Exact zero modes in twisted Kitaev chains

We study the Kitaev chain under generalized twisted boundary conditions, for which both the amplitudes and the phases of the boundary couplings can be tuned at will. We explicitly show the presence of exact zero modes for large chains belonging to the topological phase in the most general case, in spite of the absence of “edges” in the system. For specific values of the phase parameters, we rigorously obtain the condition for the presence of the exact zero modes in finite chains, and show that the zero modes obtained are indeed localized. The full spectrum of the twisted chains with zero chemical potential is analytically presented. Finally, we demonstrate the persistence of zero modes (level crossing) even in the presence of disorder or interactions.

Figure 1

The schematic representation of the chain with $\mathsf{a}=\mathsf{b}=1,\mu =0,t=\mathrm{\Delta }$, and ${\varphi }_1={\varphi }_2=\pi /2$. The red lines represent the coupling between Majorana operators with the amplitude $it$. The Majorana operator $a_1$ is isolated from the rest, and hence does not enter the Hamiltonian. The other Majorana operator that commutes with the Hamiltonian is $\left(b_L-a_2\right)/\sqrt{2}$.

Figure 2

Examples of the zeros of $detM$ for large chains. The solution of $detM=0$, i.e., the solution of $tcos{\varphi }_1+\mathrm{\Delta }cos{\varphi }_2=0$, is represented on a ${\varphi }_1-{\varphi }_2$ plane (${\varphi }_1,{\varphi }_2\in \left[0,2\pi \right)$, gray curves), which separates $(0,0)$ (red points) and $(\pi ,\pi )$ (blue points). (a) $\mathrm{\Delta }=0.8t$, (b) $\mathrm{\Delta }=t$, (c) $\mathrm{\Delta }=1.2t$.

Figure 3

Phase diagram of the Majorana chain with open boundaries. The solution curves of Eq. (25) with $L=9$ are represented by red curves, which exist only in the region ${(\mathrm{\Delta }/t)}^2+{(\mu /2t)}^2\le 1$. Outside the region, exact zero modes never appear even if the chain belongs to the topological phase.

Figure 4

The schematic representation of the chain with $\mu =0$. (a) Even $L$: one Möbius ring with two defects. (b) Odd $L$: two decoupled chains, each of which has one defect.

Figure 5

The graphic representation of the functions which determine the quantized wave numbers corresponding to the eigenvalues. The allowed wave numbers are obtained by the intersection between the above graphs and the constant calculated by the parameters. (a) Even $L$: $f\left(q\right)=sinqM/\left[sinq\left(M+1\right)-{\mathsf{b}}^{2}sinq\left(M-1\right)\right]$ for $L=8$ and $\mathsf{b}=1$. (b) Odd $L$: $f\left(q\right)=sinqN/sinq\left(N-1\right)$ for $L=9$.

Figure 6

Spectrum of the interacting Hamiltonian $H_{\mathrm{int}}{|}_{\mathsf{a}=\mathsf{b}=1;(\varphi ,\varphi )}$ ($0\le \varphi \le \pi$), for the topological phase and the trivial phase. Calculations are performed for $L=16$ and the lowest ten eigenenergies are shown. (a) Topological phase ($t=4,\mathrm{\Delta }=4,U=2,\mu =6$): the level crossing between the two lowest-lying states occurs at $\varphi \simeq \pi /2$. (b) Trivial phase ($t=4,\mathrm{\Delta }=4,U=2,\mu =18$): the spectral gap above the ground state never closes at any $\varphi$.

Figure 7

Dependence of the level crossing angle $|\varphi /\pi -1/2|$ on the system size. Fitting functions are shown by solid curves, which demonstrate the exponential decay of $|\varphi /\pi -1/2|$ with increasing $L$. The values of the parameters $t,\mathrm{\Delta },U,\mu$ are taken within the region of the topological phase.