Effective theory and emergent $\text{SU}\left(2\right)$ symmetry in the flat bands of attractive Hubbard models

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 $\text{SU}\left(2\right)$ 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 $D\ge {\mathcal{W}}^2$ between the Drude weight $D$ and the winding number $\mathcal{W}$, which guarantees ballistic transport for topologically nontrivial flat bands in one dimension.

Figure 1

The Creutz ladder: The green box indicates the $\mathrm{i}\mathrm{th}$ unit cell containing two orbitals depicted by an empty and a full circle. The two sublattices are labeled by $\alpha$. The hopping amplitudes for the spin-$\uparrow$ fermions are given on corresponding links. The arrows indicate the sense of the complex hoppings. The blue box indicates the maximally localized Wannier function for the upper/lower $(\pm )$ flat band, also called plaquette states; the numbers correspond to the respective amplitudes which are nonzero only for the sites inside the blue box.

Figure 2

The power-law dependence of $c_1\left(L\right)$ (top plot) and $c_2\left(L\right)$ (bottom plot) on $1/L$ plotted in log-log scale. The dots represent the DMRG results and the lines correspond to fitted function. Different colors correspond to different strengths of the attractive Hubbard interaction. Here, we set $t=1$. In the legends the values of $c_{1\left(2\right)}^{\infty }$ and $c_{1\left(2\right)}^{\mathrm{SW}}$ are presented for a direct comparison.