Relaxation toward an Ideal Chern Band through Coupling to a Markovian Bath

We propose a microscopic, weak-coupling mechanism by which generic Chern bands asymptotically relax toward ideal bands. We consider coupling interacting electrons to a Caldeira-Leggett-like Ohmic bosonic bath. Using the Born-Markov approximation, we analytically show that, upon taking the leading order contribution in momentum of the form factor, Slater determinant states of a Chern band under Hartree-Fock approximation evolve toward Slater determinant states corresponding to an ideal Chern band. We also numerically validate our proposal by performing simulation of a massive Dirac model with the extended Hubbard interaction, showing that the Berry curvature and quantum metric co-evolve toward saturation of the trace condition. Our proposal provides a concrete dissipative mechanism for driving Chern bands toward ideal quantum geometry, a fundamental building block for the stabilization of fractional Chern insulators.