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.
The phase transitions in the Bose-Fermi-Hubbard model are investigated.The boson Green's function is calculated in the random phase approximation (RPA) and the formalism of the Hubbard operators is used.The regions of existence of the superfluid and Mott insulator phases are established and the (µ, t) (the boson chemical potential vs. hopping parameter) phase diagrams are built at different values of boson-fermion interaction constant (in the regimes of fixed chemical potential or fixed concentration of fermions).The effect of temperature change on this transition is analyzed and the phase diagrams in the (T, µ) plane are constructed.The role of thermal activation of the ion hopping is investigated by taking into account the temperature dependence of the transfer parameter.The reconstruction of the Mott-insulator lobes due to this effect is analyzed.
Phase transitions are investigated in the Bose-Fermi-Hubbard model in the mean field and hard-core boson approximations for the case of infinitely small fermion transfer and repulsive on-site boson-fermion interaction.The behavior of the Bose-Einstein condensate order parameter and grand canonical potential is analyzed as functions of the chemical potential of bosons at zero temperature.The possibility of change of order of the phase transition to the superfluid phase in the regime of fixed values of the chemical potentials of Bose-and Fermi-particles is established.The relevant phase diagrams are built.
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 present a method for calculating the Yang-Lee partition function zeros of a translationally invariant model of lattice fermions, exemplified by the Hubbard model. The method rests on a theorem involving the usual single-electron self-energy ${\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k},i{\ensuremath{\omega}}_{n}|\ensuremath{\mu})$ with chemical potential $\ensuremath{\mu}$, in the imaginary time Matsubara formulation. The theorem maps the Yang-Lee zeros to a set of wave vector and spin labeled virtual energies ${\ensuremath{\xi}}_{\stackrel{P\vec}{k}\ensuremath{\sigma}}$. These, thermodynamically derived virtual energies, are solutions of the set of equations ${\ensuremath{\xi}}_{k\ensuremath{\sigma}}={\ensuremath{\varepsilon}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k})\ensuremath{-}\frac{1}{2}U+{\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k},i\frac{\ensuremath{\pi}}{\ensuremath{\beta}}|{\ensuremath{\xi}}_{k\ensuremath{\sigma}}+\frac{1}{2}U\ensuremath{-}i\frac{\ensuremath{\pi}}{\ensuremath{\beta}})=0$. Examples of the method in simplified situations are provided.