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

Bruno Mera, Anwei Zhang, Nathan Goldman

SciPost Phys. 12, 018 (2022) · published 12 January 2022

- doi: 10.21468/SciPostPhys.12.1.018
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### 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.

### Cited by 21

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}^{2}Bruno Mera, -
^{3}^{4}Anwei Zhang, -
^{5}Nathan Goldman

^{1}Universidade de Lisboa / University of Lisbon^{2}Instituto de Telecomunicações [IT]^{3}University of Hong Kong [HKU]^{4}아주대학교 / Ajou University^{5}Université Libre de Bruxelles [ULB]

- European Research Council [ERC]
- Fonds De La Recherche Scientifique - FNRS (FNRS) (through Organization: Fonds National de la Recherche Scientifique [FNRS])
- Fundação para a Ciência e a Tecnologia (through Organization: Fundação para a Ciência e Tecnologia [FCT])