Spectral Gap and Exponential Decay of Correlations

TL;DR

This paper establishes that a spectral gap in quantum systems guarantees exponential decay of correlations, with results applicable to both short-range and long-range interactions, and includes cases with symmetry considerations.

Contribution

It provides a comprehensive analysis linking spectral gaps to correlation decay in quantum systems, extending to systems with long-range interactions and symmetry properties.

Findings

Spectral gap implies exponential decay of correlations for anticommuting observables.

Connected correlations decay exponentially under the spectral gap assumption.

Correlation functions decay exponentially in systems with certain self-similarity or translational invariance.

Abstract

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D<2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions.