Projected Entangled Pair States Research Articles

Generating Function for Projected Entangled-Pair States

Generating Function for Projected Entangled-Pair States

Tensor network variational optimizations for real-time dynamics: Application to the time-evolution of spin liquids

Within the Projected Entangled Pair State (PEPS) tensor network formalism, a simple update (SU) method has been used to investigate the time evolution of a two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench [Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational frameworks to describe the time dynamics of SU(2)-symmetric translationally-invariant PEPS, aiming to improve the accuracy. In one approach, after using a Trotter-Suzuki decomposition of the time evolution operator in term of two-site elementary gates, one considers a single bond embedded in an environment approximated by a Corner Transfer Matrix Renormalization Group (CTMRG). A variational update of the two tensors on the bond is performed under the application of the elementary gate and then, after symmetrization of the site tensors, the environment is updated. In the second approach, a cluster optimization is performed on a finite (periodic) cluster, maximizing the overlap of the exact time-evolved state with a symmetric finite-size PEPS ansatz. Observables are then computed on the infinite lattice contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational schemes outperform the SU method and remain accurate over a significant time interval before hitting the entanglement barrier. Studying the spectrum of the transfer matrix, we find that the asymptotic correlations are very well preserved under time evolution, including the critical nature of the singlet correlations, as expected from the Lieb-Robinson (LR) bound theorem. Consistently, the system (asymptotic) boundary is found to be described by the same Conformal Field Theory of central charge c = 1c=1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.

Optimal Sampling of Dynamical Large Deviations in Two Dimensions via Tensor Networks.

We use projected entangled-pair states (PEPS) to calculate the large deviation statistics of the dynamical activity of the two-dimensional East model, and the two-dimensional symmetric simple exclusion process (SSEP) with open boundaries, in lattices of up to 40×40 sites. We show that at long times both models have phase transitions between active and inactive dynamical phases. For the 2D East model we find that this trajectory transition is of the first order, while for the SSEP we find indications of a second order transition. We then show how the PEPS can be used to implement a trajectory sampling scheme capable of directly accessing rare trajectories. We also discuss how the methods described here can be extended to study rare events at finite times.

Multi-Layered Projected Entangled Pair States for Image Classification

Tensor networks have been recognized as a powerful numerical tool; they are applied in various fields, including physics, computer science, and more. The idea of a tensor network originates from quantum physics as an efficient representation of quantum many-body states and their operations. Matrix product states (MPS) form one of the simplest tensor networks and have been applied to machine learning for image classification. However, MPS has certain limitations when processing two-dimensional images, meaning that it is preferable for an projected entangled pair states (PEPS) tensor network with a similar structure to the image to be introduced into machine learning. PEPS tensor networks are significantly superior to other tensor networks on the image classification task. Based on a PEPS tensor network, this paper constructs a multi-layered PEPS (MLPEPS) tensor network model for image classification. PEPS is used to extract features layer by layer from the image mapped to the Hilbert space, which fully utilizes the correlation between pixels while retaining the global structural information of the image. When performing classification tasks on the Fashion-MNIST dataset, MLPEPS achieves a classification accuracy of 90.44%, exceeding tensor network models such as the original PEPS. On the COVID-19 radiography dataset, MLPEPS has a test set accuracy of 91.63%, which is very close to the results of GoogLeNet. Under the same experimental conditions, the learning ability of MLPEPS is already close to that of existing neural networks while having fewer parameters. MLPEPS can be used to build different network models by modifying the structure, and as such it has great potential in machine learning.

