Correlations in Quantum Spin Systems from the Boundary Effect

TL;DR

This paper introduces the boundary effect function as a new way to understand correlations in quantum spin systems, linking boundary effects to correlation decay, entanglement laws, and spectral gaps.

Contribution

It defines the boundary effect function and proves its relationship to correlation decay, entanglement area laws, and spectral gaps in local spin systems.

Findings

Exponential decay of the boundary effect function implies exponential clustering of correlations.

Gapped systems with nondegenerate ground states typically exhibit exponential boundary effects.

Moderately decaying boundary effects can lead to area laws even in gapless systems.

Abstract

We introduce the boundary effect on the ground state as an attribute of general local spin systems that restricts the correlations in the ground state. To this end, we introduce what we call a boundary effect function, which characterises not only the boundary effect, but also the thermodynamic limit of the ground state. We prove various aspects of the boundary effect function to unfold its relationship to other attributes of the system such as a finite spectral gap above the ground state, two-point correlation functions, and entanglement entropies. In particular, it is proven that an exponentially decaying boundary effect function implies the exponential clustering of two-point correlation functions in arbitrary spatial dimension, the entanglement area law in one dimension, and the logarithmically corrected area law in higher dimension. It is also proven that gapped local spin systems with nondegenerate ground states ordinarily fall into that class. In one dimension, the area law can also result from a moderately decaying boundary effect function, in which case the system is thermodynamically gapless.