Vanishing k-space fidelity and phase diagram's bulk-edge-bulk correspondence

TL;DR

This paper demonstrates that the k-space fidelity between states deep inside different quantum phases vanishes at gapless points, linking fidelity to phase boundaries and providing a unified framework for various topological and condensed matter models.

Contribution

It establishes a general connection between fidelity vanishing and gapless points in phase diagrams, extending the understanding of quantum phase transitions across multiple models.

Findings

Fidelity vanishes at gapless points in various models

Sufficient conditions for gapless points based on fidelity

Explicit counter-example showing conditions are not necessary

Abstract

The fidelity between two infinitesimally close states or the fidelity susceptibility of a system are known to detect quantum phase transitions. Here we show that the k-space fidelity between two states far from each other and taken deep inside (bulk) of two phase s, generically vanishes at the k-points where there are gapless points in the energy spectrum that give origin to the lines (edges) separating the phases in the phase diagram. We consider a general case of two-band models and present a sufficient condition for the existence of gapless points, given there are pairs of parameter points for which the fidelity between the corresponding states is zero. By presenting an explicit counter-example, we showed that the sufficient condition is not necessary. Further, we showed that, unless the set of parameter points is suitably constrained, the existence of gapless points generically imply the accompanied pairs of parameter points with vanishing fidelity. Also, we showed the connection between the vanishing fidelity and gapless points on a number of concrete examples (topological triplet superconductor, topological insulator, 1d Kitaev model of spinless fermions, BCS superconductor, Ising model in a transverse field, graphene and Haldane Chern insulator), as well as for the more general case of Dirac-like Hamiltonians. We also briefly discuss the relation between the vanishing fidelity and gapless points at finite temperatures.