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

TL;DR

This paper investigates how a phase gradient in the pairing term of a one-dimensional Kitaev model influences its phase diagram, revealing a ground state with off-diagonal long-range order and topological phases characterized by Majorana modes.

Contribution

It introduces the effect of a uniform phase gradient on the Kitaev chain, showing the emergence of ODLRO in the ground state and analyzing phase transitions with an exact solution.

Findings

Phase diagram includes trivial and nontrivial topological phases.

Maxima of the order parameter occur at triple critical points.

Ground state exhibits exact off-diagonal long-range order at critical points.

Abstract

We study a one-dimensional Kitaev model with uniform phase gradient pairing term. We show that the gradient constant dramatically 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 (BCS)-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.