (2.2) where E (x) is the spatial profile of the electric field amplitude, ω is the laser frequency and ˆɛ is the polarization vector. The interaction of the field with the atom can be described using the standard dipole approximation, where the interacting part of the Hamiltonian takes the form H dip = µ E (x, t) + h.c. (2.3) with µ the dipole operator of the atom, proportional to the position operator. This approximation is valid as long as the field is not extremely intense, which is not the case in the present situation, and also varies slowly compared to the atomic size. In the following we will consider a two level atom for simplicity, bearing in mind that the procedure can be easily generalized to more than two levels. Let us denote the ground state g and the excited state e (Fig. 2.1). To treat the problem we will first shift to the interaction picture by rotating the Hamiltonian with respect to the non-interacting part H I = e ih At He ih At (2.4) Inserting the identity operator I = g g + e e twice, we obtain the so-called dipole Hamiltonian in the interaction picture after eliminating the fast rotating terms using the rotating wave approximation (RWA). This is a standard technique in quantum optics, valid whenever ω e ω ω e + ω, due to the fact that highly oscillatory terms give negligible contributions. H dip,i = Ω (x) 2 eiδt e g + h.c. (2.5) where δ = ω e ω is the detuning and Ω (x) the Rabi frequency, Ω (x) = 2E (x) e µ ˆɛ g (2.6) 6

15 2.1. Optical Potentials The total Hamiltonian in this picture is then H I = p2 + δ e e + 2m ( Ω (x) 2 eiδt e g + h.c. ) (2.7) In the case of a large detuning δ, transitions from the ground to the excited state are very unlikely, so we can adiabatically eliminate the latter considering only these transitions as virtual second order processes. Following the procedure of Appendix (A.1) we obtain an effective Hamiltonian for the atom in the ground state H eff = p2 + V (x) (2.8) 2m Ω (x) 2 V (x) = (2.9) 4δ The resulting Hamiltonian only acts in configuration space. Moreover, we can see how the potential is proportional to the intensity profile of the field E (x) 2, by modifying it we can obtain the desired potential. If the laser is red-detuned (δ > 0) the atom is attracted towards the high intensity regions, whereas for a blue-detuned laser (δ < 0) the potential minima correspond to the low intensity regions. A comment is now in order before proceeding. In the former derivation we have neglected the effect of spontaneous emission. It can be seen [6] that this effect is more important when using red-detuned lasers compare to the blue-detuned ones, since in the first case the atoms are localized around high intensity regions. In any case, these effects are negligible as long as δ V 0, so in the following spontaneous emission will not be considered Periodic Potentials Our final goal is to create a lattice structure, and for this we need some kind of a periodic potential. We can archive that by using counter-propagating lasers of the same polarization forming standing waves with E (x) = E 0 cos(kx). If we do this in all three dimensions of space we will have a potential of the form V (x) = V 0x cos 2 (kx) + V 0y cos 2 (ky) + V 0z cos 2 (kz) (2.10) The main parameters characterizing this 3D cubic lattice are the potential depth V 0, usually expressed in units of the recoil energy E r = 2 k 2 /2m (2.11) and the lattice spacing d = λ/2 with λ = 2π/k the wavelength of the lasers. In the following we will refer to the potential minima as the lattice sites. Cubic lattices are the simplest example of optical lattices, but many more geometries and dimensionalities can be archived by modifying the direction, intensity, phase, frequency and other parameters of the laser beams [7]. On top of the lattice structure, several kinds of interactions between the atoms can be devised, enabling the simulation of a great variety of many-body Hamiltonians. We will enter in detail about these topics in the next sections, but first, we will consider the behaviour of non-interacting particles in such periodic potentials, since the results can be used to describe the situation where interactions are present. 7

16 2. Ultracold Atoms in Optical Lattices 2.2. One-Particle Picture: Band Structure Consider a neutral atom in an periodic optical lattice. This is equivalent to a well known problem from solid-state physics, where the spectrum of non-interacting electrons in crystalline lattice potentials is studied [74]. Using Bloch s theorem for periodic potentials, a description of the electrons is given in terms of Bloch waves, with energy levels consisting of bands separated by a finite gap. We shall treat atoms in a similar fashion, starting from the corresponding Schrödinger equation [ ] p 2 2m + V (x) Φ (n) q (x) = E (n) q Φ (n) q (x) (2.12) where the one dimensional case is considered for simplicity, without loss of generality. According to Bloch s theorem, the solutions of an equation such as this, for a particle of mass m moving in a periodic potential V (x) = V (x+d), are wavefunctions with a specific structure Φ (n) q (x) = e iqx u (n) q (x) (2.13) where u (n) q (x) = u (n) q (x + d) has the same periodicity as the potential. The index n characterized the infinite discrete set of energy bands, while q is the so called quasimomentum. These are the so-called Bloch waves, which are invariant by adding any vector of the reciprocal lattice to the quasimomentum, for this reason it is enough to consider it inside the first Brillouin zone [74]. Inserting the Bloch wave into the Schrödinger equation we get [ (p + q) 2 2m ] + V (x) u (n) q (x) = E (n) q u (n) q (x) (2.14) Taking the Fourier transform of the (sinusoidal) potential and the wavefunction V (x) = (V 0 /4)(e 2ik Bx + e 2ik Bx + 2) (2.15) u (n) q (x) = l c (n,q) l e 2ilk Bx (2.16) we obtain an equation for the Fourier coefficients and the energy bands with l H l,l c (n,q) l (2l + q/ k) 2 E r if l = l H l,l = V 0 /4 if l l = 1 0 otherwise = E q (n) c (n,q) l (2.17) (2.18) The eigenvalues and eigenvectors can be then obtained by truncating the sum at some fixed and large l. In figure 2.2 we can see the band structure for a 1D optical lattice for different potential depths. There is no gap when the potential is not present, 8

17 2.2. One-Particle Picture: Band Structure V0 =0Er -1.0 V0 = Er E HEr L V0 =5Er E HEr L q HkB L V0 =10Er E HEr L E HEr L q HkB L q HkB L q HkB L -2 Figure 2.2.: Band structure of a 1D optical lattice for different potentials depths V0. The energy is represented (in units of the recoil energy) as a function of the quasimomentum in the first Brillouin zone. For any finite potential depth, an energy gap formed between different bands, which become more flat when the depth increases. but as soon as there is a finite potential depth, a gap opens between different bands. As the potential is increased, the gap gets bigger and the bands become more flat. In the previous discussion a sinusoidal optical potential was used, but the method can be used to obtain the band structure of any periodic potential. It is interesting to represent the single particle wavefunctions in a different basis, called the Wannier basis, formed by localized functions around the lattice minima. The Wannier functions are defined as the Fourier transform of the Bloch wavefunctions 1 X iqxi (n) wn (x xi ) = e Φq (x) M q (2.19) where M is a normalization constant. This is an orthonormal set of functions, each representing a single lattice site. One of the interesting properties of these functions is that they are more and more localized around lattice sites when one increases the lattice depth. In particular, they tend, in this limit, towards the eigenfunctions of the harmonic oscillator with frequency ωt = 4V0 Er [6]. In the next section, we will make use of these functions to express the Hamiltonian in a second quantized form using operators acting only on lattice sites. 9

18 2. Ultracold Atoms in Optical Lattices 2.3. Two-Body Interactions Another important step in the construction of many-body Hamiltonians is the description of the interaction between the atoms. We will see that, under some special circumstances [7], most of the details of the interaction are not relevant, and it can be replaced by a pseudopotential characterized by just one parameter. To see this, let us first recall the basic concepts of scattering theory used in the description of collisions between quantum particles [75] Scattering Theory Consider two particles in the center of mass frame that interact during a short period through a spherically symmetric central potential V (r), where r is the distance between the particles [75]. The particles are considered as free long times before and after the interaction takes place. A free particle moving in the positive direction of the z-axis is described by a plane wave such as Ψ = e ikz (2.20) Far away from the interacting regime the wavefunction has the asymptotic form Ψ e ikz + f(θ) eikr (2.21) r where the second factor represents an outgoing spherical wave. f(θ) is called the scattering amplitude and θ is the angle between the z-axis and the direction of the scattered particle. There is no dependence on the azimuthal angle φ since the system presents a cylindrical symmetry around the z-axis. We introduce now one of the most important quantities in scattering theory, the differential cross-section. It is defined as the ratio between the probability per unit time for the particle to pass through a differential surface element ds = r 2 dω and the current density in the incident wave. It can be expressed as dσ = f(θ) 2 dω (2.22) It is shown [75], using symmetry arguments, that the scattering amplitude can be written as a superposition of Legendre polynomials characterized by different values l of the angular momentum (partial wave approximation) f(θ) = 1 2ik (2l + 1)f l (k)p l (cosθ) (2.23) l=0 Integrating the differential cross-section for all solid angles and using the properties of the Legendre polynomials, we obtain the following expression for the total crosssection σ = dω f(θ) 2 = 4π k 2 sin 2 δ l (2.24) Consider now the low-energy limit. This is the case when the velocities of the particles are so small that their wavelength is large compared to the effective range l=0 10

19 2.3. Two-Body Interactions of the potential, R (i.e. kr 1) and their energy is small compared with the field inside this radius. One can prove that in this case f l k 2l (2.25) hence, in the low-energy limit, all the partial wave amplitudes f l (k) are negligible compared to the s-wave, the one corresponding to l = 0. This situation is regarded as s-wave scattering. The scattering process can now be characterized by just one parameter, the scattering lenth, defined as a = lim k 0 f 0 (k) k which can be positive or negative. The total scattering length takes then the form and the scattering amplitude is (2.26) σ = 4πa 2 (2.27) f(k) = a 1 + ika (2.28) The actual scattering length can be derived for the specific type of atom-atom interaction [76], and can be remarkably tuned in an actual ultracold atom experiment using the Feshbach resonance technique [7], very helpful in the design of the desired many-body Hamiltonian. Information about this technique can be found in the Appendix B. In the following we will assume that every scattering length can be tuned to a desired value Pseudopotential The previous discussion referred to the general case of two-body collision processes between quantum particles. For the specific case of atoms, in particular bosonic ones, we should study the potential interaction to obtain the scattering length. This is done in [76] for different potentials. A useful model that capture the important properties of the interaction between neutral atoms is the the cut-off Van der Waals potential, V (r) = { C/r 6 if r > r c (2.29) if r r c with Van der Waals forces acting at large distances and a hard-core potential at a distance r c of the order of the atomic dimension. The scattering length of this potential can be analytically obtained and depends only on the constants r c and C. This potential provides a good description of interactions between atoms in optical lattices provided that no long-range interactions are present, as in the case of dipolar gases [9], which will not be considered here. But we can simplify things even more. We have seen that in the low energy limit the two-body collisions are characterized just by the scattering length. Then we could use, instead of the cut off Van der Waals potential, an effective one that results in 11

20 2. Ultracold Atoms in Optical Lattices the same scattering amplitude (2.28), as was proven first for a hard-core spherical potential in 1957 [77]. This pseudopontential has the form V (x)(...) = 2πa m δ(x) (r...) (2.30) r which can be reduced when acting on functions that are regular at r = 0 to V (x) = 2πa δ(x) gδ(x) (2.31) m This is the potential that we will use to describe interactions between ultracold atoms in optical lattices. For positive scattering length it will be attractive, and repulsive for negative ones. Although this is a very simplify description, in the proper regime it accounts perfectly for two-body collisions between the atoms Ultracold Limit In order to talk about the low-energy limit of scattering processes, we need to specify in which regime this is a valid approximation, when dealing with an actual experiment of cold atoms [7]. Consider again the Van der Waals potential (2.29). For states with l 0 the potential contains a centrifugal barrier with an approximate height of E 2 l 3 /ma 2 c, with a characteristic length a c = (2mC/ 2 ) 1/4. This energy scale provides us with an estimate for the low-energy regime in which only the s-wave scattering is important. Converting this to temperatures, one obtains, for typical atoms, that this limit is reached around 1 mk. Below this temperature the scattering channels with l 0 are effectively frozen. Therefore the ultracold regime is define for temperatures in the sub-milikelvin regime. To reach these temperatures, several experimental techniques are available, such as laser or evaporative cooling [71] Second Quantization We have now all the ingredients to build our first many-body Hamiltonian on the lattice. The first-quantized Hamiltonian for ultracold atoms in the optical potential consists, as we have seen, of a free kinetic energy term for each atom, an effective potential due to the interaction with the off-resonant laser fields and a two body pseudo-potential interaction between different atoms. It was shown in a seminal paper by Jaksch et al. [5] that the second-quantized form of this Hamiltonian is none other than the Bose-Hubbard one. We will analyze this example in detail as it is the simplest many-body Hamiltonian that can be obtained using ultracold atoms in optical lattices, as well as the first one that was realize experimentally [25]. We will also see in this section how this model can be generalized to include more atomic species, in particular, what happens when the internal levels of the atoms are considered, in what it is known as spinor gases. Finally, fermionic atoms will be discussed The Bose-Hubbard Model To show how the Bose-Hubbard model is obtained, let us first express the ultracold atomic Hamiltonian in second-quantized form: 12

21 2.4. Second Quantization Ĥ(t) = ] dr ˆΨ (r, t) [ 2 2m 2 + V ext (r) ˆΨ(r, t) dr dr ˆΨ (r, t) ˆΨ (r, t)v (r r ) ˆΨ(r, t) ˆΨ(r, t) (2.32) The first term corresponds to the single-body part, whereas the second represents the two-body interactions between the atoms. The external potential accounts for both the dipole interaction V dip (r), and a global trapping potential V T (r). V (r r ) is the contact pseudo-potential described in the previous section and ˆΨ (r, t), ˆΨ(r, t) are the local field operators, acting on the Fock space, with canonical commutation relations [ ˆΨ(r, t), ˆΨ (r, t )] = δ(t t )δ (3) (r r ) (2.33) We will now expand the field operators in terms of creation and annihilation operators acting on lattice sites. We can do this by using the Wannier functions defined previously (2.19) and recalling their properties. In particular, since we are in the lowenergy limit (low temperatures), and assuming that the potential is sufficiently deep, we can consider the atoms lying in the first energy band, since there is not enough energy to excite them to higher bands. The Wannier functions form a complete basis and, since we can disregard higher energy bands, the field operators can be expanded in the following way ˆΨ(r, t) = i ˆbi w(r r i ) (2.34) Where w(r r i ) are the Wannier functions corresponding to the first energy band (no band index will be used in the following) and ˆb i (ˆb i ) are bosonic operators destroying (creating) a particle on lattice site i that lies in the first energy band. These bosonic operators obey the usual canonical commutation relations [ˆb i, ˆb j ] = δ ij (2.35) We now insert these expressions for the field operators into equation (2.32). To obtain the usual Bose-Hubbard model, recall again the localization properties of the Wannier functions. In the deep potential limit we will assume that the overlap between Wannier functions that are not nearest-neighbours is negligible, thus limiting the first term of the Hamiltonian only to nearest-neighbors (n.n.) hopping terms between lattice sites. The second term in the Hamiltonian involves an overlap integral of four Wannier functions. In this case we will only consider the terms that involve four functions corresponding to the same site as non-zero. After these approximations (tight binding) the Hamiltonian takes the form H = i,j t ij (ˆb iˆb j + h.c.) + U 2 ˆn i (ˆn i 1) + i i ɛ iˆn i (2.36) The first term corresponds to nearest-neighbours hopping and the second to on-site interactions, with ˆn i = ˆb iˆb i the local bosonic occupation number. The parameters t ij and U are the tunneling matrix elements and the interaction strength, respectively. The last term accounts for the interaction with the global trapping potential, with ɛ i 13

22 2. Ultracold Atoms in Optical Lattices an offset energy term that can be seen as a chemical potential. All these parameters are expressed by integrals over the Wannier functions t i,j = ] dr w (r r i ) [ 2 2m 2 + V ext w(r r j ) (2.37) U i = g dr w(r r i ) 4 (2.38) ɛ i = dr V T (r)w(r r i ) V T (r i ) (2.39) These are precisely the ones that do not vanish according to the previous discussion. For an homogeneous and isotropic external potential both t ij and ɛ i are regarded as site independent. This is a good approximation if the trapping potential varies very slowly across the system. The Hamiltonian we have obtained corresponds to the so-called Bose-Hubbard model, introduced in 1989 [78] as the bosonic version of the Hubbard model, which originated in solid-state physics as an approximate description of a system of electrons moving between the atoms of a crystalline solid. The Bose-Hubbard model is interesting as a toy model in condensed matter physics because, apart from helping in the study magnetic properties of materials, it is the simplest many-body (interacting) Hamiltonian that can not be reduced to a single-particle Hamiltonian. Notice that the parameters in the Hamiltonian are totally determined once we know the scattering length and the Wannier functions of the system. While discussing the band structure, we saw how to obtain the latter from the Bloch waves, depending, in the end, only on the optical lattice parameters, this is, the lattice depth V 0 (intensity) and the wavelength of the laser λ. This shows the power of quantum simulations using ultracold atoms: the parameters of the Hamiltonian can be changed at will by fine-tuning a few easily controllable experimental quantities. To get a feeling on how the Bose-Hubbard parameters depend on the experimental quantities we can approximate, in the limit of deep lattices, the Wannier functions by Gaussian wave functions, since these are the corresponding ground states of the quantum harmonic oscillator. By doing so we get an estimate on the tunneling rate and the interaction strength t 4 ( ) ( 3/4 V0 E r exp 2 π E r ( V0 E r ) 1/2 ) (2.40) ( V0 ) 3/4 (2.41) U 8πa k π E r E r Particularly interesting is the ratio between the two quantities, since it characterizes, as we will see, the phase diagram of the system at zero temperature. U/J a k π exp (2 ( V0 E r ) 1/2 ) (2.42) 14

23 2.4. Second Quantization Therefore, by only changing the lattice depth we can drive the system through a critical point, with the corresponding phase transition. This was shown experimentally by Greiner et al. [25] in one of the first realizations of a many-body Hamiltonian using ultracold atoms. Phase Diagram The phase diagram for the Bose-Hubbard model was first studied by Fisher et al. [78]. In the grand canonical ensemble the Hamiltonian takes the form H = t (ˆb iˆb j + h.c.) + U ˆn i (ˆn i 1) µ ˆn i (2.43) 2 i,j i i where the tunneling rate t is considered to be uniform for all sites and directions and the chemical potential µ fixes the average number of particles. We will not include here a full study of the phase diagram of this model, but will rather simply outline some if its general properties due to its historical importance. At zero temperature, and for integer values of the density of particles per lattice site, the model exhibits two distinct quantum phases characterized by the ratio t/u, with a quantum critical point at some value (t/u) c separating them. When the ratio is very small, thus the tunneling rate is much smaller than the interaction strength, a Mott insulator ground state appears. This is characterized by the presence of n localized particles per lattice site, zero compressibility and a gapped spectrum due to the cost of moving a particle from one site to another. The system posses a finite correlation length and the ground state wave-function can be written just as a product state Ψ Mott (ˆb i )n 0 (2.44) i On the other hand, when the ratio is very big, so the tunneling is much bigger than the interaction strength, the ground state of the system behaves as a superfluid. In this case the number of particles per site is not fixed and fluctuates largely. Also the correlation length diverges and the spectral gap vanishes. Since the particles are totally delocalized the ground state can be expressed as Ψ SF = 1 N ( i ˆb i ) n 0 (2.45) which in the limit of U = 0 is just a product of coherent states in each lattice site. Note that if the density of particles per site is not an integer number the system behaves for all values of the parameters as a superfluid. The phase diagram is usually represented in a t/u µ/u plot, where the so-called Mott lobes can be appreciated (Fig. 2.3). According to equation (2.42), the ratio U/J increases exponentially when increasing the lattice depth V 0, so if we start from a system in the superfluid regime in a shallow lattice potential we would be able to drive the system to the Mott insulating regime, crossing the quantum critical point, just by continuously increasing the lattice depth V 0. This was precisely demonstrated experimentally in 2002 by Greiner et al. [25], where the transition between these two phases was observed in a system of ultracold 87 Rb atoms. This seminal experiment opened the door to the quantum simulation of strongly correlated many-body systems using ultracold atoms. 15

