Spin liquid and quantum phase transition without symmetry breaking in a frustrated three-dimensional Ising model

Transcription

1 Spin liquid and quantum phase transition without symmetry breaking in a frustrated three-dimensional Ising model Masterarbeit zur Erlangung des akademischen Grades Master of Science vorgelegt von Julia Röchner geboren in Herne Lehrstuhl für Theoretische Physik I Fakultät Physik Technische Universität Dortmund 2016

2 1. Gutachter : Prof. Dr. Kai P. Schmidt 2. Gutachter : Prof. Dr. Leon Balents Datum des Einreichens der Arbeit: 30. September 2016

3 Abstract In this thesis the transverse field Ising model on the three-dimensional pyrochlore lattice is investigated. This model is highly frustrated in the low-field limit which entails an extensively large ground-state degeneracy. Quantum fluctuations in the ground-state manifold at T = 0 are present and give rise to a Coulomb quantum spin liquid phase. Based on this knowledge the quantum phase transition without symmetry breaking between the polarized phase and the Coulomb quantum spin liquid phase is located quantitively. This is done by investigating the high- and low-field limit via perturbative continuous unitary transformation which allows to determine quantities exactly as series up to a certain order in a perturbation parameter. We show by extrapolating the bare series of the elementary excitation gap of the high-field limit that the model does not exhibit a second-order phase transition. Strong evidence for a first-order phase transition is gained by comparing and locating the crossing of the groundstate energies of both limits. Kurzfassung In dieser Arbeit wird das Ising-Modell im transversalen Magnetfeld auf dem dreidimensionalen Pyrochlorgitter untersucht. Dieses Modell ist im Limes kleiner Felder hoch frustriert, was eine extensiv große Grundzustandsentartung zur Folge hat. Quantenfluktuationen in der Grundzustandsmannigfaltigkeit bei T = 0 sind der Grund für die Existenz einer Coulomb Quanten-Spinflüssigkeitsphase. Basierend darauf wird der Quanten-Phasenübergang ohne Symmetriebrechung zwischen der polarisierten Phase und der Coulomb Quanten-Spinflüssigkeitsphase quantitativ bestimmt, indem der Hoch- und Niedrigfeldlimes mittels perturbativer kontinuierlicher unitärer Transformation untersucht wird, was eine exakte Reihenentwicklung von Größen bis zu einer bestimmten Ordnung im Perturbationsparameter ermöglicht. Durch Extrapolieren der nackten Reihe der elementaren Anregungslücke im Hochfeldlimes wird gezeigt, dass das Modell keinen Phasenübergang zweiter Ordnung aufweist. Stattdessen wird ein Phasenübergang erster Ordnung gefunden, der durch den Kreuzungspunkt der Grundzustandsenergien im Hoch- und Niedrigfeldfall lokalisiert wird.

4 IV

5 Contents Table of Contents V 1 Introduction 1 2 Quantum spin ice Water ice Spin ice Quantum effects Transverse Field Ising Model on the pyrochlore lattice Model Particle representation High-field limit Low-field limit Series expansion methods Perturbative Continuous Unitary Transformation Continuous Unitary Transformation Perturbative CUT Cluster decomposition and cluster additivity Extrapolation Padé-approximation DlogPadé-extrapolation V

6 5 Full graph decomposition Graph representation Graph generation Reduced energies Graph embedding in lattice Ground state Hopping elements Loop graphs Results High-field limit Low-field limit Comparison of both limits Experimental realization Conclusion and outlook 56 A One-particle gap of antiferromagnetic TFIM on kagome lattice 60 B One-particle gap of ferromagnetic TFIM on kagome lattice 61 Bibliography 67 VI

7 Chapter 1 Introduction In condensed matter physics the study of phase transitions in strongly correlated systems displays an exciting field in theory as well as in experiment. Different phases in materials are distinguishable by their macroscopic properties. The likely most popular example of a phase transition is the one when water freezes to ice. In the fluid phase the H 2 0 molecules are disordered, while for temperatures below the critical value T c = 0 C, the system orders and the molecules form a crystalline structure. The terms "order" and "disorder" can also be defined in the realm of magnetism on a spin level, where "spin" corresponds to the orientation of magnetic moments caused by atoms in a compound. If we consider ferromagneticly coupled spins residing on the sites of a square lattice, the system is in the disordered paramagnetic phase for high temperatures. Below a critical value T c, the system orders and takes one of the ground states, where all spins follow a parallel ordering according to the ferromagnetic coupling (see Fig. 1.1). The ordered state spontaneously breaks the symmetry, since it has lower symmetry than the Hamiltonian. Note that the phase transitions described above are classical because they are induced by temperature change. In contrast to that, there can also occur phase transitions at T = 0, if materials show strong quantum effects. Those transitions are then called quantum phase transitions, which are for example generated by intrinsic or applied electric or magnetic fields. While phase transitions engendered by symmetry breaking are well known and extensively studied, phases without broken symmetries receive high attention, since they are prone to host fascinating and exotic properties. The most common examples are phase transitions generated by topological order [1 5]. Phase transitions without symmetry breaking in some other materials can be traced back to occurring frustration [6]. Frustration happens, if competing interactions in materials cannot be simultaneously satisfied usually due to the 1

