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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 Kai Zapp: Matrix Product States for Lattice Gauge Theories, © October 2015 supervisors: Jun.-Prof. Román Orús Univ.-Prof. Harvey B. Meyer ABSTRACT 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 CONTENTS i introduction 1 1 introduction and motivation 3 2 basic concepts 5 2.1 Entanglement 5 2.1.1 Entanglement and the EPR Paradox 5 2.1.2 Entanglement and Quantum Information Theory 6 2.1.3 Bipartite Entanglement and the Schmidt Decomposition 7 2.1.4 The Reduced Density Matrix 7 2.1.5 Entanglement Entropy 8 2.1.6 Entanglement in Quantum Many-Body Systems 2.2 The Variational Principle 10 ii tensor network theory 13 3 tensor networks 15 3.1 Why Tensor Networks 15 3.2 Tensors, Tensor Networks and Graphical Notation 15 3.3 Tensor Network Representation of Quantum Many-Body States 17 3.4 Matrix Product States (MPS) 19 3.5 Construction of an MPS 21 3.6 Matrix Product Operators 26 3.7 Ground State Calculations in one Dimension 27 3.7.1 Infinite Time Evolving Block Decimation (iTEBD) 3.7.2 Infinite Density Matrix Renormalization Group (iDMRG) 29 4 the ising model in a transverse field 39 4.1 Ground State Properties 39 4.2 Results of the iTEBD Calculations 41 iii the schwinger model 45 5 the schwinger model 47 5.1 The Schwinger Model as a Lattice Field Theory 5.2 Chiral Symmetry and Chiral Condensate 50 6 calculation of the chiral condensate 53 6.1 Gauge Term vs. Thermodynamic Limit 53 6.2 iDMRG with Gauge Invariant MPS 55 7 conclusion and outlook 67 a appendix 69 a.1 The Singular Value Decomposition 69 9 27 48 v vi contents a.2 Continuum Extrapolations of the Subtracted Chiral Condensate 70 bibliography 75 LIST OF FIGURES Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Figure 24 Figure 25 Figure 26 A neutral pion at rest decays into an electronpostitron pair. 5 The entanglement entropy between A and B scales with the size of the boundary ∂A between the two subsystems. 10 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 Graphical notation of tensors 16 Tensor networks 16 TN diagram: Trace of six matrices 17 TN representation of a quantum many-body state 18 Matrix Product State: OBC 19 Matrix Product States: PBC and thermodynamic limit 20 Canonical form of an MPS 21 Canonical Form: Expectation value of a singlesite observable 22 Left- and right-canonical form 22 Mixed canonical form: Single-site expectation value 22 Successive Schmidt Decomposition 24 Onion-like structure of the Hilbert space. 24 MPS and the Valence Bond Picture 26 Matrix Product Operator 26 MPO acting on MPS 27 iTEBD: Application of UAB on an infinite MPS with 2-site translational invariance 29 iTEBD: Detailed update process 30 Transformation of a tensor into a vector and effective Hamiltonian 31 iDMRG: Approximative environments and effective Hamiltonian 34 iDMRG: odd step 34 iDMRG: even step 35 2-site iDMRG 36 Groundstate energy for the transverse Ising model: Exact and iTEBD calculation 42 vii Figure 27 Figure 28 Figure 29 Figure 30 Figure 31 Figure 32 Figure 33 Figure 34 Figure 35 Figure 36 Figure 37 Figure 38 Figure 39 Figure 40 Figure 41 Figure 42 Figure 43 Figure 44 Transverse Ising Model: Error in ground state energy 42 Transverse Ising Model: Magnetization mz for different bond dimensions 43 Imaginary Time Evolution with an MPO 54 MPO of the local Schwinger model Hamiltonian 57 Schwinger Model with one-site iDMRG 59 Computed chiral condensate for m/g=0.25 60 Subtracted chiral condensate for m/g=0.25 61 Subtracted chiral condensate for m/g=0.25. Focus on small lattice constants. 61 Chiral condensate: extrapolation in the bond dimension for x = 100 62 Chiral condensate: extrapolation in the bond dimension for x = 500 63 Continuum extrapolation of the chiral condensate for m/g = 0.25 63 Continuum extrapolation of the chiral condensate for m/g = 0.25 64 Continuum extrapolation of the chiral condensate for m/g = 0. 70 Continuum extrapolation of the chiral condensate for m/g = 0 71 Continuum extrapolation of the chiral condensate for m/g = 0.125 71 Continuum extrapolation of the chiral condensate for m/g = 0.125 72 Continuum extrapolation of the chiral condensate for m/g = 0.5 72 Continuum extrapolation of the chiral condensate for m/g = 0.5 73 L I S T O F TA B L E S Table 1 Table 2 viii Simulation parameters in the one-site iDMRG algorithm. 60 Comparison: subtracted chiral condensate in the continuum. 64 List of Tables Table 3 Results: subtracted chiral condensate in the continuum. 68 ix Part I INTRODUCTION 1 I N T R O D U C T I O N A N D M O T I VAT I O N 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 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 1flavour 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. 2 BASIC CONCEPTS 2.1 entanglement The aim of this section is to familiarize the reader with the notion of entanglement1 . 2.1.1 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 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 configuration2 1 |ψe− e+ i = √ (|↑− i |↓+ i − |↓− i |↑+ i) . 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. 2.1.2 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. 2.1 entanglement will discuss them in subsection 2.1.6. But first, we have to introduce some basic notions. 2.1.3 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 |ψAB i be a pure state living in the tensor product Hilbert space HAB = HA ⊗ HB of the composite system. It is called separable or product state if it can be written as the product of pure states, i.e. |ψAB i = |ψA i ⊗ |ψB i , with |ψA i ∈ HA , |ψB i ∈ HB . (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. 2.1. 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 decomposition3 : Let |ψAB i ∈ HAB be a pure state of a composite system AB. Then there exist an orthonormal basis {|αA i} of subsystem A, and an orthonormal basis {|αB i} of subsystem B such that |ψAB i = χ X α=1 λα |αA i |αB i , (2.3) P 2 where λα > 0, χ α=1 λα = 1, and χ = min {dim HA , dim HB }. The non-negative numbers λα are the Schmidt coefficients, and χ is called Schmidt rank. Since the Schmidt basis {|αA i ⊗ |αB i} 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. 2.1.4 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). 7 8 basic concepts its complement is left unobserved4 . For subsystem A it is obtained by tracing out the degrees of freedom of B in the joint state |ψAB i, i.e. ρA = trB (|ψAB i hψAB |) ≡ dim HB X i=1 hiB | (|ψAB i hψAB |) |iB i , (2.4) where trB denotes the partial trace over system B, and {|iB i} 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: ρA = ρB = χ X α=1 χ X α=1 λ2α |αA i hαA | , (2.5) λ2α |αB i hαB | . (2.6) The eigenvalues of ρA and ρB 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. 2.1.5 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 (ρA ) = − tr ρA log2 ρA . (2.7) Or, in terms of the Schmidt basis: S (ρA ) = − χ X λ2α log2 λ2α = S (ρB ) , (2.8) α=1 where λα 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 1 |ψAB i = √ (|↑A i |↓B i − |↓A i |↑B i) . 2 (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. 2.1 entanglement It can be easily seen that this state is already √ in its Schmidt decompostion with Schmidt coefficients λ1 = λ2 = 1/ 2, and Schmidt rank χ = 2. The entanglement entropy is 1 1 1 1 S = − log2 − log2 = log2 2. 2 2 2 2 (2.10) It can be shown that the Schmidt rank χ provides an upper bound for the entanglement entropy, namely S 6 log2 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. |ψAB i = |↑A i |↑B i , (2.12) there is always only one Schmidt coefficient with λ1 = 1, and the entanglement entropy will be S = 0. It is quantum correlations that make the entropy of reduced states become non-vanishing. 2.1.6 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]. 