M AT R I X P R O D U C T S TAT E S F O R L AT T I C E G A U G E T H E O R I E S. kai zapp

Transcription

1 M AT R I X P R O D U C T S TAT E S F O R L AT T I C E G A U G E T H E O R I E S kai zapp October 2015

2 Kai Zapp: Matrix Product States for Lattice Gauge Theories, October 2015 supervisors: Jun.-Prof. Román Orús Univ.-Prof. Harvey B. Meyer

3 A B S T R A C T In this thesis the matrix product states formalism is used to calculate the chiral condensate in the massive 1-flavour Schwinger model for different fermion masses. To this end, we use the one-site infinite density matrix renormalization group algorithm applied on gauge invariant matrix product states. The results obtained are in agreement with previous studies and can be seen as a proof of concept that an matrix product ansatz can describe the relevant physical states in a Hamilton lattice gauge theory in (1+1)D. iii

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5 C O N T E N T S i introduction 1 1 introduction and motivation 3 2 basic concepts Entanglement Entanglement and the EPR Paradox Entanglement and Quantum Information Theory Bipartite Entanglement and the Schmidt Decomposition The Reduced Density Matrix Entanglement Entropy Entanglement in Quantum Many-Body Systems The Variational Principle 10 ii tensor network theory 13 3 tensor networks Why Tensor Networks Tensors, Tensor Networks and Graphical Notation Tensor Network Representation of Quantum Many-Body States Matrix Product States (MPS) Construction of an MPS Matrix Product Operators Ground State Calculations in one Dimension Infinite Time Evolving Block Decimation (itebd) Infinite Density Matrix Renormalization Group (idmrg) 29 4 the ising model in a transverse field Ground State Properties Results of the itebd Calculations 41 iii the schwinger model 45 5 the schwinger model The Schwinger Model as a Lattice Field Theory Chiral Symmetry and Chiral Condensate 50 6 calculation of the chiral condensate Gauge Term vs. Thermodynamic Limit idmrg with Gauge Invariant MPS 55 7 conclusion and outlook 67 a appendix 69 a.1 The Singular Value Decomposition 69 v

6 vi contents a.2 Continuum Extrapolations of the Subtracted Chiral Condensate 70 bibliography 75

7 L I S T O F F I G U R E S Figure 1 A neutral pion at rest decays into an electronpostitron pair. 5 Figure 2 The entanglement entropy between A and B scales with the size of the boundary A between the two subsystems. 10 Figure 3 The quantum many-body states obeying an area law for the scaling of entanglement entropy correspond to a tiny manifold in huge manybody Hilbert space. 10 Figure 4 Graphical notation of tensors 16 Figure 5 Tensor networks 16 Figure 6 TN diagram: Trace of six matrices 17 Figure 7 TN representation of a quantum many-body state 18 Figure 8 Matrix Product State: OBC 19 Figure 9 Matrix Product States: PBC and thermodynamic limit 20 Figure 10 Canonical form of an MPS 21 Figure 11 Canonical Form: Expectation value of a singlesite observable 22 Figure 12 Left- and right-canonical form 22 Figure 13 Mixed canonical form: Single-site expectation value 22 Figure 14 Successive Schmidt Decomposition 24 Figure 15 Onion-like structure of the Hilbert space. 24 Figure 16 MPS and the Valence Bond Picture 26 Figure 17 Matrix Product Operator 26 Figure 18 MPO acting on MPS 27 Figure 19 itebd: Application of U AB on an infinite MPS with 2-site translational invariance 29 Figure 20 itebd: Detailed update process 30 Figure 21 Transformation of a tensor into a vector and effective Hamiltonian 31 Figure 22 idmrg: Approximative environments and effective Hamiltonian 34 Figure 23 idmrg: odd step 34 Figure 24 idmrg: even step 35 Figure 25 2-site idmrg 36 Figure 26 Groundstate energy for the transverse Ising model: Exact and itebd calculation 42 vii

8 Figure 27 Transverse Ising Model: Error in ground state energy 42 Figure 28 Transverse Ising Model: Magnetization m z for different bond dimensions 43 Figure 29 Imaginary Time Evolution with an MPO 54 Figure 30 MPO of the local Schwinger model Hamiltonian 57 Figure 31 Schwinger Model with one-site idmrg 59 Figure 32 Computed chiral condensate for m/g= Figure 33 Subtracted chiral condensate for m/g= Figure 34 Subtracted chiral condensate for m/g=0.25. Focus on small lattice constants. 61 Figure 35 Chiral condensate: extrapolation in the bond dimension for x = Figure 36 Chiral condensate: extrapolation in the bond dimension for x = Figure 37 Continuum extrapolation of the chiral condensate for m/g = Figure 38 Continuum extrapolation of the chiral condensate for m/g = Figure 39 Continuum extrapolation of the chiral condensate for m/g = Figure 40 Continuum extrapolation of the chiral condensate for m/g = 0 71 Figure 41 Continuum extrapolation of the chiral condensate for m/g = Figure 42 Continuum extrapolation of the chiral condensate for m/g = Figure 43 Continuum extrapolation of the chiral condensate for m/g = Figure 44 Continuum extrapolation of the chiral condensate for m/g = L I S T O F TA B L E S Table 1 Table 2 Simulation parameters in the one-site idmrg algorithm. 60 Comparison: subtracted chiral condensate in the continuum. 64 viii

9 List of Tables ix Table 3 Results: subtracted chiral condensate in the continuum. 68

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11 Part I I N T R O D U C T I O N

