Correlations in Quantum Spin Systems from the Boundary Effect

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.