9 10 basic concepts ∂A B 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 E0 of a system, described by a Hamilitionian H, is always less than or equal to the expectation value of H in any normalized state |ψi, i.e. E0 6 hψ| H |ψi , where hψ |ψi = 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 |ψi can be written as X |ψi = ck |φk i , with H |φk i = Ek |φk i . (2.14) k The eigenstates themselves are assumed to be orthonormalized, hφk |φl i = δkl . Hence, we get XX X hψ| H |ψi = c∗k El cl hφk |φl i = Ek |ck |2 . (2.15) k l k 2.2 the variational principle Since the ground state energy corresponds, by defintion, to the smallest eigenvalue, i.e. E0 6 Ek for all k, it follows X hψ| H |ψi > E0 |ck |2 = E0 , (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. 11 Part II T E N S O R N E T W O R K T H E O RY 3 TENSOR NETWORKS 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 2N 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 qubits1 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 (vα ) is a rank-1 tensor, and a matrix Aαβ 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 Cαβ = D X Aαγ Bγβ , (3.1) γ=1 1 A qubit is a quantum mechanical two-level system. 15 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. Eαβγδ = Dρ Dσ Dω Dν X X X X Aανρω Bβσν Cσρ Dωγδ , (3.2) ν=1 ρ=1 σ=1 ω=1 where the index ι ∈ {ν, ρ, σ, ω} can take Dι 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 B β (b) α E δ ν γ = α σ ρ A ω D C 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 3.3 tensor network representation of quantum many-body states 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 |ψi that describes its physical properties can be written as d X |ψi = i1 ,i2 ,...iN =1 Ci1 i2 ...iN |i1 i ⊗ |i2 i ⊗ · · · ⊗ |iN i , (3.3) where {|ir i} is an individual basis for the single particle states of each particle r = 1, . . . , N. The dN complex numbers Ci1 i2 ...iN encoding the many-body wave function can be considered as the coefficients of a high-rank tensor C with N indices i1 i2 . . . in , 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 dN 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 |ψi in terms of a TN is computationally efficient, since typically it depends on a polynomial number of parameters. In general, the number of parameters mtot to determine a tensor network is mtot = NT X m (Ti ) , (3.4) i=1 where m (Ti ) is the number of parameters for tensor Ti , and NT denotes the number of tensors in the TN under consideration. For every practical tensor network its number of tensors NT is sub-exponential 17 18 tensor networks (a) (b) C (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. NT = O (poly(N)). Each of the individual tensors Ti has a number of parameters given by rank(Ti ) Y m (Ti ) = O D (ι) , (3.5) ι=1 where the product is taken over all the different indices ι = 1, 2, . . . , rank (Ti ) of the tensor, D (ι) is the number of different values the index ι can take, and rank (Ti ) is the rank, or equivalently, the number of indices of the tensor Ti . If we denote with DTi the maximum of all D (ι), then we have m (Ti ) = O (DTi )rank(Ti ) . (3.6) All in all, the total number of parameters is mtot = NT X i=1 O (DTi )rank(Ti ) = O (poly (N) poly (D)) , (3.7) where D is the maximum of DTi taken over all tensors {Ti } of the considered TN. Here we also assumed that the rank of each tensor is bounded by a constant. 3.4 matrix product states (mps) C i1 i2 A1 A 2 iN i1 i2 AN iN 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 free2 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. 19 20 tensor networks (a) A1 A 2 AN i1 iN i2 (b) A A A A A 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 |ψi across that index, i.e: |ψi = χ X α R λα φL α ⊗ φα . (3.8) R Here λα denote the Schmidt coefficients, and φL are the α , φα 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: [1]i [1] [2]i [2] [3]i [3] [N−1] [N]i Ci1 i2 ...iN = Γα1 1 λα1 Γα1 α22 λα2 Γα2 α33 λα3 · · · λαN−1 ΓαN−1N , (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 3.5 construction of an mps (a) (b) Γ1 λ1 Γ2 λ2 Γ3 λ3 Γ4 Γ λ λ λ Γ Γ 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. 13. 3.5 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. 21 22 tensor networks (a) Γ1 λ1 Γ2 λ2 Γ3 λ3 Γ4 λ4 Γ5 λ2 Γ3 = O λ3 O Γ1* λ1 Γ2* λ2 Γ3* λ3 Γ4* λ4 Γ5* λ2 Γ3* λ3 Γ λ (b) λ Γ λ Γ λ Γ λ Γ = O Γ* λ Γ* λ Γ* λ Γ Γ* λ Γ* λ O Γ* λ 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. 3.5 construction of an mps 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 |ψi = d X i1 ,i2 ,...iN =1 Ci1 i2 ...iN |i1 i ⊗ |i2 i ⊗ · · · ⊗ |iN i (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, |ψi = d d X X i1 =1 α1 =1 E [2,...,N] [1]i [1] . Γα1 1 λα1 |i1 i ⊗ φα1 (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 [2,...,N] i in terms of the baby expanding the right Schmidt vectors | φα1 sis 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 [3,...,N] i according to a Schmidt decompostion between sites [1, 2] | φα2 and [3, . . . , N], i.e. d d2 E X X [2,...,N] [2] [3,...,N] [2]i i = Γα1 α22 λα2 |i2 i ⊗ | φα2 φα1 (3.12) i2 =1 α2 =1 If we substitute Eq. 3.12 in Eq. 3.11, we obtain |ψi = d X d d X X 2 i1 ,i2 =1 α1 =1 α2 =1 [1]i [1] [2]i [2] [3,...,N] Γα1 1 λα1 Γα1 α22 λα2 |i1 i ⊗ |i2 i ⊗ | φα2 i. (3.13) By iterating the above procedure for the right Schmidt vectors [3,...,N] [4,...,N] [N] | φα2 i, | φα3 i , . . . , | φαN−1 i 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 N space, namely up to d 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. 23 24 tensor networks i1 Γ1 i2 i iN Schmidt decomposition 3 λ1 i1 i2 i3 iN Γ1 λ1 Γ2 λ2 Γ3 i1 i2 i3 λN-1 ΓN λ3 Γ4 iN 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. Many-body Hilbert Space N/2 χ=d χ=1000 1D Area-law states χ=100 χ=10 N/40 χ=d N/20 χ=d 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. 3.5 construction of an mps 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 X 1 √ |αi |αi χ (3.14) α=1 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= χ d X X i=1 α,β=1 [i] Aαβ |ii hα| hβ| . (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 log2 χ, and therefore we can conclude that the entanglement entropy between the considered subsystem and its complement is S = 2 log2 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]. 25 26 tensor networks (a) (c) ↵ X 1 p |↵i |↵i ↵=1 (b) (d) ↵ i P = d X X i=1 ↵, =1 1 [i] p A↵ |ii h↵| 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. d X O= i1 ,i2 ,...iN , j1 ,j2 ,...jN =1 Wi1 i2 ...iN j1 j2 ...jN |i1 i2 . . . iN i hj1 j2 . . . jN | . (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 j1 j2 jN j1 j2 jN iN i1 i2 iN W i1 i2 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 3.7 ground state calculations in one dimension (b) (a) 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). 3.7.1 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)i = exp (−iHt) |ψ (0)i , (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 |ψgs i can be found via exp (−Hτ) |ψ0 i , τ→∞ kexp (−Hτ) |ψ0 ik |ψgs i = lim hψgs |ψ0 i 6= 0, (3.19) where |ψ0 i is an arbitrary initial state that has non-zero overlap with the ground state4 . The idea of iTEBD is to implement the imaginary (or real) time evolution on MPS5 . As a first step we discretize the time and split the time-evolution operator U into m small imaginary time steps δτ, m U (τ) = e−τH = e−Hδτ = U (δτ)m , (3.20) 4 This can be easily seen by an eigenfunction expansion of |ψ0 i in terms of energy eigenfunctions. 5 Other TN states like PEPS could also be considered. 