Locality estimes for complex time evolution in 1D

TL;DR

This paper extends locality estimates for 1D quantum systems with exponential tails in interactions, proving analyticity of time-evolution operators and exponential decay of correlations at high temperatures, with implications for spectral gap problems.

Contribution

It generalizes Araki's locality estimates to include exponential tails, establishing analyticity and correlation decay in 1D quantum systems with broader interaction types.

Findings

Proves analyticity of time-evolution operators with exponential tails

Shows exponential decay of correlations above a temperature threshold

Applies results to spectral gap problems in 2D PEPS models

Abstract

It is a generalized belief that there are no thermal phase transitions in short range 1D quantum systems. However, the only known case for which this is rigorously proven is for the particular case of finite range translational invariant interactions. The proof was obtained by Araki in his seminal paper of 1969 as a consequence of pioneering locality estimates for the time-evolution operator that allowed him to prove its analiticity on the whole complex plane, when applied to a local observable. However, as for now there is no mathematical proof of the abscence of 1D thermal phase transitions if one allows exponential tails in the interactions. In this work we extend Araki's result to include exponential (or faster) tails. Our main result is the analyticity of the time-evolution operator applied on a local observable on a suitable strip around the real line. As a consequence we obtain that thermal states in 1D exhibit exponential decay of correlations above a threshold temperature that decays to zero with the exponent of the interaction decay, recovering Araki's result as a particular case. Our result however still leaves open the possibility of 1D thermal short range phase transitions. We conclude with an application of our result to the spectral gap problem for Projected Entangled Pair States (PEPS) on 2D lattices, via the holographic duality due to Cirac et al.