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International Symposium on Lattice Field Theory (LATTICE)

Entanglement filtering and improved coarse-graining on two dimensional tensor networks including fermions

Ryo Sakai, Muhammad Asaduzzaman, Simon M. Catterall, Yannick Meurice, Goksu Can Toga

Published Feb 1, 2023• 0 citations
Publisher

Abstract

Tensor renormalization group (TRG) has attractive features like the absence of sign problems and the accessibility to the thermodynamic limit, and many applications to lattice field theories have been reported so far. However it is known that the TRG has a fictitious fixed point that is called the CDL tensor and that causes less accurate numerical results. There are improved coarse-graining methods that attempt to remove the CDL structure from tensor networks. Such approaches have been shown to be beneficial on two dimensional spin systems. We discuss how to adapt the removal of the CDL structure to tensor networks including fermions, and numerical results that contain some comparisons to the plain TRG, where significant differences are found, will be shown. The detailed discussion of this work is given in ref. [1].

DOI
10.22323/1.430.0034

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Tensor renormalization group (TRG) has attractive features like the absence of sign problems and the accessibility to the thermodynamic limit, and many applications to lattice field theories have been reported so far. However it is known that the TRG has a fictitious fixed point that is called the CDL tensor and that causes less accurate numerical results. There are improved coarse-graining methods that attempt to remove the CDL structure from tensor networks. Such approaches have been shown to be beneficial on two dimensional spin systems. We discuss how to adapt the removal of the CDL structure to tensor networks including fermions, and numerical results that contain some comparisons to the plain TRG, where significant differences are found, will be shown. The detailed discussion of this work is given in ref. [1].
Coarse-graining renormalization aims to reformulate a tensor network model with a coarse-grained one at a larger scale. It has attracted particular attention in recent years because it opens a new avenue to unveil the entanglement structure of a tensor network model under the scaling transformation. This chapter reviews and compares the tensor renormalization group (TRG) and other coarse-graining methods developed in the past two decades. The methods can be divided into two groups according to whether or not the renormalization effect of the environment tensors is incorporated in the optimization of local tensors. The local optimization methods include TRG, HOTRG (a variant of TRG based on the higher-order singular value decomposition), tensor network renormalization (TNR), and loop-TNR. The global optimization methods include the second renormalized TRG and HOTRG, referred to as SRG and HOSRG, respectively. Among all these coarse-graining methods, HOTRG and HOSRG are the only two that can be readily extended and efficiently applied to three-dimensional classical or two-dimensional quantum lattice models.
Abstract We show how to apply renormalization group algorithms incorporating entanglement filtering methods and a loop optimization to a tensor network which includes Grassmann variables which represent fermions in an underlying lattice field theory. As a numerical test a variety of quantities are calculated for two dimensional Wilson-Majorana fermions and for the two flavor Gross-Neveu model. The improved algorithms show much better accuracy for quantities such as the free energy and the determination of Fisher’s zeros.
We propose a real-space renormalization group algorithm for accurately coarse-graining two-dimensional tensor networks. The central innovation of our method lies in utilizing variational boundary tensors as a globally optimized environment for the entire system. Based on this optimized environment, we construct renormalization projectors that significantly enhance accuracy. By leveraging the canonical form of tensors, our algorithm maintains the same computational complexity as the original tensor renormalization group (TRG) method, yet achieves higher accuracy than existing approaches that do not incorporate entanglement filtering. Our work offers a practical pathway for extending TRG methods to higher dimensions while keeping computational costs manageable.
We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.
We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization approach that removes local entanglement and produces a coarse-grained lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors Tinv{T}_{\text{inv}}Tinv​ plus the symmetry group Gsym{G}_{\text{sym}}Gsym​ of the tensors (i.e., the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by two-dimensional (2D) statistical Ising model, 2D statistical loop-gas model, and 1+1D1+1\text{D}1+1D quantum spin-1/2 and spin-1 models. In particular, using such a (Gsym,Tinv)({G}_{\text{sym}},{T}_{\text{inv}})(Gsym​,Tinv​) characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry-protected topological phase. The (Gsym,Tinv)({G}_{\text{sym}},{T}_{\text{inv}})(Gsym​,Tinv​) characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.
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.