8 2 Chapter 1. Introduction Figure 1.1: Sketch of phase diagram of ferromagneticly coupled spins on square lattice. For low temperatures T < T c the system is ordered while for high temperatures T > T c the system is disordered and in a paramagnetic phase. geometric arrangement of the magnetic moments. This entails, that not only one or a few classical spin configurations in the material minimize the energy, but instead the number of ground states is extensively large inducing highly entangled states in case quantum effects play a significant role. Generically, frustration gives rise to exotic phenomena in materials and can lead, under certain conditions, to a classical or quantum spin liquid state, similar to the disorder of molecules in real liquids. Spin liquids are disordered states with fractional excitations which do not break any symmetries [7]. Due to the large ground-state degeneracy of classical spin liquids, fluctuations, mainly in the ground-state manifold, occur down to very low temperatures, which can be either classical or quantum. Especially if the origin lies in quantum effects, one expects to observe exotic properties. The study of frustrated magnets therefore aroused great interest in the past. There has been much work on two-dimensional frustrated systems, for example antiferromagneticly coupled spins on the triangular or kagome lattice [8,10]. In turn, three-dimensional frustrated systems are much more complicated, since their dimensionality makes a microscopic investigation difficult, which is simultaneously the same reason why it has become an exciting and very challenging field of research. The most storied three-dimensional examples are the spin ice pyrochlores Ho 2 Ti 2 O 7 or Dy 2 Ti 2 O 7, where strong geometric frustration among coupled magnetic moments gives rise to classical spin liquids with defects that behave like magnetic monopoles [11,12]. However, quantum effects are essentially zero in these spin-ice compounds,

9 3 which are described accurately by classical Ising models, i. e. only the (local) σi z component of the spins appears in the Hamiltonian. However, theoretically, a quantum version of spin ice is highly desirable. One expects the presence of a so-called Coulomb quantum spin liquid (CQSL) [7, 13, 14] with gapped electric and magnetic excitations as well as an emergent gapless photon. Quantum fluctuations may be introduced by additional exchange interactions involving spin flips (e. g. XY or more complex couplings), which naturally occur in some other pyrochlores like Yb 2 Ti 2 O 7 [15,16]. However, such quantum exchange models are quite complex, and their phase diagrams contain many other ground states in addition to the desired CQSL [18, 19]. At the model Hamiltonian level, a simpler route to "quantum-ize" classical spin ice is to add a transverse field. The low-energy physics of such systems is expected to be described to a good extent by the transverse-field Ising model (TFIM) on the three-dimensional pyrochlore lattice, which is depicted in Fig Figure 1.2: Illustration of the pyrochlore lattice, which consists out of alternating up- and down pointing corner-sharing tetrahedra. The TFIM is one of the archetypal models used in various areas in physics and is known to host a plethora of interesting physical phenomena, especially on highly frustrated lattices [8 10,20]. At the same time the theoretical treatment of three-dimensional frustrated systems including TFIMs represents a notable challenge and quantitative results are hard to extract since common standard methods, for example exact diagonalization, are inefficient due to the rapidly growing cluster. In turn, series expansions operating in the thermodynamic limit scale well with the dimension and are well suited to treat TFIMs on three-dimensional lattices. [20 22]. In this work we apply low- and high-field series expansions to determine the phase diagram of the three-dimensional pyrochlore TFIM quantitatively. More precisely, the quantum phase

10 4 Chapter 1. Introduction transition without symmetry breaking between the polarized phase and the CQSL phase is located and compared to the mean-field result (h/j) c = 0.7 in Ref. [17], where J is the nearest-neighbor coupling of spins and h is the transverse magnetic field. The next chapter deals with spin ice models itself and how they were developed based on experimental discoveries. The subsequent chapter is about the TFIM on the pyrochlore lattice, which displays a proper model to describe some interesting features of quantum spin ice materials and which is used to locate the quantum phase transition between the polarized and the CQSL phase. The fourth chapter introduces the series expansion methods used in order to investigate the low- and high-field limit of the TFIM on the pyrochlore lattice. It turns out, that all calculations can be done on topological equivalent graphs, which makes it possible to reduce the computational effort and thus displays the cornerstone in order to establish an efficient computer program. This is explained in chapter five. Results are presented in chapter six and summarized in the last chapter.

11 Chapter 2 Quantum spin ice Quantum spin ices have been of great interest in the past few years because of their exotic phases and excitations [13, 23 25]. In particularly, we investigate in this thesis the quantum phase transition occurring between the polarized phase and the CQSL phase of the TFIM on the pyrochlore lattice. The TFIM model is suitable in order to study phase transitions in quantum spin ice. This chapter is about spin ice and later quantum spin ice in general, starting with water ice, which forms the origin of the nomenclature "spin ice". A more detailed description of water and spin ice can be found in [26]. 2.1 Water ice In the early 1930s, William Giauque and co-workers detected by specific heat measurements a non-vanishing entropy S 0 = 0.82 ± 0.05 Cal/deg mol in water ice in the low temperature regime [27, 28]. This was remarkable since it seemed to violate the third law of thermodynamics. The phenomenon was exposed by Linus Pauling, who traced it back to macroscopic proton (H + ) configurations caused by the crystalline structure of water ice and the unsymmetric bonding of the hydrogen ions [29]. He calculated theoretically a zero point entropy of S 0 = 2N 0 k B ln(3/2) = 0.81 Cal/deg mol, where N 0 is the number of water molecules, which is very close to the experimental value. In water ice, the oxygen ions O 2 form a diamond lattice with H + protons located in between the O O bonds. The H + protons itself reside on a pyrochlore lattice, which consists out of corner-sharing up- and down-pointing tetrahedra. The structure of the H 2 O molecule in the solid phase is maintained because of their strong chemical binding energy of 221 kcal/mol. This entails that the H + protons are not located 5

