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

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 ${\mathbb{Z}}_2$ 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 ${\mathbb{Z}}_2$ 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 $\pi$-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 ${\mathbb{Z}}_2$ trivial and nontrivial phases.

Figure 1

Square lattice and link vectors ${\mathit{e}}_{\mu }$ with $\mu =0,1,2,3$ emanating from a site on the A sublattice (open circle) to a neighboring site on the B sublattice (filled circle). The dashed lines indicate a unit cell.

Figure 2

Phase diagram in the parameter space $(J_0,J_1,J_2,J_3)$. Shaded regions are gapped phases and the other area is gapless phase.

Figure 3

Four-spin interaction terms. Left: The ${\mathbb{Z}}_2$ gauge fields obtained from the two paths connecting next-nearest-neighbor sites. Right: Two directions are labeled by $\alpha =1,2$, respectively.

Figure 4

(a) Phase diagram in the $J_0/J-J_1/J$ plane. The region between the dashed lines $J_1=J_0\pm 2J$, including the isotropic point $J_1=J_0=J$, is the gapless phase when the next-nearest-neighbor hopping terms are absent (“B phase”). The gapless phase is turned into a topologically nontrivial phase when the second-nearest-neighbor hopping is turned on. Both the gapped phases for $J_1>J_0+2J$ and for $J_1<J_0-2J$ remain topologically trivial upon including the second-nearest-neighbor hopping terms (“A phase”). The numbered solid circles indicate the parameter sets for which the energy spectra are calculated and shown in Fig. 5. (b) Strip geometry with edges along the vector ${\mathit{e}}_0$.

Figure 5

Energy spectra of the one-dimensional strip [Fig. 4b] as a function of the momentum along the edge. The energy spectra are calculated for the following parameter sets [see also Fig. 4a]: $J_{\mu }/J=(4,1,1,1)$ in (1a) and (1b); $J_{\mu }/J=(1,1,1,1)$ in (2a) and (2b); $J_{\mu }/J=(3,3,1,1)$ in (3a) and (3b); $J_{\mu }/J=(1,4,1,1)$ in (4a) and (4b). The ${\mathbb{Z}}_2$ gauge is fixed as $u^{\mu }=(-1,1,1,1)$. The second-nearest-neighbor hopping $K_i^{\alpha }=0$ ($\alpha =1,2$ and $i=z,x$) in the left figures [(1a), (2a), (3a), (4a)], $K_i^{\alpha }/J=0.15$ in the right figures [(1b), (2b), (3b), (4b)].

Figure 6

(a) The number of states in the topological phase for $20\times 42$ system with a pair of vortices separated by 20 lattice spacings. Periodic boundary conditions are imposed in the $x$ and $y$ directions. The parameters in the Hamiltonian are $J^{\mu }=1$ and $K_{x,z}^{1,2}=0.3$. (b) The energy difference $\Delta E$ between the positive and negative energy eigenvalues of bound states, as a function of the distance between the vortices.

Figure 7

One-dimensional system on a two-leg cylindrical lattice.

Figure 8

Phase diagram of the one-dimensional cylindrical lattice model in the parameter space $(J_0+J_2,J_1,J_3)$. The phase in which one of the horizontal bonds $J_0$ and $J_2$ is greater than the other three bonds is a topologically trivial phase. The shaded regions (tetrahedra) are a topological phase, where each end of the ladder has twofold-degenerate zero-energy Majorana states.

Figure 9

String operators of Jordan-Wigner fermions that represent Dirac matrices for (a) Majorana fermion made of $c_1$ ($U_j$), and (b) those made of $c_2$ ($V_j$).