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13 I N T R O D U C T I O N A N D M O T I VAT I O N 1 Gauge theories have revolutionized our understanding of fundamental interactions. In particular, the Standard Model of particle physics which is based on gauge theories is presently the best description of three of the four fundamental forces: the electromagnetic force, the strong force and the weak force. Within the framework of the Standard Model, forces between elementary particles are mediated by gauge fields corresponding to a particular gauge symmetry. In a perturbative treatment, one uses series expansions of transition amplitudes to obtain physical predictions. At a pictorially level the coefficients of these power series expansions can be represented by the well known Feynman diagrams which can be classified by the order of the coupling constant of the considered theory. Applied to the fundamental theory of electromagnetism, namely quantum electrodynamics (QED) this approach was extremly successful, and led to very precise predictions like the Lamb shift or the magnetic moment of the electron [38]. However, perturbation theory as a calculational tool fails once interactions become strong, as it is the case for instance in low-energy quantum chromodynamics (QCD) 1. In this non-perturbative regime, Lattice QCD, which is based on Monte Carlo evaluation of the discretized Euclidean path integral, has become a most powerful quantitative tool. For example Lattice QCD calculations for the light hadron mass spectrum have reached an impressive agreement with experimental data [14, 20]. But despite being a highly mature field, as a Monte Carlo method it suffers from the so-called sign problem which makes calculations for systems with large fermionic densities computationally inaccessible. Furthermore, the use of Euclidean time instead of real time presents a serious barrier for the understanding of out-of-equilibrium dynamics. It is therefore of primary importance to search for tools which overcome these problems, and can serve as a complementary ansatz for the numerical simulation of lattice gauge theories. This thesis deals with so-called tensor network methods as such an alternative approach. Tensor network states provide an efficient description of quantum many-body states based on entanglement properties. Their main limitation is very different compared to other numerical techniques: it is defined by the amount and structure of entanglement in quantum many-body states. 1 QCD is the theory of the strong interactions. 3

14 4 introduction and motivation We focus here on the application of so-called matrix product states, which are tensor network states for one dimensional systems, to the 1- flavour massive Schwinger model in its formulation as a Hamiltonian lattice gauge theory. The Schwinger model, or QED in two space-time dimensions, is a popular toy model for QCD, since both theories share different features like confinement or chiral symmetry breaking. We aim to study the model directly in the thermodynamic limit, since the possibility of applying algorithms for infinite-size systems would allow us to estimate the properties without the burden of finite-size scaling effects.

15 B A S I C C O N C E P T S entanglement The aim of this section is to familiarize the reader with the notion of entanglement Entanglement and the EPR Paradox Quantum mechanics has features that are radically different from those known from the classical description of Nature. For example, one may think of superposition of quantum states, interference, or tunneling. All these well known examples have one thing in common: they can already be observed in single-particle systems. But there are further purely quantum-mechanical phenomena that manifest themselves in systems that are comprised of at least two subsystems. Perhaps one of the most interesting and puzzling features of quantum mechanics is associated to such composite systems, namely entanglement. The characteristics of an entangled state is the fact that the wavefuctions of the individual particles are not well defined. This is fundamentally different to our classical description of Nature, where full knowledge of the parts is equivalent to full knowledge of the whole system. In dealing with entangled states we may even reach paradoxical conclusions by using classical lines of thought. A famous example of such a paradox was formulated by Einstein, Podolsky and Rosen in 1935 [15]. It rests on the classical assumption of locality, which in particular states that no influence can propagate faster than the speed of light. A simplified version of this so called EPR paradox, illustrated with spin-half particles, was introduced by Bohm and Aharonov [8]. The argument reads as follows: Let us consider the decay of a system of zero total angular momentum into two spin-1/2 particles. For concreteness, we can think of the decay of a neutral pi meson in its rest frame into an electron and a positron (Fig. 1). e -! 0 e + Figure 1: A neutral pion at rest decays into an electron-postitron pair. 1 Entanglement is the English translation of the German word Verschränkung, which was first introduced by Schrödinger [37]. 5

16 6 basic concepts Using Clebsch-Gordan coefficients to ensure conservation of angular momentum, we see that the spin vector of the electron-positron system has to be in the singlet configuration 2 ψ e e + = 1 2 ( + + ). (2.1) This is a paradigmatic example of an entangled state; the z-component of the spin of each particle in this state is not well defined. However, the situation changes when measurements of the spin of one of the particles are made. For example, by measuring the spin of the electron it becomes either up or down. Furthermore, then also the spin of the positron is immediately determined. It must be either down or up respectively. No matter how far electron and positron are apart, the measurement on one particle has an instantaneous (i.e. faster-thanlight) effect on the other. It was this, in Einstein s words, spooky action at a distance that led Einstein, Podolsky, and Rosen to the conclusion that the quantum mechanical description of physical reality in terms of wavefunctions had to be incomplete. In order to save locality, a number of so called hidden variable theories, which sought to supplement quantum mechanics, were introduced. Later, however, in 1964 Bell published his famous theorem (Bell s theorem) [5], and proved that these theories must yield to predictions that are inconsistent with quantum mechanics. In particular, he was able to show that all local hidden variable theories must satisfy Bell s inequalities. Since 1982 a great number of experiments confirmed the violation of Bell s inequalities, e.g., Refs. [3, 50], demonstrating that Nature itself is fundamentally nonlocal Entanglement and Quantum Information Theory Over the recent years our understanding and knowledge of entanglement, although far from being complete, made significant progress. For example, in the context of quantum information theory entanglement was identified as a useful resource, like energy, which can be used to perform tasks that could not be achieved with classical states. Among the applications are quantum cryptography [17], superdense coding [6], and quantum teleportation [7]. Furthermore, it has become clear that concepts and methods of entanglement theory which originally emerged from quantum information theory, can lead to new insights in the context of many-body systems. Important examples are the so-called area-laws for the entanglement entropy, i.e. laws that characterize entanglement in physically relevant many-body states. Since these area-laws support the theoretical framework of the tensor network methods used in this thesis, we 2 Sometimes this state is also called a Bell state, or EPR pair.

