Quantum geometry in many-body systems with precursors of criticality

TL;DR

This paper investigates the geometric structure of ground states in many-body quantum systems near phase transitions, highlighting finite-size precursors, the role of diabolic points, and limitations of common approximations, using a Lipkin-Meshkov-Glick model.

Contribution

It provides a detailed analysis of quantum geometry near criticality, revealing the significance of diabolic points and demonstrating the inadequacy of standard approximations in describing geometric features.

Findings

Finite-size systems show precursors of geometric singularities at QPT boundaries.

Diabolic points are key in first-order QPT formation and cause irregular geodesic behavior.

Standard mean field and two-level approximations fail to accurately capture the geometric structure.

Abstract

We analyze the geometry of the ground-state manifold in parameter-dependent many-body systems with quantum phase transitions (QPTs) and describe finite-size precursors of the singular geometry emerging at the QPT boundary in the infinite-size limit. In particular, we elucidate the role of diabolic points in the formation of first-order QPTs, showing that these isolated geometric singularities represent seeds generating irregular behavior of geodesics in finite systems. We also demonstrate that established approximations, namely the mean field approximation in many-body systems composed of mutually interacting bosons and the two-level approximation near a diabolic point, are insufficient to provide a reliable description of geometry. The outcomes of the general analysis are tested and illustrated by a specific bosonic model from the Lipkin-Meshkov-Glick family.