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BCS-BEC crossover in an optical lattice

We model fermions with an attractive interaction in an optical lattice with a single-band Hubbard model away from half-filling with on-site attraction $U$ and nearest neighbor hopping $t$. Our goal is to understand the crossover from BCS superfluidity in the weak attraction limit to the BEC of molecules in the strong attraction limit, with particular emphasis on how this crossover in an optical lattice differs from the much better studied continuum problem. We use a large-$N$ theory with Sp(2N) symmetry to study the fluctuations beyond mean field theory. At T=0, we calculate across the crossover various observables, including chemical potential, gap, ground state energy, speed of sound and compressibility. The superfluid density $n_s$ is found to have non-trivial $U/t$ dependence in this lattice system. We show that the transition temperature $T_c$ scales with the energy gap in the weak coupling limit but crosses over to a $t^2/U$ scaling in the BEC limit, where phase fluctuations controlled by $n_s$ determine $T_c$. We also find, quite contrary to our expectations, that in the strong coupling limit, the large-$N$ theory gives qualitatively wrong trends for compressibility. A comparison with a simple Hartree shifted BCS theory, which takes into account both pairing and Hartree shifts, and correctly recovers the atomic limit and the right qualitative trend for compressibility, reveals that the large-$N$ theory on the lattice, although considers a larger number of diagrams, is in fact inferior to the simpler Hartree shifted BCS theory. The failure of the large-$N$ approach is explained by noting (i) the importance of Hartree shift in lattice problems, and (ii) inability of the large-$N$ approach to treat particle-particle and particle-hole channels at equal footing at the saddle point level.