17 2.1 entanglement 7 will discuss them in subsection But first, we have to introduce some basic notions Bipartite Entanglement and the Schmidt Decomposition Let us start with a formal definition of entanglement. Here we focus on bipartite quantum systems AB, i.e. systems which can be decomposed into two different subsystems, A and B. Let ψ AB be a pure state living in the tensor product Hilbert space H AB = H A H B of the composite system. It is called separable or product state if it can be written as the product of pure states, i.e. ψ AB = ψ A ψ B, with ψ A H A, ψ B H B. (2.2) A state that cannot be written as such a product is called an entangled state. An example of an entangled state was already given in Eq In general, it is not evident whether a state is separable or entangled. However, for pure states this separabilty problem is easy to handle due to the so-called Schmidt decomposition 3 : Let ψ AB H AB be a pure state of a composite system AB. Then there exist an orthonormal basis { α A } of subsystem A, and an orthonormal basis { α B } of subsystem B such that ψ AB = χ λ α α A α B, (2.3) α=1 where λ α > 0, χ α=1 λ2 α = 1, and χ = min {dim H A, dim H B }. The non-negative numbers λ α are the Schmidt coefficients, and χ is called Schmidt rank. Since the Schmidt basis { α A α B } consists of separable states, the information of entanglement of a given state is encoded in its (unique) Schmidt coefficients. Pure product states correspond to those states whose Schmidt decompositions have exact one Schmidt coefficient. Otherwise, if there are at least two Schmidt coefficients different from zero, the state is entangled. In fact, the number of Schmidt coefficients is what is called a (discontinuous) measure of entanglement. The larger χ is, the larger the amount of entanglement in the considered state is The Reduced Density Matrix The Schmidt coefficients can be related to the eigenvalues of the reduced density matrix of either A or B. The reduced density matrix turned out to be a very usefel tool for the description of the individual subsystems of a composite system. It describes all the properties or outcomes of measurements of the considered subsystem, given that 3 The Schmidt decomposition is essentially a restatement of the singular value decomposition (see App. A.1).

18 8 basic concepts its complement is left unobserved 4. For subsystem A it is obtained by tracing out the degrees of freedom of B in the joint state ψ AB, i.e. ρ A = tr B ( ψ AB ψ AB ) dim H B i=1 i B ( ψ AB ψ AB ) i B, (2.4) where tr B denotes the partial trace over system B, and { i B } is a basis in B. Note, the partial trace maps operators defined on the Hilbert space of the composite system to operators defined on the Hilbert space of the considered subsystem. 5 The reduced density matrix for subsystem B is analogously defined. By choosing the Schmidt basis to evaluate the reduced density matrices for A and B they can be diagonalized: ρ A = ρ B = χ λ 2 α α A α A, (2.5) α=1 χ λ 2 α α B α B. (2.6) α=1 The eigenvalues of ρ A and ρ B are identical, and are given by the squares of the Schmidt coefficients. Therefore, the reduced density matrix is also a useful tool for finding the Schmidt decompostion Entanglement Entropy We already mentioned that the Schmidt rank χ quantifies entanglement in bipartite pure states, and is therefore one possible measure of entanglement. Here we want to introduce the so-called entanglement entropy of a subsystem, which is the von Neumann entropy of its reduced density matrix. For subsystem A it is given by S (ρ A ) = tr ( ρ A log 2 ρ A ). (2.7) Or, in terms of the Schmidt basis: S (ρ A ) = χ λ 2 α log 2 λ 2 α = S (ρ B ), (2.8) α=1 where λ α are the Schmidt coefficients. Thus the entanglement entropy is for both subsystems the same. This reflects the intuition that entanglement, as a correlation between A and B, is a common property. The entanglement entropy is a continuous measure of entanglement. To illustrate this, let us look again at the singlet state ψ AB = 1 2 ( A B A B ). (2.9) 4 For a detailed introduction, including the density matrix formalism, we may refer the reader to [27]. 5 Different to the trace, which maps operators to scalars.

19 2.1 entanglement 9 It can be easily seen that this state is already in its Schmidt decompostion with Schmidt coefficients λ 1 = λ 2 = 1/ 2, and Schmidt rank χ = 2. The entanglement entropy is S = 1 2 log log = log 2 2. (2.10) It can be shown that the Schmidt rank χ provides an upper bound for the entanglement entropy, namely S log 2 χ, (2.11) and therefore the singlet state is a so-called maximally entangled state. In contrast to this, for a product state, like e.g. ψ AB = A B, (2.12) there is always only one Schmidt coefficient with λ 1 = 1, and the entanglement entropy will be S = 0. It is quantum correlations that make the entropy of reduced states become non-vanishing Entanglement in Quantum Many-Body Systems As already indicated, the study of the entanglement properties has led to useful insights in the context of quantum many-body systems. In particular, we want to discuss here qualitatively the so-called arealaw scaling of the entanglement entropy. 6 Let us consider a connected subsystem A of a quantum many-body system and its complement B. Then a natural question might be how the entanglement entropy between A and B scales. Since the entropy is an extensive quantity, one could expect that it scales with the volume of the subystem. And indeed, for a quantum state picked at random from the many-body Hilbert state, this will be most likely true. However, many important Hamiltonians in Nature tend to be local, with interactions limited to close neighbors. It turns out that this locality of interactions has important consequences. For example, it can be proven that the low-energy states of gapped many-body Hamiltonians with such local interactions obey an area law. That means the entanglement entropy of a region of space A and its complement B is proportional to the area of the boundary separating both regions, see Fig. 2. That is, low-energy states of gapped models are (much) less entangled than they actually could be. If we aim to study these states, this huge constraint on the entanglement properties identifies the relevant, although exponentially small, corner of quantum states in the many-body Hilbert space, see Fig. 3. This will be key to the understanding of Tensor Network methods introduced in the next chapter. 6 For a detailed review on area-laws the reader may be referred to Ref. [16].

