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arXiv:1807.05758 · Journal of Physics A Mathematical and Theoretical

Topological phase transitions in random Kitaev α -chains

Cécile Monthus

Published Oct 17, 2018• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/aae5db

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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 a periodically driven one-dimensional Kitaev model in the presence of disorder. In the clean limit our model exhibits four topological phases corresponding to the existence or nonexistence of edge modes at zero and $\ensuremath{\pi}$ quasienergy. When potential disorder is added, the system parameters get renormalized and the system may exhibit a topological phase transition. When starting from the Majorana $\ensuremath{\pi}$ mode (MPM) phase, which hosts only edge Majoranas with quasienergy $\ensuremath{\pi}$, disorder induces a transition into a neighboring phase with both $\ensuremath{\pi}$ and zero modes on the edges. We characterize the disordered system using (i) exact diagonalization, (ii) Arnoldi mapping onto an effective tight-binding chain, and (iii) topological entanglement entropy.
The topological properties of the Kitaev chain model with extended-range interactions are investigated, focusing on cases where the topological winding number is preserved. We assume that the pairing and hopping terms decay algebraically in space with exponents ααα and βββ, respectively. We show that in the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. In addition, an explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. Besides the analytical description, we introduce a new physical insight into the topological excitations of the ground-state by measuring the spatial distribution of the edge modes with the Majorana average position. Taking the next-nearest neighbor Kitaev chains as a probe, various numerical calculations of Majorana edge states were performed in finite-size clusters to determine the sensitivity of the topological zero energy modes to the parameters of interest. The occupation of edge-to-edge non-local fermion states is computed and defined as an effective parity. Such an effective parity exhibits new interesting features beyond the energetic exchange from the ground-state fermion parity switches, which are related to the distribution of the respective edge modes. Our calculations show a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position.
We study the competition of disorder and superconductivity for a generalized Kitaev model in incommensurate potentials. The generalized Kitaev model describes one dimensional spinless fermions with long-range p-wave superconducting pairing, which decays with distance l as a power law ∼[Formula: see text]. We focus on the transition from the topological superconducting phase to the topologically trivial Anderson localized phase, and effects of the exponent α on this phase transition. In the topological superconducting phase, for a system under open boundary condition the amplitude of zero-mode Majorana fermion has a hybrid exponential-algebraic decay as the distance increases from the edge. In the Anderson localized phase, some single-particle states remain critical for very strong disorders and the number of critical states increases as α decreases. In addition, except for critical disorders, the correlation function always has an exponential decay at the short range and an algebraic decay at the long range. Phase transition points are also numerically determined and the topological phase transition happens earlier at a smaller disorder strength for a system with smaller α.
In this work, we study the topological phase transitions of a Kitaev chain generalized by the addition of nearest-neighbor Coulomb interaction. We show the presence of a robust topological phase as a function of the interaction strength and of the on-site energy with associated non-zero energy Majorana states localized at the chain edges. We provide an effective mean-field model that allows for the self-consistent computation of the mean value of the local particle number operator, and we also perform Density Matrix Renormalization Group numerical simulations based on a tensor network approach. We find that the two methods show a good agreement in reporting the phase transition between trivial and topological superconductivity. Temperature robustness within a physically relevant threshold has also been demonstrated. These findings shed light on an entire class of topological interacting one-dimensional systems in which the effects of residual Coulomb interactions play a relevant role.
We study the effects of disorder on a Kitaev chain with longer-range hopping and pairing terms which is capable of forming local zero energy excitations and, hence, serves as a minimal model for localization-protected edge qubits. The clean phase diagram hosts regions with 0, 1, and 2 Majorana zero-modes (MZMs) per edge. Using a semianalytic approach corroborated by numerical calculations of the entanglement degeneracy, we show how phase boundaries evolve under the influence of disorder. While in general the 2 MZM region is stable with respect to moderate disorder, stronger values drive transition towards the topologically trivial phase. We uncover regions where the addition of disorder induces local zero-modes absent for the corresponding clean system. Interestingly, we discover that disorder destroys any direct transition between phases with zero and two MZMs by creating a tricritical point at the 2-0 MZM boundary of the clean system. Finally, motivated by recent experiments, we calculate the characteristic signatures of the disorder phase diagram as measured in dynamical local and nonlocal ``qubit'' correlation functions. Our work provides a minimal starting point to investigate the coherence properties of local qubits in the presence of disorder.
Topological edge states are celebrated for their robustness against disorder, yet the interplay between disorder and system size remains poorly understood. We use large deviations theory as a framework to study finite-size effects beyond the central limit theorem. We analyze Lyapunov exponent fluctuations in the static and periodically driven disordered Kitaev chain and find an asymmetry in the large deviations statistics that makes stronger edge localizations of Majorana zero modes exponentially more likely than weaker ones. We demonstrate that this fluctuation asymmetry is not tied to the topological phase. This asymmetry endows topological edge states with an additional protection against disorder and persists across a broad class of disorder distribution. We show how to use our framework to find the minimum system size required to satisfy topological quantum computing constraints.