12 6 Chapter 2. Quantum spin ice in the middle of the O 2 ions, but favor a position, in which they are closer to one of the O 2 ions. This is depicted in Fig Figure 2.1: Depiction of the position of the hydrogen ions H + (red circles) in water ice. Two are located closer to the central O 2 ion (gray circle), two favor a "far away" position. The bond length between two distinct O 2 ions is 2.76Å, the covalent O H bond of the water molecule is 0.96Å [26]. This phenomenon is coined the Bernal-Fowler ice-rules which indicate that two H + protons should be close to one O 2 ions, while the other two favor a "far away" position. From an electrostatic point of view, the H + protons want to be located as far as possible from each other resulting into strong frustration and an extensive ground-state degeneracy, which implicate the residual zero point entropy. A similar residual entropy was detected in so-called spin ice compounds, which is presented in the next section. 2.2 Spin ice Spin ice has aroused great interest since it displays a novel class of frustrated ferromagnetic Ising systems. They exhibit intriguing properties at very low temperatures due to their frustration like a non-vanishing entropy [30, 31]. Spin ices are systems with spins located on a pyrochlore lattice (see Fig. 2.2). The ground state is formed by two spins pointing in and two spins pointing out of each tetrahedron and is therefore highly degenerate since six different spin configurations per tetrahedron follow this constraint. This exponentially large number of degenerate ground states Ω 0 entails an extensive residual, or zero point, entropy S 0 = k B ln(ω 0 ) = 0.81 Cal/deg mol as it was the case for water ice. The name "spin

13 2.2. Spin ice 7 Figure 2.2: Illustration of spin ice taken from [35]. Spins are located on a pyrochlore lattice. The tetrahedra pointing upwards form an fcc Bravais lattice. White circles display spins that point into downward tetrahedra, black circles illustrate spins pointing out of downward tetrahedra. A ground state is formed by two spins pointing in and two spins pointing out of each tetrahedron. The gray bold line indicates a hexagon which contains alternating spins. ice" originates from the direct analogy between the H + protons in water ice and the spin configuration in spin ice itself. In nature, rare earth pyrochlore oxides are candidates for realizing spin ice, such as Ho 2 Ti 2 O 7 and Dy 2 Ti 2 O 7, where Ho 3+ and Dy 3+ are located on a pyrochlore lattice carrying the magnetic moments. Their f-electron spins are large and thus they behave classical. The unit cell can be chosen cubic with up-pointing (or down-pointing) tetrahedra forming an fcc Bravais lattice (see Fig. 2.2). Harris, Bramwell and collaborators [30] studied the material Ho 2 Ti 2 O 7 and determined a positive Curie-Weiss temperature Θ CW +1,9K. Although this suggests overall ferromagnetic coupling, they could not prove any transition to long-range magnetic order down to 0.35K (and down to 0.05 K via muon spin relaxations [32]), which is surprising since long-range order is expected when dealing with a three dimensional cubic system with ferromagnetic interactions below the Curie-Weiss temperature. This gives rise to the emergence of frustration preventing the system from ordering which, at a first glance, seems to be inconsistent with the presence of ferromagnetic interactions. The term "spin liquid" was coined to describe the disorder in the temperature regime below Θ CW characterizing a state without conventional order that does not break any symmetries and which is smoothly connected to the paramagnetic phase [7]. Let us consider the main forces which play a role in spin ice compounds in order to explain the phenomenon of the occurring frustration. In rare earth ions the spin-orbit interaction is

14 8 Chapter 2. Quantum spin ice large and J = L + S represents a good quantum number. The physics can be described by H = H cf + H Z + H int (2.1) where H cf captures the crystal field originating from electrostatic and covalent bonding effects and which lifts the ion ground-state (2J + 1)-fold electronic degeneracy. The Zeeman energy H Z describes the interaction of the rare earth magnetic moments with an applied magnetic field B and H int is the inter-ion interaction Hamiltonian, which includes exchange coupling and long-range magnetostatic dipole-dipole interaction. As for spin ices, H cf is the dominant term in H with an energy scale separating the ground state from the first excited state. Since the energy scale of H cf is much larger than the energy scale of H int and H Z, which means that the high-energy sector is well separated from the low-energy sector, we can find an effective Hamiltonian which describes the low-energy physics by neglecting the excited states of H cf. Thus it appears convenient to describe the problem in the basis of the crystal-field eigenstates. We find from inelastic neutron measurements in the spin ice material Ho 2 Ti 2 O 7, that magnetic moments can only point parallel or antiparallel to the local [111] direction assuming the prior mentioned cubic unit cell of the pyrochlore lattice [33]. Apparently, the crystal field forces the spins to point along the axis which connects two neighboring centers of tetrahedra (see Fig. 2.2). This means we are dealing with an 111 pyrochlore Ising model. In other words, the magnetic moments are Ising-like but each moment has its own local axis. We assume, that the low-energy physics can be described by the interaction term in (2.1) leading to the effective classical 111 pyrochlore Ising model or "dipolar spin ice model" (DSM) for the low-energy sector H DSM = J 2 ij (ẑ i ẑ j )σ z i i σz j j + D 2 i,j (ẑ i ẑ j 3ẑ i ˆr ij ˆr ij ẑ j ) (r ij /r nn ) 3 σ z i i σz j j (2.2) where J is the exchange coupling between neighboring spins, D the dipolar interaction, ẑ i the quantization axis of J i pointing in or out of a tetrahedron, r ij the distance between spin i and spin j and r nn denotes the distance between nearest-neighbor spins. J > 0 (J < 0) indicates ferromagnetic (antiferromagnetic) exchange coupling. If we set D = 0 for a moment, then H nn = J ij (ẑ i ẑ j )σ z i i σz j j (2.3) describes a material with no dipolar interactions. In pyrochlore it counts ẑ i ẑ j = 1/3 and

