Abstract We study the connection between topological order and degeneracy of the entanglement spectrum by explicitly solving the two-dimensional dimerized quantum Heisenberg model in the form of tensor product state ansatz. By evaluating the topological entanglement entropy, we identify a new phase with topological order in this model, in which the entanglement spectrum is doubly degenerate. Degeneracy of the entanglement spectrum is robust against various types of perturbations, in accordance with our expectation for topological order. Our results support the connection among topological order, long range entanglement and the dominant degenerate singular values of entanglement spectrum. In the context of tensor product state ansatz, the numerical evaluation of entanglement spectrum costs far less computation power than the one for topological entanglement entropy. Our results provide a more viable way to numerically identify the topological order for the generic frustrated systems.
We study the correspondence between boundary excitation distribution spectrum of nonchiral topological orders on an open surface
M with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed surface. The closed surface is bipartitioned into two subsystems, one of which has the same topology as
M. Specifically, we focus on the case of generalized string-net models and discuss the cases where
M is a disk or a cylinder. When
M has the topology of a cylinder, different combinations of boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. When both boundaries are charge (smooth) boundaries, the entanglement spectrum can be identified with the boundary excitation distribution spectrum at infinite temperature and constant fugacities. Examples of toric code,
ZN theories, and the simple non-Abelian case of doubled Fibonacci are demonstrated.
We study the "entanglement spectrum" (a presentation of the Schmidt decomposition analogous to a set of "energy levels") of a many-body state, and compare the Moore-Read model wave function for the nu=5/2 fractional quantum Hall state with a generic 5/2 state obtained by finite-size diagonalization of the second-Landau-level-projected Coulomb interactions. Their spectra share a common "gapless" structure, related to conformal field theory. In the model state, these are the only levels, while in the "generic" case, they are separated from the rest of the spectrum by a clear "entanglement gap", which appears to remain finite in the thermodynamic limit. We propose that the low-lying entanglement spectrum can be used as a "fingerprint" to identify topological order.