24 2. Ultracold Atoms in Optical Lattices Figure 2.3.: Zero temperature phase diagram of the Bose-Hubbard model. The Mott insulator lobes appear for integer particle density n, outside them the system behaves as a superfluid Ultracold Spinor Atomic Gases The description of the ultracold atomic systems considered so far assumes atoms without any internal structure. In this picture, atoms are just point particles affected by an external potential and colliding among themselves. Consequently, scalar operators are enough, in this context, to describe the second-quantized Hamiltonian (2.32) that characterizes the dynamics of the system. However, due to the spin degrees of freedom, each atom possesses a very rich internal structure. This can be used to simulate systems that are more complicated than the ones we have considered so far using the point particle picture, which is correct nevertheless when every atom lies in the same internal level. In this section, we will see how the Bose-Hubbard Hamiltonian (2.36) is generalized when the atomic spin is taken into account, and how the structure of the interactions is specified by the corresponding symmetry. The resulting ultracold atomic system is thus called a spinor atomic gas [5, 9, 79]. Usual ultracold atomic experiments use alkali atoms, which possess a single electron in the outer energy level, corresponding to an s-orbital with zero orbital angular momentum and spin S = 1/2, due to the internal electronic spin. The total atomic angular momentum F = I + J (2.46) is composed of the total nuclear spin I and the total electronic angular momentum J = L+S. The latter depends on the electronic spin S and the orbital angular momentum L. The total electronic angular momentum for alkali atoms is therefore J = 1/2, whereas the nuclear spin depends on each atomic species. In this case the total atomic angular momentum F can take only two values: I + 1/2 and I 1/2. The states corresponding to each one of these are regarded as hyperfine levels, being degenerate 16

25 2.4. Second Quantization in the absence of a magnetic field for different values of the magnetic quantum number m F = F,..., F. In the second-quantized formalism a spinor operator ˆΨ α, with index α designating each state m F, is used to characterize atoms belonging to the state manifold given by a particular hyperfine angular spin F. The Hamiltonian will fulfill a SU(2F + 1) symmetry, corresponding to rotations of the spinor operator in the angular momentum space. A comment is now in order before continuing. In some ultracold atomic experiments magnetic fields are used to trap the atoms or to change the scattering length via Feshbach resonances. In that case, the spin rotational symmetry is broken and the system can not be described using the spinor picture. In all the situations that we will consider this is not the case. We assume no magnetic fields are present, in particular the trapping is due only to the dipolar potential induced by lasers, as explained at the beginning of this chapter. The Feshbach resonances are as well considered to be optically induced (Appendix B). The most general second-quantized Hamiltonian that we can write generalizing (2.32) for many different bosonic species is [79] Ĥ(t) = α,β + αβδγ dr ˆΨ α(r, t) (δ ( αβ m 2 + V α ext(r) ) ) + Ω αβ (r) ˆΨ β (r, t) dr dr ˆΨ α (r, t) ˆΨ β (r, t)v αβδγ (r r ) ˆΨ δ (r, t) ˆΨ γ (r, t) (2.47) For the present discussion we consider only one type of atom with different hyperfine states m F, but the previous Hamiltonian can also be used when more than one type of atom is present. In this case the Greek indices represent then the different internal levels and the sums run from F to F. The Hamiltonian contains a non-interacting part equal to (2.32) corresponding to each level, where the external potential can be different for each case. This can be archived by creating state-dependent optical lattice structures [5, 80], with minima sitting in the same position or not. The Hamiltonian also accounts for species-changing transitions among different levels characterized by the Rabi frequency Ω αβ. These kind of processes are called Raman transitions and can be induced experimentally [5]. The second part of the Hamiltonian accounts for the collisions among every pair of atoms. As we have seen, the interatomic potential V αβδγ (r r ) can be approximated by a contact potential proportional to a scattering length. In principle there is a different scattering length for each possible combination of atomic levels participating in the collision, this is, N 4 with N = 2F + 1. If we take into account the symmetries present in the system, however, this number of free parameters is largely reduced. For example, particle exchange symmetry implies g αβδγ = g βαδγ, whereas time reversal symmetry makes g αβδγ = g δγαβ. For spinor gases the number is reduced even more. Imagine that all the atoms have the same hyperfine spin F. During interatomic collisions, as a consequence of the spin rotational symmetry, the total spin f = F 1 +F 2 of the two atoms must be conserved. Its possible values go from f = 0 to f = 2F, with only even values allowed for identical bosons (odd for fermions). Therefore there are only F + 1 independent scattering lengths. The complete interaction potential 17

26 2. Ultracold Atoms in Optical Lattices can be expressed as a sum over the possible total spin values V (r r ) = 4π m δ(r r ) f a f P f (2.48) with P f the total spin projector operator. It can be proven [81] that this sum can also be written using spin operators ( F ) V (r r ) = α n ( F 1 F 2 ) n δ(r r ) (2.49) n=0 As an example, consider the case F = 1, which is one of the hyperfine ground states of 23 Na and 87 Rb. For identical bosons the value of f can only be either 0 or 2. The structure of the interatomic potential in this case is V (r r ) = (α 0 + α 1 F1 F 2 ) δ(r r ) (2.50) with α 0 = 4π m 2a f=2 + a f=0 3 (2.51) and the total Hamiltonian is Ĥ(t) = α,β + αβδγ + g 2 α 1 = 4π m a f=2 a f=0 3 ) ) dr ˆΨ α(r, t) (δ ( αβ 2 2m 2 + Vext(r) α + Ω αβ (r) ˆΨ β (r, t) η 1 2 dr dr [g 0 ˆΨ α (r, t) ˆΨ β (r, t) ˆΨ α (r, t) ˆΨ β (r, t) ( ) ( ) ] ˆΨ α (r, t)(f η ) α,β ˆΨ β (r, t) ˆΨδ (r, t)(f η ) δ,γ ˆΨγ (r, t) (2.52) (2.53) Just as in the Bose-Hubbard model, we can use the Wannier basis to write the Hamiltonian using creation and annihilation operators acting on the lattice vertices for each atomic species Ĥ = α,β i,j t α,β i,j ˆb i,αˆb j,β + i,j,k,l α,β,δ,γ U α,β,δ,γ i,j,k,l ˆb i,αˆb j,βˆb k,δˆbl,γ (2.54) where U α,β,δ,γ i,j,k,l have a specific structure whenever the rotational spin symmetry is present, as we have seen. This is the most general lattice Hamiltonian that we are going to need for the purposes of this thesis. It will provide the basis to simulate a great variety of many-body Hamiltonians. 18

27 2.4. Second Quantization Fermions So far, we have only considered atomic gases composed of bosons, this is, atoms with a total integer spin F. This is enough for the purposes of this thesis, since only multispecies ultracold bosonic gases will be considered, such as the ones described by the Hamiltonian (2.54). The fermionic case, as well as the fermion-boson atomic mixtures, is an almost straightforward generalization. However, some details are different from the bosonic system, due to the nature of particle statistics. We will comment on them briefly for completeness. As a consequence of the Pauli exclusion principle, two identical fermions can not be found in the same position. This causes the s-wave scattering channel to vanish. Therefore, the lower non-vanishing order in the partial wave decomposition for scattering processes is the one corresponding to l = 1, the p-wave channel. In the zero temperature limit, the fermions can actually be considered as non-interacting. Nevertheless, the potential interaction can still be approximated by a contact potential characterized by some scattering length, although the origin of such a length will be different from the bosonic case [7]. The fact that fermions do not interact at very low temperatures has consequences in the way they are cooled down, as well. One of the most successful techniques for cooling atomic gases is the one known as evaporative cooling, used in bosonic systems to achieve low enough temperatures for the observation of Bose-Einstein condensation [72, 73]. This technique requires the thermalization of atoms via elastic collisions, so in the fermionic case lower temperatures are harder to archive [7]. Nevertheless, this problem can be overcome by using different fermionic species, a strategy that allowed to reach low enough temperatures to observe degeneracy of fermionic gases [82]. As an example of ultracold fermionic gas on the lattice we could consider a system composed of 1/2-spin atoms. Following a derivation similar to the case of bosons, such a system can be described in second-quantization using the Fermi-Hubbard model, Ĥ F H = t i,j,σ( ˆf iσ ˆf jσ + h.c.) + U i ˆf i ˆf i ˆf i ˆfi iσ µ iσ ˆf iσ ˆf iσ (2.55) The fermionic creation and annihilation operators ˆf iσ, ˆf jσ act on different lattice sites i, j, for different spin states σ, σ, fulfilling the canonical anticommutation relations { ˆf iσ, ˆf jσ } = δ i,j δ σ,σ. This is perhaps one of the most important models in condensed matter physics, describing a system of interacting electrons inside a material, where the lattice corresponds to the crystalline structure of the solid, with atoms in the vertices. The Fermi-Hubbard model is also important because is believed to be related to the phenomenon of high-tc superconductivity [83]. This and other interesting condensed matter models involving electrons, such us magnetic systems [9], can be simulated using ultracold fermionic atoms in optical lattices. As a last remark, let us consider the most general ultracold atomic Hamiltonian, containing both bosonic and fermionic atoms. Apart from the terms we have already considered, corresponding to both bosons and fermions separately, one must include 19

28 2. Ultracold Atoms in Optical Lattices a term that accounts for collision between a boson and a fermion, i,j,k,l α,β,δ,γ Ũ α,β,δ,γ i,j,k,l ˆf i,α ˆf j,βˆb k,δˆb l,γ (2.56) where the structure of the constants Ũ α,β,δ,γ i,j,k,l depends, again, on the symmetries that are present in the system. This Hamiltonian is important for quantum simulations of high energy physics since, in theories such as lattice QED or QCD, both fermionic and bosonic atoms are needed to simulate the corresponding matter and gauge degrees of freedom [59]. 20

29 3. Lattice Gauge Theories Lattice gauge theories are formulations of gauge theories on a discretized space (or space-time) [50, 51, 84 86]. They describe many-body quantum systems that are invariant under some local (gauge) transformation group. As approximated versions of continuous theories, they allow for simpler calculations of important quantities, in particular in the non-perturbative regime using, for example, numerical techniques, such as Monte Carlo methods. The lattice structure also provides a natural regularization scheme for field theories, where the inverse of the lattice spacing serves as an energy cutoff. Representing gauge theories with a discrete set of degrees of freedom arising from the discrete nature of space-time is also very convenient to perform quantum simulations in high energy physics, using, in particular, a Hamiltonian formulation for the corresponding lattice theory [87]. By doing this, a more straightforward mapping between the simulated and the simulating system can be achieved [88]. In this chapter, we will review the main properties of lattice gauge theories, focusing on the role that gauge invariance plays in their construction, in particular for abelian theories in the abscence of dynamical matter. We will start by discussing the concept of gauge symmetry for continuous theories. Next, following an historical approach, we will explain the main motivation that led to the discretization of space-time in quantum field theory: the quark confinement problem [49]. We will derive, explicitly, the lattice version of the quantum electrodynamics action. The transfer matrix method will be introduced in order to obtain the quantum mechanical Hamiltonian that describes this theory, starting from the corresponding classical action. The latter is the celebrated Kogut-Susskind Hamiltonian [87]. In the last part of the chapter, its Hilbert space structure, along with the description of confinement in the Hamiltonian formulation of abelian gauge theories, will be presented in detail Gauge Theories in the Continuum Quantum field theories provide, in high energy physics, the mathematical framework to describe elementary particles and their interactions. Here, as well as in other branches of theoretical physics, the concept of symmetry plays an important role. The study of the structures and properties of many theories is, to a large extent, simplify by the use of symmetry arguments. Among them, local symmetries are particularly prominent. They refer to transformations whose parameters are functions of spacetime, under which the theory remains invariant. As in the case of other symmetries appearing in physics, the set of such transformations forms a group, called gauge group in this case. The theories invariant under such transformation are regarded as gauge theories [45 48]. It is important to notice that gauge symmetries are not physical symmetries, like global ones, but a consequence of the redundancies present in the mathematical description of the theory. In particular, states connected by a 21

30 3. Lattice Gauge Theories gauge transformation can not be physically distinguished, as opposed to those related by global transformations. Some important examples of gauge theories are quantum electrodynamics (QED) and quantum chromodynamics (QCD), both present at the core of the Standard Model of particle physics [46]. Studying the properties of gauge groups is, therefore, an important strategy regarding the construction of relevant theories in modern physics. In this section, we will follow such route to find the Lagrangian of QED starting just from symmetry arguments, stressing the important role played by gauge invariance and the corresponding gauge group, U(1). The argument can be generalized to more complicated groups, such as SU(N), enabling the construction of the so-called Yang- Mills theories, of which QCD is an important example. Here, we will not treat the non-abelian case. For a detailed derivation see [45]. Finally, we will introduce a gaugeinvariant observable, the Wilson loop, which will be relevant to study the confinement problem in later sections Quantum Electrodynamics The goal here is to find a suitable Lagrangian to describe the interaction between charged fermionic particles. These are represented, in quantum field theory, by complex-value Dirac fields ψ(x) [45], where x R d+1 is a point in a d + 1 dimensional space-time. We start by imposing the condition that our theory, or equivalently, the Lagrangian, is invariant under local U(1) transformations, represented by complex phases α(x), ψ(x) e iα(x) ψ(x) (3.1) The simplest gauge-invariant term that we can write in the Lagrangian is the fermionic mass term, m ψ(x)ψ(x) (3.2) If we want now to include derivatives we face a problem, since they involve the comparison of the field at different points n µ 1 µ ψ = lim [ψ(x + ɛn) ψ(x)] (3.3) ɛ 0 ɛ consequently, these terms are not gauge invariant. One way to construct meaningful gauge-invariant derivatives is to introduce a scalar quantity, U(y, x), that allow us to compare fields at different points. This can be achieve if this comparator satisfies the transformation law U(y, x) e iα(y) U(y, x)e iα(x) (3.4) The derivative is, then, substituted by the so-called covariant derivative, n µ 1 D µ ψ = lim [ψ(x + ɛn) U(x + ɛn, x)ψ(x)] (3.5) ɛ 0 ɛ Without loss of generality, U(y, x) can be regarded as a phase, and, for points separated by an infinitesimal distance, expanded using a vector field A µ (x), U(x + ɛn, x) = 1 ie ɛn µ A µ (x) + O(ɛ 2 ) (3.6) 22

31 3.1. Gauge Theories in the Continuum Figure 3.1.: The electromagnetic field tensor (F µν ) is extracted from the gaugeinvariant product of four comparators, each one associated to one side (of length ɛ) of a square loop on the (1, 2) plane, in the limit ɛ 0. where e is an arbitrary constant (which will turn out to be the electric charge unit). The vector field A µ is called a connection, and we can write the covariant derivative in terms of it, D µ ψ(x) = µ ψ(x) + iea µ (x)ψ(x) (3.7) Using the previous definitions, we can find the transformation law for A µ (x), A µ (x) A µ (x) 1 e µα(x) (3.8) The covariant derivative transform in the same way as the field, D µ ψ(x) e iα(x) D µ ψ(x) (3.9) and can be used to construct gauge-invariant terms. The vector field A µ (x) is also called gauge field, and, in the case of QED, is nothing but the field associated to the photon. Here, we note that its very existence is required in order to write gaugeinvariant Lagrangians involving derivatives of the field ψ(x). The next step is finding a kinetic term for the gauge field, a gauge-invariant term that depends only on A µ (x) and its derivatives. To motivate how to do this, consider the product of four different comparators on the four different sides of a square loop (Fig. 3.1), U(x) = U(x, x + ɛˆ2)u(x + ɛˆ2, x + ɛˆ1 + ɛˆ2) (3.10) U(x + ɛˆ1 + ɛˆ2, x + ɛˆ1)u(x + ɛˆ1, x) where ˆ1 and ˆ2 are two orthogonal unit vector of a two dimensional plane. This term is gauge invariant by construction. In the limit ɛ 0 it can be shown to take the form [45] U(x) = 1 iɛ 2 e [ 1 A 2 (x) 2 A 1 (x)] + O(ɛ 3 ) (3.11) From this, we can conclude that the term F µν = µ A ν ν A µ (3.12) is gauge invariant. This is nothing else than the electromagnetic field tensor,which can be seen as the simplest gauge-invariant term that depends only on the gauge field 23

32 3. Lattice Gauge Theories and its derivatives. The reason for this is that any pure gauge-invariant term can be written in terms of it [45]. If one takes into account the conditions for writing a renormalizable Lagrangian, as well as other symmetries that should be fulfilled, such as parity, time-reversal and Lorentz invariance, it can be shown [45] that the only gauge-invariant Lagrangian in four dimensions involving Dirac fields is L = ψ(iγ µ D µ m)ψ 1 4 F µνf µν (3.13) which is exactly the usual QED Lagrangian. Here, we have obtained it starting from just symmetry arguments involving local transformations Wilson Loop Before proceeding to discuss the lattice version of gauge theories, we will finish this section by constructing a gauge-invariant quantity, the Wilson loop, that will prove to be useful when trying to distinguish between different phases of the theory. So far, we have just used the infinitesimal version of the comparator U(y, x) (3.6), writing it in terms of the gauge field A µ (x). We can also write an expression for points separated by a finite distance. It is easy to check that the expression [ ] U P (z, y) = exp ie dx µ A µ (x) P (3.14) satisfies the required properties, where P is a path that goes from y to z. This object is called Wilson line, and its value depends on the actual path taken to go from y to z. If we take a closed path, we will obtain the so-called Wilson loop, [ ] U P (z, y) = exp ie dx µ A µ (x) P (3.15) This quantity is gauge invariant, which it is easy to check. Wilson loops can be generalized to construct gauge-invariant quantities in non-abelian theories [45] Lattice Formulation of Gauge Theories In this section, we will construct gauge theories on a discretized space-time [50, 51, 84 86]. As in the continuum case, the main requirement will be the invariance of the theory s Lagrangian under gauge transformations. After motivating the introduction of these theories to study the confinement of quarks, we will study how field theories can be discretized, in such a way that the correct classical continuum limit is obtained when the lattice spacing is taken to zero. In particular, we will derive the lattice version of the QED Lagrangian in the absence of dynamical matter. Finally, the transfer matrix method will be introduced, and the Hamiltonian formulation of lattice gauge theories discussed, focussing on the Hilbert space structure and the description of charge confinement in this setup. 24

33 3.2. Lattice Formulation of Gauge Theories Figure 3.2.: A quark and an antiquark can form a bound state, interacting through the exchange of gluons. When the distance between the two particles increases, the gluon tube connecting them elongates, increasing the energy of the system. At some point, it is energetically more favorable to break the tube and create a quark-antiquark pair from the vacuum, resulting in two new composite particles, preventing individual quarks to become isolated Motivation: The Quark Confinement Problem Lattice gauge theories were introduced by K. Wilson, in 1975, as a tool to study the quark confinement problem in quantum chromodynamics [49]. This non-perturbative phenomenon refers to the impossibility of detecting free quarks or gluons in nature. These elementary particles are always confined in bound states, forming hadrons, composite particles made of different quarks [46], or glueballs, hypothetical particles made solely of gluons [89]. This property is not restricted to QCD. In fact, for any non-abelian Yang-Mills theory, the confinement of its elementary constituents, both matter particles and gauge bosons, is present. This contrasts with abelian gauge theories, such us QED, where states made of free electrons or photons are possible. The difference comes from the non-linear behaviour of non-abelian gauge theories. For such theories, the self-interaction of the gauge fields causes the effective coupling strength to grow with the distance between particles [46]. This results, for example, in a quark-antiquark static potential that increases linearly with the separation between the particles, V (r) σr (3.16) where σ is called string tension. This situation makes the existence of free quarks impossible, since it is energetically more favorable to create another quark-antiquark pair, rather than continue increasing the distance between them (Fig. 3.2). Photons, on the other hand, do not interact among themselves. For this reason, the Coulomb potential between electric charges decreases with the distance as 1/r, allowing electrons and positrons to become unbound we will see, however, how confinement arises, also, in the lattice version of abelian theories, making the study of these theories interesting to understand the confinement of matter. Quark confinement is an example of a non-perturbative phenomenon, which makes 25