15 2.2. Spin ice 9 we can therefore write (2.3) as H nn = J( 1 3 ) ij = J 3 ij σ z i i σz j j (2.4) σ z i i σz j j. (2.5) From this, we can extract a very interesting property. Obviously, a ferromagnetic nearestneighbor 111 pyrochlore Ising model results into an effective antiferromagnetic Ising model on the pyrochlore lattice. A ground state is formed by two up and two down pointing spins per tetrahedron. The ground state is highly frustrated and degenerate since one can find an extensive large number of ground states fulfilling the "two-up-two-down" rule. A ferromagnetic 111 pyrochlore Ising model is therefore frustrated due to geometric arrangement of the spins while an antiferromagnetic nearest-neighbor 111 Ising model would form a conventional long-range Neel ordered ground state. However, in the materials Ho 2 Ti 2 O 7 and Dy 2 Ti 2 O 7, D 0 is of the same order of magnitude as J or even larger and thus not negligible. Monte-Carlo simulations revealed a positive value for the nearest-neighbor dipole-dipole interaction D nn and a negative value for J, which implicates antiferromagnetic nearest-neighbor exchange coupling [34]. Still there was no transition to a long-range ordered state found down to 0.05K [32] and 0.35K [30]. Why do those materials behave like spin ices despite their antiferromagnetic exchange coupling? In [34] they showed that it depends on the ratio J/D nn. For values larger than J/D nn 0.91, the ferromagnetic nearest-neighbor dipole-dipole interaction and the antiferromagnetic nearest-neighbor exchange coupling result into an effective ferromagnetic nearest-neighbor coupling, which leads to frustration and spin-ice behavior. Even though experimental data did not show any tendencies for an ordered state of spin-ice materials down to very low temperatures for a long time, Melko and co-workers [35] predicted an ordered state with zero total magnetization in spin ice below a critical value T c and thus a vanishing residual entropy in contrast to water ice. They state, that long-range dipolar interactions, which have not been incorporated in this section so far, lift the degeneracy of the spin-ice ground-state manifold for sufficiently low temperatures. The system orders and takes a unique ordered ground state. This was recently consolidated experimentally by Pomaranski and co-workers [36], who managed to verify the absence of a residual entropy in a thermally equilibrated Dy 2 Ti 2 O 7 compound. Summarized, it can be said, that classical spin ice is in a paramagnetic phase for T > Θ CW and in a spin liquid regime for T c < T < T CW, which is not a separate phase but is smoothly

16 10 Chapter 2. Quantum spin ice connected to the paramagnetic phase. In the latter the system is frustrated due to an effective ferromagnetic nearest-neighbor coupling, preventing the system from ordering. Due to the low temperature, the system fluctuates almost entirely within the ice-rule manifold, but still spin flips, known as magnetic monopoles in spin-ice materials [11], can occur as temperatureinduced excitations. As the temperature decreases, spin dynamics lower and it becomes more and more unlikely, that the system fluctuates between different spin-ice ground states due to the energy, which has to be brought up for flipping spins. At least six alternating spins resided on a hexagon within the pyrochlore lattice (see Fig. 2.2) need to be flipped in order to get from one ground state to another. The system freezes and takes one of the disordered ice-rule ground states. For even lower temperatures T < T c, the system orders due to long range dipolar interactions which lift the degeneracy of the ground-state manifold. Classical spin ice and their exotic excitations like magnetic monopoles have been investigated extensively [11, 12]. Quantum effects are absent in these materials which is the reason why at sufficiently low temperatures, classical spin ices freeze because flipping spins is energetically too expensive and tunnelling between different ground states is not likely due to the weak quantum effects [7]. However, quantum spin ices are highly desirably, since they are prone to exhibit exotic quantum phenomena. Therefore quantum spin ices are introduced in the subsequent section. 2.3 Quantum effects It has recently been investigated, that in the spin ice compound Yb 2 Ti 2 O 7 anisotropic exchange coupling predominates over long-range dipolar interactions [23]. Still, a first-order phase transition for Yb 2 Ti 2 O 7 at T c 0.24mK could be detected [37]. Moreover, it was shown in Ref. [38] that there exists no long-range magnetic order below the critical value. Instead the transition corresponds to a change of the fluctuation rate of the Yb 3+ spins. Above T c one observes thermal fluctuations whereas below T c the spin fluctuation rate appears to be temperature-independent indicating the existence of quantum spin fluctuations. It seems like the anisotropic exchange coupling induces quantum phenomena and suppresses long-range magnetic ordering. Theoretically, these quantum spin ices can be investigated by using for example the S = 1/2 Heisenberg antiferromagnet on the pyrochlore lattice H H = ij { Jzz S z i S z j J ± ( S + i S j + S i S+ j ) } (2.6)

