Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers

TL;DR

This paper establishes exact relations between the quantum metric and topological invariants like Chern and winding numbers in Dirac Hamiltonians, revealing how quantum geometry dictates topological properties across various quantum systems.

Contribution

It introduces a general theoretical framework linking the quantum metric to topological invariants in Dirac Hamiltonians, applicable to diverse topological materials.

Findings

Topological indices are bounded by the quantum volume from the quantum metric.

The framework applies to topological insulators and semimetals in any dimension.

Clarifies the role of the Fubini-Study metric in topological states.

Abstract

Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and many-body quantum physics. In this context, two fundamental geometric quantities are known to play complementary roles: the Fubini-Study metric, which introduces a notion of distance between quantum states defined over a parameter space, and the Berry curvature associated with Berry-phase effects and topological band structures. In fact, recent studies have revealed direct relations between these two important quantities, suggesting that topological properties can, in special cases, be deduced from the quantum metric. In this work, we establish general and exact relations between the quantum metric and the topological invariants of generic Dirac Hamiltonians. In particular, we demonstrate that topological indices (Chern numbers or winding numbers) are bounded by the quantum volume determined by the quantum metric. Our theoretical framework, which builds on the Clifford algebra of Dirac matrices, is applicable to topological insulators and semimetals of arbitrary spatial dimensions, with or without chiral symmetry. This work clarifies the role of the Fubini-Study metric in topological states of matter, suggesting unexplored topological responses and metrological applications in a broad class of quantum-engineered systems.