The topological order is encoded in the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order for topological band models through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice models with bosonic particles. We identify multiple independent minimal entangled states (MESs) in the ground state manifold on a torus. The extracted modular S matrix from MESs faithfully demonstrates the Ising anyon or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings unambiguously demonstrate the topological nature of the quantum states for these flatband models without using the knowledge of model wave functions.The topological order is encoded in the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order for topological band models through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice models with bosonic particles. We identify multiple independent minimal entangled states (MESs) in the ground state manifold on a torus. The extracted modular S matrix from MESs faithfully demonstrates the Ising anyon or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings unambiguously demonstrate the topological nature of the quantum states for these flatband models without using the knowledge of model wave functions.

arXiv:1308.3878Physical Review LettersMar 4, 201428 citations

Topological Order in Three Dimensions and Entanglement Entropy of Gapped Phases

Tarun Grover, Ari M. Turner, Ashvin Vishwanath

Bulletin of the American Physical SocietyMar 1, 20120 citations

Ground-state degeneracy of non-Abelian topological phases from coupled wires

Thomas Iadecola, Titus Neupert, Claudio Chamon +1 more

We construct a family of two-dimensional non-Abelian topological phases from coupled wires using a non-Abelian bosonization approach. We then demonstrate how to determine the nature of the non-Abelian topological order (in particular, the anyonic excitations and the topological degeneracy on the torus) realized in the resulting gapped phases of matter. This paper focuses on the detailed case study of a coupled-wire realization of the bosonic $su{(2)}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ Moore-Read state, but the approach we outline here can be extended to general bosonic $su{(2)}_{k}^{\phantom{\rule{0.16em}{0ex}}}$ topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases.We construct a family of two-dimensional non-Abelian topological phases from coupled wires using a non-Abelian bosonization approach. We then demonstrate how to determine the nature of the non-Abelian topological order (in particular, the anyonic excitations and the topological degeneracy on the torus) realized in the resulting gapped phases of matter. This paper focuses on the detailed case study of a coupled-wire realization of the bosonic $su{(2)}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ Moore-Read state, but the approach we outline here can be extended to general bosonic $su{(2)}_{k}^{\phantom{\rule{0.16em}{0ex}}}$ topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases.

arXiv:1703.03418Physical review. B./Physical review. BJun 20, 201921 citations

Edge theory approach to topological entanglement entropy, mutual information, and entanglement negativity in Chern-Simons theories

Xueda Wen, Shunji Matsuura, Shinsei Ryu

We develop an approach based on edge theories to calculate the entanglement entropy and related quantities in (2+1)-dimensional topologically ordered phases. Our approach is complementary to, e.g., the existing methods using replica trick and Witten's method of surgery, and applies to a generic spatial manifold of genus $g$ , which can be bipartitioned in an arbitrary way. The effects of fusion and braiding of Wilson lines can be also straightforwardly studied within our framework. By considering a generic superposition of states with different Wilson line configurations, through an interference effect, we can detect, by the entanglement entropy, the topological data of Chern-Simons theories, e.g., the $R$ symbols, monodromy, and topological spins of quasiparticles. Furthermore, by using our method, we calculate other entanglement/correlation measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent noncontractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders.We develop an approach based on edge theories to calculate the entanglement entropy and related quantities in (2+1)-dimensional topologically ordered phases. Our approach is complementary to, e.g., the existing methods using replica trick and Witten's method of surgery, and applies to a generic spatial manifold of genus $g$, which can be bipartitioned in an arbitrary way. The effects of fusion and braiding of Wilson lines can be also straightforwardly studied within our framework. By considering a generic superposition of states with different Wilson line configurations, through an interference effect, we can detect, by the entanglement entropy, the topological data of Chern-Simons theories, e.g., the $R$ symbols, monodromy, and topological spins of quasiparticles. Furthermore, by using our method, we calculate other entanglement/correlation measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent noncontractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders.

arXiv:1603.08534Physical review. B./Physical review. BJun 20, 2016110 citations

su{(2)}_{2}^{\phantom{\rule{0.16em}{0ex}}}

Moore-Read state, but the approach we outline here can be extended to general bosonic

su{(2)}_{k}^{\phantom{\rule{0.16em}{0ex}}}

topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases.

g

, which can be bipartitioned in an arbitrary way. The effects of fusion and braiding of Wilson lines can be also straightforwardly studied within our framework. By considering a generic superposition of states with different Wilson line configurations, through an interference effect, we can detect, by the entanglement entropy, the topological data of Chern-Simons theories, e.g., the

R

symbols, monodromy, and topological spins of quasiparticles. Furthermore, by using our method, we calculate other entanglement/correlation measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent noncontractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders.