We elucidate the topological features of the entanglement entropy of a region in two-dimensional quantum systems in a topological phase with a finite correlation length $\ensuremath{\xi}$. First, we suggest that simpler reduced quantities, related to the von Neumann entropy, could be defined to compute the topological entropy. We use our methods to compute the entanglement entropy for the ground-state wave function of a quantum eight-vertex model in its topological phase and show that a finite correlation length adds corrections of the same order as the topological entropy which come from sharp features of the boundary of the region under study. We also calculate the topological entropy for the ground state of the quantum dimer model on a triangular lattice by using a mapping to a loop model. The topological entropy of the state is determined by loop configurations with a nontrivial winding number around the region under study. Finally, we consider extensions of the Kitaev wave function, which incorporate the effects of electric and magnetic charge fluctuations, and use it to investigate the stability of the topological phase by calculating the topological entropy.We elucidate the topological features of the entanglement entropy of a region in two-dimensional quantum systems in a topological phase with a finite correlation length $\ensuremath{\xi}$. First, we suggest that simpler reduced quantities, related to the von Neumann entropy, could be defined to compute the topological entropy. We use our methods to compute the entanglement entropy for the ground-state wave function of a quantum eight-vertex model in its topological phase and show that a finite correlation length adds corrections of the same order as the topological entropy which come from sharp features of the boundary of the region under study. We also calculate the topological entropy for the ground state of the quantum dimer model on a triangular lattice by using a mapping to a loop model. The topological entropy of the state is determined by loop configurations with a nontrivial winding number around the region under study. Finally, we consider extensions of the Kitaev wave function, which incorporate the effects of electric and magnetic charge fluctuations, and use it to investigate the stability of the topological phase by calculating the topological entropy.

arXiv:0709.0729Physical Review BDec 19, 200749 citations

Entanglement Spectrum In Topological Phasess

Bogdan Andrei Bernevig

Bulletin of the American Physical SocietyMar 1, 20120 citations

Accuracy of Topological Entanglement Entropy on Finite Cylinders

Hong‐Chen Jiang, Rajiv R. P. Singh, Leon Balents

Topological phases are unique states of matter which support nonlocal excitations which behave as particles with fractional statistics. A universal characterization of gapped topological phases is provided by the topological entanglement entropy (TEE). We study the finite size corrections to the TEE by focusing on systems with a Z2 topological ordered state using density-matrix renormalization group and perturbative series expansions. We find that extrapolations of the TEE based on the Renyi entropies with a Renyi index of n≥2 suffer from much larger finite size corrections than do extrapolations based on the von Neumann entropy. In particular, when the circumference of the cylinder is about ten times the correlation length, the TEE obtained using von Neumann entropy has an error of order 10(-3), while for Renyi entropies it can even exceed 40%. We discuss the relevance of these findings to previous and future searches for topological ordered phases, including quantum spin liquids.Topological phases are unique states of matter which support nonlocal excitations which behave as particles with fractional statistics. A universal characterization of gapped topological phases is provided by the topological entanglement entropy (TEE). We study the finite size corrections to the TEE by focusing on systems with a Z2 topological ordered state using density-matrix renormalization group and perturbative series expansions. We find that extrapolations of the TEE based on the Renyi entropies with a Renyi index of n≥2 suffer from much larger finite size corrections than do extrapolations based on the von Neumann entropy. In particular, when the circumference of the cylinder is about ten times the correlation length, the TEE obtained using von Neumann entropy has an error of order 10(-3), while for Renyi entropies it can even exceed 40%. We discuss the relevance of these findings to previous and future searches for topological ordered phases, including quantum spin liquids.

arXiv:1304.0780Physical Review LettersSep 5, 201318 citations

Finite-size scaling of entanglement entropy in one-dimensional topological models

Yuting Wang, Tobias Gulden, Alex Kamenev

We consider scaling of the entanglement entropy across a topological quantum phase transition for the Kitaev chain model. The change of the topology manifests itself in a subleading term, which scales as ${L}^{\ensuremath{-}1/\ensuremath{\alpha}}$ with the size of the subsystem $L$ , here $\ensuremath{\alpha}$ is the R\'enyi index. This term reveals the scaling function ${h}_{\ensuremath{\alpha}}(L/\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the correlation length, which is sensitive to the topological index. The scaling function ${h}_{\ensuremath{\alpha}}(L/\ensuremath{\xi})$ is independent of model parameters, suggesting some degree of its universality.We consider scaling of the entanglement entropy across a topological quantum phase transition for the Kitaev chain model. The change of the topology manifests itself in a subleading term, which scales as ${L}^{\ensuremath{-}1/\ensuremath{\alpha}}$ with the size of the subsystem $L$, here $\ensuremath{\alpha}$ is the R\'enyi index. This term reveals the scaling function ${h}_{\ensuremath{\alpha}}(L/\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the correlation length, which is sensitive to the topological index. The scaling function ${h}_{\ensuremath{\alpha}}(L/\ensuremath{\xi})$ is independent of model parameters, suggesting some degree of its universality.

arXiv:1609.03998Physical review. B./Physical review. BFeb 1, 201714 citations

The entanglement structure of topological quantum phases

Michael Mariën

Jan 1, 20160 citations

L

, here $\ensuremath{\alpha}$ is the R\'enyi index. This term reveals the scaling function ${h}_{\ensuremath{\alpha}}(L/\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the correlation length, which is sensitive to the topological index. The scaling function ${h}_{\ensuremath{\alpha}}(L/\ensuremath{\xi})$ is independent of model parameters, suggesting some degree of its universality.