Symmetry breaking and criticality in tensor-product states

TL;DR

This paper investigates phase transitions in tensor-product states of the transverse-field Ising chain, revealing a crossover from first-order to continuous mean-field criticality with increasing system size and tensor dimension.

Contribution

It introduces a variational approach using small matrix-product states to analyze symmetry breaking and critical behavior in quantum spin chains and 2D tensor networks.

Findings

Finite systems show first-order transitions with symmetry-broken states.

As system size increases, a continuous transition with mean-field criticality emerges.

In 2D tensor-product states, asymptotic mean-field behavior is observed.

Abstract

We discuss variationally optimized matrix-product states for the transverse-field Ising chain, using D*D matrices with small D=2-10. For finite system size N there are energy minimums for symmetric as well as symmetry-broken states, which cross each other at a field value hc(N,D); thus the transition is first-order. A continuous transition develops as N->infinity. The asymptotic critical behavior is then always of mean-field type (the magnetization exponent beta=1/2), but a window of field strengths where true Ising scaling holds (beta=1/8) emerges with increasing D. We also demonstrate asymptotic mean-field behavior for infinite-size two-dimensional tensor-product (iPEPS) states with small tensors.