Projected entangled pair states at finite temperature: Iterative self-consistent bond renormalization for exact imaginary time evolution

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 two-dimensional 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 (ST) decomposition, the renormalization in principle has to take into account full environment made of the new tensors with the bond dimension $k\ifmmode\times\else\texttimes\fi{}D$. Here, we propose a self-consistent renormalization procedure operating with the original bond dimension $D$, but without compromising the accuracy of the ST 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.