34 3. Lattice Gauge Theories it very hard to study with the usual quantum field theory tools, based on the perturbative expansion of path integrals in terms of the coupling strength. Finding a solution to this, and other related problems, motivated Wilson to develop a description of quantum field theories in a discretized space-time. This approach has several advantages. From a computational point of view, it allows the application of numerical techniques, such as Monte Carlo methods, in a very efficient way. Moreover, it makes a fruitful connection to statistical physics, clarifying important concepts such as renormalization, scaling or universality [51] Space-Time Discretization Let us discuss how the basic constituents of continuous theories change upon discretization of space and time. In the path integral formalism of quantum field theory, one usually encounters functional integrals, such as the the partition function [48], Z = Dφ e is[φ] (3.17) with S[φ] = d d+1 x L[φ(x)] (3.18) the action of the theory in d spatial and one temporal dimensions. The possible fields that appear in the action are represented by φ(x). It is very convenient, in practice, to make an analytic continuation to imaginary time, also called Wick s rotation, by performing the substitution t it. The term euclidan comes from the fact that, after the rotation, space and time are treated on equal footing, as oppose to the Minkowski space-time. The integrand in the imaginarytime path integral, e S E, is real and bounded from below. This makes numerical calculation and theoretical analysis much easier. It makes also evident the connection between quantum field theory and classical statistical physics in d+1 dimensions [51]. Let us restrict the discussion to d + 1 = 4. Consider a hypercubic lattice of size L = Na, where a is the lattice spacing, and N is the number of discrete points. The total volume of the system is L 4. The allowed values for the position x µ, in the direction µ (= 1, 2, 3, 4), are x µ = n µ a, n µ = 0, 1,..., N 1 (3.19) The field φ(x) reduces to a discrete set of points φ x, whereas its derivatives are expressed using finite differences, µ φ(x) 1 a (φ x+aˆµ φ x ) (3.20) where ˆµ is a unit vector in the µ direction. The integrals that may appear in the action are replaced by sums over lattice positions, x N 1 a 4 n 1 =0... N 1 n 4 =0 a 4 n (3.21) 26

35 3.2. Lattice Formulation of Gauge Theories such that, in the continuum limit with N, a 0 L = Na fixed one obtains f(x) d 4 x f(x) (3.22) x for smooth enough functions f(x) [51]. We will refer to this as the classical limit. Finally, the functional integral in (A.26) reduces to a product of conventional integrals, Dφ x ( a 2π dφ x ), x n (3.23) The factor 1/ 2π is conventional. Having all the necessary ingredients, we can write continuous actions and functional integrals in the lattice. Taking the classical continuum limit will restore such discretized actions to their continuous versions, for sufficiently smooth fields φ x compared to the lattice spacing a. The typical field configurations, however, are not necessarly smooth. This makes the continuum limit of lattice theories a delicate topic, related to the renormalization of the theory [47, 50], which will not be addressed here Wilson s Action Let us generalized the discussion to include fermions and gauge fields. The first approach to obtain a lattice version of QED (3.13) would be to substitute, as in the scalar case, the derivatives by differences and the integrals by sums. This will render, however, an action that is not gauge invariant for non-zero lattice spacings a, being the continuum limit presumably not gauge invariant either [49]. Another strategy is to construct a lattice theory that is explicitly gauge invariant for all values of a. This alternative was the one introduced by K. Wilson [49], and it is the one we are going to follow here. First, consider the discretized version of the Dirac action in imaginary time, describing fermionic fields in a d + 1 dimensional lattice, S E = a d+1 n m ψ n ψ n ad n,µ ψ n γ µ (ψ n+ˆµ ψ n ˆµ ) (3.24) where the variables ψ n, placed at the vertices of the lattice (Fig. 3.3), are discretized versions of the field ψ(x), related to the latter, in the continuum limit, through the identification ψ n = ψ(na) (3.25) This naive action encounters problems when taking the continuum limit, due to the phenomenom of fermion doubling, causing the appearence of spurious fermionic states. This can be solved using different strategies, such as the introduction of staggered fermions [90]. Here, however, we will not consider this, since our goal is to obtain the proper action for lattice gauge theories in the absence of matter fields, being the previous action enough to show how gauge invariance is achieve in that case, and eliminating the fermions in the end. In the next chapter, we will introduced scalar fields as dynamical matter, where this problem is not present. 27

36 3. Lattice Gauge Theories Figure 3.3.: In the lattice formulation of gauge theory, the matter fields are placed on the vertices of the lattice, whereas the gauge fields correspond to the links joining different vertices. Gauge transformations, corresponding to the group U(1), are applied to the vertices of the lattice. The fermionic fields transform as follows, ψ n V n ψ n ψ n ψ n V n (3.26) where V n are elements of the gauge group, which can be chosen to be different for different vertices of the lattice. To make the action gauge invariant we introduce, as in the continuum theory, a connector U n,µ, which transform as U n,µ V n U n,µ V n+ˆµ (3.27) These variables are placed at the links of the lattice, corresponding U n,µ to the link that joins the vertices n and n + ˆµ (Fig. 3.3). They can be expressed using angular variables, U n,µ = e iθn,µ (3.28) The variables θ n,µ are related to the gauge fields in the continuum theory through the identification [50] θ n,µ = aga a µ(na) (3.29) where g is the coupling strength (a notation for the electric charge e of QED in the lattice). Making use of the connectors, we can write a lattice gauge-invariant action for the Dirac field, S E = a d+1 n m ψ n ψ n ad n,µ ( ψn γ µ ψ n+ˆµ e iθn,µ ψ n+ˆµ γ µ ψ n e iθn,µ ) (3.30) As in the continuum theory, we have to add a kinetic term for the gauge field. The most local gauge-invariant terms, made solely of gauge fields, rendering the correct classical continuum limit, are the plaquette terms (Fig. 3.4), U n,µ U n+ˆµ,ν U n+ˆν,µ U n,ν (3.31) 28

37 3.2. Lattice Formulation of Gauge Theories Figure 3.4.: The plaquette terms contain the product of four connectors, corresponding to the four link of a plaquette. It is inspired in the construction of the gauge-invariant kinetic term in the continuum (3.10). where the trace is computed in the group space. The complete lattice theory for Dirac fields, invariant under U(1) transformation, is given by the action S E =a d+1 n ad n,µ m ψ n ψ n ad 3 ) (U 2g 2 n,µ U n+ˆµ,ν U n+ˆν,µ U n,ν + h.c n,µ,ν ( (3.32) ψn γ µ U n,µ ψ n+ˆµ ψ n+ˆµ γ µ U n,µ n) ψ The factor in front of the plaquette term is chosen, so that, in the continuum limit, the correct factor for the pure-gauge term in the QED action is recovered. Continuum Limit Let us compute the classical continuum limit of the action (3.32). We will use equation (3.29) to show the explicit dependence with the lattice spacing a. First, consider the hopping term, 1 ( 2 ad ψn γ µ e iagaµ(na) ψ n+ˆµ ψ ) n+ˆµ γ µ e iagaµ(na) ψ n (3.33) n,µ We expand the exponentials in terms of a. Using the identification (3.25) and the following definition for the continuous derivative, ψ(na + ˆµa) = ψ(na ˆµa) + 2a µ ψ(na) + O(a 2 ) (3.34) the hopping term can be written as a d+1 n,µ ψ(na)γ µ [ (ψ(na + ˆµa) ψ(na ˆµa)) 2a ] + iga µ (na)ψ(na) (3.35) up to higher order terms in a. These will vanish after taking the continuum limit, a 0, n, with x = na fixed. By doing so, we obtain the correct expression for the covariant derivative, d d+1 x ψ(x)γ µ D µ ψ(x) (3.36) 29

38 3. Lattice Gauge Theories Consider now the plaquette interaction, ad 3 2g 2 n,µ,ν ( ) e iθn,µ e iθ n+ˆµ,ν e iθ n+ˆν,µ e iθn,ν + h.c Using the discrete derivative of the gauge field A ν, (3.37) A ν (na + ˆµa) = A ν (na) + a µ A ν (na) + O(a 2 ) (3.38) as well as the definition of F µν (3.12), we obtain ad 3 2g 2 n,µ,ν ( e ia2 gf µν+o(a 3) ) + h.c (3.39) Expanding the exponential and taking the continuum limit a 0, the expression simplifies to 1 d d+1 x F µν F µν (3.40) 4 up to non-important constants, and the kinetic term of the euclidean theory in the continuum is recovered Hamiltonian Formalism Feynman s path integral formalism of quantum mechanics [91] shows how the transition amplitude of a quantum process can be expressed as the sum over all possible paths the system could take, x 1 Û(t 1, t 0 ) x 0 = Dx e is[x] (3.41) The weight factor e is[x] depends on the action of a particular path, S[x] = t1 t 0 dt L(x(t)) (3.42) where L(x(t)) is the classical Lagrangian of the system, and the boundary conditions x(t 0 ) = x 0, x(t 1 ) = x 1, with x 0 and x 1 fixed, are assumed. The first term of equation (3.41) is the transition probability of a system, initially at state x 0, to be found at the final state x 1, after a time t 1 t 0. The unitary evolution operator Û(t 1, t 0 ), solution to the Schrödinger equation, takes a simple exponential form when the system s Hamiltonian Ĥ is time independent, Û(t 1, t 0 ) = e iĥ(t 1 t 0 ) (3.43) Thus, equation (3.41) provides the connection between to different formulations of quantum mechanics. Note how the fundamental objects in each one of them are mathematically different. On the one hand, we have the Hamiltonian formalism, with a Hamiltonian operator acting on a Hilbert space. On the other hand, the path integral approach makes use of the Lagrangian, which is a functional over fields, 30

39 3.3. Hamiltonian Formalism therefore, a classical object. In the latter, the theory is quantized by summing over all possible paths, including, in this way, the corresponding quantum fluctuations around the classical solution, characterized by δs = 0. Feynman showed how the classical Lagrangian can be obtained from the quantum mechanical Hamiltonian, for non-relativistic systems [91]. In quantum field theory, the path integral formalism is the one that is mainly used, as we have been considering. In the later case, the theory is constructed starting directly from the classical Lagrangian, and defining the transition probabilities as the corresponding path integrals [45, 46]. In both cases, Feyman s approach can be reversed, obtaining the quantum mechanical Hamiltonian corresponding to a particular action, using the method known as transfer matrix (see Appendix A.3). For many applications, working in the Hamiltonian formalism is more convenient. This is clear when one tries to simulate lattice gauge theories using other quantum mechanical systems, such as ultracold atoms in optical lattices, since the latter are represented by a second-quantized many-body Hamiltonian [88]. In this section, we will obtain the Hamiltonian formulation of lattice QED, the so-called Kogut-Susskind Hamiltonian [87], using the transfer matrix method. This model will be studied, paying spatial attention to the implications that gauge invariance has in the structure of the corresponding Hilbert space. Finally, the phenomenon of charge confinement in abelian lattice gauge theories will be reviewed using the Hamiltonian formalism Transfer Matrix Method for Gauge Theories Consider the lattice gauge action (3.32), invariant under the abelian gauge group U(1). The connectors on the links of the lattice are simply complex phases, U n,µ = e iθn,µ (3.44) We will use the transfer matrix method (Appendix A.3) to obtain the Hamiltonian operator corresponding to the pure-gauge part of the action, S = β n,µν cos θ n,µν (3.45) where θ n,µν = θ n+ˆµ,ν θ n,ν θ n+ˆν,µ + θ n,µ (3.46) First, we separate the spatial and temporal directions, by introducing anisotropic couplings, S = β τ cos θ n,0k β cos θ n,ik (3.47) n,ik n,k where the spatial links are labelled with Latin indices, i and k, and the temporal direction τ with the index 0. The coupling β and β τ will be fixed, in the end, so that the correct continuum theory is recovered. Since the theory is gauge invariant, we have the freedom to fix a gauge condition. We will do this only for the links in the temporal direction, θ n,0 = 0. This gauge is called the temporal gauge. Note that τ-independent gauge transformation are still 31

40 3. Lattice Gauge Theories possible, there is still gauge invariance in the spatial links. Due to the gauge condition, the first term in the action simplifies, θ n,0k = θ n+ˆ0,k θ n,k (3.48) In the end, we will take the continuum limit in the temporal direction, a τ 0, therefore, remembering the relation (3.29), we can approximate the first term by cos (θ n+ˆ0,k θ n,k ) (θ n+ˆ0,k θ n,k )2 (3.49) Now, the action has the same structure as (A.39), up to irrelevant constants, S = 1 2 β τ (θ θ n+ˆ0,k n,k )2 β cos θ n,ik (3.50) n,k n,ik Choosing the coupling strengths to take the values β = a d a τ /g 2, β τ = a d 2 /(g 2 a τ ), the Hamiltonian of the theory is obtained by following the same reasoning as in (A.39). In the limit a τ 0, we get the so-called Kogut-Susskind Hamiltonian [87], Ĥ = g2 2 a2 d Ên,k 2 1 g 2 ad 4 cos (ˆθ ˆθ n+î,k n,k ˆθ n+ˆk,i + ˆθ n,i ) (3.51) n,k n,ik where ˆθ n,k and ˆL n,k are canonically conjugate operators, with commutation relations [ˆθ n,k, Ên,k ] = iδ n,n δ k,k (3.52) The two different terms in the Hamiltonian will be called the electric and the magnetic parts, respectively, for reasons that will be clear when we take the continuum limit of the spatial lattice. A generalized discussion of the Hamiltonan formulation of non-abelian lattice gauge theories can be found in [87] Kogut-Susskind Hamiltonian Let us discuss the main properties of the Kogut-Susskind Hamiltonian (3.51), without dynamical matter. In the next chapter we will include dynamical scalar fields that couple to the gauge fields. A discussion involving the Kogut-Susskind Hamiltonian including fermions can be found in [87]. In the following, we will consider a = 1. Gauge Symmetry In order to obtain the Hamiltonian formulation of the pure-gauge action (3.45), we fixed the gauge in the temporal direction, θ n,0 = 0. However, we have not fixed the gauge in the spatial directions. Therefore, the Kogut-Susskind Hamiltonian is still gauge invariant under a restricted subset of local transformations applied to the vertices of the spatial lattice, V n = e iαn (3.53) 32

41 3.3. Hamiltonian Formalism From now on, we will call the latter just gauge transformations, keeping in mind that they refer only to the spatial lattice. Recall the form of the U(1) connectors U n,k (3.44), defined on the links of the lattice. In the Hamiltonian formalism, they correspond to unitary operators, behaving, under gauge transformations, as Û n,k = e iˆθ n,k (3.54) Û n,k V n Û n,k V n+ˆk (3.55) The commutation relations between these and the operators Ên,k, can be obtained from (3.52), [Ên,k, U n,k ] = δ k,k δ n,n U n,k (3.56) In the Hamiltonian formalism, the gauge transformations are generated by operators G n, defined on the vertices, that commute with the Hamiltonian, [Ĝn, Ĥ] = 0 (3.57) Any operator  transforms, under gauge transformations, in the following way,  e i n αnĝnâe i n αnĝn (3.58) The transformation is characterized by the same parameters α n as in (3.53). Since the Hamiltonian Ĥ commute with the generators, it remain invariant under such transformations, e i n αnĝnĥe i n αnĝn = Ĥ (3.59) This is the definition of a gauge-invariant theory in the Hamiltonian formulation. Consider now the Kogut-Susskind Hamiltonian, expressed in terms of the unitary operators Ûn,k, Ĥ = g2 2 n,k Ên,k 2 1 ) 2g (Ûn,i 2 Û n+î,k Û Û + h.c. n+ˆk,i n,k In this case, the generators G n are defined using the operators Ên,k, Ĝ n = k n,ik (Ên,k Ên ˆk,k) (3.60) (3.61) in such a way that they commute with the Hamiltonian, which is easy to see in this case. Applying the transformation (3.58) to Ûn,k, it is easy to verify, using the commutation relations (3.56), that it transforms as Û n,k e iα n Ûn,k e iα n (3.62) The result is the expected gauge transformation for the operators in the link (3.55). 33

42 3. Lattice Gauge Theories The eigenvalues of the operators Ĝn are called static charges q n. The Hilbert H of the system is divided into sectors, each one associated to a different static charge configuration, H = {q n} H({q n }) (3.63) such that Ĝ n ψ({q n }) = q n ψ({q n }), ψ({q n }) H({q n }) (3.64) Since the Hamiltonian commutes with the generators of the gauge transformations, the dynamics can not mix different sectors. The above equation is known as the Gauss s law. The reason for this will be clear once we consider the continuum limit of the theory. Hilbert Space Let us study now how the different operators that appear in the Kogut-Susskind Hamiltonian (3.60) act inside a given gauge-invariant sector of the Hilbert space, characterized by a set of static charges. The lattice version of QED that we have considered so far makes use of angular variables, θ n,k, related in the continuum limit to the gauge fields A k. This formulation of lattice QED, called compact (denoted cqed), is not the only possibility that renders the correct continuum limit [47]. It is interesting, however, because it gives rise, for finite lattice spacings, to confinement of abelian charges, as we will see as well as leading, naturally, to the quantization of dynamical charges [47]. The angular variables θ n,k, representing the gauge fields, are periodic, with period 2π. In the Hamiltonian formulation, the corresponding canonically conjugated operators, Ê n,k, behave like angular momenta. The commutation relations (3.56) imply that the unitary operators, Û n,k and Û n,k, are raising and lowering operators, with respect to Ên,k. Consider the Hilbert space of a given link (n, k). In the strong coupling limit g 1, an interesting basis is the one formed by the eigenstates m of the operator Ên,k, Ê n,k m = m m (3.65) where m are integers numbers, positive and negative [50]. We shall refer to them as the electric field on the links, since Ên,k is related to the electric field operator in the continuum theory. As opposed to the latter, the lattice version of the electric field has an integer spectrum. Using Gauss s law, one can see how this fact leads to the quantization of charges [47]. The operators Ûn,k and Û n,k act on these states by adding and subtracting, respectively, one quantum of electric field m, Û n,k m = m + 1, Û n,k m = m 1 (3.66) The effect of the plaquette terms in (3.60) is to increase the electric field in two links of the corresponding plaquette, and lowering it in the other two. According to Gauss s law (3.61), this can only be done in two different ways (Fig. 3.5). 34