Locality Estimates for Complex Time Evolution in 1D

It is a generalized belief that there are no thermal phase transitions in short range 1D quantum systems. However, the only known case for which this is rigorously proven is for the particular case of finite range translationally invariant interactions. The proof was obtained by Araki in his seminal paper of 1969 as a consequence of pioneering locality estimates for the time-evolution operator that allowed him to prove its analyticity on the whole complex plane, when applied to a local observable. However, as for now there is no mathematical proof of the absence of 1D thermal phase transitions if one allows exponential tails in the interactions. In this work we extend Araki’s result to include exponential (or faster) tails. Our main result is the analyticity of the time-evolution operator applied on a local observable on a suitable strip around the real line. As a consequence we obtain that thermal states in 1D exhibit exponential decay of correlations above a threshold temperature that decays to zero with the exponent of the interaction decay, recovering Araki’s result as a particular case. Our result however still leaves open the possibility of 1D thermal short range phase transitions. We conclude with an application of our result to the spectral gap problem for Projected Entangled Pair States (PEPS) on 2D lattices, via the holographic duality due to Cirac et al.

Complete characterization of non-Abelian topological phase transitions and detection of anyon splitting with projected entangled pair states

It is well-known that many topological phase transitions of intrinsic Abelian topological phases are accompanied by condensation and confinement of anyons. However, for non-Abelian topological phases, more intricate phenomena can occur at their phase transitions, because the multiple degenerate degrees of freedom of a non-Abelian anyon can change in different ways after phase transitions. In this paper, we study these new phenomena, including partial condensation, partial deconfinement and especially anyon splitting (a non-Abelian anyon splits into different kinds of the new anyon species) using projected entangled pair states (PEPS). First, we show that anyon splitting can be observed from the topologically degenerate ground states. Next, we construct a set of PEPS describing all possible degrees of freedom of the same non-Abelian anyon. From the overlaps of this set of PEPS of a given non-Abelian anyon with the ground state, we can extract the information of partial condensation. Then, we construct a central object, a matrix defined by the norms and overlaps among the PEPS in that set. The information of partial deconfinement can be extracted from this matrix. In particular, we use it to construct an order parameter which can directly detect anyon splitting. We demonstrate the power of our approach by applying it to a range of non-Abelian topological phase transitions: From $D(S_3)$ quantum double to toric code, from $D(S_3)$ quantum double to $D(Z_3)$ quantum double, from Rep($S_3$) string-net to toric code, and finally from double Ising string-net to toric code.

Real-time dynamics of a critical resonating valence bond spin liquid

Implementation of the hardcore-dimer Hilbert space in cold Rydberg-atom simulators opens a new route of investigating real-time dynamics of dimer liquids under Hamiltonian quench. Here, we consider an initial Resonating Valence Bond (RVB) state on the square lattice realizing a critical Coulomb phase with algebraic and dipolar correlations. Using its representation as a special point of a broad manifold of SU($2$)-symmetric, translationnally invariant, Projected Entangled Pair States (PEPS), we compute its non-equilibrium dynamics upon turning on inter-site Heisenberg interactions. We show that projecting the time-evolution onto the PEPS manifold remains accurate at small time scales. We also find that the state evolves within a PEPS sub-manifold characterized by a U($1$) gauge symmetry, suggesting that the Coulomb phase is stable under such unitary evolution.

Scaling Hypothesis for Projected Entangled-Pair States.

We introduce a new paradigm for scaling simulations with projected entangled-pair states (PEPS) for critical strongly correlated systems, allowing for reliable extrapolations of PEPS data with relatively small bond dimensions D. The key ingredient consists of using the effective correlation length ξ for inducing a collapse of data points, f(D,χ)=f(ξ(D,χ)), for arbitrary values of D and the environment bond dimension χ. As such we circumvent the need for extrapolations in χ and can use many distinct data points for a fixed value of D. Here, we need that the PEPSs have been optimized using a fixed-χ gradient method, which can be achieved using a novel tensor-network algorithm for finding fixed points of 2D transfer matrices, or by using the formalism of backwards differentiation. We test our hypothesis on the critical 3D dimer model, the 3D classical Ising model, and the 2D quantum Heisenberg model.

