TY - JOUR

T1 - Relating the topology of Dirac Hamiltonians to quantum geometry

T2 - When the quantum metric dictates Chern numbers and winding numbers

AU - Mera, Bruno

AU - Zhang, Anwei

AU - Goldman, Nathan

N1 - Publisher Copyright:
Copyright B. Mera et al.

PY - 2022/1

Y1 - 2022/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85124082110&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85124082110&partnerID=8YFLogxK

U2 - 10.21468/SciPostPhys.12.1.018

DO - 10.21468/SciPostPhys.12.1.018

M3 - Article

AN - SCOPUS:85124082110

SN - 2542-4653

VL - 12

JO - SciPost Physics

JF - SciPost Physics

IS - 1

M1 - 018

ER -