Unlocking the general relationship between energy and entanglement spectra via the wormhole effect

Based on the path integral formulation of the reduced density matrix, we develop a scheme to overcome the exponential growth of computational complexity in reliably extracting low-lying entanglement spectrum from quantum Monte Carlo simulations. We test the method on the Heisenberg spin ladder with long entangled boundary between two chains and the results support the Li and Haldane's conjecture on entanglement spectrum of topological phase. We then explain the conjecture via the wormhole effect in the path integral and show that it can be further generalized for systems beyond gapped topological phases. Our further simulation results on the bilayer antiferromagnetic Heisenberg model with 2D entangled boundary across the (2 + 1)D O(3) quantum phase transition clearly demonstrate the correctness of the wormhole picture. Finally, we state that since the wormhole effect amplifies the bulk energy gap by a factor of beta, the relative strength of that with respect to the edge energy gap will determine the behavior of low-lying entanglement spectrum of the system. It was shown that the entanglement spectrum of topological systems is related to the energy spectrum of edge states, but only for gapped phases. Here the authors explain this relationship in terms of the wormhole effect in the path integral of the reduced density matrix and extend it beyond gapped phases.

Automated summary

Based on the path integral formulation of the reduced density matrix, the authors develop a scheme to extract low-lying entanglement spectrum from quantum Monte Carlo simulations. The method is tested on the Heisenberg spin ladder and supports the conjecture on the entanglement spectrum of topological phase. Furthermore, the authors explain the conjecture via the wormhole effect and extend it to systems beyond gapped topological phases.