SU(2) slave fermion solution of the Kitaev honeycomb lattice model

We apply the SU(2) slave fermion formalism to the Kitaev honeycomb lattice model. We show that both the toric code phase (the A phase) and the gapless phase of this model (the B phase) can be identified with $p$-wave superconducting phases of the slave fermions, with nodal lines which, respectively, do not or do intersect the Fermi surface. The non-Abelian Ising anyon phase is a $p+ip$ superconducting phase that occurs when the B phase is subjected to a gap-opening magnetic field. We also discuss the transitions between these phases in this language.

Figure 1

Fermionic spectra for the dispersing mode of the fermionized Hamiltonian, in the gapless $B$ phase (a), and gapped $A$ phase (b). (The remaining 6 bands of the Hamiltonian (6) are flat, and lie at energies $\pm J_x,\pm J_y$, and $\pm J_z$). In the gapless phase the Fermi surface consists of nodes at two distinct points in the hexagonal Brillouin zone. The loci of these nodes varies with the relative magnitudes of $J_x,J_y$, and $J_z$. The transition to the gapped $A$ phase occurs when the pair of nodes come together and annihilate, as shown in (c).

Figure 2

Schematic phase diagram of the Hamiltonian (1).

Figure 3

The product of spin operators conserved separately on each plaquette by the Kitaev Hamiltonian (1). The Hamiltonian (1) distinguishes between three types of links on the honeycomb lattice, which we call $x$, $y$, and $z$ links (color-coded red, green, and blue, respectively, here). On $x$ links the spin-spin interaction term is $S_i^{\left(x\right)}{S}_j^{\left(x\right)}$, and similarly for $y$ and $z$ links. The product of spin operators shown here—a product around a plaquette of the spin variable associated with the external edge at each vertex—commutes with the spin Hamiltonian.

Figure 4

The three-spin interaction that breaks $\widehat{\mathbf{T}}$ but commutes with the conserved six-spin product on each plaquette (see Fig. 3) (Ref.24). The operator is constructed by taking the product of spin operators on three adjacent vertices, where the direction of the central spin is that associated with the external edge at the vertex, while the two external spins match the edges joining their associated vertices to the central vertex. In the figure, the central vertex is at the upper left, and the operator is the product of ${\sigma }_z$ at the lower left vertex, ${\sigma }_y$ at the upper left vertex, and ${\sigma }_x$ at the top vertex.

Figure 5

Schematic phase diagram of the Kitaev honeycomb model, and the corresponding superconducting phases. The phase is determined by the ratio $J/J_z$, and by whether the coefficient $J^{\prime }$ of the three-spin interaction is non-vanishing.