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arXiv:1806.05025 · Physical review. E

Detecting topological phases via survival probabilities of edge Majorana fermions

Yucheng Wang

Published Oct 15, 2018• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.98.042128

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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.
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.
Evolution of Majorona zero-energy edge states in a T2=−1T^2 = -1T2=−1 symmetry protected 1D topological superconductor with dominant spin-orbit coupling
We consider a 1D topological superconductor (TSC) constructed by coupling a pair of Kitaev's Majorana chains with opposite spin configurations. Such a 1D lattice model is known to be protected by a T2=−1T^2 = -1T2=−1 time-reversal symmetry. Furthermore, we consider a modeled Rashba spin-orbit coupling on such a system of T2=−1T^2=-1T2=−1 time-reversal symmetric TSC. The Rashba spin-orbit coupling together with the chemical potential engineered the phase transitions of the edge states in the system and consequently the number of Majorona's zero-energy edge modes (MZM's) emerging at the edge of the coupled chains. Correspondingly, the topological nature of the system is described by a phase diagram consisting of three different phases. The three phases are characterized by a topological winding number, W=1\mathcal{W}=1W=1, 222 (with one and two MZM's: topological phases) and W=0\mathcal{W}=0W=0 (devoid of any MZM: trivial insulating phase).
Abstract The topological phases of random Kitaev α -chains are labelled by the number of localized edge Majorana zero modes. The critical lines between these phases thus correspond to delocalization transitions for these localized edge Majorana zero modes. For the random Kitaev chain with next-nearest couplings, where there are three possible topological phases , the two Lyapunov exponents of Majorana zero modes are computed for a specific solvable case of Cauchy disorder, in order to analyze how the phase diagram evolves as a function of the disorder strength. In particular, the direct phase transition between the phases n = 0 and n = 2 is possible only in the absence of disorder, while the presence of disorder always induces an intermediate phase n = 1, as found previously via numerics for other distributions of disorder.
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 emergent topological gapless phases in the square-lattice Kitaev model with additional hopping terms. In the presence of nearest-neighbor hopping only, the model is known to exhibit gapless phases with two topological gapless points. When the strength of the newly added next-nearest-neighbor hopping is smaller than a certain value, qualitatively the same phase diagram persists. We find that further increase of the extra hopping results in a new topological phase with four gapless points. We construct a phase diagram to clarify the regions of emergent topological gapless phases as well as topologically trivial ones in the space of the chemical potential and the next-nearest-neighbor hopping strength. We examine the evolution of the gapless phases in the energy dispersions of the bulk as the chemical potential varies. The topological properties of the gapless phases are characterized by the winding numbers of the present gapless points. We also consider the ribbon geometry to examine the corresponding topological edge states. It is revealed that Majorana-fermion edge modes exist as flat bands in topological gapless phases. We also perform the analytical calculation as to Majorana-fermion zero-energy modes and discuss its implications on the numerical results.
We consider the temporal evolution of a zero-energy edge Majorana of a spinless p-wave superconducting chain following a sudden change of a parameter of the Hamiltonian. Starting from one of the topological phases that has an edge Majorana, the system is suddenly driven to the other topological phase or to the (topologically) trivial phases and to the quantum critical points (QCPs) separating these phases. The survival probability of the initial edge Majorana as a function of time is studied following the quench. Interestingly when the chain is quenched to the QCP, we find a nearly perfect oscillation of the survival probability, indicating that the Majorana travels back and forth between two ends, with a time period that scales with the system size. We also generalize to the situation when there is a next-nearest-neighbor hopping in a superconducting chain and there results in a pair of edge Majorana at each end of the chain in the topological phase. We show that the frequency of oscillation of the survival probability gets doubled in this case. We also perform an instantaneous quenching of the Hamiltonian (with two Majorana modes at each end of the chain) to an another Hamiltonian which has only one Majorana mode in equilibrium; the MSP shows oscillations as a function of time with a noticeable decay in the amplitude. On the other hand for a quenching which is reverse to the previous one, the MSP decays rapidly and stays close to zero with fluctuations in amplitude.