Singularities in fidelity surfaces for quantum phase transitions: a geometric perspective

TL;DR

This paper explores how the geometric properties of fidelity surfaces, specifically Gaussian curvature, reveal quantum phase transitions and critical behavior in quantum lattice systems, exemplified by the transverse Ising model.

Contribution

It introduces a geometric perspective using Gaussian curvature to identify and analyze quantum phase transitions, including finite size scaling and critical exponent extraction.

Findings

Gaussian curvature becomes singular at transition points.

Finite size scaling confirms the critical exponent matches conformal invariance.

Fidelity surface geometry effectively detects quantum criticality.

Abstract

The fidelity per site between two ground states of a quantum lattice system corresponding to different values of the control parameter defines a surface embedded in a Euclidean space. The Gaussian curvature naturally quantifies quantum fluctuations that destroy orders at transition points. It turns out that quantum fluctuations wildly distort the fidelity surface near the transition points, at which the Gaussian curvature is singular in the thermodynamic limit. As a concrete example, the one-dimensional quantum Ising model in a transverse field is analyzed. We also perform a finite size scaling analysis for the transverse Ising model of finite sizes. The scaling behavior for the Gaussian curvature is numerically checked and the correlation length critical exponent is extracted, which is consistent with the conformal invariance at the critical point.