Gauge symmetry in Kitaev-type spin models and index theorems on odd manifolds

Abstract

We construct an exactly soluble spin-1 2 model on a honeycomb lattice, which is a generalization of Kitaev model. The topological phases of the system are analyzed by study of the ground state sector of this model, the vortex-free states. Basically, there are two phases, A phase and B phase. The behaviors of both A and B phases may be studied by mapping the ground state sector into a general p-wave paired states of spinless fermions with tunable pairing parameters on a square lattice. In this p-wave paired state theory, the A phase is shown to be the strong paired phase, an insulating phase. The B phase may be either gapped or gapless determined by the generalized inversion symmetry is broken or not. The gapped B is the weak pairing phase described by either the Moore-Read Pfaffian state of the spinless fermions or anti-Pfaffian state of holes depending on the sign of the next nearest neighbor hopping amplitude. A phase transition between Pfaffian and anti-Pfaffian states are found in the gapped B phase. Furthermore, we show that there is a hidden SU(2) gauge symmetry in our model. In the gapped B phase, the ground state has a non-trivial topological number, the spectral first Chern number or the chiral central charge, which reflects the chiral anomaly of the edge state. We proved that the topological number is identified to the reduced eta-invariant and this anomaly may be cancelled by a bulk Wess-Zumino term of SO(3) group through an index theorem in 2+1 dimensions.

Splitting of a Cooper Pair by a Pair of Majorana Bound States

“…1, top panel). If the bound states are widely separated, the qubit is robust against local sources of decoherence and provides a building block for topological quantum computation [1,2].While Majorana bound states have not yet been demonstrated experimentally, there is now a variety of candidate systems. In an s-wave superconductor, zero-FIG.…”

“…1, top panel). If the bound states are widely separated, the qubit is robust against local sources of decoherence and provides a building block for topological quantum computation [1,2].…”

“…The positive cross-correlation of the current pulses in the two leads ensures that the total transferred charge is 2e. This "splitting" of a Cooper pair is a highly characteristic signature of a Majorana qubit, reminiscent of the h/e (instead of h/2e) flux periodicity of the Josephson effect [1,25,26]. Notice that for any stochastic process the crosscorrelator is bounded by the auto-correlator,…”

The Yang-Baxter paradox

“…These include algebraic [57], analytic [49], functional [27], thermodynamic [56], offdiagonal [13], double-row transfer matrix constructions [53], and by using separation of variables [52]. Moreover there are other techniques available to yield exact solutions, such as the Jordan-Wigner transform [36], those used in Kitaev-type models [33,64], and those used in Rabi-type models [12,31,45,65]. These somewhat confuse attempts to provide an unambiguous definition for what constitutes exact-solvability, and to identify its relationship to integrability.…”

Realization of the exactly solvable Kitaev honeycomb lattice model in a spin-rotation-invariant system

“…2,3 The analogy between the non-Abelian Ising vortices and vortices in p + ip superconductors has been raised in several works. [4][5][6][7] Exact diagonalization has been used to study the Kitaev model on small lattices. 8 And perturbative expansion methods have been developed to study the gapped phases of the Kitaev-type models.…”