Different temperature dependence for the edge and bulk of the entanglement Hamiltonian

Abstract

We propose a physical picture based on the wormhole effect of the path-integral formulation to explain the mechanism of the entanglement spectrum (ES), such that our picture not only explains the topological state with bulk-edge correspondence of the energy spectrum and ES (the Li and Haldane conjecture), but is generically applicable to other systems independent of their topological properties. We point out it is ultimately the relative strength of bulk energy gap (multiplied with inverse temperature β=1/T) with respect to the edge energy gap that determines the behavior of the low-lying ES of the system. Depending on the circumstances, the ES can resemble the energy spectrum of the virtual edge, but can also represent that of the virtual bulk. We design models both in one and two dimensions to successfully demonstrate the bulklike low-lying ES at finite temperatures, in addition to the edgelike case conjectured by Li and Haldane at zero temperature. Our results support the generality of viewing the ES as the wormhole effect in the path integral and the different temperature dependence for the edge and bulk of ES.