17 2.3. Quantum effects 11 where t runs over all tetrahedra and ij over all nearest-neighbor spins. Assuming J ± J zz allows a perturbative treatment of the low-energy physics. In Ref. [13] they investigated this model and it was found out, that those quantum spin ices enter a fractionalized quantum spin liquid phase for sufficiently small J ± with emergent U(1) gauge structure and exotic excitations like gapped "spinons", gapped "magnetic" monopoles and gapless "artificial photons". This phase is called Coulomb quantum spin liquid phase due to the similarities to conventional quantum electrodynamics. However, for the sake of completeness, it should be emphasized, that this is only a theoretical model and that in real quantum spin ice materials even more couplings play a role leading to a Hamiltonian which comprises more terms than (2.6). parameters of the model H H = ij { Jzz S z i S z j J ± ( S + i S j + S i S+ j ) In Ref. [15] they extracted all + J ±± (γ ij S + i S+ j + γ ijs i S j ) [ ( + J z± S z i ζij S j + + ζ ijsj ) ]} + i j (2.7) with matrices γ ij and ζ ij consisting of unimodular complex numbers, from high-field inelastic neutron scattering experiments with a Yb 2 Ti 2 O 7 sample. Results showed that J ±± and J z± are non-negligible. Preliminary work actually suggests, that the J z± interaction further stabilizes the quantum spin liquid phase. A schematic phase diagram has been derived as shown in Fig In the scope of this thesis, we take the antiferromagnetic Ising model on the pyrochlore lattice at T = 0 and apply a transverse magnetic field, which is an easier model in order to study quantum spin ice theoretically. The Hamiltonian reads H TFIM = J N σi z σj z + h σi x (2.8) i,j where N denotes the number of sites of the lattice, J > 0 is the antiferromagnetic nearestneighbor exchange coupling, h is an applied transverse magnetic field and σ z and σ x are Pauli matrices. As was shown earlier, the bare Ising term captures the frustration in spin ice. The transverse magnetic field induces quantum effects and lifts the extensive groundstate degeneracy in the system and thus makes it an appropriate model in order to expose some interesting properties of quantum spin ice. In case of high magnetic fields h, the system is in the conventional polarized phase, where all spins point into the direction of the transverse field. Since the model enters a CQSL phase for sufficiently small h, exactly the i=1

18 12 Chapter 2. Quantum spin ice Figure 2.3: Illustration of phase diagram of quantum spin ice taken from [15]. temperature and H the transverse magnetic field inducing quantum fluctuations. T denotes the one discussed for the antiferromagnetic S = 1/2 Heisenberg model (2.6), it has to undergo a phase transition. Note that this phase transition does not exist due to symmetry breaking. Instead the wave function of the polarized phase is a product state, whereas the CQSL phase is a disordered phase which cannot be described by such a product state. This gives rise to an existing quantum phase transition as a function of h. This phase transition has only been investigated once in the past in [17] using mean-field theory which can detect second-order phase transitions. They derived the critical value (h/j) c = 0.7. In this thesis the critical point (h/j) c of the TFIM on the pyrochlore lattice is investigated via perturbative continuous unitary transformation (pcut), which is a powerful method in order to find second-order phase transitions. Since it turns out, that this method does not provide any indications of a second-order transition, a first-order transition is found by comparing the perturbed ground-state energies from the low- and high-field regime. The next section therefore presents the TFIM on the pyrochlore lattice and provides necessary preliminary work in order to use pcut.

19 Chapter 3 Transverse Field Ising Model on the pyrochlore lattice The antiferromagnetic TFIM is chosen to investigate spins on the pyrochlore lattice. It is used commonly to examine magnetism and quantum phase transitions in spin systems [39]. Although the TFIM represents one of the easier models in theoretical condensed matter physics, it can get very challenging when studying three-dimensional systems. In this chapter the model is introduced as well as suitable particle representations in the high- and low-field limit, which makes it easier to understand the processes going on in the low-energy regime. 3.1 Model The antiferromagnetic TFIM Hamiltonian reads H = J N σi z σj z + h σi x (3.1) i,j where J > 0 is the coupling constant between two spins, h > 0 is the applied transverse magnetic field and N the number of lattice sites. The first sum runs over all nearest-neighbor spins which are located on sites of a lattice. This model is only exactly solvable in one dimension with the help of the Jordan-Wigner- Transformation [40, 41]. In all other cases it can only be approximately solved for instance it can be investigated in the high-field limit (h J) and in the low-field limit (h J) which makes a perturbative treatment possible as it is done in this thesis. As a starting point both i=1 13

20 14 Chapter 3. Transverse Field Ising Model on the pyrochlore lattice limits are discussed qualitatively. In general there are two kinds of lattices on which the antiferromagnetic TFIM can be investigated: Frustrated and unfrustrated systems. In case of a vanishing coupling constant J = 0, the only state which minimalizes the energy is the one where all spins point into the direction of the magnetic field h and thus forms the unique ground state independent of the underlying lattice (frustrated and unfrustrated). This phase is called polarized phase. An unfrustrated system undergoes a second-order phase transition at a critical value (h/j) c and is in the so-called ordered phase for sufficiently small fields h. In the ordered phase the system forms a classical antiferromagnet and the ground state for h = 0 is twofold degenerate which is also known as the Neel order. By selecting one of these ground states the symmetry of the Hamiltonian is broken which is the reason for the existence of the phase transition. An example for such an unfrustrated model is the antiferromagnetic TFIM on the bipartite square lattice. Frustrated systems in turn do not necessarily exhibit two phases or a phase transition, respectively. The Hilbert space of the ground state for h = 0 is extensively large and ground states are "disordered". By turning on an infinitesimal small field h some systems order ("order by disorder") and some stay disordered ("disorder by disorder"). Systems that follow the order by disorder phenomenon always have two phases and thus a phase transition as for instance the TFIM on the triangular lattice [42]. Some of the materials which are proved to stay disordered for infinitesimal applied fields h do not necessarily exhibit a phase transition as it was shown for instance in Ref. [10, 42], [43] and [8, 44] for the TFIM on the kagome lattice, on the diamond chain and on the sawtooth chain. The reason for the absence of a phase transition is that states for infinitesimal small h are adiabatically connected to the ground state of high fields h J so that these systems are therefore immediately in the polarized phase. In turn, the antiferromagnetic TFIM on the pyrochlore lattice, which is the topic of this thesis, is an example for a disorder by disorder scenario which exhibits a quantum phase transition not due to symmetry breaking but because of an emergent CQSL phase for low magnetic fields h. We will now specify the phenomenon of frustration in greater detail which occurs in case of h = 0. As already mentioned, applying the antiferromagnetic Ising model on the square lattice does not lead to frustration because spins are able to arrange in a way such that all neighboring spins follow an antiparallel orientation. If we now try to arrange the spins analogously on a triangle, we will observe that we cannot satisfy every bond because there will always be two nearest-neighbor spins that point into the same direction and thus cause a ferromagnetic bond. This is called frustration and is depicted in Fig. 3.1 (a). Within the scope of this thesis, the spins are located on a pyrochlore lattice which consists out of corner-sharing tetrahedra. If we consider four spins occupying the corners of a tetrahedron and interacting via an antiferromagnetic nearest-neighbor Ising coupling, one will notice that

