Reentrant topological phase transition in a bridging model between Kitaev and Haldane chains

We present a reentrant phase transition in a bridging model between two different topological models: Kitaev and Haldane chains. This model is activated by introducing a bond alternation into the Kitaev chain [A. Y. Kitaev, Phys. Usp. 44, 131 (2001)]. Without the bond alternation, the finite pairing potential induces a topological state defined by the zero-energy Majorana edge mode, while finite bond alternation without the pairing potential makes a different topological state similar to the Haldane state, which is defined by the local Berry phase in the bulk. The topologically ordered state corresponds to the Su-Schrieffer-Heeger state, which is classified as the same symmetry class. We thus find a phase transition between the two topological phases with a reentrant phenomenon, and extend the phase diagram in the plane of the pairing potential and the bond alternation by using three techniques: recursive equation, fidelity, and Pfaffian. In addition, we find that the phase transition is characterized by both the change of the position of Majorana zero-energy modes from one edge to the other edge and the emergence of a string order in the bulk, and that the reentrance is based on a sublattice U(1) rotation. Consequently, our paper and model not only open a direct way to discuss the bulk and edge topologies but demonstrate an example of the reentrant topologies.

Figure 1

Phase diagram of the Kitaev chain with bond alternation. The solid line $\left|\lambda \right|=\left|\gamma \right|$ and dashed line $\left|\lambda \right|={\left|\gamma \right|}^{-1}$ show the phase boundary obtained by the recursive equation and the Pfaffian. There are four phases that are distinguished by colored regions corresponding to the four cases mentioned in the main text: green, phase 1 (case 1); purple, phase 2 (case 2); red, phase 3 (case 3); and blue, phase 4 (case 4). Letters such as “L:R” denote the positions of the $a$-type and $b$-type MZMs, which are separated by a colon, i.e., “L:R” means that the $a$-type MZM is located on the left edge and the $b$-type MZM is on the left edge. The term “odd” (“even”) in the bracket denotes that $N$ is odd (even). The phases without the bracket give the same result; both cases of the odd and even number are the same regardless of even or odd number. $\mathcal{M}$ is the Majorana number obtained by the Pfaffian.

Figure 2

Amplitude of Majorana edge mode $u_{k_0,2i+1}$ ($i=0,1,...,99$) for fixed (a) $\gamma =0$ and (b) $\gamma =0.75$ in a $N=199$ sites system. Inset: Fidelity $\mathrm{Fd}(\lambda ,\gamma ;\delta \lambda =0.01,\delta \gamma =0)$.

Figure 3

Correlation function of the string order for various $\lambda$ with fixed $\gamma =-0.75$.