Abstract The BCS–BEC crossover of the large N attractive Fermi–Hubbard model is studied on a one-dimensional lattice using the mean field approximation in the presence of an imaginary chemical potential. It is demonstrated that the crossover is governed by three parameters. The imaginary chemical potential i θ , the temperature via a thermal kernel g ( β E k , β θ ) and the parameter δ u whose sign controls the weak and strong coupling regimes. At the unitarity point ( U = U c ), a thermal window ϕ = β θ = 2 π / 3 , 4 π / 3 is identified 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 awaiting changes in the coupling to be selected as the dominant regime. These results are expected to unveil a better understanding of pairing correlations in lattice many-body physics.
We find a mapping between the attractive Fermi-Hubbard model and the repulsive Bose-Hubbard model at finite temperature and at imaginary chemical potential
μ=iθ. We show, by using a large
N-expansion, that the partition functions of the two models are related by a simple shift
θ→θ+π. This condition maps the BCS--BEC crossover of attractive fermions to a Bose--Fermi crossover (fermion-like occupation) of repulsive bosons. Central feature of this correspondence plays the thermal kernel
g(βE,ϕ), whose analytic continuation
gB(βE,ϕ)=gF(βE,ϕ+π) governs the bosonic and fermionic sectors. Interestingly, we are able to find that the special angles
ϕ=2π/3,4π/3 for fermions correspond to
ϕ=π/3,5π/3 for bosons, marking the boundaries of a universal thermal window. We further argue that the present mechanism shows that fermionization can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. This suggests that it is a new way of fermionization (not a change in statistics but a fermion-like behaviour) unlike the Tonks--Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Essentially, the phase
ϕ is a statistical parameter; by twisting the thermal phase, it generates fermion-like behaviour without hard-core constraints or infinite repulsion but only by using thermodynamics. We derive the gap equation and number equation for the bosonic model, highlighting the role of the imaginary chemical potential as a statistical regulator. Our results provide a unified framework for understanding crossovers in interacting lattice systems.
A formal thermodynamic mapping is established between the attractive Fermi–Hubbard model and the repulsive Bose–Hubbard model at finite temperature and at imaginary chemical potential μ=iθ. By utilizing a large N-expansion, it is shown that the partition functions of the two models are related by a plain shift θ→θ+π. This condition maps the BCS–BEC crossover of attractive fermions to a Bose–Fermi crossover (fermion-like occupation) of repulsive bosons. A central feature of this correspondence is the thermal kernel g(βE,ϕ) (with β the inverse absolute temperature, E the energy scale, and ϕ the phase angle), whose analytic continuation gB(βE,ϕ)=gF(βE,ϕ+π) governs the bosonic (B) and fermionic (F) sectors. Interestingly, the particular angles ϕ=2π/3 and 4π/3 for fermions correspond to ϕ=π/3 and 5π/3 for bosons, marking the boundaries of an universal thermal window. It is further argued that the present mechanism shows how an emergent, fermionization-like phenomenon can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. It is emphasized that this does not imply a transmutation of quantum statistics at the operator level, but rather a thermodynamic exclusion-like behavior driven by the imaginary chemical potential, unlike the Tonks–Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Effectively, the phase ϕ is a statistical parameter; by twisting the thermal phase, it generates fermion-like behavior without hard-core constraints or infinite repulsion through purely thermodynamic mechanisms. The gap equation and number equation for the bosonic model are derived, highlighting the role of the imaginary chemical potential as a statistical regulator. The results obtained here provide a unified framework for understanding crossovers in interacting lattice systems.
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
ns is found to have non-trivial
U/t dependence in this lattice system. We show that the transition temperature
Tc scales with the energy gap in the weak coupling limit but crosses over to a
t2/U scaling in the BEC limit, where phase fluctuations controlled by
ns determine
Tc. 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.
Abstract We address the thermodynamics of the zero temperature BCS-BEC crossover of a balanced two component mixture of a finite range interacting fermions within a mean field approach. For the analysis we consider two wide types of finite range potentials describing the interaction between different Fermi species: purely attractive, as the square well, the exponential and the Yukawa potentials, and others having both a repulsive wall at the origin as well as a finite attractive tail, as Van der Waals and dipolar potentials. We study the energy gap, the chemical potential and the pressure as functions of the scattering length for all the interacting potentials considered. The system spatial structure is further analyzed via the the pair functions associated to each potential. We find that for those with pure attractive contributions the crossover can be recognized either, when the chemical potential changes its sign or when the scattering length diverges. In contrast, the pair functions linked to potentials with both repulsive and attractive terms, show wide crossover regions without sharp distinctions at the sign change of the chemical potential nor at unitarity.
A Fermi liquid with weak attractive interaction undergoes a BCS transition to a superconductor with reducing temperature. With increasing interaction strength, the thermal transition is progressively modified as the high temperature `metallic' phase develops a pseudogap due to pairing fluctuations and the resistivity above T_c shows insulating behaviour. The crossover to insulating character occurs much before the system can be considered to be in the BEC regime of preformed fermion pairs. We use a new Monte Carlo tool to map out the BCS-BEC crossover in the attractive Hubbard model on large two dimensional lattices and explicitly compute the resistivity to demonstrate how the metal to superconductor (MS) thermal transition at weak coupling crosses over to an insulator to superconductor (IS) transition at intermediate coupling. Our high resolution access to the single particle and optical spectrum at finite temperature allows us to completely describe the transport crossover in this longstanding problem.