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

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. Nevertheless, 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 the 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, if a simple uniform pairing condition on the single-particle states is satisfied, 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 W 2 between the Drude weight D and the winding number W, which guarantees ballistic transport for topologically nontrivial flat bands in one dimension.

Flat-band many-body localization and ergodicity breaking in the Creutz ladder

“…We also expect similar phenomena in other flat-band models, such as the saw-tooth, diamond and Lieb lattice models, which are to be realized in experiments [70][71][72][73][74], and also some studies in the Creutz ladder in the clean limit [75,76] pointed out the presence of conserved quantities. Such conjecture may support our results.…”

Temperature and doping induced instabilities of the repulsive Hubbard model on the Lieb lattice

“…The self-energies for all sites, i.e., A, B, and C, are diverging for ω n → 0, a key feature of Mott insulators. Non-Fermi-liquid behavior in the presence of a flat band has been discussed for a multiband lattice Hamiltonian in the presence of an attractive Hubbard interaction using perturbation theory [66]. We conclude that the presence of the flat band, causing singular behavior at the Fermi level, leads to the NFL behavior at finite temperature in the nonmagnetic weakly interacting regime.…”

Superconductivity in repulsively interacting fermions on a diamond chain: Flat-band-induced pairing

“…We can thus envisage dramatic, possibly non-perturbative phenomena from the electronelectron interaction on these macroscopically degenerate manifolds of single-particle states. Beside the ferromagnetism, the flat band systems have attracted recent attentions for possible realization of topological insulators with non-trivial Chern numbers [8][9][10][11][12] . The next goal, in our view, is to realize superconductivity in flat-band systems.…”