Motivated by recent experiments on ultracold fermionic spin-1/2 6Li atoms in a Lieb lattice at various Hubbard repulsion U and filling fractions n (Lebrat et al., arXiv:2404.17555), we conduct a density matrix renormalization group (DMRG) analysis of the Hubbard model on a two-leg Lieb ladder. At half-filling, we find a ferrimagnetic Mott insulating ground state, consistent with Lieb's theorem. Away from half-filling, a state with finite total spin S2=0 and vanishing charge gap persists down to filling nc≈2/3. For lower, incommensurate fillings, the system is described by a Luttinger liquid with one charge and one spin mode. Intriguingly, in a narrow window near nc=2/3, close to the onset of ferromagnetic order, we identify a superconducting Luther-Emery phase with dominant sxy-wave pairing.
Existing quantum Monte Carlo studies have investigated the properties of fermions on a Lieb (CuO2) lattice interacting with on-site or near-neighbor electron-electron coupling. Attention has focused on the interplay of such interactions with the macroscopic degeneracy of local zero-energy modes, from which Bloch states can be formed to produce a flat band in which energy is independent of momentum. The resulting high density of states, in combination with the Stoner criterion, suggests that there should be pronounced instabilities to ordered phases. Indeed, a theorem by Lieb rigorously establishes the existence of ferrimagnetic order. Here we study the charge density wave phases induced by electron-phonon coupling on the Lieb lattice, as opposed to previous work on electron-electron interactions. Our key result is the demonstration of charge density wave phases at one third and two thirds fillings, characterized by long-range density-density correlations between doubly occupied sites on the minority or majority sublattices, and an accompanying gap. We also compute the transition temperature to the ordered phase as a function of the electron-phonon coupling.
The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite Tcdw in two dimensions. Quantum Monte Carlo calculations have determined Tcdw for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in d=2 is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine Tsc for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, where t is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher T. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.