Quantum critical scaling of the geometric tensors

TL;DR

This paper unifies geometric approaches to quantum phase transitions by analyzing the critical behavior of a complex tensor, revealing conditions for divergence and validating results with exact diagonalization.

Contribution

It demonstrates that quantum geometric tensor singularities underpin phase transitions and clarifies when superextensive divergence occurs, extending previous understanding.

Findings

Critical singular behavior of the quantum geometric tensor near phase transitions.

Conditions under which the tensor divergence becomes superextensive.

Validation of theoretical analysis through exact diagonalization of the XXZ chain.

Abstract

Berry phases and the quantum-information theoretic notion of fidelity have been recently used to analyze quantum phase transitions from a geometrical perspective. In this paper we unify these two approaches showing that the underlying mechanism is the critical singular behavior of a complex tensor over the Hamiltonian parameter space. This is achieved by performing a scaling analysis of this quantum geometric tensor in the vicinity of the critical points. In this way most of the previous results are understood on general grounds and new ones are found. We show that criticality is not a sufficient condition to ensure superextensive divergence of the geometric tensor, and state the conditions under which this is possible. The validity of this analysis is further checked by exact diagonalization of the spin-1/2 XXZ Heisenberg chain.