Thermal Phase Structure of the Attractive Fermi Hubbard Model with Imaginary Chemical Potential

We study the BCS--BEC crossover of the large $N$ attractive Fermi-Hubbard model on a one-dimensional lattice using the mean field approximation in the presence of an imaginary chemical potential. We show that the crossover is governed by three parameters. The imaginary chemical potential $i\theta$, the temperature via a thermal kernel $g(\beta E_k,\beta\theta)$ and the parameter $\delta_u$ whose sign controls the weak and strong coupling regimes. At the unitarity point ($U=U_c$), we find a thermal window $\phi=\beta\theta=2\pi/3,4\pi/3$ where the gap vanishes while the fermion number $N_f$, which quantifies the balance between particle-like and hole-like excitations, has a local maximum/minimum. Inside this thermal window BCS and BEC physics are await changes in the coupling to be selected as the dominant regime. We expect that our results will unveil a better understanding of pairing correlations in lattice many-body physics.