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arXiv:1707.08430 · Physical review. B./Physical review. B

Characterization of topological phases of dimerized Kitaev chain via edge correlation functions

Yucheng Wang, Jian-Jian Miao, Hui-Ke Jin, Shu Chen

Published Nov 20, 2017• 39 citations
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Abstract

We study analytically topological properties of a noninteracting modified dimerized Kitaev chain and an exactly solvable interacting dimerized Kitaev chain under open boundary conditions by analyzing two introduced edge correlation functions. The interacting dimerized Kitaev chain at the symmetry point $\mathrm{\ensuremath{\Delta}}=t$ and the chemical potential $\ensuremath{\mu}=0$ can be exactly solved by applying two Jordan-Wigner transformations and a spin rotation, which permits us to calculate the edge correlation functions analytically. We demonstrate that the two edge correlation functions can be used to characterize the trivial, Su-Schrieffer-Heeger-like topological and topological superconductor phases of both the noninteracting and interacting systems and give their phase diagrams.

DOI
10.1103/physrevb.96.205428

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We study analytically topological properties of a noninteracting modified dimerized Kitaev chain and an exactly solvable interacting dimerized Kitaev chain under open boundary conditions by analyzing two introduced edge correlation functions. The interacting dimerized Kitaev chain at the symmetry point \ensuremathΔ=t\mathrm{\ensuremath{\Delta}}=t\ensuremathΔ=t and the chemical potential \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=0 can be exactly solved by applying two Jordan-Wigner transformations and a spin rotation, which permits us to calculate the edge correlation functions analytically. We demonstrate that the two edge correlation functions can be used to characterize the trivial, Su-Schrieffer-Heeger-like topological and topological superconductor phases of both the noninteracting and interacting systems and give their phase diagrams.
It was recently shown that an interacting Kitaev topological superconductor model is exactly solvable based on two-step Jordan-Wigner transformations together with one spin rotation. We generalize this model by including the dimerization, which is shown also to be exactly solvable. We analytically determine the topological phase diagram containing seven distinct phases. It is argued that the system is topological when a fermionic many-body Majorana zero-energy edge state emerges. It is intriguing that there are two tetracritical points, at each of which four distinct phases touch.
It was recently shown that an interacting Kitaev topological superconductor model is exactly solvable based on two step Jordan-Wigner transformations together with one spin rotation. We generalize this model by including the dimerization, which is also exactly solvable. In this extended model there are two topological indices associated with the sublattice symmetry and the particle-hole symmetry. We analytically determine the topological phase diagram containing seven distinct topological phases. There are two tetra-critical points, at which four distinct phases touch. It is intriguing that a topological charge-density-wave state and a topological Schr\odinger-cat state emerge for strong interactions. We confirm various topological phases by examining the presence of zero-energy edge states.
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system at zero chemical potential as well as in the symmetric case of Δ = t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion interaction. We suggest and examine an edge correlation function of Majorana fermions to characterize the long range order in the topological superconducting states and study the phase diagram of the interating Kitaev chain.
We investigate the evolutions of edge Majorana fermions (MFs), and we show that they can be used to characterize different topological phases and study topological phase transitions. For some limiting cases of the evolution process for the Kitaev chain model and the Su-Schrieffer-Heeger (SSH) model, we give analytical expressions of the survival probabilities (SPs) of the edge MFs. For a general case, we consider a dimerized Kitaev model as well as the Kitaev chain with disorder chemical potential, and we numerically calculate the SPs of two edge correlation MFs. Our results show that both of them are equal to zero, one of them is equal to zero at some times, or neither of them is equal to zero, which corresponds to the SSH-like topological, topological superconductor, and trivial phases, respectively.
The Kitaev chain model with a nearest neighbor interaction U is solved exactly at the symmetry point Δ=t and chemical potential μ=0 in an open boundary condition. By applying two Jordan-Wigner transformations and a spin rotation, such a symmetric interacting model is mapped onto a noninteracting fermion model, which can be diagonalized exactly. The solutions include a topologically nontrivial phase at |U|<t and a topologically trivial phase at |U|>t. The two phases are related by dualities. Quantum phase transitions in the model are studied with the help of the exact solution.
Topological insulators and superconductors are characterized by their gapless boundary modes. In this paper, we develop a recursive approach to the boundary Green function, which encodes this nontrivial boundary physics. Our approach describes the various topologically trivial and nontrivial phases as fixed points of a recursion and provides direct access to the phase diagram, the localization properties of the edge modes, as well as topological indices. We illustrate our approach in the context of various familiar models such as the Su-Schrieffer-Heeger model, the Kitaev chain, and a model for a Chern insulator. We also show that the method provides an intuitive approach to understand recently introduced topological phases which exhibit gapless corner states.