Gradient flow towards quantum states with ideal quantum geometry

Abstract:We propose a gradient-flow method for quantum states in lattice models, generated by an action consisting of the quantum-metric and the square of the Berry curvature. These two terms drive the spectral projector toward Bogomolny saturation and uniform Berry curvature, respectively. We show that, due to a no-go theorem for finite-dimensional projectors, the two conditions cannot in general be satisfied simultaneously in lattice models. Hence the flow is expected to approach a nontrivial fixed point that balances the two geometric requirements. For the Wilson-Dirac model, we demonstrate that the flowed projector exhibits almost uniform Berry curvature while remaining close to the Bogomolny bound. We further construct a short-range truncated flattened Hamiltonian from the flowed projector and obtain a lattice model with nearly flat bands and nearly uniform Berry curvature. We also apply the method to the Hofstadter model and confirm the roles of the metric and Berry-curvature terms in a Chern band with higher Chern number.

Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

arXiv:2608.19770 [quant-ph]

(or arXiv:2608.19770v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.19770 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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