Majorana zero modes in a quantum Ising chain with longer-ranged interactions

A one-dimensional Ising model in a transverse field can be mapped onto a system of spinless fermions with $p$-wave superconductivity. In the weak-coupling BCS regime, it exhibits a zero-energy Majorana mode at each end of the chain. Here, we consider a variation of the model, which represents a superconductor with longer-ranged kinetic energy and pairing amplitudes, as is likely to occur in more realistic systems. It possesses a richer zero-temperature phase diagram and has several quantum phase transitions. From an exact solution of the model, we find that these phases can be classified according to the number of Majorana zero modes of an open chain: zero, one, or two at each end. The model possesses a multicritical point where phases with zero, one, and two Majorana end modes meet. The number of Majorana modes at each end of the chain is identical to the topological winding number of the Anderson pseudospin vector that describes the BCS Hamiltonian. The topological classification of the phases requires a unitary time-reversal symmetry to be present. When this symmetry is broken, only the number of Majorana end modes modulo 2 can be used to distinguish two phases. In one of the regimes, the wave functions of the two phase-shifted Majorana zero modes decay exponentially in space but in an oscillatory manner. The wavelength of oscillation is identical to that in the asymptotic connected spin-spin correlation of the $XY$ model in a transverse field, to which our model is dual.

Figure 1

The region ${\lambda }_2>1+{\lambda }_1$ is ordered in the original spin representation and the boundary of it is a critical line where the gap at $k=0$ collapses. The region ${\lambda }_2<1-{\lambda }_1$ is disordered as well and the boundary corresponds to a critical line where the gap at $k=\pi$ collapses. The point ${\lambda }_1=0$ and ${\lambda }_2=1$ is a special multicritical point[24] with an emergent $\mathrm{U}\left(1\right)$ symmetry, most transparently seen in the dual representation (see below). In the same dual representation, the region enclosed by ${\lambda }_1^2=-4{\lambda }_2$ is an oscillatory ferromagnetically ordered phase separating from an ordered phase for ${\lambda }_2<0$, as determined by the spatial decay of the instantaneous spin-spin correlation function. Note that duality exchanges the ordered and disordered phases. Here $n=0,1,2$ correspond to regions with $n$ Majorana zero modes at each end of an open chain.

Figure 2

The two Majorana zero modes for ${\lambda }_1=0.05$ and ${\lambda }_2=1.5$. The row number 1 shows $u_{01}\left(i\right)$ and $v_{01}\left(i\right)$ corresponding to the the first Majorana zero mode. The second row corresponds to the second Majorana zero mode which is orthogonal to the first. The third row corresponds to the probability distributions $P_{01}\left(i\right)$ and $P_{02}\left(i\right)$ for the two respective Majorana modes. The numerical diagonalization was carried out for a lattice of $N=100$ sites with open boundary condition. For lattices larger than 150, one quickly loses numerical control because of the ${10}^{16}$ difference in the orders of magnitude of the largest eigenvalue and the zero mode.

Figure 3

The two Majorana zero modes for ${\lambda }_2=-1.2$ and ${\lambda }_1=1$. The first row depicts $u_{02}\left(i\right)$ and $v_{02}\left(i\right)$ corresponding to the first Majorana zero mode. The second row shows the second Majorana zero mode which is orthogonal to the first. The third row corresponds to the probability distributions $P_{01}\left(i\right)$ and $P_{02}\left(i\right)$ for the two respective Majorana modes.

Figure 4

Splitting of $n=2$ Majorana zero modes for the complex Hermitian BdG equation. Left: ${\lambda }_2=2.5$. Right: ${\lambda }_2=-2.0$. The magnitude of ${\varepsilon }_0$ is the smallest eigenvalue. The slight rounding in the proximity of the quantum critical point is due to the finite size of the lattice: $N=200$. The quantum critical point for the infinite system is at ${\lambda }_1=1.5$ for the left panel and ${\lambda }_1=3.0$ for the right panel. The $n=1$ Majorana zero mode survives intact. Note that at ${\lambda }_1=0$ the chain splits into two independent chains, and hence there is a zero mode irrespective of the fact that the ${\lambda }_2$ pairing amplitude is purely imaginary.

Figure 5

Topological phase transition across the point ${\lambda }_1=1,{\lambda }_2=0$. (a) ${\lambda }_1=1^{-},{\lambda }_2=0$. Of the two solid curves, the one with greater dispersion is ${\epsilon }_k$ and the other is the quasiparticle energy. Of the two dashed lines, the horizontal line is $\mu$, and the other is ${\Delta }_k$. (b) The same quantities plotted for ${\lambda }_1=1^{+},{\lambda }_2=0$. The associated Anderson pseudospin vector $\widehat{d}\left(k\right)$ is drawn schematically below each plot. It is clear that in (a) the pseudospin does not wind along the 1D Brillouin zone, i.e., $W=0$, whereas in (b) it winds once, i.e., $W=1$.

Figure 6

Topological phase transition in the vicinity of the point ${\lambda }_1=0,{\lambda }_2=1$. (a) ${\lambda }_2=1^{-},{\lambda }_1=0$; (b) ${\lambda }_2=1^{+},{\lambda }_1=0$. It is clear that in (a) the pseudospin does not wind along the 1D Brillouin zone, i.e., $W=0$, whereas in (b) it winds twice, i.e., $W=2$. The labels are the same as in Fig. 5.

Figure 7

Topological phase transition in the vicinity of the point ${\lambda }_1=1.5,{\lambda }_2=-1$. (a) ${\lambda }_2=-1^{+},{\lambda }_1=1.5$; (b) ${\lambda }_2=-1^{-},{\lambda }_1=1.5$. In (a), the pseudospin does not wind along the 1D Brillouin zone, i.e., $W=0$, whereas in (b) it winds twice, i.e., $W=2$. The labels are the same as in Fig. 5.

Figure 8

Topological phase transition along the line ${\lambda }_2=-1$. (a) ${\lambda }_1=1.6,{\lambda }_2=-1$. (b) ${\lambda }_1=1.8,{\lambda }_2=-1$. (c) ${\lambda }_1=2.0,{\lambda }_2=-1$. (d) ${\lambda }_1=2.2,{\lambda }_2=-1$. The associated Anderson pseudospin vector is not shown since it is not defined at points where the quasiparticle energy gap vanishes. However, the system in (d) is fully gapped and has $W=1$, which is consistent with the analysis of previous sections. The labels are the same as in Fig. 5.