Effective theory and emergent $SU(2)$ symmetry in the flat bands of attractive Hubbard models

TL;DR

This paper develops an effective low-energy theory for flat band Hubbard models with attractive interactions, revealing an emergent SU(2) symmetry, the exact BCS ground state, and conditions for ballistic transport in topological flat bands.

Contribution

It introduces a perturbative approach to derive an effective Hamiltonian for flat bands, demonstrating the emergent SU(2) symmetry and its breaking, and establishes a fundamental inequality linking Drude weight and winding number.

Findings

BCS wave function is the exact ground state of the projected interaction Hamiltonian.

Emergent SU(2) symmetry leads to diverging compressibility.

Ballistic transport guaranteed by inequality D ≥ W² in topological flat bands.

Abstract

In a partially filled flat Bloch band electrons do not have a well defined Fermi surface and hence the low-energy theory is not a Fermi liquid. Neverethless, under the influence of an attractive interaction, a superconductor well described by the Bardeen-Cooper-Schrieffer (BCS) wave function can arise. Here we study the low-energy effective Hamiltonian of a generic Hubbard model with a flat band. We obtain an effective Hamiltonian for the flat band physics by eliminating higher lying bands via perturbative Schrieffer-Wolff transformation. At first order in the interaction energy we recover the usual procedure of projecting the interaction term onto the flat band Wannier functions. We show that the BCS wave function is the exact ground state of the projected interaction Hamiltonian and that the compressibility is diverging as a consequence of an emergent $SU(2)$ symmetry. This symmetry is broken by second order interband transitions resulting in a finite compressibility, which we illustrate for a one-dimensional ladder with two perfectly flat bands. These results motivate a further approximation leading to an effective ferromagnetic Heisenberg model. The gauge-invariant result for the superfluid weight of a flat band can be obtained from the ferromagnetic Heisenberg model only if the maximally localized Wannier functions in the Marzari-Vanderbilt sense are used. Finally, we prove an important inequality $D \geq \mathcal{W}^2$ between the Drude weight $D$ and the winding number $\mathcal{W}$, which guarantees ballistic transport for topologically nontrivial flat bands in one dimension.