Simulating Chiral Spin Liquids with Projected Entangled-Pair States.

Doubts have been raised on the representation of chiral spin liquids exhibiting topological order in terms of projected entangled pair states (PEPSs). Here, starting from a simple spin-1/2 chiral frustrated Heisenberg model, we show that a faithful representation of the chiral spin liquid phase is in fact possible in terms of a generic PEPS upon variational optimization. We find a perfectly chiral gapless edge mode and a rapid decay of correlation functions at short distances consistent with a bulk gap, concomitant with a gossamer long-range tail originating from a PEPS bulk-edge correspondence. For increasing bond dimension, (i)the rapid decrease of spurious features-SU(2) symmetry breaking and long-range tails in correlations-together with (ii)a faster convergence of the ground state energy as compared to state-of-the-art cylinder matrix-product state simulations involving far more variational parameters, prove the fundamental relevance of the PEPS ansatz for simulating systems with chiral topological order.

Generalized Gibbs ensemble description of real-space entanglement spectra of (2+1) -dimensional chiral topological systems with SU(2) symmetry

We provide a quantitative analysis of the splittings in low-lying numerical entanglement spectra (ES), at given momentum, of a number of quantum states that can be identified, based on "Li-Haldane state-counting", as ground states of (2+1)-dimensional chiral topological phases with global SU(2) symmetry. The ability to account for numerical ES splittings solely within the context of conformal field theory (CFT) is an additional diagnostic of the underlying topological theory, of finer sensitivity than "state-counting". We use the conformal boundary state description of the ES, which can be viewed as a quantum quench. In this language, the ES splittings arise from local conservation laws in the chiral CFT besides the energy, which we view as a Generalized Gibbs Ensemble (GGE). Global SU(2) symmetry imposes strong constraints on the number of such conservation laws, so that only a small number of parameters can be responsible for the splittings. We work out these conservation laws for chiral SU(2) Wess-Zumino-Witten CFTs at levels one and two, and for the latter we notably find that some of the conservation laws take the form of local integrals of operators of fractional dimension, as proposed by Cardy for quantum quenches. We analyze numerical ES from systems with SU(2) symmetry including chiral spin-liquid ground states of local 2D Hamiltonians and two chiral Projected Entangled Pair States (PEPS) tensor networks, which exhibit the "state-counting" of the SU(2)-level-one and -level-two theories. We find that the low-lying ES splittings can be well understood by the lowest of our conservation laws, and we demonstrate the importance of accounting for the fractional conservation laws at level two. Thus the states we consider, including the PEPS, appear chiral also under our more sensitive diagnostic.

Variational methods for contracting projected entangled-pair states

The norms or expectation values of infinite projected entangled-pair states (PEPS) cannot be computed exactly, and approximation algorithms have to be applied. In the last years, many efficient algorithms have been devised -- the corner transfer matrix renormalization group (CTMRG) and variational uniform matrix product state (VUMPS) algorithm are the most common -- but it remains unclear whether they always lead to the same results. In this paper, we identify a subclass of PEPS for which we can reformulate the contraction as a variational problem that is algorithm independent. We use this variational feature to assess and compare the accuracy of CTMRG and VUMPS contractions. Moreover, we devise a new variational contraction scheme, which we can extend to compute general N-point correlation functions.

