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

We propose a physical picture based on the wormhole effect of the path-integral formulation to explain the mechanism of 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 $\beta=1/T$) 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 1D and 2D to successfully demonstrate the bulk-like low-lying ES at finite temperatures, in addition to the edge-like 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.