A two component model of negative U centers coupled with the Fermi sea of itinerant fermions is discussed in connection with high-temperature superconductivity of cuprates, and superfluidity of atomic fermions. We examine the phase transition and the condensed state of this boson-fermion model (BFM) beyond the ordinary mean-field approximation in two and three dimensions. No pairing of fermions and no condensation are found in two-dimensions for any symmetry of the order parameter. The expansion in the strength of the order parameter near the transition yields no linear homogeneous term in the Ginzburg-Landau-Gor'kov equation and a zero upper critical field in any-dimensional BFM, which indicates that previous mean-field discussions of the model are flawed. Normal and anomalous Green's functions are obtained diagrammatically and analytically in the condensed state of a simplest version of 3D BFM. A pairing of bosons analogous to the Cooper pairing of fermions is found. There are three coupled condensates in the model, described by the off-diagonal single-particle boson, pair-fermion and pair-boson fields. These results negate the common wisdom that the boson-fermion model is adequately described by the BCS theory at weak coupling.
We reveal a quantum coherent state characterized by composite bosonic trions, wherein paired fermions further bind with bosons, in one-dimensional Bose-Fermi mixtures.This phase emerges in two separate models, both featuring onsite boson-fermion attraction that induces negative binding energy for the composite trions. The first is the pair-hopping model, in which spinless fermions undergo pair hopping to form preformed pairs. The second is the extended Bose-Fermi Hubbard model, describing dipolar bosons and fermions with density-density interactions. Notably, the formation of composite trions is independent of the specific pairing interactions of fermions. Extensive density matrix renormalization group calculations demonstrate the quasi-condensation of the composite trions, evidenced by algebraically decaying correlations of trions, gapped single-particle excitations, and suppressed fermion pair correlations. Our findings provide valuable insights into the three-body pairing mechanism in mixed-particle systems.
Boson-Fermion models of superconductivity are getting attention as they are able to explain some of the high temperature superconductor's properties. Here we report on the condensation energy of a 3D non-interacting mixture of paired fermions (electrons) as Cooper pairs assumed to be composite bosons, which are responsible for carrying superconductivity, plus unpaired fermions both trapped in a periodic multilayer structure like that of the cuasi-two dimensional High-Temperature superconductor planes, generated by applying an external Dirac's comb potential in the direction perpendicular to the planes where superconductivity preferably occurs, while in the other two directions parallel to the planes the mixture moves freely. For bosons we give the Bose-Einstein condensation critical temperature, which we assume is equal to the superconducting critical one, while for both bosons and fermions we give the chemical potential, the internal energy and the entropy, all of them as functions of temperature, in order to calculate the Helmholtz free energy, which we use to obtain the condensation energy of a mixture of
NF ideal fermions (electrons), which after turning on the attractive pair interaction become
NB=NF/2 ideal bosons. For several plane impenetrability magnitudes, we calculate the condensation energy as the difference between the free energies of the fermions which are in the normal state minus that of the bosons in the condensed state, where we observe that as the plane impenetrability increases: the condensation energy increases; the critical temperature decreases, as expected for example for cuprate superconductors, and the behavior of the entropy and the internal energy show a dimensional crossover from 3D to 2D.