BCS-BEC Crossover on the Two-Dimensional Honeycomb Lattice

The attractive Hubbard model on the honeycomb lattice exhibits, at half filling, a quantum critical point between a semimetal with massless Dirac fermions and an $s$-wave superconductor (SC). We study the BCS-BEC crossover in this model away from half filling at zero temperature and show that the appropriately defined crossover line (in the interaction-density plane) passes through the quantum critical point at half filling. For a range of densities around half filling, the “underlying Fermi surface” of the SC, defined as the momentum space locus of minimum energy quasiparticle excitations, encloses an area which changes nonmonotonically with interaction. We also study fluctuations in the SC and the semimetal, and show the emergence of an undamped Leggett mode deep in the SC. Finally, we consider possible implications for ultracold atoms in optical lattices and the high temperature SCs.

Figure 1

Left: Honeycomb lattice showing basis vectors and the two sites per unit cell, as well as the hopping matrix elements in model (1). Right: Quasiparticle dispersion with Dirac nodes in first Brillouin zone of the noninteracting semimetal.

Figure 2

(a) Plot of the mean-field gap ${\Delta }_0/t$ versus $n$ and $U/t$ (for $t^{\prime }=-0.15t$). White horizontal line at $n=1$ is the semimetal phase terminating in a semimetal-SC transition at $U_c/t\approx 2.13$. Dashed lines indicate the BCS-BEC crossover discussed in the text. The thick black line indicates the parameters at which the underlying Fermi surface (UFS), discussed in text and panels (b) and (c), shrinks to a point. The light arrow indicates a possible trajectory of the Hamiltonian discussed in the text. (b) Mean-field chemical potential $\mu /t$ versus $U/t$ for indicated fillings ($t^{\prime }=-0.15t$). The dashed line at $\mu =3t^{\prime }$ is a guide to the eye. (c) Schematic underlying Fermi surface (see text), roughly circular loci around the nodes $\pm \mathbf{K}$, drawn for parameters indicated by arrows in panel (b).

Figure 3

(a) Plot of the SC transition temperature $T_c=\min (T_c^0,T_{\mathrm{KT}}^{*})$ (see text). Shaded region is the crossover regime between $T_c\sim T_c^0$ (“BCS”) and $T_c\sim T_{\mathrm{KT}}^{*}$ (“BEC”). Region marked “Leggett,” beyond the dashed line, indicates parameters where an undamped Leggett mode emerges. The solid horizontal line at $n=1$ is the semimetal terminating at a QCP. (b) Spectral function (in arbitrary units) of SC fluctuations in the semimetal at $U=2.1t$. Small $\mathbf{q}$ fluctuations appear critically damped, with the damping threshold being strongly $\mathbf{q}$ dependent. Large $\mathbf{q}$ fluctuations can propagate as undamped modes when close to the QCP.