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

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 phases, 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.

Entanglement and Fidelity: Statics and Dynamics

“…This typically occurs as we vary some parameter across a critical point (gapless point in energy space) which leads to a vanishing contribution to the fidelity (see Ref. [156] for a discussion of the necessary and sufficient conditions to associate a vanishing k-space fidelity with the crossing of one or several quantum critical points along a certain line in the phase diagram). The condition for gapless points is…”

“…Considering for instance the Sato and Fujimoto model [106] simplified to a topologically equivalent case with no s-wave pairing and no spin-orbit coupling, the transitions between the various phases occur at the momentum points k = (0, 0), (0, π), (π, π) (and equivalent points). At k = (0, 0), and similarly for other momentum values [156], the gapless condition implies that 4t + µ + M z = 0, where t is the hopping, µ is the chemical potential, and M z is the magnetization (or external magnetic field). When…”

“…As we discussed above, in equilibrium, an efficient way to detect a quantum phase transition between different phases is to use the fidelity [12,14]. Furthermore, the transition line may be detected considering the overlap between states that correspond to points that are far from the transition line [155,156] or considering the 2D fidelity map on overlap between states at different driving parameters [179].…”

Interferometric geometry from symmetry-broken Uhlmann gauge group with applications to topological phase transitions

“…Therefore, it is crucial to understand the effect of temperature on topological phase transitions, specially with regards to applications to quantum computers, such as those involving Majorana modes in topological superconductors [20]. To approach this problem, the fidelity and the associated Bures metric and, in addition, the Uhlmann connection, the generalization of the Berry connection to the case of mixed states, have been probed for systems that exhibit zero temperature symmetry protected topological phases [21][22][23][24][25].…”

Information Geometry in the Analysis of Phase Transitions

“…Fidelity is an information theoretical quantity which has been widely used in the study of phase transitions [9][10][11][12]. Recently, the problem of the effect of temperature on topological phase transitions has been addressed.…”