Kähler geometry and Chern insulators: Relations between topology and the quantum metric

Abstract

We study Chern insulators from the point of view of Kähler geometry, i.e. the geometry of smooth manifolds equipped with a compatible triple consisting of a symplectic form, an integrable almost complex structure and a Riemannian metric. The Fermi projector, i.e. the projector onto the occupied bands, provides a map to a Kähler manifold. The quantum metric and Berry curvature of the occupied bands are then related to the Riemannian metric and symplectic form, respectively, on the target space of quantum states. We find that the minimal volume of a parameter space with respect to the quantum metric is π|C|, where C is the first Chern number. We determine the conditions under which the minimal volume is achieved both for the Brillouin zone and the twistangle space. The minimal volume of the Brillouin zone, provided the quantum metric is everywhere non-degenerate, is achieved when the latter is endowed with the structure of a Kähler manifold inherited from the one of the space of quantum states. If the quantum volume of the twist-angle torus is minimal, then both parameter spaces have the structure of a Kähler manifold inherited from the space of quantum states. These conditions turn out to be related to the stability of fractional Chern insulators. For two-band systems, the volume of the Brillouin zone is naturally minimal provided the Berry curvature is everywhere non-negative or non-positive, and we additionally show how the latter, which in this case is proportional to the quantum volume form, necessarily has zeroes due to topological constraints.

Topological Lattice Models with Constant Berry Curvature

“…In this section we prove a no-go theorem, showing that in a two-band model the fluctuations of the Berry curvature have a finite lower bound. This result has also been proved recently in the case of a single site per unit cell [35]. Here we give a more detailed proof and generalize the statement to systems where the unit cell has spatial structure so that the Bloch Hamiltonian is not necessarily periodic in reciprocal lattice vectors.…”

Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric

“…[89]. We will show its importance in the band topology of pH phases by generalizing some of the results previously presented in the Hermitian cases [50,[90][91][92][93][94][95][96][97][98][99][100][101]. Importantly, we also show that the NH skin effects will appear after breaking the pseudo-Hermiticity in the original pH models [102].…”

Universal semiclassical equations based on the quantum metric

“…The angle η here can be obtained from the wavevector change rate dk x /dt using the fact that any non-zero change of the parameters of the Hamiltonian leads to a finite non-adiabaticity described by η and given by the quantum metric along the wave vector evolution direction [30]: valid for all 2-band Hamiltonians with a Berry cuvature of a constant sign [26]. The two approaches indeed give the same contribution to the transverse velocity.…”

“…The potential roles of the QM have been more recently underlined in the calculations in quantum informatics, quantum phase transitions [16], magnetic susceptibility [17,18], excitonic levels [19], superfluidity in flat bands [20,21]. The QM is now explicitly accounted for in the design and engineering of topological systems [22,23], and its integral is linked with the Chern number [18,[24][25][26]. Experimental measurements of the QM in different systems also start to appear [27][28][29].…”

“…The value of the QM at its extremum has a very fundamental nature. Since the integral of the QM is often quantized, representing a topological invariant [26] similar to the Chern number, this maximal value also determines the extension of the metric in the parameter space, that is, the characteristic scale at which the changes occur (for example, level crossing). It determines both the maximal amplitude of the ZBW oscillations and of the anomalous Hall drift (even though the latter is an integral quantity).…”