Abstract

We study Chern insulators from the point of view of K\"ahler 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 K\"ahler 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 $\pi |\mathcal{C}|$, where $\mathcal{C}$ is the first Chern number. We determine the conditions under which the minimal volume is achieved both for the Brillouin zone and the twist-angle 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 K\"ahler 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 K\"ahler 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 nonpositive, and we additionally show how the latter, which in this case is proportional to the quantum volume form, necessarily has zeros due to topological constraints.

Highlights

  • The notion of topological phases has drastically changed our understanding of gapped phases of matter

  • In this paper we have studied the geometry of Chern insulators using the natural Kähler geometry of the space of orthogonal projectors of a given rank, the latter being associated with the number of occupied bands

  • We have established important results, namely Theorem 1, which renders the saturation of the Cauchy-Schwarz-like inequality equivalent to a Kähler map condition for the Fermi projector, and Theorem 2, which determines the conditions of minimal quantum volume both in the Brillouin zone and in the twist-angle space

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Summary

INTRODUCTION

The notion of topological phases has drastically changed our understanding of gapped phases of matter. In the particular case of two spatial dimensions, in the absence of other symmetries, the occupied bands may have a nontrivial topological twist determining what is called a Chern insulator. The minimal volume of the Brillouin zone, provided the quantum metric is everywhere nondegenerate, is achieved when the latter is endowed with the structure of a Kähler manifold induced from the one of the space of quantum states. If the quantum volume of the twist-angle torus is minimal, both tori have the structure of a Kähler manifold inherited from the space of quantum states These conditions are related to the geometric stability conditions for fractional Chern insulators presented in Refs.

GEOMETRY OF CHERN INSULATORS
MAIN RESULTS
PHYSICAL INTERPRETATION OF THE KÄHLER MAP CONDITION
CONCLUSIONS
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