27 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, X H= hi,i+1 . (3.21) i One can then decompose the Hamiltonian into a sum of even and odd parts, X X H= hi,i+1 + hi,i+1 = Heven + Hodd , (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 e−Hδτ = e−(Heven +Hodd )δτ = e−Heven δτ e−Hodd δτ + O δτ2 , (3.23) where e−Heven δτ = Y i,even e−Hodd δτ = Y i,odd e−hi,i+1 δτ ≡ UAB , e−hi,i+1 δτ ≡ UBA . (3.24) (3.25) Eqs. 3.23 and 3.25 show that the time evolution can be traced back to a sequence of two-body gates, where the two-body gate gi,i+1 between site i and i + 1 is given by gi,i+1 = e−hi,i+1 δτ . (3.26) The imaginary-time evolution can finally be simulated by m 1 repetitions of the operator Y Y U (δτ) = gi,i+1 gi,i+1 = UAB UBA . (3.27) i,even i,odd The application of the operator UAB 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 , λA , λB 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 |ψ0 i with bond dimension χ, one has to repeat the following steps: 1. Infinitesimal evolution (even part): apply UAB on the MPS, getting a new MPS |ψ 0 i with bond dimension χ 0 > χ. 2. Truncation: compress the MPS |ψ 0 i from bond dimension χ 0 to χ. 3.7 ground state calculations in one dimension λB ΓA λA ΓB λB ΓA λA ΓB λB ΓA λA ΓB λB g g g ~ ~ ~ ~ ~ ~ ~ ~ ~ λB ΓA λA ΓB λB ΓA λA ΓB λB ΓA λA ΓB λB Figure 19: Two diagrammatical representations of UAB |ψi: (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 UBA on the MPS |ψ 0 i with bond dimension χ, getting a new MPS |ψ 00 i with bond dimension χ 00 > χ. 4. Truncation: compress the MPS |ψ 00 i from bond dimension χ 00 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 d2 χ2 and O d3 χ3 . 3.7.2 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 [|ψi] = hψ| H |ψi hψ |ψi (3.28) over the family of MPS with bond dimension χ or equivalently, using a Lagrange multiplier λ enforcing normalization, by finding min (hψ| H |ψi − λ hψ |ψi) . |ψi∈MPS (3.29) 29 30 tensor networks λB ΓA λA ΓB λB γ α (i) Θ γ α j i X -1 λB (λB) X i γ α i (iv) γ -1 ~A λ' Y (λB) λB [jγ] (iii) ~A λ' Y α (v) Θ [αi] j i g (ii) ~A λ' Y X [jγ] [αi] j (vi) ~ ~ ~ λB ΓA λA ΓB λB α i j γ j Figure 20: (i) First we contract the tensors into a single tensor Θαijγ , 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 P A Θ = , (iv) and reshape the matrices X X λe0 Y [αi][jγ] β [αi]β β β[jγ] and Y into rank-3-tensors by undoing the index fusion. (v) We introduce λB back in the tensor network and (vi) form new tensors −1 −1 A Γ ˜A = λB X , Γ ˜A = Y λB . We also truncate λe0 containing β the χ 0 Schmidt coefficents back to bond dimension χ by keeping the χ largest values. 3.7 ground state calculations in one dimension (a) γ α [αiγ] i (b) |ψ i H iψ | α i γ [βjδ] β j [αiγ] δ 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. 3.29 can be written as ~ † Heff A ~ − λA ~ † NA ~ . (3.30) min (hψ| H |ψi − λ hψ |ψi) = min A A A ~ In the above equation, all coefficients of A are arranged as a vector A as shown in Fig. 21(a), Heff 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 hψ| H |ψi and hψ |ψi respectively, but written in matrix form (see e.g. Fig. 21(b)). The minimization condition ∂ ~ † ~ − λA ~ † NA ~ =0 A Heff A (3.31) ~† ∂A leads to the generalized eigenvalue problem ~ = λNA. ~ Heff A (3.32) 31 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. 3.32 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 LH and RH representing the left and right half, with respect to the added tensor A, of the TN for hψ| H |ψi (see Fig. 22(a)), one has to repeat ~ corresponding to the min1. Relaxation: compute the eigenvector A ~ = λA ~ and imal eigenvalue of the eigenvalue problem6 Heff A 6 We choose A as the center cite for the mixed canonical form of the MPS. 3.7 ground state calculations in one dimension ungroup its index to return to its original rank-3 shape. The effective Hamiltionian is shown in Fig. 22(b). 2. Absorption (odd step): at an odd simulation step, the optimized tensor is contracted into the left environment LH . In detail: a) merge the first bond index and the physical index of A to form a matrix, and compute the singular value decompositon A = UΣV † (see Fig. 23(a)). b) Undo the index fusion for the left index of U to get back to a rank-3 shape (see Fig. 23(a)) and compute EH as defined in Fig. 23(b). c) Refine the approximation for the left environment LH by contracting EH into it, i.e. LH := LH · EH , as shown in Fig. 23(c). 3. Absorption (even step): at an even simulation step, the optimized tensor is analoguesly contracted into right environment RH (see Fig. 24). In detail: a) merge the second bond index and the physical index of A to form a matrix, and compute the singular value decompositon A = UΣV † . b) Undo the index fusion for the right index of V † to get back to a rank-3 shape and compute the analogue of the tensor EH . c) Refine the approximation for the right environment RH by contracting EH into it, i.e. RH := EH · RH . Since U and V are isometries the mixed canonical form of the MPS is preserved at every simulation step. To check for convergence it is useful to calculate the desired expectation value after, e.g., each first or second simulation step. For a single-site operator acting on the added site this is easily done as shown in Fig. 13. The main computationalcost is given by the eigenvalue problem and scales therefore as O χ3 . Two-site Infinite DMRG If only a single site is added at every simulation time, the bond dimension χ of the MPS is fixed from the beginning, since it is always an upper bound for the number of non-negative singular values7 . However, one may think of situations in which it would be advantageous to increase the bond dimension during the calculation. This limitation can be avoided by a slight modification of the previously introduced 7 According to the SVD theorem, the maximal number of non-zero singular values for a m × n matrix is min (m, n). One may compare with Fig. 23(a) and Fig. 24(a) to draw the conclusion. 33 34 tensor networks (a) A A |ψi H α i [βjδ] RH j RH A* γ LH β LH iψ | A* (b) ≈ [αiγ] Heff δ Figure 22: (a) Definition of the approximative environments LH and RH . (b) Definition of the effective Hamiltonian. (a) A A γ α U γ [αi] V† Σ [αi] γ i (b) U [αi] (c) U β α i EH U = EH β LH LH H U* Figure 23: Odd step: (a) SVD of the optimized tensor A. (b) Definition of EH . (c) Refinement of the left environment. 3.7 ground state calculations in one dimension (a) A γ α U A α α [iγ] Σ V† [iγ] i (b) V† [iγ] β β V† i † EH V = (c) γ EH RH RH H V†* Figure 24: Even step: (a) SVD of the optimized tensor A. (b) Definition of EH . (c) Refinement of the right environment. algorithm, namely one has to add two sites at each simulation step, see Fig. 25. The infinite DMRG algorithm is then as follows: ~ corresponding to the min1. Relaxation: compute the eigenvector Θ ~ = λΘ, ~ where imal eigenvalue of the eigenvalue problem Heff Θ ~ the effective Hamiltionian Heff and the vector Θ are defined as shown in Fig.25(c) and Fig.25(b), respectively. 2. Absorption: the optimized tensor is simultaneously contracted into the left environment LH and into the right environment RH . In detail: a) compute the singular value decomposition Θ = UΣV † (see Fig. 25(d)) b) Undo the index fusion for the left index of U and for the right index of V † . c) Compute the tensors EHL and EHR as defined in Fig. 25(e). d) Refine the approximations for the left environment LH and for right environment RH by the contractions LH := LH · EHL and RH := EHR · RH shown in Fig. 25(f). The crucial point is that, if one adds two sites at a time, the center matrix becomes a square matrix of increased dimension md × md as can be seen in Fig. 25 (b). This allows, in principle, for a SVD truncation in simulation step 2.(d), see also Fig. 25(d). This is especially useful if one tries to implement symmetries on the level of the tensors, since one can take account of them by giving the bond index a multiple index structure. Therefore, the SVD truncation on the bond index may change the symmetry sectors one keeps. In practice this means that 35 36 tensor networks (a) A (b) B A B LH Θ γ α [jγ] j i RH [αi] Θ A* (c) α i [αijγ] B* j γ LH RH β (d) k Heff U [jγ] EHL U = [αijγ] δ Θ [αi] (e) l [βklδ] H Σ V† [αi] [jγ] (f) EHL LH LH U* V† EHR = H EHR RH RH V†* Figure 25: Modifications for the Two-site iDMRG algorithm. 3.7 ground state calculations in one dimension the algorithm can readapt itself to more relevant symmetry sectors, which have more weight in terms of Schmidt coefficients. This may lead to an improved accuracy. 