Exactly soluble spin- $\frac{1}2$ models on three-dimensional lattices are proposed by generalizing Kitaev model on honeycomb lattice to three dimensions with proper periodic boundary conditions. The simplest example is spins on a diamond lattice which is exactly soluble. The ground state sector of the model may be mapped into a p-wave paired state on cubic lattice. We observe for the first time a topological phase transition from a gapless phase to a gapped phase in an exactly soluble spin model. Furthermore, the gapless phase can not be gapped by a perturbation breaking the time reversal symmetry. Unknotted and unlinked Wilson loops arise as eigen excitations, which may evolute into linked and knotted loop excitations. We show that these closed string excitations obey abelian statistics in the gapped phase and non-abelian statistics in the gapless phase.Exactly soluble spin-$\frac{1}2$ models on three-dimensional lattices are proposed by generalizing Kitaev model on honeycomb lattice to three dimensions with proper periodic boundary conditions. The simplest example is spins on a diamond lattice which is exactly soluble. The ground state sector of the model may be mapped into a p-wave paired state on cubic lattice. We observe for the first time a topological phase transition from a gapless phase to a gapped phase in an exactly soluble spin model. Furthermore, the gapless phase can not be gapped by a perturbation breaking the time reversal symmetry. Unknotted and unlinked Wilson loops arise as eigen excitations, which may evolute into linked and knotted loop excitations. We show that these closed string excitations obey abelian statistics in the gapped phase and non-abelian statistics in the gapless phase.

arXiv:0709.1302Sep 10, 200710 citations

On Freedman's Lattice Models for Topological Phases

James Brink, Zhenghan Wang

arXiv:math-ph/0303018Quantum Information ProcessingApr 1, 20032 citations

Exactly soluble lattice models for non-Abelian states of matter in two dimensions

Maciej Koch-Janusz, Michael Levin, Ady Stern

Following an earlier construction of exactly soluble lattice models for Abelian fractional topological insulators in two and three dimensions, we construct here an exactly soluble lattice model for a non-Abelian $\ensuremath{\nu}=1$ quantum Hall state and a non-Abelian topological insulator in two dimensions. We show that both models are topologically ordered, exhibiting fractionalized charge, ground-state degeneracy on the torus, and protected edge modes. The models feature non-Abelian vortices which carry fractional electric charge in the quantum Hall case and spin in the topological insulator case. We analyze the statistical properties of the excitations in detail and discuss the possibility of extending this construction to three-dimensional non-Abelian topological insulators.Following an earlier construction of exactly soluble lattice models for Abelian fractional topological insulators in two and three dimensions, we construct here an exactly soluble lattice model for a non-Abelian $\ensuremath{\nu}=1$ quantum Hall state and a non-Abelian topological insulator in two dimensions. We show that both models are topologically ordered, exhibiting fractionalized charge, ground-state degeneracy on the torus, and protected edge modes. The models feature non-Abelian vortices which carry fractional electric charge in the quantum Hall case and spin in the topological insulator case. We analyze the statistical properties of the excitations in detail and discuss the possibility of extending this construction to three-dimensional non-Abelian topological insulators.

arXiv:1306.2789Physical Review BSep 19, 201316 citations

Exactly Solvable Models for Symmetry-Enriched Topological Phases

Yang Qi, Meng Cheng, Zheng‐Cheng Gu +1 more

Bulletin of the American Physical SocietyMar 15, 20171 citation

Extended Hubbard Model with Ring Exchange: A Route to a Non-Abelian Topological Phase

Michael Hartley Freedman, Chetan Nayak, Kirill Shtengel

We propose an extended Hubbard model on a 2D kagome lattice with an additional ring exchange term. The particles can be either bosons or spinless fermions. We analyze the model at the special filling fraction 1/6, where it is closely related to the quantum dimer model. We show how to arrive at an exactly soluble point whose ground state is the "d-isotopy" transition point into a stable phase with a certain type of non-Abelian topological order. Near the "special" values, d=2cos(pi/(k+2), this topological phase has anyonic excitations closely related to SU(2) Chern-Simons theory at level k.We propose an extended Hubbard model on a 2D kagome lattice with an additional ring exchange term. The particles can be either bosons or spinless fermions. We analyze the model at the special filling fraction 1/6, where it is closely related to the quantum dimer model. We show how to arrive at an exactly soluble point whose ground state is the "d-isotopy" transition point into a stable phase with a certain type of non-Abelian topological order. Near the "special" values, d=2cos(pi/(k+2), this topological phase has anyonic excitations closely related to SU(2) Chern-Simons theory at level k.

arXiv:cond-mat/0312273Physical Review LettersFeb 14, 200560 citations

Non-Abelian topological phases

Michael Hartley Freedman

Bulletin of the American Physical SocietyMar 15, 20060 citations

Exactly soluble spin-

\frac{1}2

models on three-dimensional lattices are proposed by generalizing Kitaev model on honeycomb lattice to three dimensions with proper periodic boundary conditions. The simplest example is spins on a diamond lattice which is exactly soluble. The ground state sector of the model may be mapped into a p-wave paired state on cubic lattice. We observe for the first time a topological phase transition from a gapless phase to a gapped phase in an exactly soluble spin model. Furthermore, the gapless phase can not be gapped by a perturbation breaking the time reversal symmetry. Unknotted and unlinked Wilson loops arise as eigen excitations, which may evolute into linked and knotted loop excitations. We show that these closed string excitations obey abelian statistics in the gapped phase and non-abelian statistics in the gapless phase.