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arXiv:2212.02694 · Physical review. B./Physical review. B

Off-diagonal long-range order in the ground state of the Kitaev chain

E. S. Ma, Z. Song

Published May 10, 2023• 9 citations
arXivHTMLPDF

Abstract

We study a one-dimensional Kitaev model with uniform phase gradient pairing term. We show that the gradient constant significantly affects the phase diagram, which consists of topologically trivial and nontrivial phases, associated with Majorana edge modes. Based on the exact solution, a Bardeen-Cooper-Schrieffer-pair order parameter is introduced to characterize the phase diagram by its value and nonanalytic behavior at phase boundaries. We find that this order parameter obtains its maxima at the triple critical points, at which the pairing phase gradient suppresses the single-particle scattering process due to the coherent destructive interference. In particular, we show that the ground state at such a point possesses exact off-diagonal long-range order (ODLRO), in the thermodynamic limit. Our result provides an example of a gapless $p$-wave superconducting ground state possessing ODLRO.

DOI
10.1103/physrevb.107.205117

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We study a one-dimensional Kitaev model with uniform phase gradient pairing term. We show that the gradient constant significantly affects the phase diagram, which consists of topologically trivial and nontrivial phases, associated with Majorana edge modes. Based on the exact solution, a Bardeen-Cooper-Schrieffer-pair order parameter is introduced to characterize the phase diagram by its value and nonanalytic behavior at phase boundaries. We find that this order parameter obtains its maxima at the triple critical points, at which the pairing phase gradient suppresses the single-particle scattering process due to the coherent destructive interference. In particular, we show that the ground state at such a point possesses exact off-diagonal long-range order (ODLRO), in the thermodynamic limit. Our result provides an example of a gapless ppp-wave superconducting ground state possessing ODLRO.
In this paper, we have considered the extended version of the Kitaev model in one dimension, i.e., a long-range p-wave superconducting wire. In the long-range Kitaev chain, the superconducting hopping and pairing terms in the Hamiltonian decay, independently, in a power-law fashion , where l is the distance between the two sites and x is some positive constant. We have studied the appearance of Majorana zero-energy edge modes and also, massive Dirac edge modes by exact diagonalization, as well as analytical computations. Exact diagonalization indicates the existence of both kinds of massless and massive edge modes in the energy spectrum. Furthermore, we obtain the phase diagram and the topological phase transitions by calculating the winding number, which is the topological invariant.
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.
We introduce a frustration-free, one-dimensional model of spinless fermions with hopping, ppp-wave superconducting pairing and alternating chemical potentials. The model possesses two exactly degenerate ground states even for finite system sizes. We present analytical results for the strong Majorana zero modes, the phase diagram, and the topological order. Furthermore, we generalize our results to include interactions.
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We present a detailed study of the topological properties of the Kitaev chain with long-range pairing terms and in the presence of an Aubry-André-Harper on-site potential.Specifically, we consider algebraically decaying superconducting pairing amplitudes; the exponent of this decay is found to determine a critical pairing strength, below which the chain remains topologically trivial.Above the critical pairing, topological edge modes are observed in the central gap.For sufficiently fast decay of the pairing, these modes are identified as Majorana zero modes.However, if the pairing term decays slowly, the modes become massive Dirac modes.Interestingly, these massive modes still exhibit a true level crossing at zero energy, which points towards an intimate relation to Majorana physics.We also observe a clear lack of bulk-boundary correspondence in the long-range system, where bulk topological invariants remain constant, while dramatic changes appear in the behavior at the edge of the system.In addition to the central gap around zero energy, the Aubry-André-Harper potential also leads to other energy gaps at nonzero energy.As for the analogous short-range model, the edge modes in these gaps can be characterized through a 2D Chern invariant.However, in contrast to the short-range model, this topological invariant does not correspond to the number of edge mode crossings anymore.This provides another example of the weakening of the bulk-boundary correspondence occurring in this model.Finally, we discuss possible realizations of the model with ultracold atoms and condensed matter systems.
Topological phases in superconductors are of great interest due to their potential use for topological quantum computing. They are usually characterized by quasiparticles located at the edges of the superconductor. We investigate how long-range interactions affect the properties of the well-known Kitaev chain, which is solved self-consistently with the mean-field approximation. We find that for weak interactions, the Majorana edge modes have finite energy that decay algebraically to zero with the long-range power law. Strong long-range interactions are a special case due to their slow decay. The solution of the thermodynamic limit without finite-size effects is a local minimum to this problem, but with a larger ground-state energy in contrast to the discussed solution. Our results show the difference between the self-consistent and the Kitaev approach, which has clear constraints on long-range interactions.