37 4 THE ISING MODEL IN A TRANSVERSE FIELD In this chapter we use the iTEBD algorithm to study some ground state properties of the one-dimensional Ising model in a transverse magnetic field. It was first introduced by de Gennes in 1963 to describe the order-disorder transition in ferroelectric crystals [13], and is presumably one of the simplest systems which exhibit a quantum phase transition1 . To the present day many physical systems have been found where it serves as a successfull description, see e.g. Refs. [34, 42]. Since the transverse Ising model is one of the rare cases of an exactly solvable many particle problem [32], it is also a popular benchmark model. We use it here to verify the validity of our iTEBD algorithm. 4.1 ground state properties Let us start by writing down the Hamiltonian. It is X X H = −J σzi σzi − h σxi , i (4.1) i where J is a coupling coupling constant, h > 0 is the strength of the external transverse magnetic field, and σα i are the Pauli matrices for the α-component of the spin at site i. The first term proportional to J describes the nearest-neighbor interaction between the spins. Here we focus on the case J > 0 where a ferromagnetic configuration, in which all spins are aligned parallel, is energetically favorable. The second term proportional to h describes an applied external magnetic field, which disturbs the preferred ordering. Therefore, it should come as no surprise that the nature of the ground state depends upon the value of the dimensionless parameter λ ≡ J/h. To specify this, let us consider two opposing limits. For h = 0 and J > 0 the ground state is either given by O O |0i = |0ii |1i = |1ii , or (4.2) i i where |0ii and |1ii are the two possible eigenstates of the Pauli matrix σzi with eigenvalues ±1. That is, the ground state is doubly generate and all spins are completely ordered with respect to the z-direction. These are both ferromagnetic states. Since the Hamiltonian in Eq. 4.1 is invariant under a Z2 -symmetry transformation, namely σz → −σz , 1 A quantum phase transition is a phase transition at zero temperature. 39 40 the ising model in a transverse field one may expect that the ground state is given by their symmetric combination 1 |ψsym i = √ (|1i + |0i) . 2 (4.3) This is, however, not the case and the system in thermodynamic limit will “choose” one of the states in Eq. 4.2 as its ground state2 . This phenomenon is called spontaneous symmetry breaking. Let us now consider the opposing limit, i.e. J = 0 and h > 0. In this case the unique ground state is O |+ii |+i = (4.4) i where |+ii = √12 (|0ii + |1ii ) is the eigenstate of the Pauli matrix σxi with eigenvalue 1. That means all spins point in the x-direction, but are “disordered” in the z-direction. We see that the nature of the ground states in the limits λ → 0 and λ → ∞ are qualitatively very different. The exact solution of the transverse Ising model shows that each limit corresponds to a different phase. The critical point, i.e. the point in the parameter space which separates both phases, is found to satisfy J λc = = 1. (4.5) h c In many cases different phases of a system can be characterized by a so-called order parameter which vanishes in one phase and is different from zero in the other phase. For the transverse Ising model an order parameter is the magnetization in z-direction, i.e. mz = hσz i . (4.6) It has a finite value in the ordered ferromagnetic phase (λ > λc ) and is zero in the disordered paramagnetic phase (λ < λc ). The ground state energy is Z J π p E0 = − 1 + h2 − 2h cos k dk. (4.7) 2π −π In order to prove the validity of our algorithm we compare our results for the ground state energy with the exact formula given in Eq. 4.7 and check for the expected behavior of the order parameter mz . 2 Of which one may be favorable due to an infinitesimal perturbation like, e.g., an infinitesimal small external magnetic field. 4.2 results of the itebd calculations 4.2 results of the itebd calculations In our calculations we set the coupling parameter J = 1, and compute the ground state for different values of the external magnetic field h. Then according to Eq. 4.5 we expect the critical point for the phase transisiton driven by the variation of the magnetic field at hc = 1. We start from a randomly chosen initial MPS in canonical form and perform the iTEBD algorithm sequentially for decreasing imaginary time steps sizes δτ ∈ {0.1, 0.05, 0.01, 0.005, 0.001, 0.0005, 0.0001}. The time step size is decreased if all the singular values are converged within a chosen tolerance . In particular, if we denote with ~λj the vector containing all singular values at time step j we assume them to be converged if ~ λj − ~λj−n < · ~1χ , n ∈ N. (4.8) ~ λ i where χ is the bond dimension of the MPS, |·| denotes the componentwise taken absolut value, k·k is the Euclidean norm, and ~1χ is the χ-dimensional vector containing only ones. For our calculations we chose n = 10, i.e. we check convergence at every time step with respect to the singular values ten time steps before. We set = 10−5 . The calculated ground state energy for χ = 40 and the exact result are shown in Fig. 26. We can see that our obtained results are in good agreement with the exact solution. In Fig. 27, we show the absolute error ∆E as a function of the external magnetic field. One can clearly see that the error is of the order of the chosen accuracy . In the vicinity of the expected critical point of the external magnetic field hc = 1 the error increases. This is expected, since MPS capture the ground states properties of gapped Hamiltonians, i.e. away from the critical point. However, this also shows that for a large enough χ they can possibly also be used to describe accurately one-dimensional systems at criticality. This behavior can also be seen in Fig. 28 in which the magnetization mz = hσz i is shown for different bond dimensions. The calculated magnetization shows the expected behavior as an order parameter for the two phases. The value at which this sudden change of the from a finite value to a zero value happens tends to move closer to the critical point h = 1 with increasing bond dimension. We conclude that our results for the ground state energy and the magnetization describe faithfully the predicted behavior from the exact solution. This shows the validity of our algorithm. 41 42 the ising model in a transverse field Figure 26: The exact groundstate energy and the ground state energy computed via iTEBD starting from a random MPS with bond dimension χ = 40. Figure 27: The absolut error for the energy as a function of the magnetic field. 4.2 results of the itebd calculations Figure 28: The calculated magnetization mz shows the expected behavior as an order parameter. As in the inset can be seen, the expected jump from a finite value to zero gets closer to the critical point at h = 1 for higher bond dimensions. 43 Part III THE SCHWINGER MODEL 5 THE SCHWINGER MODEL In this chapter, we briefly introduce the Schwinger model [39] and its equivalent theory on a lattice [23], which will be the starting point of our study with TN methods. Readers who are interested in a more detailed discussion are referred to Ref. [10], on which this section is mainly based on. The massive Schwinger Model is quantum electrodynamics in two space-time dimensions. Its Lagrangian density in the continuum reads 1 L = ψ (i∂µ γµ − m) ψ − Fµν Fµν − gψAµ γµ ψ, 4 (5.1) where Fµν = ∂µ Aν − ∂ν Aµ . (5.2) The first term is the Dirac Lagrangian density for a free fermion and the second term corresponds to the field energy of the electric field. The third term is the interaction term. It has the important feature that it arises from the constraints imposed by a local gauge transformation. That means, its shape is determined by demanding the invariance of the Lagrangian density under the following transformation ψ 0 = eigχ ψ, Aµ0 = Aµ + ∂µ χ, (5.3) where χ is an arbitrary real function of space and time 1 , i.e. χ = χ (x, t). The Schwinger model describes the interaction of one flavor of fermions ψ with mass m through a U(1) gauge field A with coupling g. In (1+1)D the Lorentz indices µ, ν run from 0 to 1 and the gamma matrices satisfy analogously to (3+1)D the Clifford algebra {γµ , γν } = 2gµν , (5.4) but due to the fact that there is no spin degree of freedom in one spatial dimension these are 2x2 matrices. Substituting the Lagrangian of the Schwinger model into Euler-Lagrange equations for the fields ψ and A results in the equations of motion γµ (i∂µ − gAµ ) ψ = 0, (5.5) ∂µ Fµν = gjν , (5.6) and 1 This is what is meant by local. 