43 3.3. Hamiltonian Formalism Figure 3.5.: The product of four operators, Û n,k and Û n,k, each one placed in one of the links forming a plaquette, is the most local gauge-invariant interaction, made solely of gauge field operators, that we can write. In order to fulfill Gauss s law, the change in the electric field must be the same in the bottom and right links, and opposite to the change in the top and left ones. Only two gauge-invariant products can be constructed in such fashion. Figure 3.6.: In any static charge sector, the physical states are generated by applying gauge-invariant products of operators to the ground state of the electric part of the Hamiltonian. The only way to achieve this is by forming closed loops in the lattice, made out of Ûn,k and Û n,k operators in a gauge-invariant way. If the loop goes through the same link twice, it breaks in two smaller loops. The total Hilbert space of the system is the tensor product, over all the links of the lattice, of the individual Hilbert spaces on each link. However, this space contains too many states. The physical Hilbert space is formed just by the gauge invariant states belonging to a given sector. These can be constructed by applying gauge-invariant operators to the vacuum 0. This is defined as the tensor product of the ground states of the electric part of the Hamiltonian (3.60) in each link, this is, the state with zero electric field (m = 0). Note that the vacuum is not the ground state of the whole Hamiltonian, due to the plaquette terms. The only non-trivial gauge-invariant operators one can construct are the lattice versions of the Wilson loops (3.14), formed by products of operators Ûn,k and Û n,k, acting on links along a closed path in the lattice, in such a way that the Gauss s law is preserved (Fig. 3.6). If no static charges are present, the physical Hilbert space can be generated by applying all possible Wilson loop operators to the vacuum, representing states with closed electric flux lines. 35

44 3. Lattice Gauge Theories Figure 3.7.: Two separated static charges are joined, in the strong coupling limit, by an electric flux line. When the coupling constant starts to decrease, more and more pure-gauge loops can appear from the vacuum. If some link in the loop coincides with the electric flux line between the charges (in the opposite direction), it will produce a deformation of the flux line. Le us study now a sector characterized by a non-zero static charge configuration. The sum of the generators (3.61) for all links is zero, due to the cancellation of the electric field operators. As a consequence, the sum of all static charges must be zero as well. Consider, for example, two static charges of opposite signs, +1 and 1, separated by R links. They are placed, for simplicity, in the same lattice direction, with positions n and n + Rˆk (Fig. 3.7). The ground state of the electric part of the Hamiltonian includes, in this case, non-zero values for the electric field in the links between the static charges, Û n,k... Ûn+(R 1)ˆk,k 0 (3.67) which corresponds to an electric flux line connecting them. The creation of this line is a consequence of gauge invariance. The rest of the physical states are obtained by applying closed Wilson loop operators to this one, as in the case of zero static charge, producing closed loops of electric flux away from the static charge configuration, as well as deformations of the electric flux line joining them (Fig. 3.7). Charge Confinement in Abelian Theories The phenomenon of charge confinement appears in quantum field theories, defined in a continuous space-time, only when these are invariant under non-abelian gauge groups [45, 46]. For lattice gauge theories, however, it can appear even if the gauge group is abelian. This fact allows us to study charge confinement in simpler theories, which can be regarded as toy models, providing useful insight for the study of more complicated and realistic cases (QCD). Although the cause of the phenomenon may be different in these models, compared to the continuum case, the techniques used to study it are similar [47]. Here, we will study the confinement of static charges for the pure-gauge version of cqed. In the next chapter, this phenomenon will be reviewed with the inclusion of dynamical matter. In the strong coupling limit (g 1) the electric part of the Hamiltonian (3.60), H E, dominates over the magnetic one, H B (plaquette terms), which can be regarded as 36

45 3.3. Hamiltonian Formalism Figure 3.8.: The relation between the expectation value of the Wilson loop and the potential between two static charges of opposite sign can be shown, qualitatively, by calculating the former for a rectangular loop in a plane formed by the temporal and one spatial dimension. a perturbation. The ground state of the unperturbed Hamiltonian, H 0 = H E, when two static charges are present, is formed by the shortest possible tube of electric field excitations joining the two charges, as we have seen (3.67). The energy of such configuration is E 0 (R) = g2 2 R (3.68) This energy, stored in the electric flux tube, can be regarded as the potential between the two charges, increasing linearly with the distance. This potential gives rise to a force between them that is independent on the separation. Such a force is enough to confine the charges [87]. When the perturbation H B is considered, the state (3.67) is not an eigenstate of the system any longer. The flux line starts fluctuating, and corrections of order O(g 2 ) are added to the energy. For weak couplings, perturbation theory breaks. This is precisely the reason why Wilson introduced lattice gauge theories in the first place (3.14). In order to study the presence of confinement, also at weak couplings, he computed the expectation value of what later were to be called Wilson loops, for different values of g [49]. This gauge-invariant quantities can distinguish between different phases of a lattice gauge theory, in the same way as order parameters are used to distinguish between different phases in condensed matter [83]. Wilson showed how lattice cqed have two distinct phases in dimensions, separated by a critical point at g = g c a confining one for g > g c, and a nonconfining, or Coulomb phase, for g < g c. In dimensions, however, the theory confines for all values of the coupling constant, an effect that is thought to have a topological origin [47] (more related to the origin of confinement in non-abelian theories). Confinement in other theories, such as non-abelian Yang-Mills theories, can be studied using similar techniques [84]. In the Hamiltonian formalism, the relation between the Wilson loop and the confinement mechanism can be argued, qualitatively, by calculating the expectation value of such quantity for a simple square loop in one temporal and one spatial dimension (Fig. 3.8), 37

46 3. Lattice Gauge Theories W (RT ) = Ω e ie P dxµ A µ(x) Ω (3.69) In this formalism, we can use the temporal gauge, A 0 (x) = 0. The integral along the closed path may be decomposed in two integrals, both in the z direction, one for t = 0 and another for t = T, W (RT ) = Ω e ie 0 R dz A 1(z,T ) e ie R 0 dz A 1(z,0) Ω (3.70) We change now from the Heisenberg to the Schrödinger picture, W (RT ) = Ω e iht e ie 0 R dz A 1(z,T ) e iht e ie R 0 dz A 1(z,0) Ω (3.71) Setting the ground state energy to zero and defining the state Ψ = e ie R 0 dz A 1(z,T ) Ω (3.72) we can expressed the expectation value of the Wilson loop as W (RT ) = Ψ e iht Ψ (3.73) If we perform a gauge transformation, A 1 (z) A 1 (z) 1 e zα(z), it is easy to see that the state ψ transforms as ψ e iα(r) e iα(0) ψ (3.74) corresponding to a configuration of two static charges of opposite sign, separated by a distance R. This state can be written in terms of the eigenstates of H, Ψ = n n Ψ n, H n = E n n (3.75) Finally, changing to imaginary time, t it, and taking the limit T, we get W (RT ) = n n Ψ 2 e EnT 0 Ψ 2 e E 0(R)T (3.76) where only the term in the sum corresponding to the minimum energy survives. The energy E 0 (R) can be regarded as the potential V (R) energy between the two charges. Therefore, if we find that lim log ( W (RT ) ) RT (3.77) T we can conclude that V (R) R, and the charges are confined. This scaling with the area of the loop is called the Area law behaviour, and it is a criterion to distinguish confined phases from non-confined ones, being the latter characterized by a scaling with the perimeter of the loop [49]. The argument generalizes for non-abelian gauge theories [47, 85]. 38

47 3.3. Hamiltonian Formalism Classical Continuum Limit To finish our discussion of the pure-gauge Kogut-Susskind Hamiltonian, we will check, for consistency, its classical continuum limit, taking the lattice spacing to zero. In the Lagrangian formulation, we related the classical field θ n,k to the vector potential A k (3.29). Here, we will do the same for the corresponding operators, ˆθ n,k = Âk(na) (3.78) Since, in the Hamiltonian formulation of quantum electrodynamics, the vector potential is canonically conjugated with the electric field operator, it is natural to relate the latter, in the continuum limit, with the operator Ên,k. Imposing the relation Ê n,k = ad 1 Ê k (na) (3.79) g we obtain the correct commutation relations between the vector potential and the electric field, when a 0, [Âk(na), Êk (n a)] = ia d δ n,n δ k,k [Â(x), Ê(x )] = iδ(x x )δ k,k (3.80) The second term in the Hamiltonian (3.51) can be related to the magnetic field by recalling the relation between this quantity and the vector potential, B k = ɛ ijk j A k (3.81) Using this relation, and the definition of j A k (3.38), we can rewrite the second term in the Hamiltonian as 1 g 2 ad 4 cos(a 2 gb k (na) + O(a 3 )) = 1 ( g 2 ad ) 2 a4 g 2 Bk 2 (na) + O(a5 ) n,k n,k (3.82) which, using a d n dx d, becomes, up to constant terms, 1 d d x ˆB k 2 (x) (3.83) 2 k when we take a 0. Analogously, the first term in the Hamiltonian (3.51) converges to 1 d d x Êk 2 (x) (3.84) 2 k By denoting Ê2 = k Ê2 k and ˆB 2 = ˆB k k 2, the continuum limit of the Kogut- Susskind Hamiltonian takes the form Ĥ = 1 d d (Ê2 x (x) + 2 ˆB (x)) 2 (3.85) which is just the Hamiltonian operator for QED. It can be interpreted as the energy stored in the electromagnetic field. Finally, let us consider the continuum limit of Gauss s law. Using equation (3.61) and the relation (3.79), we get (Êk ˆk)) (na) Êk(na k = 0 Ê(x) = 0 (3.86) which is the usual Gauss s law in electrodynamics in the absence of matter, hence the name. 39

48

49 4. Abelian-Higgs Theory So far, we have discussed different aspects of lattice gauge theories in cases where only pure-gauge degrees of freedom were considered. The introduction of dynamical matter fields can exert a dramatic effect on the behaviour of the gauge theory. The simple explanation of confinement of static charges, for example, introduced in the previous chapter in terms of unbreakable electric flux lines, must be modified when matter particles and antiparticles are allowed to pop out from the vacuum. But not only this phenomenon changes upon the introduction of dynamical matter. In this chapter, we will study new phases appearing in lattice gauge theories, such as those related to the Higgs mechanism and the spontaneous breaking of symmetries. We will only consider theories where scalar matter fields are present, since these are enough to observe a rich and broad plethora of new phenomena, and are easier to study than their fermionic counterparts. In particular, we will focused on the lattice version of the so-called Abelian-Higgs theory [92]. Studies of lattice gauge theories with dynamical fermionic fields can be found in [47, 49 51, 87, 90]. The structure of the chapter will be as follows. First, we will review the Higgs mechanism for continuum gauge theories, since this phenomenon appears when scalar matter fields are added to lattice gauge theories. We will motivate, as well, the use of the Abelian-Higgs theory as a toy model to help us understand this mechanism. The lattice version of such theory is achieve by adding scalar fields to the vertices of a lattice gauge theory, coupling the scalar and gauge degrees of freedom in a proper way, such that gauge invariance is satisfied. The different phases of this theory, in the case where the radial part of the scalar field is fixed, will be discussed for different representations of the abelian U(1) gauge group. Finally, the corresponding Hamiltonian will be computed and studied, setting the proper framework to perform, in the next chapter, quantum simulations of high energy phenomena, such as charge confinement and the Higgs mechanism, using ultracold atoms in optical lattices The Brout-Englert-Higgs Mechanism: A Toy Model The Brout-Englert-Higgs mechanism will be reviewed for quantum field theories in the continuum. First, the spontaneous breaking of a global symmetries will be considered, along with Goldstone s theorem. The breaking of the symmetry is connected to a non-zero expectation value of the field in the ground state. We will explain the consequences of this effect when the symmetry is promoted to a local one (Brout- Englert-Higgs mechanism), using the Abelian-Higgs toy model. Even though the situation seems analogous, in the latter case, to the spontaneous breaking of the local (gauge) symmetry, an alternative argument can be presented, where this is not necessary to explain the mechanism. 41

50 4. Abelian-Higgs Theory Spontaneous Breaking of Global Symmetries Spontaneous symmetry breaking (SSB) refers to a situation where the Lagrangian of the theory is invariant under some symmetry, whereas the ground state of the system is not. The concept is often associated to phase transitions between phases characterized by an invariant ground state to others where this is not the case any longer. A prominent example in condensed matter are ferromagnetic materials [83]. The magnetization of such systems, referring to the average direction where the individual spins are pointing, acquires a non-zero value below some critical temperature, showing explicitly that the state of the system is not invariant under rotations any more. Here we will only consider the case of continuous symmetries. Goldstone s theorem shows that one massless particle appear for each broken generator of such symmetry [93]. These particles are called Goldstone bosons, once the theory is quantized. A good way to see this mechanism in action is by analyzing one of the simplest field theories where it occurs, the φ 4 scalar field theory. Consider the following Lagrangian, L = ( µ φ )( µ φ) + m 2 φ φ λ 4 (φ φ) 2 (4.1) where φ is a complex scalar field, self-interacting through a quartic term. This Lagrangian is invariant under global U(1) transformations, φ(x) e iα φ(x), with constant α. To analyze the symmetry breaking in this model, we will employ a semiclassical approximation. To know whether the ground state 0 is symmetric we just have to check if the expectation value of the field 0 φ 0 is zero or not, since a symmetric state necessary implies a zero value for this quantity. This is an example of an order parameter, used to distinguish between different phases of a system, similar to the magnetization for the ferromagnetic case. It can be shown [48] that this quantity is equal, up to quantum corrections, to the classical ground state of the system, this is, the solution of the Euler-Lagrange equations of motion. Neglecting the corrections allow us to understand the breaking of the symmetry from a qualitative point of view. The Lagrangian of this theory is composed of a kinetic part (including derivatives) and a potential energy term, V (φ) = m 2 φ 2 + λ 4 φ 4 (4.2) The classical solution is the one that minimize the energy of the system. This is achieved by an homogeneous field (such that the kinetic part vanishes) that minimizes the potential energy. For m 2 < 0 this happens for the zero field configuration φ = 0. On the other hand, when m 2 > 0, this state is unstable and the ground state corresponds to a constant field with modulus φ = 2 λ 2m. The situation is represented in figure (4.1). The system chooses, randomly, a new state from the degenerate ground state. The degeneration comes from the freedom in choosing the phase of the field. Neither of the new ground states is invariant under U(1) transformations. In this situation, one speaks about spontaneous breaking of the symmetry. Let us study now the quantum fluctuations around the new ground state [46]. In order to do this, we write the complex field φ(x) in terms of two real fields, σ(x) and π(x), 42

51 4.1. The Brout-Englert-Higgs Mechanism: A Toy Model Figure 4.1.: Mexican hat potential V (φ) = m 2 φ 2 + λ 4 φ 4 as a function of the complex field φ. The set of vacuum states correspond to the bottom of the hat. Massive excitations correspond to fluctuations in the radial value of the field, whereas massless ones correspond to angular fluctuations. Figure from [94]. φ(x) = ( ) 2m 2 λ + 1 σ(x) e i λπ(x) 2m (4.3) 2 Introducing this representation into the Lagrangian gives us the following expression L = 1 2 ( µσ) ( 2m) 2 σ ( µπ) 2 + non-quadratic terms (4.4) This Lagrangian corresponds to the linear sigma model. The field σ(x) is massive with mass 2m. There is no mass term, however, for the field π(x). This is the massless Goldstone boson from Goldsone s theorem, corresponding to angular fluctuations in the phase of φ. The massive field, on the other hand, correspond to radial fluctuations around the vacuum (Fig. 4.1) SSB in the Presence of Local Symmetries Let us consider now the local symmetry case. As we have seen in section (3.1.1), to promote the global U(1) symmetry to a local one, we need to introduce a gauge field A µ. This field enters into the Lagrangian by changing the derivatives to their covariant version (3.7). If we do this for the φ 4 -scalar field theory (4.1), we obtain the following Lagrangian, L = 1 4 F µνf µν + ( µ φ + iea µ φ )( µ φ iea µ φ) + m 2 φ φ λ 2 (φ φ) 2 (4.5) This is called the Abelian-Higgs theory [46]. As for the global symmetry case, the 2m expectation value of the field in the vacuum is 0 φ 0 = 2 λ v 2, for m 2 > 0. The complex field φ(x) can be expanded around this value, using, as before, two real fields σ(x) and π(x), ( ) v + σ(x) φ(x) = e i π(x) v (4.6) 2 43

52 4. Abelian-Higgs Theory If we plug this expression into the Lagrangian, we obtain, again, a massive σ field, with mass 2m, and a massless π field. Checking the pure-gauge terms we find L = 1 4 F µνf µν e2 v 2 A µ A µ +... (4.7) suggesting that the gauge field has become massive, with mass m A = ev. Let us consider now a simplified situation. In the limit m, λ, with v fixed, the Lagrangian reduces to L = 1 4 F µν F µν m2 AA µ A µ µ π µ π m A A µ µ π (4.8) We still have the gauge freedom, so we can transform both fields, A µ and π, A µ A µ (x) + 1 e µα(x), π(x) π(x) vα(x) (4.9) Using the unitary gauge, α(x) = π(x)/v, we can eliminate the massless field π(x) from the theory. We are left, therefore, with a massive scalar field σ(x) and a massive gauge field A µ (x). The gauge boson has swallowed the Goldstone boson, which no longer appears in the theory. The Goldstone theorem breaks, therefore, for local symmetries. The process we have just described is the celebrated Higgs mechanism. It was first proposed in 1962 by P.W Anderson [95], without working out an explicit relativistic model. This was done independently, in 1964, by three groups: Robert Brout and Franois Englert [96]; Peter Higgs [97]; and Gerald Guralnik, Carl Richard Hagen, and Tom Kibble [98]. Here, we have consider the abelian gauge group U(1), but the mechanism can also be applied to non-abelian gauge groups. For example, for the group SU(2) U(1), that describes electroweak interactions in the Standard Model, the Higgs mechanism allows the corresponding gauge bosons, W ± and Z, to acquire a non-zero mass [99]. Although the Higgs mechanism looks similar to the spontaneous symmetry breaking phenomenon, introduced for continuous global symmetries, it is conceptually different, being the two of them offen confused. In fact, as it can be shown [100], local symmetries can not be spontaneously broken. In our previous description, the symmetry was explicitly broken after fixing a particular gauge. The same conclusions (generation of massive gauge boson, and vanishing of the massless one) can be achieved, however, without breaking the gauge symmetry. To see this, we apply the following field transformation to the Abelian-Higgs Lagrangian (4.5), ( ) v + σ(x) φ(x) = e i π(x) v, Aµ = B µ ve µπ(x) (4.10) The transformation might seem the same as the one we perform before, but in this case the transformation on A µ is not a gauge transformation, since π(x) is not a fixed function, but a new field variable, same as B µ. The new Lagrangian shows, again, only a massive gauge field and a massive scalar field. Now, however, the gauge freedom is still present (since we did not fixed the gauge), and the new fields are explicitly gauge invariant. Since the phase π is no longer present, the new vacuum 0 φ 0 = 2m 2 /λ v/ 2 is not degenerate. The key to the Higgs mechanism is, therefore, the non-zero value of the expectation value of the Higgs field in the vacuum, not the breaking of the gauge symmetry. 44