21 3.1. Model 15 this arrangement is highly frustrated because only four out of six bonds can be satisfied and two end up being ferromagnetic due to frustration (Fig. 3.1 (b)).? (a) Frustrated spins on triangle (b) Frustrated spins on tetrahedron Figure 3.1: Figure (a) outlines frustrated spins on a triangle. In case the uncertain spin on the top points upwards the red colored bond is ferromagnetic and thus not satisfied. Figure (b) visualizes one arrangement of frustrated spins on a tetrahedron. The minimal number of ferromagnetic bonds is achieved if two spins point upwards and two downwards. This way there exist two ferromagnetic bonds which are marked red. One can also extract from Fig. 3.1 (b) that this arrangement is obtained by two up and two down pointing spins per tetrahedron. Hence the ground state is formed if every tetrahedron on the lattice follows the rule "two spins up, to spins down" which enables us to draw a connection to spin ice where the ground state in the low-field limit was determined by the ice-rules "two-in-two-out". The ground state for the Ising model with no applied magnetic field ends up having the same extensive degeneracy as the ground state in spin ice materials and therefore represents a suitable approximation to describe spin ice materials. To show qualitatively that the ground state is formed by two up and two down pointing spins per tetrahedron one can rewrite the Ising Hamiltonian in the following way: H Ising = J σi z σj z = J σi z σj z (3.2) i,j t i,j ( = J σi z σj z ) (σi z ) 2 (3.3) 2 t i,j t i t }{{} = const = 4 ( = J ( ) ) 2 σi z 4 (3.4) 2 t = JN + J 2 i t ( (σ ) ) z 2 tot,t t (3.5)

22 16 Chapter 3. Transverse Field Ising Model on the pyrochlore lattice where t runs over all tetrahedra and N is the number of spins. The sum i t σz i = σz tot,t calculates the total z-component of the spin of one tetrahedron. Thus the energy is minimized, if σ z tot,t of all tetrahedra is zero. Taking into account that there are 1/2 N tetrahedra in a pyrochlore lattice, one gains the unperturbed ground-state energy per site E 0 /N = J. Inducing quantum fluctuations, which means that we now consider the full Hamiltonian (3.1) on the pyrochlore lattice, leads to exotic phases and excitations in case of low magnetic fields. Before this scenario is discussed we will have a short look at the high-field case. As already mentioned above, the ground state is not degenerate and all spins are aligned in direction of the applied magnetic field h. As for the ground-state energy every tetrahedron takes the energy 4h. Every spin flip acts as an excitation and lifts the energy of a tetrahedron by +2h. Thus the energy spectrum is equidistant with the energy levels 4h, 2h, 0, +2h and +4h for each tetrahedron. Note that this property is one reason why pcut, which is presented in section 4.1, is a suitable and efficient method to describe the low energy physics of the system in case of large magnetic fields. Contrarily, in the low-field regime, where the ground state in case of h = 0 is formed by two spins up and two spins down per tetrahedron which results into an infinitely large groundstate Hilbertspace in the thermodynamic limit, it was stated in [13] that the system is in a CQSL phase. This phase exhibits intriguing and exotic excitations like gapless photons, magnetic monopoles or "spinons" which carry fractional quantum numbers. However, as for the perturbative treatment of this phase, single spin flips that are caused by the transverse magnetic field are of prior importance within the scope of this thesis. Tetrahedra can be in one of 2 4 = 16 different states with three distinct energies. In the ground state every tetrahedron takes the energy 2J which derives from four antiferromagnetic and two ferromagnetic bonds and is equivalent to a total spin σtot z = σ1 z + σz 2 + σz 3 + σz 4 = 0 where the indices 1 to 4 enumerate the unit cell sites. The elementary excitation is again a spin flip and lifts the energy of two corner-sharing tetrahedra to zero. In this state these tetrahedra have three spins pointing in one direction and one spin pointing in the opposite direction ( σ z tot = 2) which leads to three antiferromagnetic and three ferromagnetic bonds per tetrahedron and thus energy zero. A second spin flip can either transfer a tetrahedron back in a ground state again by flipping one of the three spins that are aligned equally or place it in an even higher excited state by flipping the only left spin with a different orientation which would lead to a total spin σ z tot = 4 and exhibits the energy +6J because of emerging six ferromagnetic bonds. Possible energy states of a single tetrahedron are depicted in Fig Thus the energy spectrum is not equidistant but holds the energy states 2J, 0 and +6J. We will later see that even in this case pcut can be a useful method to investigate low energy physics in this regime by artificially adding the energy levels +2J and +4J.