20 10 basic concepts B A A S~ A Figure 2: The entanglement entropy between A and B scales with the size of the boundary A between the two subsystems. Many-body Hilbert Space Area-law states Figure 3: The quantum many-body states obeying an area law for the scaling of entanglement entropy correspond to a tiny manifold in huge many-body Hilbert space. 2.2 the variational principle In this section we review the variational principle. It is the basis for so-called variational methods such as, for example, the infinite Density Matrix Renormalization Group (idmrg) algorithm, which is used in this thesis. The variational principle states that the ground state energy E 0 of a system, described by a Hamilitionian H, is always less than or equal to the expectation value of H in any normalized state ψ, i.e. E 0 ψ H ψ, where ψ ψ = 1. (2.13) In other words, the expectation value of H with respect to the chosen trial wavefunction is always an upper bound for the ground state energy. To prove this result, we use that the (unknown) eigenstates of H form a complete set. Therefore, the trial wavefunction ψ can be written as ψ = k c k φ k, with H φ k = E k φ k. (2.14) The eigenstates themselves are assumed to be orthonormalized, φ k φ l = δ kl. Hence, we get ψ H ψ = k c k E lc l φ k φ l = E k c k 2. (2.15) l k

21 2.2 the variational principle 11 Since the ground state energy corresponds, by defintion, to the smallest eigenvalue, i.e. E 0 E k for all k, it follows ψ H ψ E 0 c k 2 = E 0, (2.16) k which was to be proven. Of course, the variational principle per se does not tell us what kind of trial wave function should be used for a given Hamilitionian H. Successful variational methods rely on educated guesses on the wavefunction derived from physical insights or intuition.

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23 Part II T E N S O R N E T W O R K T H E O RY

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25 T E N S O R N E T W O R K S 3 This chapter is mainly based on Ref. [28]. 3.1 why tensor networks The description of quantum many-body systems is, in general, an extremly difficult task. This is related to the fact that the size of the Hilbert space grows exponentially with the size of the given system. For example, an arbitrary quantum many-body state of a system with N two-level subsystems already requires the specification of 2 N complex numbers. For a classical computer, this inefficient representation implies both storage and computational problems. On the other hand, it is well known that a separable state of N qubits 1 can be described with about O (N) paramters. The huge difference, which makes a general state difficult to describe compared to a separable state, lies in the complexity of quantum correlations, or entanglement. Therefore, one might intuitively suspect that the low-energy states of local Hamiltoninas which obey an area law (see subsection 2.1.6) can also be described with relatively few parameters compared to a random state in the many-body Hilbert space. At this point Tensor Networks come into play. They serve as an efficient parametrization for this small, albeit fundamental, corner of the Hilbert space. 3.2 tensors, tensor networks and graphical notation For our purposes, a tensor is defined as a multidimensional array of complex numbers. The number of indices needed to label a given tensor is called its rank. According to this definition a scalar (x) is a rank-0 tensor, a vector (v α ) is a rank-1 tensor, and a matrix ( A αβ ) is a rank-2 tensor. The sum over all possible values of common indices of a set of tensors is called an index contraction. A familiar example is the product of two matrices C αβ = D A αγ B γβ, (3.1) γ=1 1 A qubit is a quantum mechanical two-level system. 15

26 16 tensor networks (a) (b) (c) (d) Figure 4: Graphical notation of tensors: (a) scalar, (b) vector, (c) matrix and (d) rank-3 tensor where in this case the contraction is done with respect to the index γ, with γ = 1,..., D. Index contractions can become arbitrary complex, e.g. E αβγδ = D ν D ρ D σ D ω ν=1 ρ=1 σ=1 ω=1 A ανρω B βσν C σρ D ωγδ, (3.2) where the index ι {ν, ρ, σ, ω} can take D ι different values. Indices that are not contracted are referred to as open indices. By a tensor network (TN) we understand a set of tensors whose indices are partly or wholly connected according to some network pattern. Eqs. 3.1, 3.2 provide examples of TNs. At this stage, it is handy to proceed to a graphical notation for tensors and tensor networks by introducing tensor network diagrams. In these diagrams tensors and their indices are represented by shapes with outgoing legs, where the number of legs corresponds to the rank of the tensor, see Fig. 4. A contraction is represented by a connecting line between two tensors. Examples of contractions in diagrammatic notation are shown in Fig. 5. Tensor network diagrams serve as a powerful tool in dealing with TN calculations, since complex equations can be represented in a visual way that is easier to handle, and that sometimes reveals properties, which are more difficult to see in plain equations. One such example is the cyclic property of the trace of a matrix product, which becomes immediately apparent in terms of TN diagrams, see Fig. 6. (a) (c) β β ν B σ (b) α E γ = α A ρ C δ ω D γ δ Figure 5: Tensor network diagram notation: (a) matrix product, (b) scalar product of two vectors, (c) Eq. 3.2 as a TN diagram

27 3.3 tensor network representation of quantum many-body states 17 Figure 6: The cyclic property, in this case of six matrices, becomes obvious in terms of TN diagrams. 3.3 tensor network representation of quantum manybody states We now turn to explain what we are mainly interested in, namely the tensor network representation of quantum many-body states. Let us consider a quantum many-body system of N particles each one with d degrees of freedom. An arbitrary wave function ψ that describes its physical properties can be written as ψ = d i 1,i 2,...i N =1 C i1 i 2...i N i 1 i 2 i N, (3.3) where { i r } is an individual basis for the single particle states of each particle r = 1,..., N. The d N complex numbers C i1 i 2...i N encoding the many-body wave function can be considered as the coefficients of a high-rank tensor C with N indices i 1 i 2... i n, where each of the indices can take d different values. That reflects why quantum manybody systems provide a computational challenge, since already the description of the wave function by a rank N tensor with O ( d N) coefficients scales exponentially in the system size, and therefore it is computationally inefficient. The fundamental idea of tensor networks is to take the above high-rank tensor C and decompose it into tensors of smaller rank that are being contracted. Some examples in diagrammatic notation are given in Fig. 7. The number of parameters required to specify these tensors is much smaller. In fact, the representation of ψ in terms of a TN is computationally efficient, since typically it depends on a polynomial number of parameters. In general, the number of parameters m tot to determine a tensor network is N T m tot = m (T i ), (3.4) i=1 where m (T i ) is the number of parameters for tensor T i, and N T denotes the number of tensors in the TN under consideration. For every practical tensor network its number of tensors N T is sub-exponential

28 18 tensor networks (a) C (b) (c) Figure 7: Tensor network decomposition of coefficient C in (a) an MPS with open boundary conditions, (b) a PEPS with open boundary condition, and (c) an arbitrary tensor network. in the system size N, e.g. N T = O (poly(n)). Each of the individual tensors T i has a number of parameters given by rank(t i ) m (T i ) = O D (ι), (3.5) ι=1 where the product is taken over all the different indices ι = 1, 2,..., rank (T i ) of the tensor, D (ι) is the number of different values the index ι can take, and rank (T i ) is the rank, or equivalently, the number of indices of the tensor T i. If we denote with D Ti the maximum of all D (ι), then we have ( ) m (T i ) = O (D Ti ) rank(t i). (3.6) All in all, the total number of parameters is N T ( ) m tot = O (D Ti ) rank(t i) = O (poly (N) poly (D)), (3.7) i=1 where D is the maximum of D Ti taken over all tensors {T i } of the considered TN. Here we also assumed that the rank of each tensor is bounded by a constant.