53 4.2. Lattice Abelian-Higgs Action 4.2. Lattice Abelian-Higgs Action Despite its simplicity, the Abelian-Higgs theory shows the main features of the Higgs mechanism. A similar construction in a discretized space-time is possible [92, ]. This is done by adding dynamical complex scalar fields to an abelian gaugeinvariant action (3.45). The scalar Higgs-type fields are defined on the vertices of the lattice. The interaction between different vertices is mediated by coupling these fields to the gauge fields on the links. The resulting gauge-invariant lattice theory will allow us to study both the Higgs mechanism and the confinement problem for dynamical matter. To simplify our problem, we freeze out the radial mode of the Higgs field to a fixed length R. The resulting scalar fields are represented by complex phases, ρ n = R e iϕn, ϕ n [0, 2π) (4.11) This approximation is correct if the radial component is high enough [105]. The phase diagram of the Abelian-Higgs lattice theory depends crucially on whether the Higgs field transform under the fundamental representation of U(1) or not [92]. The irreducible representations of this Lie group are characterized by integer numbers q [106]. These numbers are interpreted as the physical charges of the matter fields that are coupled to the corresponding gauge fields. The fundamental representation corresponds to q = 1. For any value of q, the elements of the group are represented by complex phases, U q (θ) = e iqθ (4.12) We will study the phase diagram of the theory for different representation, observing the different phases that one can obtained depending on the chosen representation. The euclidean action for the d+1 dimensional lattice gauge theory involving radially frozen fields, transforming under the q-representation of U(1), reads [92] S E = R2 a d 1 [ ] e iϕn e iqθn,µ e iϕ n+ˆµ + h.c. ad 3 2 g 2 cos θ n,µν (4.13) n,µ The first term corresponds to the interaction between the Higgs field at different vertices, mediated by the gauge field. The second one is the usual pure-gauge kinetic term for the gauge field. The coupling constants have been chosen to obtain the correct classical continuum limit. Classical Continuum Limit Let us show how the Abelian-Higgs theory (4.8) is recovered from (4.13) with the Higgs field transforming in the fundamental representation of the gauge group after taking the classical continuum limit, a 0. The limit of the pure-gauge part of (4.13) was obtained in section (3.2.3). Consider now the interacting part of the action. The discrete and continuum Higgs fields are connected through the correspondence ϕ n = 1 π(na) (4.14) R The continuous derivatives are approximated by finite differences, µν 45

54 4. Abelian-Higgs Theory ϕ n+µ ϕ n = a R µπ(na) + O(a 2 ) (4.15) The interacting part of the action can be expressed as S E,int = R 2 a d 1 n,µ cos(ϕ n+µ ϕ n θ n,µ ) (4.16) Expanding the cosine in terms of the lattice spacing a, and using the relations (4.14), (4.15) and (3.29), we get S E,int = R 2 a [ d 1 1 a 2 ( 1 ] R µπ(na) ea µ (na)) 2 + O(a 3 ) (4.17) n,µ Calculating the squared term, and forgetting about non-important constants, the action becomes S E,int = R2 a d+1 [ 1 2 R 2 µ π(na) µ π(na) + e 2 A µ (na)a µ (na) n,µ ] (4.18) 2 e R Aµ (na) µ π(na) + O(a) Now we take the lattice spacing to zero, together with n and na = x fixed, obtaining [ 1 S E,int = dx d 2 µ π(x) µ π(x) + 1 ] 2 m2 AA µ (x)a µ (x) m A A µ (x) µ π(x) (4.19) where the sum turns to an integral, and the rest of the terms vanish in the limit. The mass of the gauge field, m A, is defined as m A = Re (4.20) which is the expected Abelian-Higgs action. As usual, the correct continuum limit of the quantum theory depends on its renormalization properties [47, 50] Phase Diagram The phase diagram for the lattice gauge theory (4.13) depends, as we have already mentioned, on the specific group representation under which the Higgs field transforms [92, 105]. In particular, one finds a phase diagram with two phases if the representation is the fundamental one (q = 1), and another one with three phases if any other representation is used. Let us discuss first the limit models that can be recovered from the complete theory. XY Model (g 0) In this case, the gauge fields are frozen to some specific configuration. We can chose the fields U n,k = e iθ n,k to be the identity, in the axial gauge. The action of the theory reduces to S E = R2 a d 1 [ ] e i(ϕn ϕn+ˆµ) + h.c. (4.21) 2 n,µ 46

55 4.2. Lattice Abelian-Higgs Action This is the well-known XY model, invariant under global U(1) transformations [83]. For d + 1 > 2, there is a phase transition at a critical point point R c, involving the spontaneous breaking of the continuous global symmetry. For R > R c, the expectation value of the field in the ground state, ρ, is different from zero, and the correlation functions has the asymptotic behaviour ρ 0 ρ n e 1 n d 1 (4.22) when n, where a power law decay is observed. On the other hand, for R < R c, the symmetry is restored. The correlation functions decay exponentially, ρ 0 ρ n e n /ξ (4.23) when n, where ξ is the correlation length, related to the mass of the particles [92]. In this case, the expectation value of the field is zero. This is true even though the radial part of the field is kept fixed. In the semiclassical approximation, the nonzero expectation value of the field would be equal to R, however, this value can be reduced to zero when quantum fluctuations are considered. For d + 1 = 2 the system presents also a phase transition, but, in this case, it is not related to the spontaneous breaking of the symmetry, since ρ = 0 for every value of R. For small values of this parameter, the correlation function decays exponentially, whereas for large values it shows, again, a power law decay. This phase transition is the well-known Berezinskii-Kosterlitz-Thouless transition, and it has a topological origin [83]. Pure-gauge Theory (R = 0) We have already commented on the main properties of the pure-gauge theory. For dimensions d + 1 4, a critical point at g c separates the Coulomb (g < g c ) and the confining phase (g > g c ). The latter is characterized by an area-law behaviour of the Wilson loop, C Γ = U ( ) n,k e A(Γ) (4.24) (n,k) Γ where A(Γ) is the area enclosed by the path Γ, and the proper operator (U n,k or U n,k ) is chosen at each link to form a gauge-invariant product. In the Coulomb phase, however, the Wilson loop decays with the perimeter of the loop, C Γ = U ( ) n,k e P (Γ) (4.25) (n,k) Γ This phase is characterized by free static charges and massless photons. Complete Theory We will describe now the properties of the possible phases one can find in the complete theory for d If the Higgs field transform according to a representation of U(1) with q > 1 the lattice theory presents three phases (Fig. 4.2a), 47

56 4. Abelian-Higgs Theory Figure 4.2.: Qualitative phase diagram of the Abelian-Higgs lattice theory in d + 1 dimensions, with d 3 for q = 1 (a) and q = 2 (b). The solid lines represent firstorder phase transitions, broken lines represent higher-order transitions. The figure is adapted from [105]. Higgs phase (large R, small g) In this phase the photon is massive (Higgs mechanism), the force between different particles is short-range, and the Wilson loop exhibits a perimeter law (no charge confinement). Coulomb phase (small R, small g) Here, the photon is massless, and the charges are free, as well. Confinement phase (large g) In this regime, the Wilson loop has an area-law behaviour, there are no free charges (confinement) and the photon has a finite mass. If the Higgs field transform under the fundamental representation of the gauge group (q = 1), the theory presents only two different phases (Fig. 4.2b), Higgs-confinement phase Massive photon characterized by a connected field-strength correlation function that decays exponentially, C( n ) = e i(θn,µν θ 0,µν) e iθn,µν 2 e µ n (4.26) for n, where µ is the mass of the photon. The forces are exponentially damped and, therefore, short-ranged. There are, also, no free charges in the spectrum. Coulomb phase Here, the photon is massless and the forces are long-range. The connected field-strength correlation function decays, asymptotically, as a power law, C( n ) = e i(θn,µν θ 0,µν) e iθn,µν 2 1 K r d+1 (4.27) 48

57 4.3. Lattice Abelian-Higgs Hamiltonian No charge confinement is present in this phase. Free charges can be found in the spectrum. It can be shown that, in the fundamental representation, there is no phase boundary between the Higgs and confining regimes [92]. The ground state energy and all Green s functions are analytic functions in a region of the (R,g) space that include these regimes. The two phases are, therefore, continuously connected. In fact, all candidates to distinguish between them, made out of products of local operators, have the same qualitatively behaviour in both of them. It is also important to notice that the Wilson loop behaviour fails as a signature of confinement in this representation [92]. For all R 0 it decays with the perimeter of the loop. Finally, in the dimensional case, just as in the pure-gauge theory, there is no Coulomb phase for finite coupling constants. In the fundamental representation, there is only one phase, the Higgs-confinement one, whereas for q > 1, there is a phase boundary separating the Higgs and confinement regimes Lattice Abelian-Higgs Hamiltonian The goal of this thesis is to propose a quantum simulation for the Abelian-Higgs theory using ultracold atoms in optical lattices. In order to map the degrees of freedom of this theory to those of the atomic system, we calculate, first, the corresponding Hamiltonian. This is achieved by using the transfer matrix method. As usual, we separate µ = (0, k) into temporal and spatial directions. The interacting part of the action is expressed, in the temporal gauge, θ n,0 = 0, as S E,int = R2 a d cos(ϕ ϕ n+ˆ0 n ) R2 a d 2 a τ cos(ϕ n+ˆk ϕ n θ n,k ) (4.28) a τ n where a τ and a are the lattice spacings in the temporal and spatial directions, respectively. Recalling the dependence of the field ϕ n with a τ (4.15), we expand the cosine. We get, disregarding irrelevant terms, S E,int = R2 a d 2a τ n,k (ϕ ϕ n+ˆ0 n )2 R 2 a d 2 a τ cos(ϕ n+ˆk ϕ n θ n,k ) (4.29) n Following the same steps as in section (A.3), we can extract the corresponding Hamiltonian, Ĥ = 1 2R 2 a d n n,k ˆQ 2 n R2 [ ] 2 ad 2 e i ˆϕn e iˆθ n,k e i ˆϕn + h.c. n,k where ˆQ n is canonically conjugate operator of ˆϕ n, with commutation relations (4.30) [ ˆQ n, ˆϕ n ] = iδ n,n (4.31) The total Hamiltonian, including the terms corresponding to the pure-gauge part of the action, reads 49

58 4. Abelian-Higgs Theory Ĥ = 1 2R 2 a d n + g2 2 n,k ˆQ 2 n R2 [ ] 2 ad 2 ˆφn Û ˆφ n+ˆk + h.c. n,k n,k Ê 2 n,k 1 2g 2 n,ik ) (4.32) Û (Ûn,i n+î,k Û Û + h.c. n+ˆk,i n,k This Hamiltonian, like the Kogut-Susskind one (3.60), is invariant under the subgroup of U(1) composed of spatial gauge transformations (3.53). Here, the Hilbert space where the Hamiltonian acts, is divided, as well, into different gauge-invariant sectors (3.63). These are generated by the eigenstates of the gauge-symmetry generators G n associated to a given static charge configuration (3.64). In this case, the local operators that commute with the Hamiltonian (4.32) are Ĝ n = (Ên,k Ên ˆk,k) + ˆQ n (4.33) k This Gauss s law, thus, relates the operators Ên,k and ˆQ n inside a gauge-invariant sector. Hilbert Space The structure of the Hilbert space associated to the Abelian-Higgs theory is similar to the one discussed for the Kogut-Susskind Hamiltonian (3.60), where the states belonging to the Hilbert space of the link (n, k) were represented using, as a basis, the eigenstates of the operator Ên,k (3.65). In that picture, U n,k and U n,k correspond to raising and lowering operators, respectively, when acting on such basis. The situation is analogous for the operators living on the Hilbert space of a given lattice vertex. The operator ˆϕ n, acting on the vertex n, is a phase operator, which results in a discrete spectrum for its canonically conjugated momentum, ˆQn, ˆQ n Q = Q n Q (4.34) where Q n are integer numbers. The state Q = n Q n contains different charges placed at each vertex. We will call ˆQn the charge operator. The operator ˆφ n = e i ˆϕn raises one unit of charge at the vertex n when acting on the state Q. On the other hand, ˆφ n = e i ˆϕn lowers one unit of charge on the corresponding vertex, ˆφ n Q n = Q n + 1 (4.35) ˆφ n Q n = Q n 1 This comes from the commutation relations, [ ˆφ n, ˆQ n ] = δ n,n ˆφ n [ ˆφ n, ˆQ n ] = δ n,n ˆφn (4.36) which are derived from (4.31). In general, if the Higgs field transforms under the q-representation of U(1), the operators ˆφ n and ˆφ n will change the charge Q n by q units, when applied to the state Q n. 50

59 4.3. Lattice Abelian-Higgs Hamiltonian Figure 4.3.: A meson-like state containing two dynamical charges of opposite sign, and joined through a electric flux line can be formed from the vacuum. Note that the roles of ˆφ n and ˆφ n on the vertices are interchanged with respect to U n,k and U n,k on the links. This is just a convention, resulting from the definition of the conjugate momentum given by the commutation relations (3.56), (4.31). The charge states Q are eigenstates of the Hamiltonian (4.32) only in the limit R 0, similar to the electric field eigenstates on the links, for g. For larger values of R, the charge is no longer a good quantum number, and the total charge starts fluctuating in every vertex, due to the interacting part of the Hamiltonian. Consider, for example, the term ˆφ n Û n,k ˆφ n+ˆk (4.37) When acting on the state Q, it raises the charge on the vertex n + ˆk and lower it on n, raising, as well, the electric flux on the link (n, ˆk) that separates them. The opposite process is achieve by the hermitian conjugate ˆφ Û ˆφ n+ˆk n,k n (4.38) All these processes are gauge invariant, which means that the system will not leave the static charge sector where it started. Let us discussed how the set of states belonging to a particular sector are generated (physical states). Consider, for simplicity, the sector where no static charges are present. As discussed for the Kogut-Susskind Hamiltonian, the vacuum state is the one with zero electric field in every link and zero charges in every vertex. Again, this is the ground state of the Hamiltonian only in the limit R 0, g. We have two types of gauge-invariant products of operators that we can apply to the vacuum to generate all the gauge-invariant states. First, we can apply gauge-invariant closed Wilson loops, made out of products of operators U n,k and U n,k (Fig. 3.6). As opposed to the pure-gauge Kogut-Susskind Hamiltonian, the presence of dynamical matter allows for a different type of gauge-invariant term. They are of the form ˆφ n Û n,k...ûn+n ˆk ˆφ,k n+n (4.39) These products, together with their hermitian conjugates, create meson-like states when acting on the vacuum. They are composed of pairs of opposite charges with an electric flux line joining them (Fig. 4.3). In summary, any physical state belonging to a given static charge sector is composed of mesons-like configurations and closed loops of electric flux. 51

60 4. Abelian-Higgs Theory Figure 4.4.: If a meson-like excitation is created along the electric flux line of an existing mesonic state, the latter can break, giving birth to two separated, and smaller, mesons Confinement in the Presence of Dynamical Matter In the previous section, we studied the confinement phenomenon, in the strong coupling limit (g 1), by calculating the potential energy between two separated charges. The linear dependence of the potential with the distance (3.68) guaranteed that the charges will remain confined, forming a bound state. This naive picture was valid because there was no mechanism that allowed the electric flux line between the charges to break. We have seen, however, that the dynamics generated by the Abelian-Higgs Hamiltonian (4.32) involve, if R is different from zero, the creation of meson-like configurations. Imagine, for example, that an excitation corresponding to the gauge-invariant operator (4.39) is created on top of the path that joins two charges (Fig. 4.4). In that case, the electric flux line between these charges can break, producing two new (and shorter) meson-like states. Notice that the charges forming the new mesons are still bounded, with a linear potential between them, therefore confined. However, checking the potential between two separated charges is no longer a good measure of confinement, since they will not be bounded to each other once the flux line breaks (they are still bounded though, but to new charges). This string-breaking behaviour also prevents the Wilson loop from being a signature for confinement in the strong coupling regime. A theoretical tool to detect charge confinement, when dynamical matter is present in the theory, is still lacking [107]. Numerical simulations of this phenomenon are also challenging, due to the so-called sign problem [52]. This problem appears, in quantum field theory, when one deals with a finite density of strongly correlated fermions. Fermionic wavefunctions change sign when two particles are interchanged, which makes the corresponding path integrals highly oscillatory, and very hard to evaluate. Quantum simulations of lattice gauge theories, on the other hand, could provide a better understanding of the confinement mechanism, thanks to the ability to measure, in real time, the breaking of flux lines between charges, for example [88]. 52

61 5. Quantum Simulations of Abelian Lattice Gauge Theories The lattice formulation of gauge theories represents a very convenient formalism to study non-perturbative phenomena, using, in particular, numerical simulations, such as the Monte Carlo method [51]. However, and in spite of the great advance in the understanding of particle physics accomplished by the latter, many problems remain hard to tackle using the current theoretical and numerical techniques. For example, the sign problem [52], appearing for large chemical potentials corresponding to fermionic matter, prevents standard numerical calculations to be performed in some regimes of QCD, such as the quark-gluon plasma or the color-superconducting phase [107]. Moreover, Monte Carlo simulations are not valid to study real-time dynamics, since only euclidean correlation functions (imaginary time) can be calculated. Using highly controllable quantum systems to perform simulations of lattice gauge theories represents a great opportunity to solve some of the mentioned difficulties, providing information about these theories beyond the capabilities of the current methods. Notwithstanding, using low-energy systems, such as ultracold atoms, to study high-energy physics is a complicated task, since the gauge and Lorentz symmetries, presented in the latter, are not natural symmetries of the former. In this chapter, the basic building blocks to simulate abelian lattice gauge theories using ultracold atoms will be reviewed (4.32). In particular, we will describe how gauge invariance is obtained from an atomic system where this symmetry is not present, as well as an effective method to get the usual plaquette interactions. Then, a complete simulation scheme for the lattice Abelian-Higgs theory using bosonic atoms will be proposed, including the experimental requirements that must be imposed on the atomic system. After introducing a mapping between the degrees of freedom of the simulating and the simulated system, we will show, theoretically, how the Hamiltonian of the latter (4.32) emerges effectively from the atomic Hamiltonian provided that the energy scales of the system satisfy certain conditions. Furthermore, this scheme comprises a new way to simulate the pure-gauge Kogut-Susskind Hamiltonian (3.60). Finaly, numerical results will be shown, backing up the theoretical conclusions, and providing new information about the limitations of the quantum simulation proposal Quantum Simulation Techniques In this section, we will review the basic techniques that are needed to simulate abelian lattice gauge theories using ultracolds atoms. First, we will see how gauge invariance can be simulated in an atomic system. Then, a method to obtain effective plaquette interactions using auxiliary particles will be introduced. Finally, we will explain how different gauge-invariant sectors are simulated by a proper initialization of the ultracold atomic system. 53

62 5. Quantum Simulations of Abelian Lattice Gauge Theories Gauge Symmetry Gauge invariance is not a fundamental symmetry for a system of ultracold atoms in an optical lattice. It is essential, however, in the description of high energy interactions, as we have seen. If one aims to correctly describe the latter, gauge symmetry must be imposed, somehow, in the simulating system. We will review two ways to achieve this [88], one where gauge invariance is obtained effectively, as an emerging, low energy symmetry, and another where it is mapped, exactly, to a fundamental symmetry of the system conservation of hyperfine angular momentum. Effective Symmetry In the Kogut-Susskind Hamiltonian formulation of lattice gauge theories, gauge invariance results in the division of the Hilbert space into different sectors (3.63), corresponding to the different eigenvalues of the symmetry generators G n (3.64). Since the gauge transformation generators commute with the Hamiltonian (3.57), the system s dynamics do not allow transitions between different sectors, which sets a superselection rule. One way to mimic this behaviour in a system where gauge invariance is not present is to add a penalty term to the Hamiltonian [88], H G = λ n G 2 n (5.1) As explained in Appendix A.1, if λ is high enough compared to the other energy scales of the system, the Hilbert space will be divided into different sectors, corresponding to the different eigenvalues of (5.1). Treating the rest of the terms in the Hamiltonian as perturbations, an effective Hamiltonian, acting separately on each sector, can be obtained. All the states that belong to this subspace satisfy the property G n ψ = q n ψ, n (5.2) which corresponds to a gauge invariant sector with some static charge configuration {q n }. In order to choose the latter we have to initialize the system in a state that belong to the corresponding sector. The way this is obtained for an ultracold atomic system will be explained in the last part of this section. Dynamical transitions between states that belong to different sectors are energetically suppressed. If the system is initialized in a given sector, it will effectively remain in it. The allowed dynamics within a sector are given by the terms in the Hamiltonian that commute with G n, together with higher order terms given by virtual processes where the system leaves the sectors and then comes back (see Appendix A.1) these contribution commute with G n as well. Using this method, gauge invariance emerges in the system as a low energy symmetry. Although not an exact symmetry, it can be considered a good approximation as long as the system satisfies the proper energy requirements. Using this idea, the quantum simulation of cqed in dimensions, both with and without dynamical matter, have been proposed using ultracold atoms [55, 56, 58]. 54

