Loop Optimization for Tensor Network Renormalization

Abstract

We introduce a tensor renormalization group scheme for coarse graining a two-dimensional tensor network that can be successfully applied to both classical and quantum systems on and off criticality. The key innovation in our scheme is to deform a 2D tensor network into small loops and then optimize the tensors on each loop. In this way, we remove short-range entanglement at each iteration step and significantly improve the accuracy and stability of the renormalization flow. We demonstrate our algorithm in the classical Ising model and a frustrated 2D quantum model. DOI: 10.1103/PhysRevLett.118.110504 Introduction.-In recent years, the tensor network (TN) approach [1,2] has become a powerful theoretical and computational [8, tool for studying condensed matter systems. Many physical quantities, including the partition function of a classical system, the Euclidean path integral of a quantum system, and the expectation value of physical observables, can be expressed in terms of tensor networks. Evaluating these quantities is reduced to the contraction of a multidimensional tensor network. In the two dimensional case, many algorithms [8,32,[37][38][39][40][41]43,[45][46][47][48][49][50][53][54][55][56][57] have been developed to implement the approximate tensor contractions. Among these, the tensor renormalization group approach introduced by Levin and Nave [38] and its generalizations [8,22,39,[43][44][45][46][47]55,56,61] have unique features: the tensor contraction is based on a fully isotropic coarse-graining procedure. Moreover, when applying the method to a system on a finite torus, the computational cost is lower than those based on matrix product states (MPS) [32,37,41,[48][49][50]53,54].However, the Levin-Nave tensor network renormalization (TRG, also referred as LN-TNR here) [38] is based on the singular value decomposition (SVD) of local tensors, which only minimizes the truncation errors of tree tensor networks. Several improvements [45][46][47] have taken into account the effect of the environments, but they are still essentially based on tree tensor networks. These approaches cannot completely remove short-range entanglements during the coarse graining process. For example, in the 2D TN calculation of a partition function (or a path integral) TNR based on simple SVD cannot simplify the corner-double-line (CDL) tensor [38], despite the CDL tensor describing a product state that should be simplified to a one-dimensional tensor. In Ref.[8], this issue was seriously discussed. The authors pointed out that to further remove short-range entanglement, it is crucial to optimize the tensor configurations that contain a loop. However, due to the computational cost, only a crude iterative method is

Gapped spin liquid with Z2 topological order for the kagome Heisenberg model

“…We find that the KAFHM ground state is a gapped QSL with the Z 2 topological order [3,74] and has a long correlation length, ξ ∼ 10 unit cells. We justify the TRG flow of modular matrices at critical points and argue that our estimation of the correlation length is valid within a distance of 30 unit cells or more, consistent with previous studies [77,78]. Such a long correlation length (ξ ∼ 10 kagome unit cells) might explain the DMRG's failure in identifying Z 2 topological order and the gapless behaviors in recent numerical simulations [69][70][71][72]79].…”

“…By studying the quantum critical state where the TRG simulation has the worst truncation errors, Refs. [77,78] found that the TRG simulation can handle a very large system size with a length of over 30 unit cells, since highly accurate critical exponents were obtained from calculations on such large systems. In the previous subsection, we directly justify this point via the TRG flow of modular matrices on a critical point.…”

Tensor network and (p-adic) AdS/CFT

“…To understand what happens, we recall works in the tensor network literature where classical partition functions of statistical models admit a tensor network description, see for example [59,60]. These tensor network representations of partition functions, which are equivalent to Euclidean path-integrals of the quantum model in one higher dimension, can also be coarse grained, which are linear maps of the constituent tensors.…”

Tensor-Network Simulations of the Surface Code under Realistic Noise

“…Generically, this problem is #P complete, and therefore, no efficient algorithm is believed to exist [9]. However, many efficient algorithms have been developed to obtain approximate solutions to this problem [10][11][12][13][14][15]. In this work, we have used both an exact, albeit inefficient, contraction algorithm and an efficient, approximate contraction algorithm for finite-sized PEPS with open boundary conditions [10].…”