Locality in Quantum and Markov Dynamics on Lattices and Networks

TL;DR

This paper proves that in gapped quantum and Markov systems, correlation functions can be approximated by exponentially decaying terms plus low-lying state matrix elements, aiding numerical and network analysis.

Contribution

It introduces a local approximation method for operators in gapped systems, enabling better understanding of correlations in quantum and Markov dynamics.

Findings

Correlation functions decompose into exponential decay and low-lying state contributions.

The method applies to both quantum and Markov systems with a spectral gap.

Applications include improved numerical simulations and network analysis.

Abstract

We consider gapped systems governed by either quantum or Markov dynamics, with the low-lying states below the gap being approximately degenerate. For a broad class of dynamics, we prove that ground or stationary state correlation functions can be written as a piece decaying exponentially in space plus a term set by matrix elements between the low-lying states. The key to the proof is a local approximation to the negative energy, or annihilation, part of an operator in a gapped system. Applications to numerical simulation of quantum systems and to networks are discussed.