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arXiv:1804.10908 · Physical review. B./Physical review. B

Disorder protected and induced local zero-modes in longer-range Kitaev chains

Simon Lieu, Derek K. K. Lee, Johannes Knolle

Published Oct 17, 2018• 28 citations

Abstract

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.

DOI: 10.1103/physrevb.98.134507

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We study the combined effects of disorder and range of the couplings on the phase diagram of one-dimensional topological superconductors. We consider an extended version of the Kitaev chain where hopping and pairing terms couple many sites. Deriving the conditions for the existence of Majorana zero modes, we show that either the range and the on-site disorder can greatly enhance the topological phases characterized by the appearance of one or many Majorana modes localized at the edges. We consider both a discrete and a continuous disorder distribution. Moreover, we discuss the role of correlated disorder which might further widen the topological regions. Finally, we show that in the purely long-range regime and in the presence of disorder, the spatial decay of the edge modes remains either algebraic or exponential, with eventually a modified localization length, as in the absence of disorder.

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In recent years, tools from quantum information theory have become indispensable in characterizing many-body systems. In this work, we employ measures of entanglement to study the interplay between disorder and the topological phase in 1D systems of the Kitaev type, which can host Majorana end modes at their edges. We find that the entanglement entropy may actually increase as a result of disorder, and identify the origin of this behavior in the appearance of an infinite-disorder critical point. We also employ the entanglement spectrum to accurately determine the phase diagram of the system, and find that disorder may enhance the topological phase, and lead to the appearance of Majorana zero modes in systems whose clean version is trivial.

Over the recent years tools from quantum information theory have become indispensable in characterizing many body systems. In this work we employ measures of entanglement to study the interplay between disorder and topology in 1D systems of the Kitaev type, which can host Majorana end modes at their edges. We find that the entanglement entropy may actually increase as a result of disorder, and identify the origin of this behavior in disorder-induced quantum criticality. We also employ the entanglement spectrum to accurately determine the phase diagram of the system, and find that disorder may enhance topology, and lead to the appearance of Majorana zero modes in systems whose clean version is trivial.

Abstract In the original Kitaev chain, Majorana zero modes (MZMs) at opposite edges have the same localization length. Here it is shown that a Hermitian finite-range extended Kitaev chain can instead support asymmetrically localized MZMs with different left and right localization lengths. Such asymmetry is excluded when all pairing amplitudes share a common phase, so the effect requires a genuinely extended model beyond the nearest-neighbor Kitaev chain. For the minimal next-nearest-neighbor case, the asymmetry originates from the inequivalence of the left and right boundary equations, which leads to different selections of decaying bulk solutions and yields a necessary-and-sufficient condition for its occurrence. A transfer-matrix reformulation further shows that the asymmetry is robust against sufficiently weak disorder preserving the block-triangular structure. For finite chains, the hybridization-induced energy splitting is exponentially small and controlled by the shorter localization length.