SciPost Phys. 12, 018 (2022) ·
published 12 January 2022
|
· pdf
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.
SciPost Phys. 10, 112 (2021) ·
published 20 May 2021
|
· pdf
We introduce a scheme by which flat bands with higher Chern number $\vert
C\vert>1$ can be designed in ultracold gases through a coherent manipulation of
Bloch bands. Inspired by quantum-optics methods, our approach consists in
creating a "dark Bloch band" by coupling a set of source bands through resonant
processes. Considering a $\Lambda$ system of three bands, the Chern number of
the dark band is found to follow a simple sum rule in terms of the Chern
numbers of the source bands: $C_D\!=\!C_1+C_2-C_3$. Altogether, our dark-state
scheme realizes a nearly flat Bloch band with predictable and tunable Chern
number $C_D$. We illustrate our method based on a $\Lambda$ system, formed of
the bands of the Harper-Hofstadter model, which leads to a nearly flat Chern
band with $C_D\!=\!2$. We explore a realistic sequence to load atoms into the
dark Chern band, as well as a probing scheme based on Hall drift measurements.
Dark Chern bands offer a practical platform where exotic fractional quantum
Hall states could be realized in ultracold gases.