29 3.4 matrix product states (mps) 19 C A A A 1 2 N i i i 1 2 N i i i 1 2 N Figure 8: Matrix Product State representation of a quantum many-body wavefunction for N particles. 3.4 matrix product states (mps) In this subsection, we aim to give a rather panoramic view on the probably most famous example of TN states, namely the family of Matrix Product States (MPS). For instance, they lie at the basis of the famous Density Matrix Renormalization Group (DMRG) algorithm [51], which has established itself as one of the most powerful numerical techniques for simulating strongly correlated quantum systems in 1D. For other families of TN states, like e.g. Projected Entangled Pair States (PEPS) 2 the interested reader is referred to Ref. [28] and references therein. Matrix Product States are TN states that are made of tensors contracted in a pattern of a one-dimensional chain. Fig. 8 shows the representation of a quantum many-body wavefunction for N particles as an MPS with open boundary conditions (OBC). Therefore, an MPS reproduces the one-dimensional physical geometry of the system. Every site in the many-body system has a corresponding tensor in the MPS. The individual tensors are glued together by the contraction of the bond indices. They can take χ different values, where χ is called the bond dimension. The open indices correspond to the physical degrees of freedom of the local Hilbert spaces and can take up to d values. Matrix Product States can be easily extendend to periodic boundary conditions (PBC), or the thermodynamic limit. In the latter case one has to choose a fundamental unit cell that is repeated infinitelymany times. Both cases are shown in Fig. 9. Note that for fixed open (physical) indices the corresponding coefficient is indeed represented as a product of matrices (rank-2 tensors) 3. This explains the name Matrix Product States. Canonical forms The representation of a quantum many-body state by a MPS is not unique, since due to the matrix product structure we have a gauge free- 2 The family of PEPS is a natural generalization of MPS to higher dimensions. 3 Except for the first and the last tensor in the case of OBC, which are vectors.

30 20 tensor networks (a) A A A 1 2 N (b) A A A A A i i i 1 2 N Figure 9: Matrix Product States of (a) a quantum many-body wavefunction for N particles with periodic boundary conditions and (b) a 1-site translational invariant system in the thermodynamic limit. dom in the bonds. In other words, between any two matrices we can insert an arbitrary invertible matrix M and its inverse without changing the state. However, there exist different choices which remove this non-uniqueness. A particular useful choice is the so-called canonical form [47, 48]. A given MPS with OBC and bond dimension χ is in its canonical form if, for every bond index α, the index corresponds to the labeling of Schmidt vectors in the Schmidt decomposition of ψ across that index, i.e: ψ = χ λ α φ L α φ R α. (3.8) α Here λ α denote the Schmidt coefficients, and { φ L α, φ R α } are the orthornormal Schmidt vectors. In the case of a finite system with N sites this means that we have the following decomposition of the coefficient of the wave function: C i1 i 2...i N = Γ [1]i 1 α 1 λ α [1] 1 Γ [2]i 2 α 1 α 2 λ α [2] 2 Γ [3]i 3 α 2 α 3 λ α [3] 3 λ [N 1] α N 1 Γ [N]i N α N 1, (3.9) where the tensors Γ correspond to changes of basis between the different Schmidt basis and the computational (spin) basis, and the diagonal matrices λ contain the Schmidt coefficients. For an infinite MPS with one-site translation invariance, the canonical form is even simpler, since only one tensor Γ and one λ is needed to describe the whole state. The TN diagrams for both cases are shown in Fig. 10. There a few properties that make the canonical form very useful for calculations. For example, it gives easy access to the eigenvalues of the reduced density matrix of different bipartitions, which are just the squares of the Schmidt coefficients. This is very useful if one is interested in calculating of e.g. entanglement spectra or entanglement entropies. Another big advantage is that calculations of expectations values of local operators simplify a lot, see Fig. 11. The biggest advantage of the canonical form is that it provides a natural truncation scheme for the bond indices of an MPS. At each simulation step one only keeps the χ largest Schmidt coefficients as an approximation. From a mathematical point this is equivalent to a well known problem, namely the low-rank approximation of a matrix. The physical

31 3.5 construction of an mps 21 (a) Γ λ Γ λ Γ λ Γ (b) Γ λ Γ λ Γ λ Γ Figure 10: Canonical form of (a) a 4-site MPS and (b) an infinite MPS with 1-site unit cell. intuition behind is that one reduces the rank of the matrices which carry the quantum corrrelations, and therefore compresses the MPS in the amount of entanglement it can support. The canonical form and this compression in entanglement will play a crucial role in the itebd algorithm presented in the next section. Besides the canonical form presented so far, there a few related choices of fixing the gauge degree of freedom. For example, one obtains the so called left-canonical (right-canonical) form by absorbing all the Schmidt coefficients into the tensors to their left (right) in the canonical form. The resulting tensors satisfy the normalization conditions depicted in Fig. 12. In practice a mixture of the the two forms turns out to be very useful. For example, this so called mixed canonical form is crucial for improving speed and numerical stability in the idmrg algorithm which will be described in the next section. The mixed canonical form is obtained by choosing a site as the orthogonality center of the MPS, and imposing that all sites to the left and right satisfy the left-canonical and right-canonical normalization conditions, respectively. For example, one advantage is that the evaluation of single-site operators at the center site involves only the operator, the center matrix and its hermitian conjugate, see Fig construction of an mps In this section, we want to discuss briefly two different ways to obtain an MPS. First we show that any pure quantum many-body state admits a MPS representation if the bond dimensions are sufficiently large. Then we explain a hypothetical preparation from maximally entangled states which allows us to understand why the class of MPS is sucessfully used to simulate the low-energy states of local one-dimensional Hamiltonians with a gap.

