Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers
Abstract
Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and manybody quantum physics. In this context, two fundamental geometric quantities are known to play complementary roles: the FubiniStudy metric, which introduces a notion of distance between quantum states defined over a parameter space, and the Berry curvature associated with Berryphase 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 FubiniStudy metric in topological states of matter, suggesting unexplored topological responses and metrological applications in a broad class of quantumengineered systems.
 Publication:

SciPost Physics
 Pub Date:
 January 2022
 DOI:
 10.21468/SciPostPhys.12.1.018
 arXiv:
 arXiv:2106.00800
 Bibcode:
 2022ScPP...12...18M
 Keywords:

 Condensed Matter  Mesoscale and Nanoscale Physics;
 Condensed Matter  Quantum Gases;
 Quantum Physics
 EPrint:
 25 pages including 5 figures and references. Revised manuscript, which includes a discussion about metrological applications. Manuscript prepared for SciPost submission