63 5.1. Quantum Simulation Techniques Exact Symmetry A different approach to obtain a gauge-invariant Hamiltonian is to use the intrinsic symmetries of the ultracold atomic system to allow only gauge-invariant terms to appear in the Hamiltonian. One way to do this is through the conservation of the total hyperfine angular momentum in atomic collisions. In order to show how this method works, let us consider the simple lattice Schwinger model, the 1+1 dimensional version of cqed with fermionic matter [59], Ĥ = m n ( 1) n ˆψ n ˆψn + g2 2 Ên 2 + ɛ n n ( ˆψ n Û n ˆψn+1 + h.c.) (5.3) where n is a integer number that denotes both the vertex n and the link connecting it with the vertex n + 1. ψ n and ψ n are fermionic annihilation and creation operators, defined on the vertices of the one dimensional lattice, and satisfying the canonical anticommutation relations, { ˆψ n, ˆψ m} = δ n,m (5.4) whereas E n and U n are the usual lattice gauge operators defined on the links (3.56). Since we are working in one spatial dimension, plaquette terms do not appear in the Hamiltonian. In order to realize this Hamiltonian using ultracold atoms we need two optical lattices, one whose minima coincide with the vertices of the simulated lattice, and another one with minima residing on the links (Fig. 5.4a). The latter will be filled with two types of bosons (a and b), with different hyperfine angular momentum m F. The total number of bosons in each link is denoted by N 0. This is a constant of motion since no interaction between bosons from different links are allowed (using, for example, a very deep lattice for the a and b species). The optical lattice corresponding to the vertices of the simulated one will be filled with fermions. The internal energy level characterized by a different hyperfine angular momentum will be different in the even and odd vertices, denoted by c and d, respectively. Due to the Pauli exclusion principle, only one fermion of each type can sit at each minima. Let us focus on the interacting part of the Hamiltonian. In order to approximate the unitary operator Ûn and Û n using bosonic creation and annihilation operators, we will represent the latter using the Schwinger algebra [108, 109], and ˆL + = â ˆb, ˆL = ˆb â (5.5) ˆL z = 1 2 (â â ˆb ˆb), 1 l = 2 (â â + ˆb ˆb) N 0 = 2 (5.6) where ˆL +, ˆL and ˆL z are standard angular momentum operators with total angular momentum l. Representing the bosonic operators in this way, we can obtain the desired unitary operators in the limit N 0 (such that l ), ˆL ± l(l + 1) l, m = 1 m(m ± 1) l(l + 1) l, m ± 1 l l, m ± 1 (5.7) 55

64 5. Quantum Simulations of Abelian Lattice Gauge Theories Figure 5.1.: Examples of two-body atomic interactions along a link of the lattice allowed by the conservation of total hyperfine angular momentum: (a) species-changing collisions and (b) non-species-changing collisions. Consider the collision terms involving fermions and bosons in the second-quantized atomic Hamiltonian (2.56). If the hyperfine angular momentum m F of the different atomic species are chosen such that m F (a) + m F (c) = m F (b) + m F (d) (5.8) the total angular momentum conservation will only allow the collision terms of the form (Fig. 5.1) ĉ nâ nˆb n ˆdn+1 + ˆd n+1ˆb n+1ân+1ĉn+2 (5.9) and (ĉ nĉ n + ˆd n+1 ˆd n+1 )(â nâ n + ˆb nˆb n ) = (ĉ nĉ n + ˆd n+1 ˆd n+1 )N 0 (5.10) The last term will become a constant, proportional to the total number of fermions, when summed over all vertices, if the scattering lengths are properly tuned using Feshbach resonances (Appendix B). The first collision term corresponds to the interaction part of the Schwinger Hamiltonian. To see this, we represent the bosonic atoms on the links using the Schwinger algebra, and perform the canonical transformation ân ˆbn ˆσ n x ân ˆbn (5.11) The collision term can be written, then, as ( ɛ n ˆψ n ˆL +,n l(l + 1) ˆψn+1 + h.c. ) (5.12) which coincides with the correct interaction in the limit N 0. The rest of the terms in the Hamiltonian (5.3) can be obtained from on-site atomic interactions [59]. Making use of the conservation of angular momentum in atomic collisions, we have obtained the correct gauge-invariant interaction. In this case, as opposed to the effective gauge symmetry introduced before, the gauge invariance is exact, making the simulation more robust against experimental imperfections. This method is also more convenient from a technical point of view, since no penalty term (5.1) has to be enforce externally on the system, which can be specially problematic for nonabelian theories [88]. Using this method, quantum simulations of both abelian and non-abelian lattice gauge theories have been proposed [59, 66]. 56

65 5.1. Quantum Simulation Techniques Figure 5.2.: The plaquette interactions are obtained as fourth order correction in the effective Hamiltonian. A virtual process consisting on one auxiliary particle hopping four times along the links of a plaquette, and returning to its original positions, results in the correct product of four unitary operators, each one attached to one link, that gives rise to the plaquette term Plaquette Interactions For dimensions d + 1, with d > 1, lattice gauge Hamiltonians, such as the Kogut- Susskind one (3.60), include plaquette-type interactions, made out of the product of four unitary operators (Fig. 3.4). Such interactions are not found in the ultracold atomic Hamiltonian. In order to simulate them, we have to obtain these type of terms effectively, following, again, the method describe in Appendix A.1. The plaquette terms are readily obtained as second order contributions when one uses the constraint (5.1) to get an effective gauge-invariant Hamiltonian [55, 56, 58]. In the second approach, a different constraint is introduced on top of the already gauge-invariant Hamiltonian, to get, in the fourth order, the desired plaquette interactions these can be considered as second order corrections, however, since no odd order contributions appear in the expansion. The advantage, in this case, is that the building blocks from which the plaquettes are constructed are gauge-invariant from the beginning. One way to get effective plaquettes is by using auxiliary particles, either fermions or bosons, subject to a strong energy penalty term [88]. Here, we will describe a method that uses hard-core bosons [59], this is, bosonic particles sitting on the vertices of the lattice, subject to the interaction H HC = λ n ˆN χ n ( ˆN χ n 1) (5.13) where ˆN n χ is the number of auxiliary bosons χ on the vertex n. We will also include a hopping term for the auxiliary bosons, mediated by the gauge bosons on the links, ɛ ) (ˆχ Û ˆχ n+ˆk + h.c. (5.14) n n,k n,k This term can be obtained making use of the conservation of hyperfine momentum, analogous to what we did to obtain the interaction between the bosons on the links and the dynamical matter on the vertices we will need, therefore, two different species of auxiliary bosons, corresponding to even and odd vertices. 57

66 5. Quantum Simulations of Abelian Lattice Gauge Theories We initialize the system with one auxiliary boson per vertex. If λ is much larger than ɛ, the hopping of auxiliary bosons to neighbouring vertices implies a large energy penalty. The effective Hamiltonian acting on the sector characterized by one boson per vertex includes higher order corrections, related to the virtual hopping of auxiliary bosons to neighbouring vertices, and the return to their original position. In the fourth order, an auxiliary boson can hop along the links of a plaquette (Fig. 5.2), resulting in a correction proportional to ɛ 4 ) λ (Ûn,i 3 Û n+î,k Û Û + h.c., i k (5.15) n+ˆk,i n,k which is the desired plaquette interaction. In the next section, the details for obtaining such effective Hamiltonian, as well as the discussion concerning the rest of the correction terms, will be worked out in detail for the lattice Abelian-Higgs theory. The use of hard-core bosons have several advantages, compared to obtaining the effective plaquettes using auxiliary fermions [59] which is the method that has been mostly used until now. First, the penalty term is easier to implement experimentally, since it is homegeneous across the vertices of the lattice. Also, as we will see when we apply it to the lattice Abelian-Higgs theory, no undesired divergence terms or inhomogeneous renormalizations will appear in the effective Hamiltonian, as opposed to [59]. Finally, the quantum simulation will only require bosonic atoms, rather than bosons and fermions, which simplifies as well the experiment Static Charges: Initializing the System Once a gauge-invariant Hamiltonian is obtained, either effectively or in an exact manner, the physical Hilbert space of the system will be divided into different sectors (3.63). If we start in a specific one, the gauge-invariant dynamics will prevent the system from leaving it. In order to choose one of them in a quantum simulation or, equivalently, to realize a specific set of static charges we have to initialize the atomic system in a state that corresponds to the desired sector in the simulated theory. Consider, for example, the sector of the Kogut-Susskind Hamiltonian (3.60) characterized by two static charges of opposite sign (Fig. 5.3). The states that belong to such sector present an electric flux line or a superposition of them that joins the two vertices where the static charges are placed, as discussed in section (3.3.2). Consequently, some of the links will be in excited states of the electric part of the Hamiltonian, whereas others remain in the ground state since, altogether, everything must satisfy all the local Gauss laws. Using Schwinger s representation (5.6), we can see how, in the atomic system, this is achieved by filling the excited links with a different number of atoms of the a and b species (Fig. 5.3), noting that their role interchange due to transformation (5.11). The rest of the links are initialized in the ground state by filling them with the same number of atoms of the two species. In general, every sector of this Hamiltonian is characterized by electric flux lines connecting pairs of static charges of opposite sign. Therefore, the difference on the number of atoms of each species placed on the links is enough to select any of them. Note that many different states belong to a given sector. In the case of two static charges, many different paths formed by excited links can join the charges (Fig. 3.7). 58

67 5.2. Simulating the Lattice Abelian-Higgs Theory Figure 5.3.: The number of bosons of species a and b, trapped initially on the vertices of the optical lattice that represent the links of the simulated one, characterizes the initial electric field states of the latter. In the figure, two static charges of opposite sign, joined by an electric flux line (blue), are simulated by placing more atoms of one species in the corresponding links (with electric field m = +1) notice that the roles of a and b interchange in alternating links due to the transformation (5.11) and the same number of a and b bosons on the rest (m = 0). Due to the plaquette interaction term in (3.60), the state of the system can be, as well, a superposition of different electric flux lines. One can reach different regimes of the theory by first initializing the system in an eigenstate of the electric part of the Hamiltonian, as explained before. Then, one can adiabatically increase the effect of the plaquette terms by tuning the corresponding scattering lengths via Feshbach resonances (Appendix B) to drive the system to the weak-coupling regime of the theory, where the electric flux lines start fluctuating due to closed electric flux loops popping out from the vacuum (Fig. 3.7). This effect can be detected by measuring the number of atoms of each species on the links of the lattice, using single-atom adressing techniques [110, 111] Simulating the Lattice Abelian-Higgs Theory In this section, a proposal to perform a quantum simulation for the lattice Abelian- Higgs theory (4.32) will be developed using ultracold bosonic atoms trapped in optical lattices. The goal is to simulate the corresponding Hamiltonian (4.32) using the one that describes the atoms. This means to impose certain experimental conditions on a highly controllable trapped atomic system, such that the Hamiltonian that describes it maps, approximately, the Hamiltonian under study. Both the parameters appearing in the latter, as well as the degree of approximation, can be controlled experimentally. This allows for the exploration of the phase diagram of the theory, and to measure properties of the system that may be hard to evaluate both analytically or with the help of numerical calculations. The simulation proposal involves three main steps. First, starting from the most general multi-species Hamiltonian that describes ultracold bosons (2.54), the experimental requirements needed to get all the ingredients for the simulation will be presented. After that, the required transformations to map the degrees of freedom of the atomic system to those of the Abelian-Higgs theory will be introduced. Finally, we will see how the desired Hamiltonian is effectively obtained, apart from correction 59

68 5. Quantum Simulations of Abelian Lattice Gauge Theories Figure 5.4.: (a) Two superimposed optical lattices are required to simulate both matter and gauge degrees of freedom, oriented with a 45 angle between their links. The vertices of one lattice (blue in the figure) represent the vertices of the simulated one, whereas the vertices of the rotated lattice (green) correspond the the links of the latter. The matter lattice is filled with four types of bosons (c), two of them on the even vertices (orange in (a)) and the other two in the odd ones (yellow in (a)), representing dynamical and auxiliary particles. The gauge lattice, on the other hand, is filled with two types of bosons on each vertex (purple in (a)). The characteristics of these lattices must be configured such that there is no interactions between bosons from different vertices of the gauge lattice using, for example, deep potentials (c) but allowing for interactions between them and the bosons from the nearest-neighbour vertices of the matter lattice. This is the case if the corresponding Wannier functions intersect (b). terms, after increasing, largely, one of the energy scales of the system compared to the rest (hard-core bosons method (5.1.2)). Some of these corrections can be made as small as necessary by changing the experimental parameters of the atomic system, keeping the freedom to move through the phase space of the simulated theory (Appendix B). This simulation scheme serves, as well, as a new method to simulate the pure-gauge Kogut-Susskind Hamiltonian. The validity of the simulations will be discussed based on the theoretical results. In the next section, we will see how numerical calculations support these claims, at least qualitatively Experimental Requirements The simulation proposal will focus on the dimensional case, which is the first non-trivial dimension, since it contains plaquette interactions. The reason for this is the large experimental effort required to simulate higher dimensional theories. As explained in section (5.1.1), we need two optical lattices to perform a lattice gauge theory simulation, one for the atoms on the vertices, and another one for the atoms on the links (Fig. 5.4a). Filling and combining two such lattices is possible in the two-dimensional case, however, it becomes very complicated in three dimensions. All 60

69 5.2. Simulating the Lattice Abelian-Higgs Theory Figure 5.5.: The hyperfine angular momenta of the six bosonic species must fulfilled certain conditions, such that the conservation of this quantity during atomic collisions permit the desire atomic interactions. the theoretical calculations are valid, nevertheless, for the d + 1 cases with d 2. To simulate the complete Abelian-Higgs Hamiltonian (4.32) we need six different bosonic species (Fig. 5.4c). Two of them will be placed on the links of the simulated lattice to represent the gauge degrees of freedom. Another pair of bosons will simulate the scalar fields on the vertices (dynamical matter), one species placed on the even vertices, and another one on the odd ones. The last two species are distributed, again, on even and odd vertices, acting as auxiliary particles used to obtain the effective plaquette interactions, as explained in (5.1.2). Each of these pairs of bosonic species corresponds to the same type of atom, but their internal level differs in the value of the hyperfine angular momentum m F. In order to obtain the correct gauge-invariant interactions, the hyperfine angular momenta of the bosons must fulfill certain conditions (Fig. 5.5), m F (a) m F (b) = m F (c) m F (d) = m F (e) m F (f) m (5.16) as explained in subsection 5.1.1, where a and b denote the bosons on the links, c and e the dynamical and auxiliary bosons on the even vertices, and d and f the corresponding ones on the odd vertices. After the canonical transformation (5.11) are applied, the bosonic operators can be renamed. The dynamical bosons will be denoted by η and the auxiliary ones by χ. In the case of the auxiliary particles, only one boson should be placed on each vertex when the simulation is initialized, as required by the hard-core bosons method. Many dynamical bosons are required, however, on each vertex, to correctly simulate the corresponding operators, as we will see. The conservation of hyperfine angular momentum allows the two types of bosons to interact inside each vertex. This will result in non-desired terms in the simulated Hamiltonian. For this reason, the system must be prepared such that this kind of interactions does not occur, using, for example, Feshbach resonances to reduce the corresponding scattering length (Appendix B). 61

70 5. Quantum Simulations of Abelian Lattice Gauge Theories Primitive Hamiltonian As we have already mentioned in (5.1.2), the plaquette interactions can be obtained effectively from the atomic Hamiltonian. The latter can be expressed using the corresponding lattice gauge theory operators, introducing a mapping between these and the atomic operators. We will call primitive Hamiltonian to the atomic Hamiltonian written in this way. The Abelian-Higgs Hamiltonian can be obtained, then, effectively from it. For this to be possible, the primitive Hamiltonian must have the form H =λ n ˆN χ n ( ˆN χ n 1) + ɛ n,k [ˆχ n Û n,k ˆχ n+ˆk + h.c. ] + µ n,k Ê 2 n,k + µ n ˆQ 2 n + ɛ n,k [ ˆφn Û n,k ˆφ n+ˆk + h.c. ] (5.17) We will focus now on obtaining this Hamiltonian from the atomic one. We will see how only an approximated Hamiltonian can be obtained. However, the level of approximation can be controlled by using a different number of atoms in the simulation. Operator Transformations In order to simulate the operators associated to the dynamical charges on the vertices, ˆQn and ˆφ n, using the bosonic operators, ˆη n and ˆη n, we apply the following transformations ˆη n = ˆφ ( n N 0,v + ˆQ ) 1/2 n ˆη n = (N 0,v + ˆQ ) 1/2 (5.18) n ˆφ n where ˆQ n = ˆη nˆη n N 0,v (5.19) and N 0,v is the mean number of η bosons on each vertex. If ˆQn and ˆφ n fulfill the commutation relations (4.36), the bosonic operators will satisfy the canonical ones. These transformations are problematic if one intends the operator ˆφ n to be unitary, which also means that it is the exponential of a self-adjoint phase ˆϕ n, ˆφ n = e i ˆϕn. This problem is related to the so-called quantum phase operator problem [ ], and lies in the fact that ˆQ n is bounded from below. A unitary operator can only be well-defined, then, in the limit N 0,v, such that the charge operator is no longer bounded. We are working with a large, but finite, number of atoms N 0,v on each vertex. The reason it is still fine, from a physical point of view, to use these definitions although they are not formally correct is that in all the interaction terms of the Hamiltonian the bosonic operators appear in pairs, corresponding to neighbouring vertices, ˆη nˆη. n+ˆk Therefore, only phase differences between different vertices are relevant. The phase difference is a well-defined quantity, since it is canonically conjugate to a number operator, ˆN = ˆN c ˆN d, which is not bounded from below. The operators associated to the gauge fields, Ê n,k and Ûn,k can be obtained from the bosonic operators on the links if we first represent them in a similar way as the 62

71 5.2. Simulating the Lattice Abelian-Higgs Theory atomic operators on the vertices, with ) 1/2 ( â n,k = Û a N0,l n,k 2 + Êa n,k ( ) 1/2 (5.20) ˆbn,k = Û b N0,l n,k 2 + Êb n,k Ê a n,k = â n,kân,k N 0,l 2 Ê b n,k = ˆb n,kˆb n,k N 0,l 2 (5.21) Again, if the commutation relations (3.56) are fulfilled, the canonical relations for the bosonic operators will be satisfied. The total number of bosons on each link, N 0,l is a conserved quantity, since only product of bosonic operators of the form â n,kˆb n,k and ˆb n,kân,k will appear in the Hamiltonian. This implies that Êa n,k = Êb n,k Ên,k. We will use to the latter as the electric field operator on the link (n, k). It can be written, as well, as E n,k = 1 2 (â n,kân,k ˆb n,kˆb n,k ) If the unitary operators acting on the links are defined as (5.22) Û n,k = Û b n,kû a n,k (5.23) they will satisfy the correct commutation relations (3.56) with the electric field (5.22). Note that, in this case, the operator E n,k is not bounded from below. Û n,k is, therefore, a well defined unitary operator. Finally, the product of two bosonic operators on the links can be written as â n,kˆb n,k = Ûn,k ( N0,l (N 0,l + 2) 4 ) 1/2 Ên,k(Ên,k + 1) ( ) ˆb n,kân,k = N0,l (N 0,l + 2) 1/2 4 Ên,k(Ên,k + 1) Û n,k (5.24) using the property introduced in Appendix (A.2), Û f(ê) = f(ê 1) Û. In the following, we will assume this property and the corresponding one for the operators on the vertices, ˆφ f( ˆQ) = f( ˆQ + 1) ˆφ to deal with non-commuting operators. Building Blocks With the atomic operators expressed in terms of the lattice gauge theory operators, we can consider now the atomic interactions that give rise to the desired primitive Hamiltonian. For this, we will apply the transformations (5.24) and (5.18) to the different terms in the atomic Hamiltonian. As we saw in section (5.1.1), the conservation of angular momentum will produce two types of interactions, apart from constant terms (Fig. 5.6). The first one involves 63