32 22 tensor networks (a) Γ λ1 Γ λ2 Γ λ3 Γ λ 4 Γ λ 2 Γ 3 λ 3 O = O (b) Γ * λ Γ * λ Γ * λ Γ * λ Γ * 4 5 Γ λ Γ λ Γ λ Γ λ Γ λ λ 2 Γ* λ 3 3 Γ λ O = O Γ* λ Γ* λ Γ* λ Γ* λ Γ* λ Γ* λ Figure 11: Expectation value of a single-site observable for an MPS in canonical form: (a) 5-site MPS and (b) infinite MPS with 1-site unit cell. (a) (b) = = Figure 12: A matrix that is part of a (a) left-canoncial MPS or (b) a rightcanonical MPS, and its hermitian conjugate contract to the identity when contracted (a) over their left indices and their physical indices or (b) over their right indices and their physical indices. O = O Figure 13: Since all tensors to the left (right) of the orthogonality center, here depicted as a red matrix, are left- (right-) canonical, the calculation of any expectation value of a single-site operator O acting on the center site requires only the contraction of the operator with the center matrix and its conjugate.

33 3.5 construction of an mps 23 MPS as successive Schmidt decompositions Let us consider a quantum many-body sytems consisting of N particles with d degrees of freedom. The coefficients of any pure quantum many-body state describing the system in a state ψ = d i 1,i 2,...i N =1 C i1 i 2...i N i 1 i 2 i N (3.10) can be represented as an MPS in canonical form by making use of successive Schmidt decompositions [49]. First we perform a Schmidt decomposition between site 1 and the complementary N 1 sites, and then expand the left Schmidt vectors in terms of the original basis, ψ = d d i 1 =1 α 1 =1 Γ [1]i 1 α 1 λ [1] α 1 i 1 φ [2,...,N] α 1. (3.11) In the above equation the matrix Γ is the corresponding change of basis matrix, and λ contains the Schmidt coefficients. We then proceed in terms of the basis given by the tensor product of the original basis on site 2, and the basis for subsystem [3,..., N] consisting of the right Schmidt vectors by expanding the right Schmidt vectors φ [2,...,N] α 1 φ [3,...,N] α 2 according to a Schmidt decompostion between sites [1, 2] and [3,..., N], i.e. φ [2,...,N] α 1 = d d 2 i 2 =1 α 2 =1 If we substitute Eq in Eq. 3.11, we obtain ψ = d d d 2 i 1,i 2 =1 α 1 =1 α 2 =1 Γ [2]i 2 α 1 α 2 λ [2] α 2 i 2 φ [3,...,N] α 2 (3.12) Γ [1]i 1 α 1 λ [1] α 1 Γ [2]i 2 α 1 α 2 λ [2] α 2 i 1 i 2 φ [3,...,N] α 2. (3.13) By iterating the above procedure for the right Schmidt vectors φ α [3,...,N] 2, φ [4,...,N] α 3,..., φ [N] α N 1 one arrives at the canonical form of an MPS, see Fig. 14. Note, this does not necessary mean that this representation is efficient. In fact, per construction the range of the bond dimension in the middle of the MPS has to be exponentially large in the system size to be able to represent any state in the Hilbert space, namely up to d N 2. However, the whole point of MPS is that low energy states of local Hamiltonians with a gap can typically be approximated very well by an MPS where the bond dimension is a constant or scales at least polynomially in the system size χ [46, 24]. The bond dimension χ can be regarded as a refinement parameter. This idea is illustrated in Fig. 15. By increasing the bond dimension we increase the subset of the many-body Hilbert space we can faithfully represent by the MPS.

34 24 tensor networks i i 1 2 i 3 in Schmidt decomposition Γ 1 λ 1 i 1 i 2 i 3 i N Γ λ1 Γ λ2 Γ λ3 Γ λ N-1 Γ N i i i i N Figure 14: Exact representation of a pure quantum many-body state by an MPS obtained by successive Schmidt decompositions and changes of basis, see text. χ=100 χ=10 Many-body Hilbert Space χ=1000 1D Area-law states χ=d N/40 χ=d N/20 χ=d N/2 Figure 15: Onion-like structure of the Hilbert space: The higher the bond dimension of an MPS, the more states are accessible in the manybody Hilbert space. For a bond dimension exponentially large in the system size, eventually, every state can be represented as an MPS.

35 3.5 construction of an mps 25 MPS and the Valence Bond Picture A less technical and physically intuitive way of thinking about MPS provides the so-called valence bond picture [45, 30]. In the valence bond picture each of the N particles with d degrees of freedom is replaced by a pair of virtual particles of dimensions χ. Every pair of neighboring virtual spins corresponding to different sites are assumed to be in a maximally entangled state, which means that the state of this pair is described by χ α=1 1 χ α α (3.14) Then we can apply a map on each site which projects two virtual systems with dimension χ into a single system with the real physical dimension d, P = d χ i=1 α,β=1 A [i] αβ i α β. (3.15) The state obtained in this way has exactly the form of an MPS made of the matrices which define the projection P (see Fig. 16). Let us mention here two nice features of the description of an MPS in terms of maximally entangled pairs. First, in this picture it becomes particularly clear that MPS satisfy a one-dimensional area-law for the entanglement entropy. Let us for example look at any connected partition of the system, e.g. the red sites in Fig. 16(d). The boundary lines, represented by the dashed black lines in the figure, will always cut exactly two of the maximally entangled states. We know that the entanglement entropy between these virtually particles is log 2 χ, and therefore we can conclude that the entanglement entropy between the considered subsystem and its complement is S = 2 log 2 χ = const. (3.16) This explains the sucess for MPS in simulation one-dimensional systems, since that is exactly the scaling behavior expected of the lowenergy states of gaped Hamiltonians in 1D. We also see that the amount of entanglement the MPS can support is determined by the bond dimension χ. As a second useful feature we want to mention that the valence bond picture provides an ansatz for a natural generalization to higher dimensional physical systems. For example, the scheme can be extended to two dimensions by replacing every physical particle with four virtualy particles. This would be the valence bond picture of a PEPS[28].

