Kitaev ring threaded by a magnetic flux: Topological gap, Anderson localization of quasiparticles, and divergence of supercurrent derivative

Abstract

We study a superconducting Kitaev ring pierced by a magnetic flux, with and without disorder, in a quantum ring configuration, and in a rf-SQUID one, where a weak link is present. In the rf-SQUID configuration, in the topological phase, the supercurrent shows jumps at specific values of the flux ${\Phi }^{\ast }=\frac{hc}{e}(1/4+n)$, with $n\in \mathbb{N}$. In the thermodynamic limit ${\Phi }^{\ast }$ is constant inside the topological phase, independently of disorder, and we analytically predict this fact using a perturbative approach in the weak-link coupling. The weak link breaks the topological ground-state degeneracy, and opens a spectral gap for $\Phi \ne {\Phi }^{\ast }$, that vanishes at ${\Phi }^{\ast }$ with a cusp providing the current jump. Looking at the quasiparticle excitations, we see that they are Anderson localized, so they cannot carry a resistive contribution to the current, and the localization length shows a peculiar behavior at a flat-band point for the quasiparticles. In the absence of disorder, we analytically and numerically find that the chemical-potential derivative of the supercurrent logarithmically diverges at the topological-to-trivial transition, in agreement with the transition being of the second order.

arXiv comment16 pages, 17 figures. Substantial rewriting for more clarity and conciseness. More precise analysis of the current-jump closure at the transition in presence of disorder, and of the localization length of quasiparticle. Version published in PRB

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