Accurate exponents from approximate tensor renormalizations

TL;DR

This paper analyzes tensor renormalization group methods for the Ising model, showing how a small subspace approximation yields accurate critical exponents and proposing improved procedures for better precision and broader applications.

Contribution

It introduces a small subspace approximation in tensor renormalization, analytically derives critical exponents, and proposes alternative blocking procedures for enhanced accuracy.

Findings

Critical exponent nu approximated as 0.964 with two-state model

Improved methods achieve nu=0.987 and 0.993

Potential to rival Monte Carlo methods for critical phenomena

Abstract

We explain the recent numerical successes obtained by Tao Xiang's group, who developed and applied Tensor Renormalization Group methods for the Ising model on square and cubic lattices, by the fact that their new truncation method sharply singles out a surprisingly small subspace of dimension two. We show that in the two-state approximation, their transformation can be handled analytically yielding a value 0.964 for the critical exponent nu much closer to the exact value 1 than 1.338 obtained in the Migdal-Kadanoff approximation. We propose two alternative blocking procedures that preserve the isotropy and improve the accuracy to nu=0.987 and 0.993 respectively. We discuss applications to other classical lattice models, including models with fermions, and suggest that it could become a competitor for Monte Carlo methods suitable to calculate accurately critical exponents, take continuum limits and study near-conformal systems in arbitrarily large volumes.