Scalar Curvature of the Quantum Exponential Family for the Transverse-Field Ising Model and the Quantum Phase Transition

TL;DR

This paper analytically investigates the scalar curvature of the quantum exponential family in the transverse-field Ising model, revealing its divergence at low temperatures and its relation to quantum criticality.

Contribution

It provides the first analytical study of scalar curvature in quantum exponential families, connecting geometric divergence to quantum phase transitions.

Findings

Scalar curvature converges to zero at high temperatures.

Scalar curvature diverges exponentially at low temperatures.

Critical behavior with exponent 1 observed at zero temperature.

Abstract

Unlike for classical many-body systems, the scalar curvature of the exponential family for quantum many-body systems has been not so investigated, and its physical meaning remains unclear. In this paper, we analytically study the scalar curvature of the space of Gibbs thermal states, belonging to the quantum exponential family, equipped with the Bogoliubov-Kubo-Mori metric for the zero- and one-dimensional transverse-field Ising model at low and high temperatures. We find that these scalar curvatures converge to zero in the high-temperature limit whereas they exponentially diverge approaching zero temperature. This divergence is a consequence of quantumness. Furthermore, if we can reconsider the criticality of the scalar curvatures at zero temperature, they both can be considered to show a critical behavior with an exponent of 1, and this critical exponent is consistent with the quantum-classical correspondence of the Ising model.