Variational methods for contracting projected entangled-pair states

Abstract

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.

Partial lifting of degeneracy in the $J_1-J_2-J_3$ Ising antiferromagnet on the kagome lattice

“…Evaluating general spin-spin correlation functions can in principle be done with tensor networks (see e.g. [82][83][84]): the complexity is similar to computing dynamical correlation functions for one-dimensional quantum systems in imaginary time 8 . The construction based on tiles, however, requires a specific implementation that is left for future investigation.…”

Simulating Chiral Spin Liquids with Projected Entangled-Pair States

“…which we evaluate using the methods from Ref. [38]. We plot this quantity through a path of the Brillouin zone for different PEPS bond dimensions, showing a smooth behavior at the X point and a vanishing value at the Γ point-exactly what one would expect for a gapped spin liquid (long-range order would manifest itself as a divergence at the X point).…”

“…This class of tensor-network states can be formulated [34] and variationally optimized [35,36] directly in the thermodynamic limit. The control parameters are the bond dimension D, controlling the amount of entanglement in the PEPS wave function, and the environment bond dimension χ, which is introduced when contracting the PEPS approximately with boundary MPS [28,37,38] or the corner-transfer matrix renormalization group (CTMRG) [39][40][41][42][43][44]. When simulating a critical model, a variational PEPS wave function typically exhibits a finite correlation length, which provides a length scale that can be used to extrapolate variational PEPS results [45][46][47][48].…”

“…The CTMRG contraction induces a second control parameter, the environment bond dimension χ. In our further calculations with the optimized PEPS, we use the CTMRG and variational uniform MPS (VUMPS) algorithms interchangeably for finding approximate environments, as these were recently shown to be equivalent [38].…”

Ground state search by local and sequential updates of neural network quantum states

“…In particular, it has been shown that, for tensor networks, local optimizations generally result in better performance than global optimizations [65]. While interesting developments in the field of tensor networks are continuously appearing [66][67][68], important limitations still exist, for example for dimensions larger than one and for long range systems.…”