Zero modes of the Kitaev chain with phase-gradients and longer range couplings

Abstract

We present an analytical solution for the full spectrum of Kitaevʼs one-dimensional p-wave superconductor with arbitrary hopping, pairing amplitude and chemical potential in the case of an open chain. We also discuss the structure of the zero-modes in the presence of both phase gradients and next nearest neighbor hopping and pairing terms. As observed by Sticlet et al, one feature of such models is that in a part of the phase diagram, zero-modes are present at one end of the system, while there are none on the other side. We explain the presence of this feature analytically, and show that it requires some fine-tuning of the parameters in the model. Thus as expected, these 'one-sided' zeromodes are neither protected by topology, nor by symmetry. IntroductionOne of the characteristic features of many topological phases is the presence of gapless boundary modes. The (fractional) quantum Hall states are a prime example [1-3], and their boundary modes provide strong evidence of the topological nature of these states. Another prime example is the Kitaev chain, whose topological p-wave superconducting phase features so-called 'Majorana zero modes' at its edges [4]. Trying to establish the existence of the topological phase is often done by trying to establish the presence of the boundary modes. This has led to strong evidence for the topological phase in for instance strongly spin-orbit coupled nano-wires that are proximity coupled to an s-wave superconductor in the presence of a magnetic field [5-9], or in chains of magnetic ad-atoms [10][11][12][13]. It has been proposed that the zero energy Majorana bound states can be used as topologically protected q-bits, for quantum information processing purposes [14,15]. By now, there exist various proposals to manipulate these q-bits, either in T-junction systems, in which the Majorana bound states can be braided explicitly [16], or in Josephson coupled Kitaev chains, in which the coupling of the various chains allows operation on the q-bits [17].Despite the intense research on the Kitaev chain models, there are still interesting features that deserve attention. In this paper, we look into one of them. It was observed by Sticlet et al [18], that the zero-modes of Kitaev chains carrying a current, i.e., in the presence of a gradient in the phase of the order parameter, have interesting properties. The most striking feature is that is it possible that at one edge of the chain, there is pair of Majorana bound states (or better, one 'ordinary' Dirac zero mode), while there is no zero mode at the other end of the chain. Clearly, from a topological point of view, this means that the chain is in a trivial phase, but it is nevertheless worthwhile to investigate these zero-modes further. In this paper, we explain the presence of these zero-modes, via an exact solution of the zero modes of an extended Kitaev chain, i.e., in the presence of both complex and next nearest-neighbor hopping an pairing terms. We show that it is necessary to fine tune the couplings in order that these 'one-sid...

Topological quantum phase transitions and criticality in a longer-range Kitaev chain

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Critical region of topological trivial and nontrivial phases in interacting Kitaev chain with spatially varying potentials

“…Driven by the demand of applications, researchers devoted great effort to the stability of the topological phase in the Kitaev chain with nearest-neighboring repulsive interactions [28][29][30]. Along this line, theoretical studies also extended to issues of dimerization [31], disorder [32][33][34][35][36][37], quasi-periodicity [35,38], long-range interactions [39][40][41], quartic interactions [42], and so on. For the interacting model without disorders [43][44][45][46], both numerical and perturbative investigations showed that MZMs can be stably present with moderate interactions [43][44][45][47][48][49], which was also found to generally broaden * Electronic address: yaoyao2016@scut.edu.cn the window of chemical potential where the system is in the nontrivial topological superconducting (TSC) phase [43-45, 47, 48].…”

Boundaries of boundaries: A systematic approach to lattice models with solvable boundary states of arbitrary codimension

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