TY - JOUR

T1 - Kähler geometry and Chern insulators

T2 - Relations between topology and the quantum metric KÄHLER GEOMETRY and CHERN INSULATORS: ... BRUNO MERA and TOMOKI OZAWA

AU - Mera, Bruno

AU - Ozawa, Tomoki

N1 - Funding Information:
B.M. acknowledges very stimulating discussions with J. P. Nunes and J. M. Mourão. B.M. and T.O. acknowledge the anonymous referees for their insightful comments, which lead to substantial improvement of this paper. T.O. acknowledges support from JSPS KAKENHI Grant No. JP20H01845, JST PRESTO Grant No. JPMJPR19L2, JST CREST Grant No. JPMJCR19T1, and RIKEN iTHEMS. B.M. acknowledges the support from SQIG - Security and Quantum Information Group, the Instituto de Telecomunicações (IT) Research Unit, Ref. UIDB/50008/2020, funded by Fundação para a Ciência e a Tecnologia (FCT), European funds, namely, H2020 project SPARTA, as well as projects QuantMining POCI-01-0145-FEDER-031826 and PREDICT PTDC/CCI-CIF/29877/2017.
Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/7/15

Y1 - 2021/7/15

N2 - 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 twist-angle space. 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 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 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.

AB - 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 twist-angle space. 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 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 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.

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U2 - 10.1103/PhysRevB.104.045104

DO - 10.1103/PhysRevB.104.045104

M3 - Article

AN - SCOPUS:85109183991

VL - 104

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 4

M1 - 045104

ER -