Biorthonormal Matrix-Product-State Analysis for Non-Hermitian Transfer-Matrix Renormalization-Group in the Thermodynamic Limit

TL;DR

This paper introduces a biorthonormal matrix-product-state framework for analyzing non-Hermitian transfer matrices in the thermodynamic limit, enabling efficient computation of bulk properties without boundary effects.

Contribution

It develops two new infinite-size biorthonormal TMRG algorithms and an efficient wave function transformation for non-Hermitian systems in the thermodynamic limit.

Findings

Both algorithms produce translationally invariant dual infinite-BMPS.

The method accurately predicts bulk properties like magnetization and spin correlations.

Computational cost becomes independent of lattice size.

Abstract

We give a thorough Biorthonormal Matrix-Product-State (BMPS) analysis of the Transfer-Matrix Renormalization-Group (TMRG) for non-Hermitian matrices in the thermodynamic limit. The BMPS is built on a dual series of reduced biorthonormal bases for the left and right Perron states of a non-Hermitian matrix. We propose two alternative infinite-size Biorthonormal TMRG (iBTMRG) algorithms and compare their numerical performance in both finite and infinite systems. We show that both iBTMRGs produce a dual infinite-BMPS (iBMPS) which are translationally invariant in the thermodynamic limit. We also develop an efficient wave function transformation of the iBTMRG, an analogy of McCulloch in the infinite-DMRG [arXiv:0804.2509 (2008)], to predict the wave function as the lattice size is increased. The resulting iBMPS allows for probing bulk properties of the system in the thermodynamic limit without boundary effects and allows for reducing the computational cost to be independent of the lattice size, which are illustrated by calculating the magnetization as a function of the temperature and the critical spin-spin correlation in the thermodynamic limit for a 2D classical Ising model.