Universal signatures of Majorana zero modes in critical Kitaev chains

TL;DR

This paper uncovers universal signatures of Majorana zero modes in the critical Kitaev chain, revealing invariant topological markers and exact analytical results across the entire critical line, advancing understanding of topological quantum matter.

Contribution

It introduces two topological markers that exhibit non-trivial signatures over the entire (1+1) Ising critical line, providing exact analytical results for MZM fidelity and occupation number.

Findings

MZM fidelity is a universal constant along the critical line.

Exact analytical expressions for MZM occupation numbers depend on Catalan's constant.

Finite-size corrections vanish at a specific ratio of pairing to hopping.

Abstract

Many topological or critical aspects of the Kitaev chain are well known, with several classic results. In contrast, the study of the critical behavior of the strong Majorana zero modes (MZM) has been overlooked. Here we introduce two topological markers which, surprisingly, exhibit non-trivial signatures over the entire (1+1) Ising critical line. We first analytically compute the MZM fidelity ${\cal{F}}_{\rm MZM}$--a measure of the MZM mapping between parity sectors. It takes a universal value along the (1+1) Ising critical line, ${\cal{F}}_{\rm MZM}=\sqrt{8}/\pi$, independent of the energy. We also obtain an exact analytical result for the critical MZM occupation number ${{\cal N}}_{\rm MZM}$ which depends on the Catalan's constant ${\cal G}\approx 0.91596559$, for both the ground-state (${{\cal N}}_{\rm MZM}=1/2-4{\cal{G}}/\pi^2\approx 0.12877$) and the first excited state (${{\cal N}}_{\rm MZM}=1/2+(8-4{\cal{G}})/\pi^2\approx 0.93934$). We further compute finite-size corrections which identically vanish for the special ratio $\Delta/t=\sqrt{2}-1$ between pairing and hopping in the critical Kitaev chain.