36 26 tensor networks (a) (c) X =1 1 p i i (b) (d) P = dx X i i=1, =1 1 p A [i] iih h Figure 16: (a) Graphical representation of a maximally entangled state. (b) The blue circle with a outgoing leg represents the map from two virtual systems into the real system on the site. (c) MPS in the valence bond picture. (d) MPS satisfy a one-dimensional area law, see text. 3.6 matrix product operators Let us now consider an operator O acting on N sites, i.e. O = d i 1,i 2,...i N, j 1,j 2,...j N =1 W i1 i 2...i N j 1 j 2...j N i 1 i 2... i N j 1 j 2... j N. (3.17) There exist a natural generalization of the MPS formalism to an operator, namely its representation as a Matrix Product Operator (MPO). An MPO consists of a contraction of rank-4-tensors, see Fig. 17. The j j 1 2 N W i i i 1 2 N j j j 1 2 N i i i 1 2 N j Figure 17: Representation of an operator as a Matrix Product operator. application of an MPO to an MPS, gives another MPS with a bond dimension that is the product of the original bond dimension of the MPS and the bond dimension of the MPO. This is shown in Fig. 18. By analogy to the construction of an MPS, one can prove that any operator can be written as an MPO using sequential SVDs. But, this conceptual way might be expontially complicated. However, many local

37 3.7 ground state calculations in one dimension 27 (a) (b) 1..χ 1..κ 1..κχ Figure 18: (a) Acting with an MPO on an MPS produces another MPS. (b) The bond dimension of the new MPS is the product of the bond dimension of the original MPS and that of the MPO. operators have often exact representations with small bond dimension. For the explicit construction we refer the reader to the literature, see e.g. Refs. [33, 11] and references therein. 3.7 ground state calculations in one dimension In the following, we want to introduce two conceptual different TN algorithms for ground state calculations of physical systems in the thermodynamic limit, namely the Infinite Time Evolving Block Decimation (itebd) algorithm [48] based on imaginary time evolution, and a variatonal ansatz called Variatonial MPS or Infinite Density Matrix Renormalization Group (idmrg) Infinite Time Evolving Block Decimation (itebd) In quantum mechanics, the time evolution of a state at an intital time t = 0 is given by ψ (t) = exp ( iht) ψ (0), (3.18) if the Hamiltonian H is independent of time. Here we are interested in the rotation to imaginary time t iτ, since an approximation of the ground state ψ gs can be found via ψ gs = lim τ exp ( Hτ) ψ 0 exp ( Hτ) ψ 0, ψ gs ψ 0 0, (3.19) where ψ 0 is an arbitrary initial state that has non-zero overlap with the ground state 4. The idea of itebd is to implement the imaginary (or real) time evolution on MPS 5. As a first step we discretize the time and split the time-evolution operator U into m small imaginary time steps δτ, U (τ) = e τh = ( e Hδτ) m = U (δτ) m, (3.20) 4 This can be easily seen by an eigenfunction expansion of ψ 0 in terms of energy eigenfunctions. 5 Other TN states like PEPS could also be considered.

38 28 tensor networks where m = τ/δτ 1. For the sake of simplicity, let us assume that the Hamiltoninan H consists of a sum of two-body nearest-neighbor terms, H = i h i,i+1. (3.21) One can then decompose the Hamiltonian into a sum of even and odd parts, H = h i,i+1 + h i,i+1 = H even + H odd, (3.22) i,even i,odd such that within both Hamiltonians all terms commute with each other. Using a first-order Suzuki-Trotter expansion[43] we can approximate the time evolultion operator to first order in δτ by where e Hδτ = e (H even+h odd )δτ = e H evenδτ e H oddδτ + O ( δτ 2), (3.23) e H evenδτ = i,even e H oddδτ = i,odd e h i,i+1δτ U AB, (3.24) e h i,i+1δτ U BA. (3.25) Eqs and 3.25 show that the time evolution can be traced back to a sequence of two-body gates, where the two-body gate g i,i+1 between site i and i + 1 is given by g i,i+1 = e h i,i+1δτ. (3.26) The imaginary-time evolution can finally be simulated by m 1 repetitions of the operator U (δτ) = g i,i+1 = U AB U BA. (3.27) g i,i+1 i,even i,odd The application of the operator U AB on an infinite MPS in canonical form with a two-site translational invariance is shown as a TN diagram in Fig. 19. Since the action of the gates preserves the two-site invariance, only the tensors Γ A, Γ B, λ A, λ B need to be updated. Let us now formulate the final itebd algorithm for calculating the ground state of an infinite 1D system. Starting from an initial MPS in canonical form ψ 0 with bond dimension χ, one has to repeat the following steps: 1. Infinitesimal evolution (even part): apply U AB on the MPS, getting a new MPS ψ with bond dimension χ χ. 2. Truncation: compress the MPS ψ from bond dimension χ to χ.

