Time-reversal symmetric Kitaev model and topological superconductor in two dimensions

Abstract

A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z2 gauge fields. The Majorana fermion model can be viewed as a model of time-reversal invariant superconductor and is classified as a member of symmetry class DIII in the Altland-Zirnbauer classification. The ground-state phase diagram has two topologically distinct gapped phases which are distinguished by a Z2 topological invariant. The topologically nontrivial phase supports both a Kramers' pair of gapless Majorana edge modes at the boundary and a Kramers' pair of zero-energy Majorana states bound to a 0-flux vortex in the π-flux background. Power-law decaying correlation functions of spins along the edge are obtained by taking the gapless Majorana edge modes into account. The model is also defined on the one-dimension ladder, in which case again the ground-state phase diagram has Z2 trivial and non-trivial phases.

Phase diagrams of SO($N$) Majorana-Hubbard models: Dimerization, internal symmetry breaking, and fluctuation-induced first-order transitions

“…[22,24] in the weakly-interacting limit. Similarly, for 𝑁 = 2, the 𝜋-flux model describes the ground-state flux sector of a generalized Kitaev spin-orbital liquid on the square lattice [23,25], perturbed by an additional Ising spin-spin interaction [32]. In these spin-orbital realizations of the SO(2) and SO(3) Majorana-Hubbard Hamiltonians, the hopping term corresponds to a generalized Kitaev spin-orbital exchange coupling [25], while the interaction terms map to Heisenberg and Ising spin-spin interactions, respectively [32].…”

“…The model is exactly solvable using a parton decomposition, in which the spin Hamiltonian is mapped to a tight-binding Hamiltonian of Majorana fermions hopping in the background of a static Z 2 gauge field. This con-struction has recently been extended to other tricoordinated lattices [14][15][16][17][18][19][20][21], as well as to systems with larger local Hilbert spaces [22][23][24][25]. In the latter cases, instead of a single Majorana fermion, an 𝑁-component vector of Majorana fermions emerges at each lattice site.…”

Entanglement of a 3D generalization of the Kitaev model on the diamond lattice

“…This class also has a Z 2 classification. Following [7], we again characterize the topological state by the Fu-Kane formula Eq. 19.…”

“…Here we obtain the topological invariant of the layers perpendicular to the s 1 direction when K z = 0 and J 1 = 0. We follow the line of reasoning in [7]. In this limit, the 3D model is effectively a set of two-dimensional systems in class DIII, each of which is classified by a Z 2 invariant.…”

Higher-dimensional Jordan-Wigner Transformation and Auxiliary Majorana Fermions

“…We conclude in Sec. VI with a discussion on how the ηχ formalism might be generalized to odd-coordinated systems, and elaborate on the relations of our approach with the design of exactly solved quantum spin liquid models [29][30][31].…”