Projected Entangled Pair States at Finite Temperature: Iterative Self-Consistent Bond Renormalization for Exact Imaginary Time Evolution

TL;DR

This paper introduces an iterative self-consistent bond renormalization method for PEPS at finite temperature, enabling accurate imaginary time evolution without increasing bond dimension, demonstrated on the quantum Ising model.

Contribution

The authors develop a novel self-consistent renormalization procedure for PEPS that maintains accuracy without enlarging bond dimension during imaginary time evolution.

Findings

Accurately computes thermal states of the quantum Ising model.

Effectively captures phase transition behavior.

Maintains computational efficiency with fixed bond dimension.

Abstract

A projected entangled pair state (PEPS) with ancillas can be evolved in imaginary time to obtain thermal states of a strongly correlated quantum system on a 2D lattice. Every application of a Suzuki-Trotter gate multiplies the PEPS bond dimension $D$ by a factor $k$. It has to be renormalized back to the original $D$. In order to preserve the accuracy of the Suzuki-Trotter (S-T) decomposition, the renormalization has in principle to take into account full environment made of the new tensors with the bond dimension $k\times D$. Here we propose a self-consistent renormalization procedure operating with the original bond dimension $D$, but without compromising the accuracy of the S-T decomposition. The iterative procedure renormalizes the bond using full environment made of renormalized tensors with the bond dimension $D$. After every renormalization, the new renormalized tensors are used to update the environment, and then the renormalization is repeated again and again until convergence. As a benchmark application, we obtain thermal states of the transverse field quantum Ising model on a square lattice - both infinite and finite - evolving the system across a second-order phase transition at finite temperature.