47 48 the schwinger model where jν = ψγν ψ. The theory is quantized using canononical quantization by imposing anti-commutation relations on the fermion fields ψ† (x, t) , ψ (x, t) = δ (x − y) (5.7) ψ† (x, t) , ψ† (x, t) = ψ (x, t) , ψ (x, t) = 0, and by imposing commutation relations on the gauge fields [E (x, t) , A1 (y, t)] = iδ (x − y) , (5.8) where the electric field E is defined by E = −F01 = F10 . (5.9) Using this definiton of the electric field in Eq. 5.6 we get analogues to Maxwell’s equations in (1+1)D: ∂E = gj0 ≡ gρ, ∂x ∂E − = gj1 ≡ gj. ∂t (Gauss’ law) (5.10) Since there is “no space” for magnetic fields in one spatial dimension, we only obtain the analogue of Gauss’ law and an equation which describes the dynamics of the electric field. 5.1 the schwinger model as a lattice field theory Starting from the Hamiltonian density H in temporal gauge, A0 = 0, 1 H = −iψγ1 (∂1 − igA1 ) ψ + mψψ + E2 , 2 (5.11) the model can be formulated on a spatial lattice using a Kogut-Susskind staggered formulation [26]. The equivalent lattice Hamiltonian is X i X † iθn (−1)n φ†n φn (5.12) H =− φn e φn+1 − h.c. + m 2a n n + ag2 X 2 L . 2 n n where a denotes the lattice spacing. In this formulation the correspondence between the fermionic lattice field φn on site n and the continuum field ψ is ! ψ ψupper upper n even φn ↔ , ψ= . (5.13) ψlower ψlower n odd 5.1 the schwinger model as a lattice field theory The gauge variables θn live on the links between the sites n and n + 1, and are connected to the vector potential via θn = −agA1n . (5.14) Their conjugate variables Ln , with [θn , Lm ] = iδnm , are related to the electric field by gLn = En . (5.15) Since θn is an angular variable, Ln will have integer charge eigenvalues pn ∈ Z. Therefore, the local Hilbert space spanned by the corresponding eigenvectors |pn i is infinite, and e±iθn are the ladder operators e±iθn |pn i = |pn ± 1i . (5.16) The lattice equivalent of Gauss’ law reads Ln − Ln−1 = φ†n φn − 1 [1 − (−1)n ] , 2 (5.17) which means excitations on odd and even sites create ∓1 units of flux, corresponding to “electron” and “positron” excitations, respectively. Using a Jordan-Wigner transformation, φn = Πk<n iσzk σ− n , where 1 ± x y σ = 2 (σ ± σ ), the fermionic degrees of freedom can be mapped to spin-1/2 degrees of freedom X g H= √ 2 x n µX (−1)n (σzn + (−1)n ) 2 n ! X iθn − +x σ+ σn+1 + h.c. . ne L2n + (5.18) n √ Here the parameters x ≡ 1/ g2 a2 and µ ≡ 2 xm/g have been introduced. The spins live on the sites of the lattice, with σzn |sn i = sn |sn i, and represent “positrons” on even sites and “electrons on odd sites. An even site with s2n = −1 corresponds to an empty state, while s2n = 1 represents an occupied positron state, and vice versa for the odd electron sites. In (1+1)D Gauss’ law, Ln − Ln−1 = 1/2 (σzn + (−1)n ), can be used to remove the gauge degrees of freedom [22]. The resulting Hamiltonian is then H =x N−2 X − σ+ n σn+1 n=0 " N−2 X + n=0 + + σ− n σn+1 N−1 µ X [1 + (−1)n σzn ] + 2 # n 1X k z (−1) + σk . 2 (5.19) n=0 (5.20) k=0 Instead of the gauge variables there is a non-local, long-range interaction term (Eq. 5.20) as opposed to Eq. 5.18. 49 50 the schwinger model 5.2 chiral symmetry and chiral condensate In this subsection, we introduce the so-called chiral condensate which will be the quantity we aim to study with TN methods. Without attempting to go into detail, we discuss two continuous symmetries of the Schwinger model of which one is broken after quantization. In this context, the chiral condensate arises as an order parameter. The Lagrangian density of the Schwinger model is invariant under global phase transformations of the Dirac field, i.e. ψ 0 = eiα ψ → L 0 = L, (5.21) where α is a real constant. Therefore, according to Noether’s theorem (see e.g. Ref. [31]), there is a conserved current jµ associated with every continuous symmetry. In this case the vector current jµ = ψγµ ψ (5.22) is conserved, i.e. ∂µ jµ = 0. (5.23) This global symmetry is known to hold at any level in fermion field theory models, although, in principle, the vacuum state could break the symmetry [19]. Let us now consider the case of massless (m = 0) fermions. Then the Lagrangian of the Schwinger model has another continuous symmetry, namely the so-called chiral symmetry. That means the Lagrangian density is invariant, if one transforms ψ into ψ 0 as ψ 0 = eiαγ5 ψ. (5.24) In the above equation γ5 ≡ γ0 γ1 anti-commutes with γµ for µ = 1, 2 and α is again a real constant. For example, in the Dirac representation the gamma matrices are given by2 ! ! ! 1 0 0 1 0 1 γ0 = , γ1 = , γ5 ≡ γ0 γ1 = . 0 −1 −1 0 1 0 (5.25) The associated Noether current for this symmetry is the so-called axial-vector current jµ 5 which is given by µ jµ 5 = ψγ γ5 ψ. (5.26) While the vector current in Eq. 5.22 is conserved in the quantized theory, the axial-vector current is not. This non-conservation of the 2 See, e.g., Ref. [19]. 5.2 chiral symmetry and chiral condensate axial-vector current is called chiral anomaly or axial anomaly. The divergence of the axial-vector current reads g µν ∂µ jµ , (5.27) 5 = 2π µν F where µν is the Levi-Civita symbol in two dimensions, see Refs.[19, 25]. As a consequence of this chiral symmetry breaking a finite chiral condensate Σ is allowed [19]. It is defined as the following vacuum expectation value Σ = ψψ . (5.28) In the case of the massless Schwinger model, the chiral condensate can be computed exactly (see e.g. Ref. [35]), and is found to be eγ (5.29) 3 ≈ 0.159929, 2π 2 where γ is the Euler-Mascheroni constant. Therefore, the chiral condensate can be regarded as an order parameter signaling chiral symmetry breaking. Σ0 = Chiral Condensate on the Lattice3 In this thesis, we aim to compute the chiral condensate on an infinite lattice for the massive Schwinger model. Written in terms of spin operators, the chiral condensate on the lattice reads √ X z x n 1 + σn (−1) Σ (x) = , (5.30) N n 2 where the expectation value is computed in the ground state. The naively computed chiral condensate is known to be UV-divergent. In particular, it diverges logarithmically in the continuum limit, a → 0. It has been argued, that this divergence comes solely from the free (g=0) theory. In the free case the chiral condensate on the lattice Σfree (x) can be computed exactly ! m 1 1 q Σfree (x) = − K , (5.31) 2 πg 1 + m2 1 + m2 g2 x g x where K (z) is the complete elliptic integral of the first kind [1]. This result can be used to subtract the divergence from the computed chiral condensate in the interacting theory, and therefore to renormalize it. In other words, we can define a so-called subtracted chiral condensate Σsub , which allows for a continuum extrapolation, by Σsub = Σ (x) − Σfree (x) , (5.32) where Σ (x) denotes the computed chiral condensate. A detailed description of our simulation and the extrapolation procedure will be given in the next chapter. 3 In preparation of this section we used i.a. Ref. [4]. 51 6 C A L C U L AT I O N O F T H E C H I R A L C O N D E N S AT E We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on. Richard P. Feynman, 1965 In this chapter, we present our study of the chiral condensate in the Schwinger Model with TN methods. First, we discuss the naive approach to simulate the model in the thermodynamic limit in the formulation of Eq. 5.20, and comment on a major problem we have been faced with due to the gauge character of the theory. Then we present an alternative ansatz which overcomes this problem by implementing the gauge symmetry on the level of MPS. Finally, we discuss the continuum extrapolation of the computed reduced chiral condensate, and compare our results to the literature. 