Fermions and nontrivial loop-braiding in a three-dimensional toric code

Abstract

We study an exactly solvable toric code type of Hamiltonian in three dimensions, defined on the diamond lattice with spin-1/2 degrees of freedom at each site. The Hamiltonian is a sum of mutually commuting plaquette operators B p , all of which have eigenvalue +1 in the ground state. The excitations are "fluxes," which are plaquettes with B p = −1. Due to certain local kinematic constraints, fluxes form loops. The elementary flux-loop excitations are fermions, in contrast to other solvable spin-1/2 models in three dimensions, where the excitations are bosons. Furthermore, the flux loops braid nontrivially, giving rise to Abelian anyonlike statistics.

Chiral spin liquids at finite temperature in a three-dimensional Kitaev model

“…W p is a conserved quantity which is odd under both T and P operations with the eigenvalues ±1 (called ±π/2 flux [20]). Similar to other 3D cases [20,22], there are local constraints on W p corresponding to the operator identities for Pauli matrices:…”

“…These possibilities make the study of 3D CSLs at finite T even more interesting, including transitions breaking parity (P) symmetry as well as T symmetry. While loop like excitations in the 3D Kitaev models and other realizations of 3D Z 2 QSL are known to trigger a thermal second-order phase transi-tion [21][22][23][24][25], rather than a crossover in the case of 2D Z 2 QSL [26], the transitions to 3D CSLs remain elusive thus far.…”

Entanglement and Majorana edge states in the Kitaev model

“…It is known that this model reduces to a problem of noninteracting Majorana fermions hopping in the presence of a background Z 2 gauge field; this reduces an otherwise quartic fermionic interaction to an effective quadratic fermionic interactions exactly 28 . All of these intriguing facts motivated a series of important studies taking the Kitaev model as a test-bed for understanding many fundamental theoretical concepts such as quenching and defect production 31 , phase transitions 32 , braid-ing statistics 33,34 , dynamics of hole vacancies 35 , to name a select few. Studies of the entanglement entropy in a particular limit of this model were undertaken early on 36,37 .…”

Topological entanglement entropy of the three-dimensional Kitaev model

“…The low-energy effective Hamiltonian for the Kitaev model in the limit J z J x , J y is a toric-code-type model defined on the diamond lattice 17,25 . Since TEE for Kitaev model is independent of the coupling parameters J α , our calculation provides TEE for the latter model as well.…”