39 3.7 ground state calculations in one dimension 29 λ B Γ A λ A Γ B λ B Γ A λ A Γ B λ B Γ A λ A Γ B λ B g g g λ B ~ ~ ~ ~ ~ ~ ~ ~ ~ Γ A λ A Γ B λ B Γ A λ A Γ B λ B Γ A λ A Γ B λ B Figure 19: Two diagrammatical representations of U AB ψ : (a) as two-site gates acting on the sites and (b) as new MPS with the same invariance under shifts by sites. 3. Infinitesimal evolution (odd part): apply U BA on the MPS ψ with bond dimension χ, getting a new MPS ψ with bond dimension χ χ. 4. Truncation: compress the MPS ψ from bond dimension χ to χ. Of course, in practical applications one has to implement a termination condition, e.g. fixing the number of time steps. A detailed diagrammatic description of the steps 1 and 2 can be found in Fig. 20. The itebd algorithm requires computational space and time that scale as O ( d 2 χ 2) and O ( d 3 χ 3) Infinite Density Matrix Renormalization Group (idmrg) In the following we introduce the so-called infinite variational MPS or infinite DMRG algorithm. Instead of simulating an evolution in imaginary time as in the itebd algorithm, the approach here relies on the variatonal principle. The educated guess on the trial wave function is based on the entanglement properties of 1D quantum many-body sytems. In particular, we want to approximate the ground state of a Hamilitonian expressed as an MPO by minimizing E [ ψ ] = ψ H ψ ψ ψ (3.28) over the family of MPS with bond dimension χ or equivalently, using a Lagrange multiplier λ enforcing normalization, by finding min ( ψ H ψ λ ψ ψ ). (3.29) ψ MPS

40 30 tensor networks α λ B Γ A λ A Γ B λ B γ (i) α Θ γ (ii) [αi] Θ [jγ] g i j (iii) i j (v) α X i λ' ~ A Y j γ (iv) [αi] X λ' ~ A Y [jγ] α λ B ( λ B ) -1 X λ' ~ A Y λ B -1 λ B ( ) γ (vi) α λ B ~ ~ ~ Γ A λ A Γ B λ B γ i j i j Figure 20: (i) First we contract the tensors into a single tensor Θ αijγ, and (ii) reshape it into a matrix Θ [αi][jγ] by an index fusion of the left (right) bond index with the left (right) physical index. (iii) Then we compute the singular value decomposition Θ [αi][jγ] = β X [αi]β λ A β Y β[jγ], (iv) and reshape the matrices X and Y into rank-3-tensors by undoing the index fusion. (v) We introduce λ B back in the tensor network and (vi) form new tensors Γ A = ( λ B) 1 X, Γ A = Y ( λ B) 1. We also truncate λ A β containing the χ Schmidt coefficents back to bond dimension χ by keeping the χ largest values.

41 3.7 ground state calculations in one dimension 31 (a) α γ [αiγ] i (b) ψ i H ψ i β α i j γ δ [βjδ] [αiγ] Figure 21: (a) Transformation of a 3-rank tensor into a vector by merging the indices. (b) Procedure to get the effective Hamiltonian for the third tensor in a 5-site MPS. Finite DMRG Before we turn to discuss the method for infinite systems, let us first briefly consider the finite case to develop our intuition and to introduce the basic tools and notions. In this case the above minimization is performed by adjusting all tensors in the MPS for all sites in order to make the expectation value of the energy the lowest possible. Ideally, this is done simultaneously. However, this global optimization problem is in general quite difficult and unfeasible. Therefore, one usually follows a sequential approach, i.e. optimizes tensor by tensor. In practice, one picks, e.g. randomly, one tensor in the MPS and minimizes with respect to its coefficients, while all other tensors remain unchanged. In terms of the chosen tensor, which we call A, the mimization problem defined by Eq can be written as min A ( ψ H ψ λ ψ ψ ) = min A ( A H eff A λ A N A). (3.30) In the above equation, all coefficients of A are arranged as a vector A as shown in Fig. 21(a), H eff is an effective Hamiltonian, and N is a normalization matrix. The effective Hamiltonian and the normalization matrix can be considered as the enviroment of tensors A and A in the two TNs for ψ H ψ and ψ ψ respectively, but written in matrix form (see e.g. Fig. 21(b)). The minimization condition ( ) A H eff A λ A N A = 0 (3.31) A leads to the generalized eigenvalue problem H eff A = λn A. (3.32)

42 32 tensor networks Once this optimization with respect to A is done, one proceeds by repeating the minimization for another tensor in the MPS. In this way, one continues sweeping through all tensors several times, until the desired convergence in expectation values is attained. Let us remark that if we start from an MPS with open boundary conditions, this algorithm is nothing else but the Density Matrix Renormalization Group (DMRG) algorithm in the language of TNs [51, 36]. In the case of open boundary conditions it is also always possible to choose an appropriate gauge for the tensors, e.g. a mixed canonical form with A as the center site, such that N = 1. Then Eq reduces to an ordinary eigenvalue problem. This is very useful for practical implementations since it avoids stability problems due to N being illconditioned, see Ref. [28]. In what follows, we always consider MPS with open boundary conditions in mixed canonical form. Infinite DMRG If we start from the very beginning with an infinite system to study systems in the thermodynamic limit, we need to modify the above procedure. The intuition that leads to our modifications is as follows [12]. Let us assume that we were given an infinitely large and translationally invariant system at absolut zero temperature, i.e. in its ground state. Then, if we were to add an additional site to the system and allow it to relax, one would expect that the new site would change to match the rest, while the other sites in the system remain unchanged. Or in the language of MPS, let us consider the case that we already had an inifinite MPS with bond dimension χ which represents the ground state of our system. Then adding a site to our system would correspond to adding another tensor in the MPS. The relaxation process could be simulated by minimizing the energy with respect to the new tensor in the environment given by the MPS which approximates the ground state. We would then obtain a tensor which looks like all of the tensors in our inifinite MPS. The idea of the algorithm is to start with a representation of the infinite systen in terms of an approximative environment. This environment is then progressively refined by embedding new sites, allowing the sites to relax, and then absorbing them. Eventually this will simulate the environment experienced by a single site in the infinite system in its ground state. The infinite-system algorithm works as follows: starting from, e.g. randomly chosen, approximative environments L H and R H representing the left and right half, with respect to the added tensor A, of the TN for ψ H ψ (see Fig. 22(a)), one has to repeat 1. Relaxation: compute the eigenvector A corresponding to the minimal eigenvalue of the eigenvalue problem 6 H eff A = λ A and 6 We choose A as the center cite for the mixed canonical form of the MPS.