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Entanglement only exists in quantum states, and this can be viewed as the fundamental distinction between classical physics and quantum physics. Entanglement spectrum (ES) is a direct measurement of the topological properties of quantum many-body systems. 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 β = 1/T) with respect to the edge energy gap that determines the behavior of the low-lying ES of the system.

In another work, we systematically compute the energy spectra of the 2D spin-1/2 Heisenberg model with long-range interactions by large-scale quantum Monte Carlo simulations. With the 1/r^α ferromagnetic and staggered antiferromagnetic interactions, we find the explicit range in α for the short-range Goldstone-type (gapless), anomalous Goldstone-type (gapless) and Higgs-type (gapped) spectra. Accompanied by the spin wave analysis, our numerical results vividly reveal how the long-range interactions alter the usual linear and quadratic magnon dispersions in 2D quantum magnets and give rise to anomalous dynamical exponents. Moreover, we find explicit cases where the gapped excitation emerges at a non-integer decay exponent α for the antiferromagnetic Hamiltonian. This work provides the first set of unbiased dynamical data of long-range quantum many-body systems and suggests that many universally accepted low-energy customs for short-range systems need to be substantially modified for long-range ones which are of immediate relevance to the ongoing experimental efforts from quantum simulators to 2D quantum moiré materials.