72 5. Quantum Simulations of Abelian Lattice Gauge Theories Figure 5.6.: The conservation of the total hyperfine angular momentum in the atomic collisions generates the desired gauge-invariant interactions between the dynamical (a) and auxiliary matter (b) on the vertices of the lattice, and the gauge degrees of freedom on the links. a collision between an auxiliary boson on a vertex and a gauge boson on a neighbour link. After applying the transformation (5.24) to this term, we get ) ɛ 0 ˆχ â ˆbn,k ˆχ = ɛ 4 1/2 (1 n n,k n+ˆk N 0,l (N 0,l + 2) (Ê2 n,k Ên,k) ˆχ Û ˆχ n n,k n+ˆk (5.25) with N0,l (N 0,l + 2) ɛ = ɛ 0 (5.26) 4 where the coupling constant have been redefined. The second possible collision process, now between one dynamical boson and one gauge boson, results, after applying (5.24) and (5.18), in ( ɛ 0 ˆη â ˆbn,k ˆη n n,k n+ˆk =ɛ ( 1 ) 1/2 ( ( N ˆQ n + 1) 0,v N 0,v ˆQn+ˆk ) 1/2 4 N 0,l (N 0,l + 2) (Ê2 n,k Ên,k)) 1/2 ˆφn Û n,k ˆφ n+ˆk where the corresponding coupling constant have been redefined again, ɛ = ɛ 0N 0,v N0,l (N 0,l + 2) 4 (5.27) (5.28) Both of these interactions are possible provided the Wannier functions (section 2.4.1) of the corresponding bosons on the lattice intersect (Fig. 5.4b). Consider now the non-interacting terms of (5.17). Both the charge and the electric field mass terms arise from on-site collisions. For the atoms in the vertices we have µ n ˆη nˆη nˆη nˆη n = µ n ˆQ 2 n + constants (5.29) where we used the fact that the total number of atoms sitting on the vertices is a conserved quantity. For the bosons on the links, different collision processes appear. All of them contribute to the electric term if the right couplings are chosen (with Feshbach resonances, Appendix B), ( µ â n,kân,kâ n,kân,k + ˆb ) n,kˆb n,kˆb n,kˆb n,k 2â n,kân,kˆb n,kˆb n,k = µê2 n,k (5.30) 64

73 5.2. Simulating the Lattice Abelian-Higgs Theory Finally the hard-core constraint for the auxiliary bosons can be obtained from onsite collision among them, as well as atomic mass terms, λ ) (ˆχ n ˆχ n ˆχ n ˆχ n ˆχ n ˆχ n = λ ˆN n χ ( ˆN n χ 1) (5.31) n n n We have to make sure that the dynamical and auxiliary bosons do not interact among them. This can be achieve by making the corresponding coupling constants small enough with the use of Feshbach resonances (Appendix B). Approximated Primitive Hamiltonian Getting together all the building blocks we can see how the atomic Hamiltonian corresponds to H =λ n ˆN χ n ( ˆN χ n 1) + µ n,k Ê 2 n,k + µ n ˆQ 2 n + ɛ n,k + ɛ n,k ( 1 [ ( ) ] 4 1/2 1 N 0,l (N 0,l + 2) (Ê2 n,k Ên,k) ˆχ Û ˆχ + h.c. n n,k n+ˆk [ ( ) 1/2 ( ( N ˆQ n + 1) ) 1/2 0,v N ˆQn+ˆk 0,v 4 Ên,k)) ] 1/2 N 0,l (N 0,l + 2) (Ê2 n,k ˆφn Û ˆφ + h.c. n,k n+ˆk (5.32) which, in the limit N 0,l and N 0,v results in (5.17). In the experiment, only a finite number of atoms is placed on each link and vertex. However, this Hamiltonian will provide a good approximation, even for small number of atoms. We will study the effect that the number of atoms have on the simulation after obtaining the effective Hamiltonian. In order to simplify the notation, we will use, from now on, the following nonunitary operators, ˆΦ n ˆΛ n,k ( N 0,v ( ˆQ n + 1)) 1/2 ˆφn ( 4 Ên,k)) 1/2 1 N 0,l (N 0,l + 2) (Ê2 n,k Ûn,k (5.33) Notice that the operators ˆΛ and ˆΛ are, respectively, proportional to the Schwinger operators ˆL and ˆL + (defined in 5.5), ˆΛ = ˆL l(l + 1), ˆΛ = ˆL + l(l + 1) (5.34) with l = N 0,l 2. This is clear by identifying Ê = ˆL z, and using the corresponding commutation relations. 65

74 5. Quantum Simulations of Abelian Lattice Gauge Theories Effective Hamiltonian To simulate the plaquette interactions of (4.32), we will apply the ideas explained in section (5.1.2) to the Hamiltonian (5.32). For this, we need to make λ large, compared to the rest of the coupling parameters, λ ɛ, ɛ, µ, µ (5.35) using, again Feshbach resonances (Appendix B). The effective Hamiltonian will be expanded perturbatively, in terms of 1 λ, following the method describe in Appendix A.1. More details about the calculations can be found in Appendix C. Here, we will present the results up to fourth order. First Order The first order contribution to the effective Hamiltonian include all the terms in the primitive one (5.32) that commute with the penalty term (5.1), H (1) eff = µ n,k Ê 2 n,k + µ n ˆQ 2 n + ɛ n,k (ˆΦn ˆΛn,k ˆΦ n+ˆk + h.c. ) (5.36) Second Order The second order contribution is just a renormalization of the electric part of the Hamiltonian, H (2) eff = ɛ2 4 Ên,k 2 (5.37) λ N 0,l (N 0,l + 2) Third Order In the third order, we get two different corrections. Again, a small renormalization of the electric field term, and a correction of the interaction term, [ H (3) eff = ɛ2 ɛ 2 ( ( )) ] 1 λ 2 N0,l 4 1 2Ên,k + 2Ê2 n,k + O ˆΦ ˆΛn,k ˆΦ N + h.c. 0,l n n+ˆk n,k µɛ2 2 λ 2 Ên,k 2 N 0,l (N 0,l + 2) n (5.38) Note that the correction of the interaction is not just a renormalization of its coupling constant, since it includes also electric field operators. It is, therefore, a new term that did not appear in the original Hamiltonian. Later, we will see that this kind of terms can be made small enough such that their effect on the simulation is negligible. Fourth Order A plethora of correction terms appears in the expansion of the effective Hamiltonian at fourth order. The most important one, and the reason why we perform this expansion, n,k 66

75 5.2. Simulating the Lattice Abelian-Higgs Theory is the plaquette interaction, 5 ɛ 4 ) ˆΛ 2 λ (ˆΛn,i 3 n+î,k ˆΛ ˆΛ + h.c. n+ˆk,i n,k n,ik (5.39) Apart from corrections to the electric part of the Hamiltonian, we obtain terms that depend on different powers of the electric field operator, [ µ 2 ɛ 2 ( ) ] 9 λ ɛ4 18 N 0,l (N 0,l + 2) λ 3 Ên,k 2 N 0,l (N 0,l + 2) + ɛ2 µ 2 8 λ 3 N 0,l (N 0,l + 2) + ɛ4 λ 3 4 N 0,l (N 0,l + 2) n ( µ 2 Ên,k 3 + ɛ2 µ 2 6 λ 3 N 0,l (N 0,l + 2) ) ɛ N 0,l (N 0,l + 2) n.n. n Ê 4 n,k n ɛ4 2 1 ] [Ên,k λ 3 3 (N 0,l (N 0,l + 2)) 2 Ê n,k 11Ê2 n,kê2 n,k n Ê n,k (5.40) The last one is an interaction between nearest-neighbour links. We also obtain interactions between the electric field on the links and the dynamical charge operators on the vertices, ɛ 2 ɛ 2 λ 3 1 N 2 0,l [ 16 n,k N0,l 2 Ên,k 2 ( ) ( N ˆQ n + ˆQ n+ˆk + 1) 0,v + 2 ( N0,v 2 Ê n,k ( ˆQ n ˆQ n+ˆk) ˆQ n ˆQn+ˆk + O ( 1 N 0,l )) ] (5.41) Some of these terms can be made negligible, as we will see. Again, as in the third order, we get corrections to the interaction term between the links and the vertices, that depend, in this case, on the electric field and the charge operators, ɛ 2 ɛ µ λ 3 1 2N 2 0,l [( n,k ( + 1 N0,l 2 ( 4 4Ên,k + 8Ê2 n,k ) + O and ɛ 2 ɛ µ λ 3 1 N 0,l (N 0,l + 2) ( 3 + 4Ên,k 4Ê2 n,k ) + 1 N 0,l (6 8Ên,k + 8Ê2 n,k ) n,k 1 N 3 0,l ) ) ˆΦ n ˆΛn,k ˆΦ n+ˆk + h.c. ] (5.42) [( ) ( 1 + 2Ên,k ˆQn ˆQ ) ] + 1 n+ˆk ˆΦn ˆΛn,k ˆΦ n+ˆk + h.c. (5.43) Finally, two new interaction terms appear in the effective Hamiltonian, consisting on applying twice the hopping term. The first one acts on the same link, ɛ 2 ɛ 2 λ 3 1 N 2 0,l [ ( n,k 1 2 N 0,l + 4 N 2 0,l (2 2Ên,k + Ê2 n,k ) + O ( 1 N 3 0,l )) ] (5.44) ) 2 (ˆΦn ˆΛn,k ˆΦ + h.c. n+ˆk 67

76 5. Quantum Simulations of Abelian Lattice Gauge Theories whereas the second involve hopping processes in two nearest-neighbour links, ɛ 2 ɛ 2 1 [( ) ) λ 3 1 2Ên,k [(ˆΦn ˆΛn,k ˆΦ n+ˆk 2N 0,l (N 0,l + 2) + h.c., ˆΦ ] ] ˆΛn,k ˆΦ + h.c n n+ˆk n.n. (5.45) and it is written in terms of the commutator between them. Notice that all the terms obtained in the effective expansion are gauge invariant, since the symmetry was exact in the primitive Hamiltonian (5.32) Pure-Gauge Theory Before discussing the whole effective Hamiltonian we have obtained, and the importance of the non-desired correction terms, let us consider a simplified case, the puregauge theory. This provides a simulation scheme for the Kogut-Susskind Hamiltonian (3.60). If we make µ = ɛ = 0, the effective Hamiltonian simplifies significantly. Most of the correction terms disappear, and we are left with H eff = [ µ + ɛ2 λ (1 µɛ2 λ 3 4 N 0,l (N 0,l + 2) ) + µɛ2 λ ɛ2 µ 2 8 λ 3 N 0,l (N 0,l + 2) + ɛ4 λ 3 4 N 0,l (N 0,l + 2) N 0,l (N 0,l + 2) ( ɛ 2 ) ] 4 λ 2 n ( µ 2 ( 2 + 9µ λ n Ê 2 n,k ) Ên,k 3 + ɛ2 µ 2 6 λ 3 N 0,l (N 0,l + 2) ) ɛ N 0,l (N 0,l + 2) n Ê 4 n,k ɛ4 2 1 ] [Ên,k λ 3 3 (N 0,l (N 0,l + 2)) 2 Ê n,k 11Ê2 n,kê2 n,k n.n. 5 ɛ 4 ) ( ) ˆΛ 2 λ (ˆΛn,i 3 n+î,k ˆΛ ˆΛ ɛ 6 + h.c. + O n+ˆk,i n,k λ 5 n,ik n Ê n,k (5.46) This Hamiltonian, although simplified, still contains some non-desired correction terms. We will study now how we can control them experimentally. Large Mean Number of Atoms First, consider the limit where a large number of atoms is placed on each link of the lattice, N 0,l. Every correction term, except from a fourth order renormalization of the electric field, disappears in this limit. The non-unitary operators (5.33) converge, as well, to the correct unitary ones, and the effective Hamiltonian becomes just the Kogut-Susskind one (3.60), H eff =µ (1 µɛ2 λ 3 ) Ên,k 2 5 ɛ 4 2 λ 3 n n,ik ) Û (Ûn,i n+î,k Û Û + h.c. n+ˆk,i n,k (5.47) up to higher order corrections, which will be, in the next orders, proportional to µɛ4 µ 3 ɛ 2 λ 4 and ɛ6 λ 5. Note that, as explained in Appendix C, no terms of order ɛn+1 λ n λ 4,, for 68

77 5.2. Simulating the Lattice Abelian-Higgs Theory n ( even, appear in the effective expansion. For this reason, it is enough to impose ɛ ) 2 λ 1 to make higher order correction negligible since correction of this type are much more important than the rest, as we will see. This applies also to the other simulations we are going to consider. We will write now the experimentally controllable parameters, ɛ and µ, in terms of the coupling constant g of the Kogut-Susskind Hamiltonian (3.60). In order to do so, since we are only interested in the ratio between the electric and the magnetic part of the Hamiltonian, we multiply everything by a constant α, which effect is just a rescale in the energy. This constant, and the coupling g must be defined, then, as g [ 2 5 ( ) λ 3 µ ɛ ɛ ) (1 ] 1/4 [ µɛ2 1 µɛ 4 λ 3, α 10 λ 3 )] 1/2 (1 µɛ2 (5.48) λ 3 The effective Hamiltonian can be expressed, after multiplying everything by α, as αh eff = g2 2 + O Ên,k 2 1 2g 2 n,ik ( 1 µ ( µ ) ) 2 g 2 + O λ ɛ n ) Û (Ûn,i n+î,k Û Û + h.c. n+ˆk,i n,k ( ) ( 1 µ 1 ( ɛ ) ) (5.49) 2 g 2 + O λ g 2 λ where the most important higher order corrections are written explicitly, allowing us to discuss their importance in the different regimes of the theory. By using the Feshbach resonance technique to fine-tune the ratios ɛ λ and µ ɛ, one can explore the phase diagram of the Kogut-Susskind Hamiltonian. Consider, in particular, two relevant limits, Strong coupling limit (g 1) To achieve this limit, we need ( ) λ 3 ɛ ɛ µ In this limit, only the electric part of the Hamiltonian is relevant. corrections are negligible as well. (5.50) All the Weak coupling limit (g 1) In this case, we need ɛ µ ( ) λ 3 (5.51) ɛ Here, the electric part vanishes and the plaquette interaction becomes relevant. The first correction in (5.49) becomes very small, since ɛ µ is very large. For the same reason, ( it can be seen that the bigger correction, in this limit, is the one of order O 1 ( ɛ ) ) 2 g 2 λ. As we will see in the finite number of atoms case, the value of N 0,l is important in the weak coupling limit, since one expects large fluctuations in the atomic number. However, since we are considering the N 0,l, the simulation is meaningful also in the weak coupling case. 69

78 5. Quantum Simulations of Abelian Lattice Gauge Theories Both ɛ λ and µ λ must be small in order to make the higher order corrections of (5.49) negligible. However, we still have freedom to change the ratio ɛ µ to achieve different values of g. Finite Number of atoms: New Approach Consider now a more realistic case, where the number of atoms on each link is finite. At the same time, a new approach to generate the electric field operators is introduced. It consists on making µ = 0, and using the renormalizations we obtained in the effective expansion as the electric field part of the Hamiltonian. Proceeding in this way, many correction terms disappear, and we are left with the following effective Hamiltonian, H eff = ɛ2 λ 4 N 0,l (N 0,l + 2) ( ɛ 2 λ 2 ) n Ên,k 2 + ɛ4 8 λ 3 (N 0,l (N 0,l + 2)) 2 ɛ4 2 1 ] [Ên,k λ 3 3 (N 0,l (N 0,l + 2)) 2 Ê n,k 11Ê2 n,kê2 n,k n.n. 5 ɛ 4 ) ( ) ˆΛ 2 λ (ˆΛn,i 3 n+î,k ˆΛ ˆΛ ɛ 6 + h.c. + O n+ˆk,i n,k λ 5 n,ik n Ê 4 n,k (5.52) Notice that, in particular, the terms proportional to the odd powers of E n,k vanish. Expressing, as before, the coupling constant in terms of the system parameters in a proper way g (( 8 5 ( λ ɛ ) ) the effective Hamiltonian can be written as αh eff = g2 2 n Ên,k g 2 (N 0,l (N 0,l + 2)) g 2 1 (N 0,l (N 0,l + 2)) 2 1 2g 2 n,ik n.n. ) 1/4 1 (5.53) N 0,l (N 0,l + 2) n Ê 4 n,k [Ên,k Ê n,k 11Ê2 n,kê2 n,k ] ) ˆΛ (ˆΛn,i n+î,k ˆΛ ˆΛ + h.c. + O n+ˆk,i n,k ( 1 ɛ 2 ) g 2 λ 2 (5.54) after multiplying it by a properly chosen constant α (different from the previous case). Since N 0,l is a fixed quantity, once the experiment starts, the coupling g is controlled with the ratio ɛ λ. Consider, again, the weak and strong coupling limits. Strong coupling limit (g 1) To access this regime, the experimental parameters must be chosen such that λ ɛ N 0,l (5.55) which will also make the higher order corrections negligible (since λ ɛ 1 will be ensured, as well). 70

79 5.2. Simulating the Lattice Abelian-Higgs Theory In this case, the electric part of the Hamiltonian dominates. Therefore, in the strong coupling regime, even for a small number of atoms in each link, the atomic Hamiltonian correctly simulates de Kogut-Susskind one. Weak coupling limit (g 1) This regime is more problematic, and the accuracy of the simulations depends strongly on the number of atoms used in the experiment. To access it, we have to impose N 0,l λ ɛ 1 (5.56) where the second condition is required to make higher order corrections negligible. By using this approach (making µ = 0), the weak regime can only be reached if many atoms are used. However, even if we could access it for a small number of atoms (using the approach where µ is not zero), the simulation would lack accuracy, since the operators ˆΛ n,k would be far from the unitary ones, Û n,k. Therefore, being the latter a more important problem, condition (5.56) does not pose an important limitation. As we commented before, the atomic number is not a good quantum number, since the corresponding operator undergo large fluctuations, but as long as condition (5.51) is satisfied, the weak limit is properly defined. In this regime, ( ) the non-desire terms become important. All of them are of the order O N 4 0,l, but E n,k can take values m, depending on the state, from N 0,l 2 to N 0,l 2. Since only the terms that give contributions of the form m4 are relevant, we can disregard the one that is proportional to Ên,kÊn,k. These non-desired terms affect mostly the boundary states with high m, in absolute value. These states are relevant when the plaquette interaction dominates over the electric part, however, their influence rapidly decreases with N 0,l. In particular, for a large number of atoms, the non-desired corrections vanish faster than ˆΛ n,k converges to Ûn,k, being the latter problem again the most important one. In the next section, we will provide numerical calculations to support these claim. Finally, the higher order corrections are of the order ( ɛ 2, λ) compared to the plaquette interactions, so they should not pose a problem, since ɛ λ is small enough. Setting µ = 0 has several advantage with respect to the more conventional approach [59]. Apart from eliminating some non-desire correction terms, it simplifies the experimental requirements of the simulation, since there is one less parameter to fine-tune in order to change the value of g. Also, the condition to enter the weak regime is easier to realize here, compared to (5.51) since ( ) λ 3 ɛ is very large. In this case, we can access the regime by making λ ɛ smaller than N 0,l. The ratio should still be larger than 1, but just enough to make the higher order corrections ( ɛ 2 λ) small. The proposed scheme to simulate the Kogut-Susskind Hamiltonian, thanks to the use of hard-core bosons instead of auxiliary fermions to effectively obtain the plaquette interactions, also eliminates the non-desired divergence terms from previous proposals [59], which survive also in the N 0,l limit. In this case, the simulated Hamiltonian is exactly the Kogut-Susskind one in the large number of atoms limit. 71