23 3.1. Model 17 E = -2J E = 0 E = +6J Figure 3.2: Depiction of some representative spin configurations for each energy state of a tetrahedron in the low-field Ising limit. The red lines illustrate ferromagnetic bonds. All states which exhibit σ z tot = 0 are in the energy state E 0 = 2J. A total spin σ z tot = 2 results into the energy E 1 = 0. Every configuration with σ z tot = 4 has the energy E 2 = +6J. One fact that makes the perturbative treatment of this phase very interesting and not trivial is the tunnelling between different ground states which is permitted by quantum fluctuations. If we restrict the series expansion to order 6 for a moment, one will notice that only tunnelling between ground states which differ only by a flipped 6-link loop are permitted. The requirement to the 6-link loop is that spins are arranged in alternating orientation. This way flipping each spin on the loop does not change the total spin of any tetrahedron and thus does not lift the energy because two neighboring loop spins always belong to the same tetrahedron leaving the ground-state configuration of "two spins up, two spins down" per tetrahedron unchanged. This scenario is shown in Fig Tunnelling (a) 6-link loop in pyrochlore (b) Tunnelling of ground states via 6-link loop Figure 3.3: Figure (a) shows a 6-link loop (marked as a bold green line) in the pyrochlore lattice. In case up- and down-spins are alternating on the sites of the loop, tunnelling between these two ground states, that only differ by one flipped 6-link loop, is permitted in order 6 and higher. This is depicted in figure (b) where the two colors green and red stand for either up pointing or down pointing spins. Using degenerate perturbation theory to calculate the ground-state energy one gains an

24 18 Chapter 3. Transverse Field Ising Model on the pyrochlore lattice effective Hamiltonian of the form H O(6) eff ( = e 0 + J ring σ + 1 σ2 σ+ 3 σ 4 σ+ 5 σ 6 + h.c.) (3.6) where the sum runs over all hexagons in the pyrochlore lattice, E 0 represents the diagonal part of the ground-state energy until order 6 and thus gives the energy of the system in case of no tunnelling. The non-diagonal term, the so-called ring exchange model, is investigated in more detail in [13, 14, 17] and was numerically solved in [14] based on quantum Monte Carlo simulations. Note that this effective Hamiltonian is valid until order 8 because there do not occur any additional processes. We will later determine explicit expression for E 0 and J ring up to order 8 in h/j. One can now find particle representations of the Hamiltonian (3.1) in both limits to describe the low energy physics which makes the treatment of these regimes easier and serves as a preparation for using pcut to find series expansions in the perturbation parameter of the ground-state energy and the dispersion. 3.2 Particle representation The particle representations of both limits is equivalent to the spin representation but will make it easier to describe the low energy physics and provides a more conceivable picture of the underlying low-energy physics in the system. In the first section the high-field limit is discussed whereas in the second section the low-field limit is represented. For both limits we will introduce appropriate quasi particles which describe the low-energy physics High-field limit The quasi particles that are used to describe the polarized phase are hardcore-bosons that correspond to spin flips in the original spin language, which behave like fermions on-site but are bosonic on different sites. As a starting point we take the Hamiltonian (3.1) and rotate the basis with the help of a unitary transformation so that the new z-axis lies on the old x-axis and the new x-axis points into the direction of the old z-axis which makes the description more convenient since everything can be described in the eigenbasis of σ z. The Hamiltonian

25 3.2. Particle representation 19 then reads H = J i,j σ x i σ x j + h N σi z (3.7) In the ground state for J = 0 all spins point downwards because of σ z = which we will now interpret as a state where no particles exist. Every spin flip is taken as a particle on the site where the spin is located. This can be summarized by i=1 i ˆ= 0 i und i ˆ= 1 i (3.8) where i can take the values 1 to N and stands for the lattice site. This way the Hamiltonian can be expressed with the help of creation and annihilation operators b i and b i, where b i creates a particle on the site i and b i annihilates a particle on the site i. Therefore the following counts: b i 0 i = 0 and b i 1 i = 0 i, (3.9) b i 0 i = 1 i and b i 1 i = 0. (3.10) Based on the knowledge that σi x engenders a spin flip and σ z is diagonal in the spin and particle representation with the eigenvalues +1 of the eigen state 1 and -1 belonging to the eigen state 0, one can also rewrite these operators and replace them by b i and b i according to the Matsubara-Matsuda transformation as introduced in [45] in the following way: σ z i = 2b i b i 1 = 2n i 1 and σ x i = b i + b i (3.11) where Q i n i is the operator that counts the number of quasi particles in the system. Applied on one single site i it thus follows the equations n i 0 i = 0 and n i 1 i = 1 i. Hence the Hamiltonian (3.1) reads H hf QP := H 2h = 1 2 N (2n i 1) + J (b 2h i + b i )(b j + b j ) (3.12) i=1 = N N 2 + n i + i=1 J }{{} 2h x i,j (b i b j + b i b j + b i b j }{{}}{{} i,j T 0, i,j T 2, i,j + bi b j ) (3.13) }{{} T +2, i,j = N 2 + Q +x (T 0 + T 2 + T +2 ) := H 0 + xv. (3.14) }{{}}{{} H 0 V

