Quantum metric on the Brillouin Zone in correlated electron systems and its relation to topology for Chern insulators

Abstract

Geometric aspects of physics play a crucial role in modern condensed matter physics. The quantum metric is one of these geometric quantities which defines the distance on a parameter space and contributes to various physical phenomena, such as superconductivity and nonlinear conductivity. Despite its importance, the quantum metric in interacting systems is poorly understood. In this paper, we introduce a generalized quantum metric(GQM) on the Brillouin zone for correlated electron systems. This quantum metric is based on the optical conductivity that is written by single-particle Green's functions. We analytically prove that this definition is equivalent to the existing definition of the quantum metric in noninteracting systems and that it is positive semi-definite as necessary for a metric. Furthermore, we point out the relationship between the GQM and the Chern number in interacting systems. We then numerically confirm these properties of the GQM in the Qi-Wu-Zhang model with and without interaction. We believe that the GQM will be a step toward generalizing the quantum metric to the interacting regime.

6 pages,6 figures

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• The numerical renormalization group method for quantum impurity systems
• A Numerical Renormalization Group approach to Green's Functions for Quantum Impurity Models
• Simplified topological invariants for interacting insulators
• Correlation effects in two-dimensional topological insulators
• Superfluid weight bounds from symmetry and quantum geometry in flat bands
• Bound on resistivity in flat-band materials due to the quantum metric
• Topological antiferromagnetic phase in a correlated Bernevig-Hughes-Zhang model
• Nontrivial quantum geometry of degenerate flat bands
• Universal semiclassical equations based on the quantum metric
• Prototypical many-body signatures in transport properties of semiconductors

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• Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization
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