80 5. Quantum Simulations of Abelian Lattice Gauge Theories Full Abelian-Higgs Theory Let us consider now the complete Abelian-Higgs theory. Following the discussion of the pure-gauge case, we will make, again, µ = 0. This makes many of the correction terms obtained in (5.2.3) disappear. We will study now the importance of the remaining ones. Again, we multiply the effective Hamiltonian by a properly chosen constant α, and define g as before (5.53). In order to get the correct coupling constanst in front of the charge term and the interaction between the dynamical charges and the electric field, we write the parameter R as R 1 g 2 5 ( ) λ 3 ɛ ɛ ɛ (5.57) Once g is fixed through λ ɛ, R can be fine-tuned by changing the ratio ɛ ɛ. Note that ɛ must be negative, since the charge term and the gauge-matter interaction have opposite signs in the Hamiltonian we are trying to simulate (4.32). We can study the importance of the correction terms, obtained in the effective expansion of the Hamiltonian, by expressing them, first, in terms of the simulated parameters. The coupling constants before the non-desired effective contributions obtained in subsection are of three different type, which, after multiplying everything by α and writing them in terms of R and g, are found to be ɛ 2 ɛ 1 λ 2 N0,l 2 α = R2 ( ɛ ) 2 1 (5.58) N0,l 2 λ ɛ 2 ɛ 2 1 λ 3 α = 5R4 g 2 ( ɛ ) 6 1 (5.59) N 0,l N 0,l λ ɛ 2 ɛ µ λ 3 α = 5 ( ɛ ) 6 4 g2 1 (5.60) λ All of them are negligible, since ɛ λ can be made small enough to compensate from the possible growth due to R and g. In general, we can say that only the corrections of order ( ɛ 4 λ) are relevant, since this is the order of the effective plaquettes, which we want to make large on the weak regime. The total effective Hamiltonian, disregarding negligible terms, reduces to αĥeff = 1 2R 2 + g2 2 n n ˆQ 2 n R2 2 ] [ˆΦn ˆΛn,k ˆΦ n+ˆk + h.c. n,k Ên,k g 2 (N 0,l (N 0,l + 2)) g 2 1 (N 0,l (N 0,l + 2)) 2 1 2g 2 n,ik n.n. n Ê 4 n,k [Ên,k Ê n,k 11Ê2 n,kê2 n,k ] ) ˆΛ (ˆΛn,i n+î,k ˆΛ ˆΛ + h.c. + O n+ˆk,i n,k ( ) ɛ 2 1 λ 2 g 2 (5.61) up to higher order effective contributions, which are negligible for all values of g provided that the conditions introduced for the pure-gauge case are fulfilled. 72

81 5.2. Simulating the Lattice Abelian-Higgs Theory We have seen how the phase diagram of the theory can be explored by changing the ratios ɛ λ and ɛ ɛ, at the same time that we maintain most of non-desired terms small enough so they do not affect the quality of the simulation. For small R and big g, the atomic system maps exactly to the Abelian-Higgs theory. When R grows and g decreases, the effect of the non-unitary character of the operators ˆΛ n,k and ˆΦ n becomes important, as well as the relevance of the non-desired electric field terms. Making N 0,l and N 0,v larger will allow us to perform better simulations, going deeper into the large R and small g regime XY Model For completeness, let us consider the limit where the XY model (4.21) is retrieved. This occurs when g 0. In that case, the electric part of the simulated Hamiltonian (5.61) vanishes, along with the non-desired correction terms. The coupling constant in front of the plaquette interaction diverges in the limit. Therefore, we can completely decouple the gauge and the matter degrees of freedom, since they act on infinitely separated energy scales. The effective Hamiltonian associated to the low-energy degrees of freedom (matter) is simply αĥeff = 1 2R 2 n ˆQ 2 n R2 2 n,k [ˆΦn ˆΦ n+ˆk + h.c. ] (5.62) Note that this Hamiltonian no longer possesses gauge invariance. Instead, a global U(1) symmetry is present. As we discusses in section (4.2.1), this model presents a phase transition at some finite value of R = R c. The two phases are characterized by different scaling behaviour of the correlations between different points ˆΦ n ˆΦ n Initialization and Measurements In section 5.1.3, we saw how the number of atoms of species a and b placed on the links of the lattice at the beginning of the simulation characterized the static charge configuration, or sector, where the system would remain along its dynamical evolution. In particular, an eigenstate of the simulated Hamiltonian in the strong coupling regime, g 1 characterized by electric flux lines between the static charges is chosen to be the starting point of the simulation (Fig. 5.7a). When dynamical matter is introduced in the system, an initial state for the vertices of the lattice must be selected as well. The initial dynamical charge configuration maps directly to the number of dynamical bosons of species c and d on the vertices (5.19). Any initial state for the dynamical bosons is an eigenstate of the total Hamiltonian (5.61) if we start the simulation with R = 0. Notice that, for this state to be gauge-invariant, electric flux lines must be created also between pairs of dynamical charges, as well as between static and dynamical charges. The ground state corresponds, in the present case, to an initial atomic system with N 0,v bosons on each vertex, such that there are no dynamical charges present at the beginning of the simulation (Fig. 5.7a). Once the atomic system is initialized in the gauge-invariant ground state of the non-interacting part of Hamiltonian (5.61), for g 1 and R = 0, other regions of the theory can be reached by reducing the value of g and increasing that of R. If we change these parameters adiabatically, the system can be driven to the ground 73

82 5. Quantum Simulations of Abelian Lattice Gauge Theories Figure 5.7.: In the figure, the state of both the simulating and the simulated systems is represented in two different situations. Figure (a) corresponds to the state of the system at the beginning of the simulation, with R = 0 and g 1. In this situation, the ground state of the simulated system, in a sector characterized by two static charges of opposite sign (q = +1 and q = 1), corresponds to a electric flux line (m = +1) between the static charges, and zero dynamical charges. The latter is achieved by placing N 0,v = 4 bosons on each vertex. The electric field on the links, on the other hand, depends on the difference between the number of atoms of species a and b. When R is adiabatically increased, the state represented in (a) is no longer an eigenstate of the Hamiltonian. In particular, the term R2 ˆΦ 2 n ˆΛ ˆΦ n,k n+ˆk ˆd nâ n,kˆb n,k ĉ n+ˆk can cause the breaking of the flux line (b), with the appearance of two dynamical charges (Q = +1 and Q = 1), leaving the system in a superposition of the states (a) and (b) notice that both of them fulfill the corresponding Gauss law at each vertex (4.33). The dynamical breaking of electric flux lines can be observed, in real time, by measuring the state of the ultracold atomic system. state of the complete Hamiltonian (including the interacting part). By measuring the state of the atomic system using, for example, single-atom addressing techniques [110, 111] we will be able to extract the ground state of the Abelian-Higgs theory throughout its phase diagram. On the other hand, measuring the state of the atomic systems as we increase R adiabatically will allow us to study the breaking of flux lines in real time (Fig. 5.7). Such phenomenon, relevant in the understanding of confinement of matter in gauge theories, is very hard to study using conventional analytical or numerical techniques, since real-time dynamics are hard to simulate using standard Monte Carlo methods. Using this quantum simulation proposal, the dynamical breaking of electric flux can be accurately measured using a highly controllable atomic system. 74

83 5.3. Numerical Study 5.3. Numerical Study In order to get an estimation for the validity of the quantum simulation, specially in the weak regime, we studied numerically the pure-gauge Hamiltonian (5.54) by means of exact diagonalization for a system composed by just one plaquette. Although a more extensive study, including larger lattices, is required to make strong statements about the accuracy of the quantum simulation given by the proposal we have just introduced, the numerical results obtained for the spectrum of the oneplaquette case can help us to detect, qualitatively, the regimes where the simulation provides valid results. We chose to study, for simplicity, the pure-gauge theory without matter. However, the results can help us to understand the complete theory, since the same non-desired correction terms appear also in the absence of matter, as we have seen (5.61). Also, the non-unitary operators ˆΛ n,k and ˆΦ n converge to the desired unitary ones in a similar way (5.33), when we increase the number of atoms used in the simulation. Therefore, the improvement in the simulation by increasing the number of atoms will be qualitatively similar for the pure-gauge case and the complete theory. Simulation Accuracy In order to check the validity of the quantum simulation, we compare the Hamiltonian (5.54) with the Kogut-Susskind Hamiltonian (3.60). To see how many atoms we need to obtain enough accuracy, we have calculated the energy differences of the first three excited states of (5.54), compared to its ground state, as a function of g. In figure 5.9 we have represented the logarithm of this energy difference for different values of l = N 0 2. In order to compare with the Kogut-Susskind Hamiltonian, we considered the approximated Hamiltonian with l = 1000, since, for such a large number of atoms, the latter gives a very good approximation to the unitary case. In the right column a broad range of values of g are represented. There, we can see how, in the strong coupling limit, the results are good enough, even for the l = 1 case. As expected, this starts to change for smaller values of g. In the left column a smaller range of coupling constant values are shown. This is the region where the results start to depart from the ideal case. For the three excited states the situation is similar. If l is bigger, we can simulate smaller values of g with good accuracy. For example, for g = 1, l = 2 is good enough. For g = 0.8 we need to go to l = 3 and, for g < 0.6 the simulation does not provide good results if l < 20 this is, the difference between the energy levels of the simulating Hamiltonian and those of the simulated ( unitary ) one is too large. These calculations give us an estimation of the number of atoms we need to get sufficiently good results in the weak regime. It is at first sight surprising to see that we get good results for low l even in the weak regime (with g 1). This may have to do with the fact that we are considering a system in dimensions with just one plaquette. Usually, the number of atoms in the system fluctuates when the value of g decreases requiring larger values of l to get meaningful results. On the other hand, the fluctuations are very small in the strong coupling limit, a fact that is related to the confinement phenomenon. In the 2+1 dimensional case, however, the confinement of charges survives in the weak limit, reducing the amplitude of the fluctuations. This is why the results are good in this 75

84 5. Quantum Simulations of Abelian Lattice Gauge Theories Figure 5.8.: Gauss s law (4.33) constrains the value of the electric field on the different links, leaving only one degree of freedom to describe the system. We choose this to be the value of the electric field on the bottom link. The state of the other links depends on the latter and the static charge configuration on the vertices. The total state of the system is denoted by m m m 1 m + 1 m + 1. case even for small values l. This situation can change for larger systems and higher dimensions where confinement disappear in the weak coupling phase. In the future, it will be interesting, therefore, to extend the numerical study to more general cases. Low-Energy Spectrum We have calculated, as well, the spectrum of the theory for different values of g. Since, for values of the coupling constant below 0.5 we need a large number of atoms to perform a good simulation, we have focussed on cases with g > 0.5, where a small number of atoms is enough to give good results. In figure 5.10, we can see the first twenty energy levels of the spectrum of the theory, and how their values converge when l is increased. We chose to represent the spectrum for different values of g, two in the weak regime, and one for a very large value of the constant. For the latter case, the simulation provides good results even for l = 1. In figure 5.11, a smaller part of the spectrum is represented for the two cases corresponding to the weak regime, providing a more accurate analysis of the convergence to the ideal case when l increases. Electric Field in the Presence of Static Charges Finally, we have studied the case where two static charges of opposite sign occupy the lower vertices of the plaquette (Fig. 5.8). We calculated the expectation value, in the ground state, of the electric field operator in the link that joins the two charges (Fig. 5.12). As expected, the value converge to 1 in the strong coupling limit, and to 0.75 when g goes to zero [117]. The case l = 1 gives bad results in the weak coupling case. However, already for l = 2 the value of the electric field is very close to the one obtained with l =

85 5.3. Numerical Study Figure 5.9.: Energy difference between the ground state and the first three excited eigenstates of the Hamiltonian (5.54) as a function of g, for different number of atoms N 0,l = 2l on each link. The right column shows the strong coupling limit, whereas the left one amplifies the region where the simulation loses its validity. 77

86 5. Quantum Simulations of Abelian Lattice Gauge Theories Figure 5.10.: Lower part of the spectrum of the simulated Hamiltonian for different values of g, and for a different number of atoms N 0,l = 2l used in the simulation. In the weak regime, the convergence to the spectrum of the unitary theory is slower than in the strong coupling case. However, in all the cases good results are obtained even for small values of l. 78

87 5.3. Numerical Study Figure 5.11.: First few energy levels of the Hamiltonian (5.54) in the weak regime, for different number of atoms N 0,l = 2l on each link. The smaller the value of the coupling constant is, the slower the result converge to the ideal case. Figure 5.12.: Expectation value of the electric field operator corresponding to one link, in a system formed by just one plaquette (lower link in Fig. 5.8), calculated in the ground state of the Hamiltonian (5.54). The expectation value is represented as a function of g for different number of atoms N 0,l = 2l used in the simulation. 79

88

89 6. Summary and Conclusion In this thesis, a quantum simulation scheme for the lattice version of the Abelian- Higgs theory was proposed using a system of ultracold bosonic atoms trapped in an optical lattice. The simulated theory serves as a toy model for the study of the Brout-Englert-Higgs mechanism. It corresponds to a complex scalar (matter) field coupled to an abelian gauge field, composing a system invariant under U(1) gauge transformations. The model simplifies when the amplitude of the field is kept fixed at a large value, leaving its phase as the only degree of freedom left for the scalar field. The interesting features of the model, however, remain in this limit. The lattice version of the Abelian-Higgs theory presents an interesting phase diagram, depending on two parameters (g and R), where, apart from the mentioned mechanism associated with the breaking of a global symmetry the confinement of scalar particles takes place in certain regimes [92]. Some of these properties can be analized using Monte Carlo calculations [92, 105]. The interest in performing a quantum simulation for the lattice Abelian-Higgs theory is twofold. On the one hand, it can serve as a benchmark to study the validity of several quantum simulation techniques, since the properties of this theory are well known from conventional theoretical and numerical calculations, and can be compared to the ones obtained from the quantum simulation. These techniques could be applied, then, to simulate more complicated lattice gauge theories, such as QCD and other non-abelian theories. The simplicity of performing a quantum simulation of the Abelian-Higgs theory lies in the fact that, as oppose to theories with fermionic matter, the dynamical scalar fields can be simulated using bosonic atoms, the same as the gauge fields. Therefore, only bosons are required in its experimental realization, instead of both fermionic and bosonic atoms. On the other hand, and despite its simplicity, this theory shows very interesting high energy phenomena, as we have mentioned. It allows the possibility to study, in the same simulation, both the Brout- Englert-Higgs mechanism which is not present, for example, in other U(1) gauge theories, such as cqed and the confinement of dynamical matter. The latter is particularly interesting, since a full theoretical characterization of this phenomenon is still lacking [107]. In High Energy Physics, quantum field theories are usually described by an action. However, for quantum simulation purposes, it is more convenient to describe them in the Hamiltonian formalism, since the atomic systems that are used to simulate them are described, as well, by a second-quantized Hamiltonian. Doing this, the degrees of freedom of the simulating and the simulated system are mapped more easily. Here, starting from the lattice action of the Abelian-Higgs theory [92], the corresponding Hamiltonian was obtained using the transfer matrix method (4.32). Such a Hamiltonian is a generalization of the pure-gauge Hamiltonian obtained by Kogut and Susskind [87], which includes, apart from the electric field and the plaquette terms (defined on the links of the lattice), a charge term corresponding to the 81

90 6. Summary and Conclusion scalar matter (defined on the vertices) and an interaction between the latter and the gauge degrees of freedom. The physical Hilbert space on which this Hamiltonian acts was analyzed in detail. The theory s gauge invariance implies a division of the Hilbert space into different sectors, characterized by a static charge configuration which are the eigenvalues of the generators of the gauge symmetry. The gauge-invariant dynamics forbid the system to move between different sectors. In the strong coupling limit (g 1), the sates that belong to a given sector correspond to pairs of charges of opposite sign joined by an electric flux line (meson-like states). This is a manifestation of the confinement phenomenon. When the value of the coupling constant g starts decreasing, new pairs of charges can pop out from the vacuum provided the value of R is high enough. This can cause the breaking of the flux lines between the initial pair of charges. The latter will still be confined with new charges, but the separation between the new pair will decrease. A quantum simulation of this regime is particularly interesting, since the usual tools to study confinement of charges (Wilson loops [49], potential between static charges [87], etc.) fail when dynamical charges are introduced in the system [107] (only the confinement of static charges is well characterized). The quantum simulation of this theory using ultracold atoms is achieved by utilizing two superimposed optical lattices, filled with six different bosonic species, corresponding to three types of atoms, each pair being referred to two different internal energy levels of the atoms. One pair simulates the gauge degrees of freedom, another one the dynamical matter, and the last pair serves as auxiliary particles, needed for obtaining the desired interactions. An intermediate primitive Hamiltonian is obtained by choosing the correct hyperfine angular momenta for the six species, and tuning the scattering lengths of the atomic interactions using Feshbach resonances which, after transforming the atomic degrees of freedom and expressing them in terms of the lattice Abelian-Higgs operators, results in a gauge-invariant Hamiltonian. The latter contains all the terms of the Abelian-Higgs Hamiltonian except for the plaquette interactions, and the unitary operators are approximated with the bosonic operators, being the approximation better if the number of bosons placed on each link and vertex increases. The primitive Hamiltonian also contains a hard-core constraint for the auxiliary bosons, from which an effective Hamiltonian can be obtained, provided that the energy scales of the systems are tuned correctly using, again Feshbach resonance. We saw how this effective Hamiltonian can simulate the Abelian-Higgs one. It contains, however, non-desired but small terms that vanish when the number of atoms is increased. The simulation scheme can be applied, as well, to simulate the pure-gauge Kogut- Susskind Hamiltonian. The validity of the simulation was studied numerically for the latter case. This showed in which regimes can the quantum simulation provide meaningful results. The strong coupling limit (g 1) can be easily reached in an actual experiment, even if the number of trapped bosons is small. If weaker regimes are to be considered, more and more bosons have to be added to the system. We calculated an estimate of the latter for different values of the coupling constant g. In the future, it will be interesting to extend the numerical calculations to consider bigger systems, as well as to include dynamical matter to study the accuracy of the quantum simulations in different regimes, and for different experimental conditions being the number of bosons the most relevant one. 82

91 A. Theoretical Tools A.1. Effective Hamiltonian Consider a Hamiltonian H 0, with eigenvalues E iα, H 0 i, α = E iα i, α (A.1) whose corresponding energy levels, i, α, are grouped into different subspaces, H (0) α, each one related to a quantum number α. They will be referred to as energy sectors. The different levels that belong to each one of them are characterized by the set of quantum numbers i. We can write the projector over a sector as P α = i i, α i, α (A.2) We consider now that the different sectors are well separated [118, 119]. This means that the spectrum of H 0 satisfies the following condition (Fig. A.1), E iα E jα E iα E jβ, for α β (A.3) Figure A.1.: Separation of the spectrum of the Hamiltonian into different sectors. The energy difference between the levels inside a given sector is much smaller than the one between levels from different sectors. The presence of well-separated sectors imply the existence of two types of degrees of freedom within the system: fast degrees of freedom, characterized by the quantum number α, and slow degrees of freedom, characterized by i. Let us include a perturbation term in the Hamiltonian, with non-zero matrix elements between states from different sectors, H = H 0 + λv (A.4) 83