The simple update (SU) and full update (FU) are the two paradigmatic time evolution algorithms for a tensor network known as the infinite projected entangled pair state (iPEPS). They differ by an error measure that is either, respectively, local or takes into account full infinite tensor environment. In this paper we test an intermediate neighborhood tensor update (NTU) accounting for the nearest-neighbor environment. This small environment can be contracted exactly in a parallelizable way. It provides an error measure that is Hermitian and non-negative down to machine precision. In the two-dimensional quantum Ising model, NTU is shown to yield stable unitary time evolution following a sudden quench. It also yields accurate thermal states despite correlation lengths that reach up to 20 lattice sites. The latter simulations were performed with a manifestly Hermitian purification of a thermal state. Both were performed with reduced tensors that do not include physical (and ancilla) indices. This modification naturally leads to two other schemes: a local SVD update and a full tensor update being a variant of FU.
The simple update (SU) and full update (FU) are the two paradigmatic time evolution algorithms for a tensor network known as the infinite projected entangled pair state (iPEPS). They differ by an error measure that is either, respectively, local or takes into account full infinite tensor environment. In this paper we test an intermediate neighborhood tensor update (NTU) accounting for the nearest-neighbor environment. This small environment can be contracted exactly in a parallelizable way. It provides an error measure that is Hermitian and non-negative down to machine precision. In the two-dimensional quantum Ising model, NTU is shown to yield stable unitary time evolution following a sudden quench. It also yields accurate thermal states despite correlation lengths that reach up to 20 lattice sites. The latter simulations were performed with a manifestly Hermitian purification of a thermal state. Both were performed with reduced tensors that do not include physical (and ancilla) indices. This modification naturally leads to two other schemes: a local SVD update and a full tensor update being a variant of FU.
Time evolution of an infinite two-dimensional (2D) many body quantum lattice system can be described by the Suzuki-Trotter decomposition applied to the infinite projected entangled pair state (iPEPS). Each Trotter gate increases the bond dimension of the tensor network, D, that has to be truncated back in a way that minimizes a suitable error measure. This paper goes beyond simplified error measures--like the one used in the full update, the simple update, and their intermediate neighborhood tensor update--and directly maximizes an overlap between the exact iPEPS with the increased bond dimension and the new iPEPS with the truncated one. The optimization is performed in a tangent space of the iPEPS variational manifold. This gradient tensor update is benchmarked by a simulation of a sudden quench of a transverse field in the 2D quantum Ising model and the quantum Kibble-Zurek mechanism in the same 2D system.
An infinite projected entangled pair state (iPEPS) is a tensor network ansatz to represent a quantum state on an infinite 2D lattice whose accuracy is controlled by the bond dimension D. Its real, Lindbladian, or imaginary time evolution can be split into small time steps. Every time step generates a new iPEPS with an enlarged bond dimension ${D}^{\ensuremath{'}}>D$, which is approximated by an iPEPS with the original D. In P. Czarnik and J. Dziarmaga, Phys. Rev. B 98, 045110 (2018), an algorithm was introduced to optimize the approximate iPEPS by maximizing directly its fidelity to the one with the enlarged bond dimension ${D}^{\ensuremath{'}}$. In this paper, we implement a more efficient optimization employing a local estimator of the fidelity. For imaginary time evolution of a thermal state's purification, we also consider using unitary disentangling gates acting on ancillas to reduce the required D. We test the algorithm simulating Lindbladian evolution and unitary evolution after a sudden quench of transverse field hx in the 2D quantum Ising model. Furthermore, we simulate thermal states of this model and estimate the critical temperature with good accuracy: 0.1 for hx=2.5 and 0.5 for the more challenging case of hx=2.9 close to the quantum critical point at hx=3.04438(2).
A typical quantum state obeying the area law for entanglement on an infinite two-dimensional (2D) lattice can be represented by a tensor network Ansatz, known as an infinite projected entangled pair state (iPEPS), with a finite bond dimension D. Its real-imaginary time evolution can be split into small time steps. An application of a time step generates a new iPEPS with a bond dimension k times the original one. The new iPEPS does not make optimal use of its enlarged bond dimension kD; hence, in principle, it can be represented accurately by a more compact Ansatz, preferably with the original D. In this work we show how the more compact iPEPS can be optimized variationally to maximize its overlap with the new iPEPS. To compute the overlap we use the corner-transfer-matrix renormalization group. By simulating sudden quench of the transverse field in the 2D quantum Ising model with the proposed algorithm, we provide a proof of principle that real-time evolution can be simulated with iPEPS. A similar proof is provided with the same model for imaginary-time evolution of purification of its thermal states.
The infinite projected entangled pair states (iPEPS) algorithm [J. Jordan et al., Phys. Rev. Lett. 101, 250602 (2008)] has become a useful tool in the calculation of ground-state properties of two-dimensional quantum lattice systems in the thermodynamic limit. Despite its many successful implementations, the method has some limitations in its present formulation which hinder its application to some highly entangled systems. The purpose of this paper is to unravel some of these issues, in turn enhancing the stability and efficiency of iPEPS methods. For this, we first introduce the fast full update scheme, where effective environment and iPEPS tensors are both simultaneously updated (or evolved) throughout time. As we shall show, this implies two crucial advantages: (i) dramatic computational savings and (ii) improved overall stability. In addition, we extend the application of the local gauge fixing, successfully implemented for finite-size PEPS [M. Lubasch et al., Phys. Rev. B 90, 064425 (2014)], to the iPEPS algorithm. We see that the gauge fixing not only further improves the stability of the method but also accelerates the convergence of the alternating least-squares sweeping in the (either ``full'' or ``fast full'') tensor update scheme. The improvement in terms of computational cost and stability of the resulting ``improved'' iPEPS algorithm is benchmarked by studying the ground-state properties of the quantum Heisenberg and transverse-field Ising models on an infinite square lattice.
We propose an improved approach to carry out the imaginary time evolution of infinite projected entangled-pair states, especially for systems with criticality. A cyclic optimal truncation is introduced to update the tensors along a closed loop, aiming to remove the redundant internal correlations. We demonstrate the algorithm by considering an elaborate evolution based on a simple update on a small plaquette. This scheme can also be applied to a full update strategy. We demonstrate their performances on simulating the ground states of the spin-1/2 antiferromagnetic Heisenberg model and the transverse field Ising model on a square lattice.
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 = 1 c = 1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.