CPD-Structured Multivariate Polynomial Optimization

We introduce the Tensor-Based Multivariate Optimization (TeMPO) framework for use in nonlinear optimization problems commonly encountered in signal processing, machine learning, and artificial intelligence. Within our framework, we model nonlinear relations by a multivariate polynomial that can be represented by low-rank symmetric tensors (multi-indexed arrays), making a compromise between model generality and efficiency of computation. Put the other way around, our approach both breaks the curse of dimensionality in the system parameters and captures the nonlinear relations with a good accuracy. Moreover, by taking advantage of the symmetric CPD format, we develop an efficient second-order Gauss–Newton algorithm for multivariate polynomial optimization. The presented algorithm has a quadratic per-iteration complexity in the number of optimization variables in the worst case scenario, and a linear per-iteration complexity in practice. We demonstrate the efficiency of our algorithm with some illustrative examples, apply it to the blind deconvolution of constant modulus signals, and the classification problem in supervised learning. We show that TeMPO achieves similar or better accuracy than multilayer perceptrons (MLPs), tensor networks with tensor trains (TT) and projected entangled pair states (PEPS) architectures for the classification of the MNIST and Fashion MNIST datasets while at the same time optimizing for fewer parameters and using less memory. Last but not least, our framework can be interpreted as an advancement of higher-order factorization machines: we introduce an efficient second-order algorithm for higher-order factorization machines.

Gapless quantum spin liquid and global phase diagram of the spin-1/2 [formula omitted] square antiferromagnetic Heisenberg model

The nature of the zero-temperature phase diagram of the spin-1/2J1-J2 Heisenberg model on a square lattice has been debated in the past three decades, and it remains one of the fundamental problems unsettled in the study of quantum many-body theory. By using the state-of-the-art tensor network method, specifically, the finite projected entangled pair state (PEPS) algorithm, to simulate the global phase diagram of the J1-J2 Heisenberg model up to 24×24 sites, we provide very solid evidences to show that the nature of the intermediate nonmagnetic phase is a gapless quantum spin liquid (QSL), whose spin-spin and dimer-dimer correlations both decay with a power law behavior. There also exists a valence-bond solid (VBS) phase in a very narrow region 0.56≲J2/J1≤0.61 before the system enters the well known collinear antiferromagnetic phase. We stress that we make the first detailed comparison between the results of PEPS and the well-established density matrix renormalization group (DMRG) method through one-to-one direct benchmark for small system sizes, and thus give rise to a very solid PEPS calculation beyond DMRG. Our numerical evidences explicitly demonstrate the huge power of PEPS for highly frustrated spin systems. Finally, an effective field theory is also proposed to understand the physical nature of the discovered gapless QSL and its relation to deconfined quantum critical point (DQCP).

Simulation of ring-exchange models with projected entangled pair states

Algorithms to simulate ring-exchange models on a square lattice using projected entangled pair states (PEPSs) are developed. We generalize the imaginary time evolution (ITE) method to optimize PEPS wave functions for the models with ring-exchange interactions. We compare the effects of different approximations of the environment. To understand the numerical instability during optimization, we introduce the ``singularity'' of a PEPS and develop a regulation procedure that can effectively reduce the singularity of a PEPS. We benchmark our method with the toric code model and obtain extremely accurate ground-state energies and topological entanglement entropy. We also benchmark our method with the two-dimensional cyclic ring-exchange model, and find that the ground state has a strong vector chiral order. The algorithms can be a powerful tool to investigate the models with ring interactions. The methods developed in this work, e.g., the regularization process to reduce the singularity, can also be applied to other models.

Projected entangled pair states study of anisotropic-exchange magnets on the triangular lattice

The anisotropic-exchange spin-1/2 model on a triangular lattice has been used to describe the rare-earth chalcogenides, which may have exotic ground states. We investigate the quantum phase diagram of the model by using the projected entangled pair state (PEPS) method, and compare it to the classical phase diagram. Besides two stripe-ordered phase, and the 120$^\circ$ state, there is also a multi-\textbf{Q} phase. We identify the multi-\textbf{Q} phase as a $Z_{2}$ vortex state. No quantum spin liquid state is found in the phase diagram, contrary to the previous DMRG calculations.

Sequential Generation of Projected Entangled-Pair States.

