Majorana Zero Modes and Bulk-Boundary Correspondence at Quantum Criticality

TL;DR

This paper explores the emergence of Majorana zero modes at quantum critical points in a one-dimensional topological superconductor with long-range interactions, revealing a novel bulk-boundary correspondence at criticality.

Contribution

It introduces a method to identify stable Majorana zero modes at criticality and proposes a scheme to separate topological invariants into fractional and integer parts for bulk-boundary correspondence.

Findings

Stable Majorana zero modes appear at criticality under certain conditions.

Topological invariant zeros validate the presence of Majorana zero modes at critical points.

A new scheme separates invariants into fractional and integer parts to establish bulk-boundary correspondence.

Abstract

Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Majorana zero modes at criticality. Trivial and non-trivial topological nature of criticality due to the presence of multicritical point cause an unusual topological transition along the critical line. We observe and investigate this unique transition in terms of eigenvalue spectrum. Appearance of MZMs at criticality demands integer value of topological invariant number in order to validate the concept of bulk-boundary correspondence. Hence we propose a scheme to separate the invariant number into fractional and integer contribution to establish bulk-boundary correspondence at criticality.