Reentrant Topological Phase Transition in a Bridging Model between Kitaev and Haldane Chains

Abstract

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. Yu Kitaev, Phys.-Usp. {\bf 44} 131 (2001)]. Without the bond alternation, finite pairing potential induces a topological state defined by 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 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 study and model do not only open a direct way to discuss the bulk and edge topologies, but demonstrate an example of the reentrant topologies.