We introduce plaquette projected entangled-pair states, a class of states in a lattice that can be generated by applying sequential unitaries acting on plaquettes of overlapping regions. They satisfy area-law entanglement, possess long-range correlations, and naturally generalize other relevant classes of tensor network states. We identify a subclass that can be more efficiently prepared in a radial fashion and that contains the family of isometric tensor network states [M. P. Zaletel and F. Pollmann, Phys. Rev. Lett. 124, 037201 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.037201]. We also show how this subclass can be efficiently prepared using an array of photon sources.

Direct sampling of projected entangled-pair states

Variational Monte Carlo studies employing projected entangled-pair states (PEPS) have recently shown that they can provide answers to long-standing questions such as the nature of the phases in the two-dimensional ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model. The sampling in these Monte Carlo algorithms is typically performed with Markov chain Monte Carlo algorithms employing local update rules, which often suffer from long autocorrelation times and interdependent samples. We propose a sampling algorithm that generates independent samples from a PEPS, bypassing all problems related to finite autocorrelation times. This algorithm is a generalization of an existing direct sampling algorithm for unitary tensor networks. We introduce an auxiliary probability distribution from which independent samples can be drawn, and combine it with importance sampling in order to evaluate expectation values accurately. We benchmark our algorithm on the classical Ising model and on variational optimization of two-dimensional quantum spin models.

Simulating both parity sectors of the Hubbard model with tensor networks

Tensor networks are a powerful tool to simulate a variety of different physical models, including those that suffer from the sign problem in Monte Carlo simulations. The Hubbard model on the honeycomb lattice with non-zero chemical potential is one such problem. Our method is based on projected entangled pair states (PEPS) using imaginary time evolution. We demonstrate that it provides accurate estimators for the ground state of the model, including cases where Monte Carlo simulations fail miserably. In particular it shows near to optimal, that is linear, scaling in lattice size. We also present a novel approach to directly simulate the subspace with an odd number of fermions. It allows to independently determine the ground state in both sectors. Without a chemical potential this corresponds to half filling and the lowest energy state with one additional electron or hole. We identify several stability issues, such as degenerate ground states and large single particle gaps, and provide possible fixes.

Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters

Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of Projected Entangled Pair States(PEPS), topological order is characterized by an entanglement symmetry of the local tensor which describes the model. This symmetry can take the form of a tensor product of group representations, or in the more general case a correlated symmetry action in the form of a Matrix Product Operator(MPO), which encompasses all string-net models. Among other things, these entanglement symmetries allow for the description of ground states and anyon excitations. Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net and its Galois conjugate, the non-hermitian double Yang-Lee (DYL) string-net. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and show that these can be mapped to certain restricted solid on solid(RSOS) models, which are equivalent to the $((5\pm\sqrt{5})/2)$-state Potts model, respectively. The known exact solutions of the statistical models allow us to locate the critical points, and to predict the critical exponents for the order parameters. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the order parameters for condensation and deconfinement.

Correlation Length in Random MPS and PEPS

Tensor network states are used extensively as a mathematically convenient description of physically relevant states of many-body quantum systems. Those built on regular lattices, i.e. matrix product states (MPS) in dimension 1 and projected entangled pair states (PEPS) in dimension 2 or higher, are of particular interest in condensed matter physics. The general goal of this work is to characterize which features of MPS and PEPS are generic and which are, on the contrary, exceptional. This problem can be rephrased as follows: given an MPS or PEPS sampled at random, what are the features that it displays with either high or low probability? One property which we are particularly interested in is that of having either rapidly decaying or long-range correlations. In a nutshell, our main result is that translation-invariant MPS and PEPS typically exhibit exponential decay of correlations at a high rate. We have two distinct ways of getting to this conclusion, depending on the dimensional regime under consideration. Both yield intermediate results which are of independent interest, namely: the parent Hamiltonian and the transfer operator of such MPS and PEPS typically have a large spectral gap. In all these statements, our aim is to get a quantitative estimate of the considered quantity (generic correlation length or spectral gap), which has the best possible dependency on the physical and bond dimensions of the random MPS or PEPS.