6.1 gauge term vs. thermodynamic limit Different to a lot of quantum many-body systems on which TN methods have been successfully applied, a lattice gauge theory has an essential new ingredient. In particular, for a gauge theory only gauge invariant states are physically relevant. That means in case of the Schwinger model that only states which obey Gauss’ law are in the physical Hilbert space. On the level of the Hamiltionian for the lattice gauge theory this constraint leads to a non-local gauge term (see Eq. 5.20): " n # N−2 X 1X k z (−1) + σk . Hg = (6.1) 2 n=0 k=0 Taking gauge invariance into account in this way will turn out to be a hard problem if one aims to work in the thermodynamic limit. First approach: Long-range Interactions with iTEBD and iDMRG Our first approach is based on a modification of the iTEBD algorithm which was introduced in Sec. 3.7.1. Since its original form considered Hamiltonians with nearest-neighbor interactions, we have to change the algorithm in such a way that it, in principle, is able to model longrange interactions. A way to realize this is to represent the imaginary time evolution operator U (δτ) (see Eq. 3.20) in terms of an MPO. 53 54 calculation of the chiral condensate λB ΓA λA ΓB λB ΓA λA ΓB λB ΓA λA ΓB λB UA UB UA UB UA UB ~B ~A ~A ~B ~B ~A ~A ~B ~B ~A ~A ~B ~B λ Γ λ Γ λ Γ λ Γ λ Γ λ Γ λ Figure 29: Application of U (δτ) in MPO form. The result is an MPS with a broken canonical form. The evolution of an imaginary time-step δτ with an MPO is shown in Fig. 29. Due to its non-unitarity, the application of the imaginary time operator breaks the canonical form of the inital MPS. However, the canonical form can be recovered by using the original iTEBD algorithm in Sec. 3.7.1 to apply a large sequence of trivial two-site gates alternatively acting on even and odd bonds1 , i.e. UAB = UBA = 1. Once the MPS is again in canonical form, it is possible to use the SVD truncation scheme to prevent the bond dimension to grow after every time step. A more detailed discussion can be found in Ref. [29]. Let us recall the full Hamiltonian of the Schwinger model (Eq. 5.20): N−2 X X µ N−1 − − + [1 + (−1)n σzn ] σ+ σ + σ σ + n n+1 n n+1 2 n=0 n=0 " n # N−2 X 1X (−1)k + σzk . + 2 H =x n=0 (6.2) (6.3) k=0 If we write the imaginary time evolution operator in a factored form U (δτ) = e−δτH = e−δτHl e−δτHg + O δτ2 , (6.4) where Hl denotes the local terms of the Hamiltonian (Eq. 6.2), one could naively expect that the imaginary time evolution becomes tractable. If one is able to find an MPO representation for the evolution operator of the gauge term2 (Eq. 6.3), then the evolution of the gauge part could be done with the algorithm above, and for the local part one could use iTEBD in its original formulation with 2-body gates. 1 A more efficient and precise procedure is described in Ref. [29]. 2 Or at least for the gauge term, since e−δτHg = 1 − Hg + O δτ2 is a good approximation for δτ sufficiently small. 6.2 idmrg with gauge invariant mps However, it turns out that the gauge term Hg depends explicitly on the system size N and is site-dependent. It is not clear whether there is a reasonable way to implement this in infinite-size methods. A naive approach could be to identify the system size with the number of added sites in the iDMRG algorithm presented in Sec. 3.7.2. We experienced that in this case the algorithm does not converge. 6.2 idmrg with gauge invariant mps Let us now present an alternative ansatz which was introduced in Ref. [9]. In this approach one starts one step before integrating out the gauge field degrees of freedom using the Gauss’ law constraint, namely with the Hamiltonian in Eq. 5.18: g H= √ 2 x X n µX (−1)n (σzn + (−1)n ) 2 n ! X + iθn − +x σn e σn+1 + h.c. . L2n + (6.5) n An obvious advantge is that this Hamiltonian is local with at most nearest-neighbor actions, and translationally invariant under two sites. Furthermore, it allows us to be a bit more general, since only in (1+1)D the gauge degrees of freedom can be integrated out. This might be quite relevant for a generalization to higher dimensions. MPO Representation for H In the following, we give an MPO representation of the Hamiltonian in Eq. 6.5 to be able to run a iDMRG simulation. For now we consider a finite lattice of N ∈ 2N sites. Further we block site n and link n into one MPS-site, such that at every MPS-site we have a fermion and a gauge field degree of freedom. Then the Hamiltonian can be regarded as the sum of a 1-site operator and a 2-site operator, i.e X H= hn + hn,n+1 , (6.6) n where hn = 1 ⊗ L2n µ n z (1 + (−1) σn ) ⊗ 1 , + 2 (6.7) and hn,n+1 = x iθn σ+ · σ− n ⊗e n+1 ⊗ 1 + − −iθn + σn ⊗ e · σn+1 ⊗ 1 (6.8) (6.9) 55 56 calculation of the chiral condensate The first factor in the tensor product ⊗ refers to the fermion degree of freedom, and the second factor to the gauge field degree of freedom at the MPS site. With · we denote here the tensor product between operators acting on different MPS-sites. The Hamiltonian can be written as an MPO with bond dimension κ = 4 where tensors are given as in Fig. 30. Imposing Gauge Invariance As a last step, we want to impose gauge invariance to enforce that our algorithm works directly within the physical subspace of the full Hilbert space. In particular, we are only interested in states |ψi that are gauge invariant, i.e. Gn |ψi = 0 ∀n, (6.10) where Gn ≡ Ln − Ln−1 − 1 z (σ + (−1)n ) . 2 n (6.11) Eq. 6.10 is nothing but the discretized version of the Gauss’ law constraint for the lattice spin-gauge system. One possibility to impose gauge invariance would be to add a penalty term to the Hamiltonian that prefers the correct subspace energetically. For example, one could simulate X H0 = H + λ G2n , (6.12) n instead of H, and consider the limit λ → ∞. However, in this case gauge invariance is only approximately realized, and one hast to extrapolate in another parameter λ. Therefore, we use here a different approach in which the Gauss’ law constraint is directly implemented on the level of the tensors in our MPS. Essentially, this implies that many tensor components in our MPS ansatz have to vanish, i.e. only components compatible with the symmetry are allowed to be different from zero. Let us again assume that we have a finite lattice of N ∈ 2N sites. Then a general, i.e. not necessarily gauge invariant, MPS ansatz for the spin-gauge system has the form: X s p s p 2N B11 C1 1 B22 C2 2 . . . Bs12N Cp (6.13) 2N |s1 , p1 , s2 , p2 . . . p2N i , sn ,pn where the matrices [Bsnn ]αβ correspond to fermionic degrees of freen dom, and the matrices Cp n αβ to gauge degrees of freedom. We denote the bond dimension with χ, i.e. the bond indices take the values α, β = 1, . . . χ. 6.2 idmrg with gauge invariant mps (a) 1 i i i 1 = 1 4 = 1 i i 4 = 1 3 i i 4 = 4 4 j (b) = j j 3 = j j j 2 2 = j i i 1 i = 4 = 2 j j = j i 3 = j (c) i = 1 i i = 4 j j = 2 j i = 3 j Figure 30: MPO tensors for (a) the bulk, (b) the left boundary, (c) the right boundary. Note, we have different tensors for even and odd sites in the bulk due to the factor (−1)n in Eq. 6.7. 