Spread Complexity and Topological Transitions in the Kitaev Chain

TL;DR

This paper demonstrates that spread complexity can effectively distinguish between trivial and topological phases in the Kitaev chain, serving as a diagnostic tool for quantum phase transitions in topological matter.

Contribution

It shows that Krylov complexity detects topological phase transitions in the Kitaev chain, providing a new diagnostic approach for quantum criticality in topological systems.

Findings

Krylov complexity differentiates between trivial and topological phases.

Spread complexity signals the quantum critical point.

Discussion on ambiguity of complexity measures in topological transitions.

Abstract

A number of recent works have argued that quantum complexity, a well-known concept in computer science that has re-emerged recently in the context of the physics of black holes, may be used as an efficient probe of novel phenomena such as quantum chaos and even quantum phase transitions. In this article, we provide further support for the latter, using a 1-dimensional p-wave superconductor - the Kitaev chain - as a prototype of a system displaying a topological phase transition. The Hamiltonian of the Kitaev chain manifests two gapped phases of matter with fermion parity symmetry; a trivial strongly-coupled phase and a topologically non-trivial, weakly-coupled phase with Majorana zero-modes. We show that Krylov-complexity (or, more precisely, the associated spread-complexity) is able to distinguish between the two and provides a diagnostic of the quantum critical point that separates them. We also comment on some possible ambiguity in the existing literature on the sensitivity of different measures of complexity to topological phase transitions.