Complex-Time Singularity and Locality Estimates for Quantum Lattice Systems

TL;DR

This paper investigates the locality bounds of complex-time dynamics in quantum lattice systems, demonstrating limitations in higher dimensions and showing conditions for finite-time blow-up of dynamics.

Contribution

It establishes a locality bound for 1D quantum spin systems and analyzes the challenges and failures of extending these bounds to higher dimensions.

Findings

Locality bound proved for 1D quantum spin systems

Extension to higher dimensions faces fundamental obstacles

Existence of interactions causing finite-time blow-up in complex dynamics

Abstract

We present and prove a well-known locality bound for the complex-time dynamics of a general class of one-dimensional quantum spin systems. Then we discuss how one might hope to extend this same procedure to higher dimensions using ideas related to the Eden growth process and lattice trees. Finally, we demonstrate with a specific family of lattice trees in the plane why this approach breaks down in dimensions greater than one and prove that there exist interactions for which the complex-time dynamics blows-up in finite imaginary time.