57 58 calculation of the chiral condensate From Eq. 6.10 and Eq. 6.11 we can see that Gauss’ law is basically a prescription how to update the electric field Ln at the right link of site n, namely 1 z (σ + (−1)n ) . (6.14) 2 n That means, if there is no charge at the site n, then Ln stays with the value Ln−1 at the left. At the same time the electric field Ln is increased/decreased by one unit, if there is a positron/electron3 at site n. This “update rule" can be implemented by giving the bond indices a multipile index structure, α → (q, αq ), and imposing the following form on the matrices in the bulk: n [Bsnn ](q,αq )(r,βr ) = bsn,q δ , (6.15) n αq ,βr q+(sn +(−1) )/2,r Ln = Ln−1 + pn n [Cp n ](q,αq )(r,βr ) = [cn ]αq ,βr δq,pn δr,pn . If one chooses the electric field to the left of the first lattice site to vanish, i.e. L0 = 0, then matrices representing the boundaries are gauge invariant if: h i s1 1 B1 (q,αq )(r,βr ) = bs1,0 δ(s−1))/2,r , (6.16) 1,βr p2N 2N C2N (q,αq )(r,βr ) = cp 2N αq ,1 δq,p2N . In the above equation, the indices q and r label the electric charge sector, and are sometimes referred to as structural or charge indices. They label the representation of the gauge symmetry group for the index. The indices αq and βr label the degeneracy subspace within each charge (symmetry) sector. Every bulk or boundary tensor which is chosen according to Eq. 6.15 or Eq. 6.16, respectively, preserves the gauge symmetry exactly. The variational freedom lies now within the n and cpn . The rather lengthy derivation of the result can matrices bsn,q n be found in Ref. [9]. We also may refer the reader to Refs. [40, 41] for details on symmetries in TN. Simulation with iDMRG We use the one-site iDMRG algortihm introduced in Sec. 3.7.2 to find a ground state approximation in the thermodynamic limit. As in the construction of the Hamiltionian, we again block a lattice site and a link into one MPS-site. This leads to an MPS ansatz with a twosite unit cell due to alternating spin-gauge systems for positrons and electrons. The initial tensors are defined according to Eq. 6.15, but are otherwise chosen randomly within the variational degree of freedom. To obtain a system that is invariant under translations of one site, we also block neighboring MPS-sites corresponding to a positron and electron spin-gauge systems together. This is illustrated in Fig. 31. 3 Recall from Sec. 5.1 that an occupied positron or electron state corresponds to sn = 1 or sn = −1, respectively, and that positrons/electrons live on even/odd sites n. 6.2 idmrg with gauge invariant mps Gauge system: (a) n n+1 Spin system: (c) (b) B C Figure 31: (a) Infinite lattice in 1D: the spins (fermions) live on the sites and the gauge variables on the links. (b) A lattice site and the link to the right a represented by one MPS-site where gauge invariance is ensured by choosing the tensors as given in Eq. 6.15. (c) Neighboring MPS-sites corresponding to a positron and electron spin-gauge systems are blocked together to make the system one-site translational invariant. A list of all simulation parameters is shown in Tab. 1. We calculate the chiral condensate for four different values of the fermion mass, m/g = 0, 0.125, 0.25, 0.5 where in each of the cases x ∈ [10, 600]. The parameter pmax > |pn | is a truncation parameter for the infinite local Hilbert space of the gauge variables. Physically it corresponds to a charge truncation. In our calculations we choose pmax = 2, i.e. we truncate the infinite dimensional Hilbert space to five dimensions4 . Further, we set N = 500 which corresponds to adding 1000 sites in the physical system due to the coarse-graining. At every simulation step we compute the expectation value of the gauge operator Gn defined in Eq. 6.11 to check the gauge invariance of the ground state approximation at that step5 . In order to get an approximation of the (subtracted) chiral condensate in the continuum, we have to perform different extrapolations. In the following, we show exemplary the procedure for m/g = 0.25. Fig. 32 shows the computed chiral √ condensate as a function of 1/ x for different bond dimensions χc and χd , and Fig. 33 shows the subtracted chiral condensate. As expected the computed results have an dependency on the chosen bond dimensions. The influence of the bond dimension seems to be bigger in case of smaller lattice parameters in contrast to smaller lattice con4 Ln |pn i = pn |pn i with pn = 0, ±1, ±2. 5 Since it is not obvious whether there is a step in the iDMRG algorithm which breaks the gauge symmetry. 59 60 calculation of the chiral condensate Parameter Description χc bond dimension of the charge index χd bond dimension of the degeneracy index pmax charge truncation N number of added sites x inverse coupling (see Sec. 5.1) m/g (dimensionless) fermion mass Table 1: Simulation parameters in the one-site iDMRG algorithm. √ Figure 32: Chiral condensate for m/g = 0.25 as a function of 1/ x for different bond dimensions. 6.2 idmrg with gauge invariant mps √ Figure 33: Subtracted chiral condensate for m/g = 0.25 as a function of 1/ x for different bond dimensions. √ Figure 34: Subtracted chiral condensate for m/g = 0.25 as a function of 1/ x for different bond dimensions. Note the cases (χc , χd ) = (3, 3) and (χc , χd ) = (5, 3) are omitted here. 61 62 calculation of the chiral condensate Figure 35: Computed chiral condensate for m/g = 0.25 and fit for the extrapolation in the bond dimension at x = 100. stants, where the computed results seem to be very well converged over the whole spectrum of the chosen bond dimensions. But, overall one could expect also convergence for small lattice constants. In order to get an approximation for the expected values, we perform for every x an extrapolation in the total bond dimension, i.e. χ = χc · χd → ∞. We find that the dependency is well described by the following fit ansatz c f1 (χ) = a exp (−bχ) + d + Σ (x, χ = ∞) , (6.17) χ where Σ (x, χ = ∞) is the extrapolated value of the computed chiral condensate for the inverse coupling x. For x = 100 or x = 500 this is shown in Fig. 35 or in Fig. 36 respectively. Finally, we perform a continuum extrapolation, i.e. a → 0 or equivalently x → ∞. As in Ref. [4], we use for the subtracted chiral condensate the following fit ansatz: log x 1 1 f (x) = Σcont. + F √ + B √ + C , (6.18) x x x where Σcont. is the extrapolated contiunuum value of the subtracted chiral condensate. As one can see in Fig. 37 this ansatz describes our data overall very well, especially in the case of larger lattice constants, where the influence of the bond dimension was also smaller. Therefore, we also perform a continuum extrapolation for x ∈ [10, 300] (see Fig.38), since we expect that our algorithm describes this region more faithfully. The obtained results for the four different fermion masses can be found in Tab. 2. The continuum extrapolations for m/g = 0, 0.125, 0.5 can be found in App. A.2 6.2 idmrg with gauge invariant mps Figure 36: Computed chiral condensate for m/g = 0.25 and fit for the extrapolation in the bond dimension at x = 500. Figure 37: Continuum limit extrapolation of the subtracted chiral condensate for m/g = 0.25 attained from all x ∈ [10, 600]. 63 64 calculation of the chiral condensate Figure 38: Continuum extrapolation of the subtracted chiral condensate for m/g = 0.25 attained from x ∈ [10, 300]. One-site iDMRG One-site iDMRG x in [10, 600] x in [10, 300] 0 0.03055 0.125 Ref. [4] Ref. [44] exact 0.15900 0.159930(8) 0.159928(1) 0.159929 0.09847 0.09425 0.092023(4) 0.092019(2) - 0.25 0.07054 0.06838 0.066660(11) 0.066647(4) - 0.5 0.04410 0.04293 0.042383(22) 0.042349 (2) - 0.75 - - - 0.03062(3) - 1 - - - 0.023851(8) - 2 - - - 0.012463 (9) - m/g Table 2: Comparison: subtracted chiral condensate in the continuum. 