26 20 Chapter 3. Transverse Field Ising Model on the pyrochlore lattice in the particle representation. The operator T 0 annihilates a particle, if one is there, and creates one on the neighboring site and thus is responsible for particle hopping. T 2 annihilates while T +2 creates two particles on neighboring sites which is the reason why they are not particle-conserving opposed to T 0. From (3.14) we gain, that the unperturbed part H 0 has an equidistant ladder energy spectrum where every creation of a particle costs energy 1. The T -operators follow the commutator relation [Q,T n ] = nt n Low-field limit As for the low-field limit one can also find an appropriate particle representation which is a bit more complex and not as straight forward as the representation in the high-field limit. Let us start with the Hamiltonian (3.1) where the Ising term is rewritten as shown in (3.5) H = JN + J 2 = E 0 + J 2 t t ( (σ z tot,t ) 2 ) + h ( (σ z tot,t ) 2 ) + h N σi x (3.15) i=1 N σi x. (3.16) As already discussed in section 3.1, the energy levels of one tetrahedron are -2J, 0 and +6J. In case the states +2J and +4J existed as well, dividing by 2J would lead to an energy spectrum without units where the difference between energy levels is 1. i=1 This is a subtle approach in order to be able to use the pcut method. Lifting the whole energy spectrum of every tetrahedron by one does not change the physics, but puts the ground-state energy to zero and thus makes it more convenient. The renormalized Hamiltonian then reads H lf := H 2J + N 2 = 1 4 t ( (σ z tot,t ) 2 ) + h 2J N σi x (3.17) where the first sum represents the unperturbed part and the second sum describes the perturbation. The energy spectrum is depicted in Fig. 3.4 which shows the three renormalized energy levels of one tetrahedron. Note that the energy spectrum of the whole system can of course be in a state with energy 2 or 3 if there are two respectively three tetrahedra with energy state 1. We do now introduce three kinds of hardcore bosons g, a and b with different flavors α, which are located in the center of the tetrahedra and which characterize the 2 4 = 16 different states the tetrahedra can take. Particle g has six flavors and is present if a tetrahedron is in the ground state. The a particles exhibits eight different flavors which cost energy 1, whereas i=1

27 3.2. Particle representation 21 E E 0 Figure 3.4: Equidistant energy spectrum of one tetrahedron. Energy levels differ by 1. Dashed lines indicate that these levels do actually not exist. particle b has energy 4 and two flavors. Therefore particles a and b exist if a tetrahedron is in an excited state and correspond to σ z tot = 2 and σ z tot = 4 in the spin language. The different particles, energy states and their degeneracy are depicted in Fig Figure 3.5: Summary of the three energy levels, their degeneracy and corresponding hardcore bosons with flavor α which map the spin model onto a quasi particle representation. The usual hardcore boson constraints count which also means that at most only one particle g, a or b with a specific flavor can exist per tetrahedron center. With the help of the new particles the perturbation h/(2j) i σx i in (3.17) can now be expressed with T-operators as in the high-field case. We remind ourselves that σ x flips a spin that belongs to two tetrahedra and hence it always changes the energy state of both of them which means that particles can only be created or annihilated in attached tetrahedra. Some of the possible states that can be

28 22 Chapter 3. Transverse Field Ising Model on the pyrochlore lattice achieved by applying σ x and thus flipping a spin are shown in Fig The numbers inside the tetrahedra specify the energy of the tetrahedron. Therefore the number in the indices of the T -operators relates to the change of energy and can be interpreted as an effective number of a particles that were created or annihilated because they carry energy 1. One will notice that only transitions where an even number of a particles net are created respectively annihilated exist, as Fig. 3.6 already suggests. The Hamiltonian (3.17) can now be expressed in second quantization and is given by HQP lf = a t,α a t,α + 4 b t,α b t,α t,α t,α }{{} + h 2J Q }{{} H 0 [ t,t α,α β,β + α,α Q U ββ αα a t,β g t,β g t,α a t,α + α,α β,β V ββ αα b t,β a t,β a t,α b t,α } {{ } T 0, t,t β,β W ββ αα a t,β a t,β g t,α g t,α + α,α β,β X ββ αα b t,β g t,β a t,α a t,α } {{ } T +2, t,t + α,α β,β + h 2J Y ββ αα a t,β b t,β g t,α a t,α } {{ } T +4, t,t }{{} x + α,α β,β Z ββ αα b t,β a t,β a t,α b t,α } {{ } T +6, t,t (T 0 + T +2 + T 2 + T +4 + T 4 + T +6 + T 6 ) } {{ } V ] +H.c. (3.18) = H 0 + xv. (3.19) This Hamiltonian now acts on the center of the tetrahedra that form a diamond lattice. The T-operators comprise the sum over all corner-sharing tetrahedra which is meant by t,t as well as the flavor α. The transition elements U ββ αα to Z ββ αα take the values 1, if the transition α,α β,β is possible or 0, if the transition cannot take place according to the original physics described in the spin language. The introduction of a flavor α is essential in order to maintain the number of states and thus not to reduce the size of the Hilbert space of the spin system. We will see in the next chapter that the ability to express the perturbed part of the Hamiltonian by particle changing T-operators is one important property which allows us to calculate energies to high orders in the perturbation parameter based on the pcut method.

29 3.2. Particle representation 23 0 T T +2 1 T T T +4 1 T T T -6 4 Figure 3.6: This figure illustrates some possible changes of energy states of two corner-sharing tetrahedra by applying σ x to the spin which connects these tetrahedra. The numbers in the center of each tetrahedron represent the energy. Particles a and b are represented as light blue and orange balls.