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 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.

A geometric perspective: experimental evaluation of the quantum Cramer-Rao bound

“…The above three-parameter model enables us to study both quasi-classical (γ = 0) and coherent (γ = 1) cases. That model, as well as the two-level model |ψ(θ, φ) = cos θ 2 |0 − sin θ 2 e −iφ |1 , is indeed one of the eigenstates of a gapless Weyl-type Hamiltonian that hosts monopoles at the origin of parameter space [27,29], leading to γ = 1 (or 0 at special points). To study intermediate cases where γ ≤ 1, one can consider different geometrical models [32] or, as done here, rely on the two-parameter subspaces of the above three-parameter model where γ is a function of α.…”

“…That model, as well as the two-level model |ψ(θ, φ) = cos θ 2 |0 − sin θ 2 e −iφ |1 , is indeed one of the eigenstates of a gapless Weyl-type Hamiltonian that hosts monopoles at the origin of parameter space [27,29], leading to γ = 1 (or 0 at special points). To study intermediate cases where γ ≤ 1, one can consider different geometrical models [32] or, as done here, rely on the two-parameter subspaces of the above three-parameter model where γ is a function of α. From the viewpoint of the Hamiltonian, this corresponds to taking a slice of the original gapless Weyl-type system, which yields a gapped topological insulator that can have γ ≤ 1.…”

“…This relation holds for any two-parameter space in a multiparameter setting. The above relation can also be derived from the fact that the QGT χ µν is a positive semidefinite matrix and has been explored in various models [25][26][27]. Reversely, generalization of the two-operator product uncertainty relations Eq.…”

Robust multipartite entanglement in dirty topological wires

“…A prominent tool to analyze ME is the quantum Fisher information (QFI) [57][58][59], which is also also central in quantum metrology [60,61]. The QFI has been investigated in topological [33,[62][63][64][65][66] as well as spin [67][68][69][70][71][72] and lattice [73,74] systems, and used to characterize interesting many-body phenomena such as quantum criticality [75][76][77], quantum chaos [78,79], quantum quenches [80,81], scrambling [82] and thermalization [83,84].…”

Experimental demonstration of topological bounds in quantum metrology

“…Besides, the imaginary part of this tensor corresponds to the Berry curvature, which plays a central role in quantum Halltype transport [39,40] and topological defects [41]. Surprisingly, inspired by the existence of correlations between the quantum metric and the Berry curvature, it has been recently suggested that the Berry curvature (and the related Chern numbers) can set bounds on quantum multi-parameter estima-tion [42,43]. Demonstrating the connection between topology and quantum metrology in experiments is highly appealing, however, accessing the limits of quantum multi-parameter estimation is a challenging task and an experimental verification has remained elusive.…”