6.2 idmrg with gauge invariant mps Discussion of the Results In the following, we discuss our results for the continuum extrapolation of the chiral condensate in comparison to previous studies and argue that they have to be regarded rather as a proof of concept than serious precision calculations. As one can see in Tab. 2 our results are in agreement with the results of Ref. [4] and Ref. [44] except for the value attained from the continuum extrapolation over x ∈ [10, 600] for m/g = 0. In the latter case the given fit ansatz in Eq. 6.18 was not able to describe our data faithfully in the region of smaller lattice constants (see Fig. 39 and Fig. 40 in App. A.2). The approach in Ref. [4] is conceptually different from ours. This study is based on finite-size DMRG calculations and used the nonlocal Hamiltonian given in Eq. 5.20. Compared to this ansatz our chosen approach is more direct, since we work directly in the thermodynamic limit without the burden of finite-size scaling effects. Furthermore, we keep the gauge degrees of freedom and implement the Gauss’ law constraint on the level of the MPS. This allows us to start from the local Hamiltonian in Eq. 5.18. We think this is quite relevant for considerations to generalize the work to higher dimensions, since the gauge degrees of freedom can only be integrated out by Gauss’ law in (1+1)D. Due to the limited time frame of a master thesis, the bond dimensions used in our calculations are comparatively small. In particular, for m/g = 0.25 we used χ = 91 as the highest total bond dimension, and in the other cases it was even smaller with χ = 63. The extrapolations in Ref. [4] were attained from calculations up to bond dimension χ = 140. Therefore, we expected our results not to be as accurate. Furthermore, we used the one-site iDMRG algorithm which has, although being more efficient, several drawbacks compared to the two-site iDMRG algorithm. As discussed in Sec. 3.7.2 the two-site version contains a SVD truncation in the bond dimension as opposed to the one-site algorithm. This can be quite advantageous, since it allows one to increase the bond dimension dynamically, i.e. as the algorithm runs. In principle, one can improve in this way the accuracy of the algorithm on the fly, but with the huge advantage that following simulations steps can build on the work of previous iterations. Moreover, the SVD truncation can lead to an extra accuracy when dealing with symmetries, since the algorithm can readapt to more relevant symmetry sectors, i.e. sectors with more weight in terms of Schmidt coefficients6 . As a consequence of these “dynamical properties” the two-site algorithm is also more robust in the sense that is not as likely as in the one-site case to get stuck in local minima, which is a potential source of error in variational methods. 6 Or, equivalently, in terms of singular values. 65 66 calculation of the chiral condensate Ref. [44] is a follow-up study of Ref. [9], where the gauge invariant MPS ansatz for the Schwinger model was introduced. Besides working with gauge invariant MPS in the thermodynamic limit, CT symmetry, i.e. invariance by a one-site translation and charge conjugation, was exploited. The ground state calculations were done with the so-called time-dependent variational principle (TDVP) by doing an imaginary time evolution [21]. In this work symmetries were treated in a more sophisticated way by distributing variational freedoms to different charge sectors. In contrast to that our approach was more “pedestrian” by just choosing gauge invariant initial tensors. In light of the points discussed above, we can conclude that our results can be seen as a further proof of concept that the MPS ansatz is able to describe the physical relevant states for a lattice gauge theory even in the thermodynamic limit. 7 CONCLUSION AND OUTLOOK The tensor network approach to lattice gauge theories was motivated by its potential to tackle problems which are inaccessible to standard techniques, like the presence of a large finite chemical potential and real time evolution. In the first part of this thesis we gave an introduction to some basic concepts on which tensor network states and tensor network methods rely on. Amongst others, we presented the modern view of entanglement as a resource in quantum information theory or as a property to characterize quantum many-body states. In particular, we discussed the so-called area laws which provide the theoretical framework for tensor network methods. In part ii, we gave an introduction on tensor network states with a focus on the familiy of MPS. We presented to conceptually different algorithms to obtain ground state approximations for infinite onedimensional systems, namely the iTEBD algorithm based on imaginary time evolution and the iDMRG algorithm as a variational method. In a case-study we investigated ground state properties of the Ising model in a transverse field, and used its exact solution to verify the validity of our iTEBD algorithm. In part iii we presented the Schwinger model and its equivalent theory on a lattice. We discussed the chiral symmetry breaking due to the chiral anomaly, and introduced the chiral condensate as an order parameter. Then we used an MPS ansatz to study the Schwinger model in its formulation as a Hamiltonian lattice gauge theory in the thermodynamic limit. We discussed how the demand for gauge invariance can lead to complications, if one takes account of it in an unfavourable way. In particular, we argued to implement the Gauss’ law constraint rather on the level of the MPS than on the level of the Hamiltonian, since in the latter case one has to deal with non-local and site-dependent terms which do not allow for direct simulation in the thermodynamic limit. Further, a detailed description of our simulation with the one-site iDMRG algorithm and the following extrapolation procedure was given. We argued that or results as shown in Tab. 3 can be seen as proof of concept that an MPS ansatz is able to describe the physical relevant states for a lattice gauge theory in (1+1)D, even with relatively simple algorithms. The work in this thesis can be extended or generalized in manifold ways. From the view of lattice gauge theory it seems natural to consider in a next step problems with finite chemical potential or to investigate non-equilibrium properties. In the context of real time 67 68 conclusion and outlook One-site iDMRG One-site iDMRG x in [10, 600] x in [10, 300] 0 0.03055 0.125 Ref. [4] Ref. [44] exact 0.15900 0.159930(8) 0.159928(1) 0.159929 0.09847 0.09425 0.092023(4) 0.092019(2) - 0.25 0.07054 0.06838 0.066660(11) 0.066647(4) - 0.5 0.04410 0.04293 0.042383(22) 0.042349 (2) - 0.75 - - - 0.03062(3) - 1 - - - 0.023851(8) - 2 - - - 0.012463 (9) - m/g Table 3: Results: subtracted chiral condensate in the continuum. evolution, one could also try to study scattering processes. Furthermore, the 2-flavour Schwinger model offers many open questions, and would therefore serve as an interesting model. One could also consider to study non-abelian gauge theories. However, the most interesting and presumably most challenging project would be to try to consider QED in (2+1)D with so-called fermionic tensor networks. We hope that in the future tensor network methods will arouse a great deal of interest in the lattice gauge theory community, and that they one day serve as a powerful tool in the, by modern standards, elusive regimes of finite fermionic densities or non-equilibrium dynamics. A APPENDIX a.1 the singular value decomposition The singular value decomposition (SVD) is a matrix decomposition that applies to any real or complex matrix. It plays a central role in many applications, rangig from e.g. image processing to optimization or statistics [18, 2]. Let M ∈ Cn×m with n > m be given. Then there exist unitary matrices U ∈ Cn×n ,V ∈ Cm×m , and a diagonal matrix Σ ∈ Rn×m such that M = UΣV † and Σr 0 0 ! 0 , (A.1) where Σr = diag(σ1 , . . . , σr ), and σ1 > σ2 > · · · > σr > 0 are the positive singular values. 69 70 appendix a.2 continuum extrapolations of the subtracted chiral condensate Figure 39: Continuum limit extrapolation of the subtracted chiral condensate for m/g = 0 attained from all x ∈ [10, 600] . A.2 continuum extrapolations of the subtracted chiral condensate Figure 40: Continuum extrapolation of the subtracted chiral condensate for m/g = 0 attained from x ∈ [10, 300]. Figure 41: Continuum limit extrapolation of the subtracted chiral condensate for m/g = 0.125 attained from all x ∈ [10, 600]. 71 72 appendix Figure 42: Continuum extrapolation of the subtracted chiral condensate for m/g = 0.125 attained from x ∈ [10, 300]. Figure 43: Continuum limit extrapolation of the subtracted chiral condensate for m/g = 0.5 attained from all x ∈ [10, 600]. 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