Classical impurities and boundary Majorana zero modes in quantum chains

TL;DR

This paper investigates how classical impurities influence boundary Majorana zero modes in quantum Ising chains, revealing their behavior across phase transitions and their connection to topological order.

Contribution

It provides a detailed analysis of impurity-induced Majorana modes, their evolution across phases, and their thermodynamic signatures as indicators of topological order.

Findings

Majorana modes remain decoupled at zero energy in finite systems.

Disordered phase shows Curie law in local susceptibility.

Topological phase exhibits continuous impurity response due to level repulsion.

Abstract

We study the response of classical impurities in quantum Ising chains. The Z2 degeneracy they entail renders the existence of two decoupled Majorana modes at zero energy an exact property of a finite system at arbitrary values of its bulk parameters. We trace the evolution of these modes across the transition from the disordered phase to the ordered one and analyze the concomitant qualitative changes of local magnetic properties of an isolated impurity. In the disordered phase, the two ground states differ only close to the impurity, and they are related by the action of an explicitly constructed quasi-local operator. In this phase the local transverse spin susceptibility follows a Curie law. The critical response of a boundary impurity is logarithmically divergent and maps to the two-channel Kondo problem, while it saturates for critical bulk impurities, as well as in the ordered phase. The results for the Ising chain translate to the related problem of a resonant level coupled to a 1d p-wave superconductor or a Peierls chain, whereby the magnetic order is mapped to topological order. We find that the topological phase always exhibits a continuous impurity response to local fields as a result of the level repulsion of local levels from the boundary Majorana zero mode. In contrast, the disordered phase generically features a discontinuous magnetization or charging response. This difference constitutes a general thermodynamic fingerprint of topological order in phases with a bulk gap.