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Boson-fermion model beyond mean-field approximation

A. S. Alexandrov

Published Jan 1, 1996• 13 citations
OpenAlex

Abstract

A model of hybridized bosons and fermions is studied beyond the mean field approximation. The divergent boson self-energy at zero temperature makes the Cooper pairing of fermions impossible. The frequency and momentum dependence of the self- energy and the condensation temperature $T_{c}$ of initially localized bosons are calculated analytically. The value of the boson condensation temperature $T_{c}$ is below $1K$ which rules out the boson-fermion model with the initially localized bosons as a phenomenological explanation of high-temperature superconductivity. The intra-cell density-density fermion-boson interaction dominates in the fermion self-energy. The model represents a normal metal with strongly damped bosonic excitations. The latter play the role of normal impurities.

Related papers

Boson - fermion model beyond the mean-field approximation

A. S. Alexandrov

A model of hybridized bosons and fermions is studied beyond the mean-field approximation. The divergent boson self-energy at zero temperature makes the Cooper pairing of fermions impossible. The frequency and momentum dependence of the self-energy and the condensation temperature of initially localized bosons are calculated analytically. The value of the boson condensation temperature is below 1 K and the long-wave spectrum is strongly damped which rules out the boson - fermion model with the initially localized bosons as a phenomenological explanation of high-temperature superconductivity. The intra-cell density - density fermion - boson interaction dominates in the fermion self-energy. The model represents a dirty normal metal, in which localized bosons play the role of impurities.

arXiv:cond-mat/9603111•Journal of Physics Condensed Matter•Sep 9, 1996•14 citations

Pairing of bosons in the condensed state of the boson-fermion model

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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.

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Hubbard model at U=∞\infty∞: Role of single and two-boson fluctuations

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Comment on 'Effect of on-site Coulomb repulsion on superconductivity in the boson-fermion model' (T. Domanski, Phys. Rev. B v.66, 134512 (2002))

A. S. Alexandrov

The two-dimensional boson-fermion model of high-temperature superconductors, numerically studied by Domanski (Phys. Rev. B v.66, 134512 (2002)) is not a superconductor. The critical temperature of the model is zero for any symmetry of the order parameter. The opposite conclusion advocated by Domanski and some other authors stems from an erroneous mean-field approximation neglecting the boson self-energy. The approximation is qualitatively incorrect even in three dimensions. Taking into account the boson self-energy changes the phase transition in 3D BFM, which is not a mean-field second-order phase transition. There are real bosons in the model at and below Tc of 3D BFM, but not only fermions and their Cooper pairs.

arXiv:cond-mat/0303608•Mar 28, 2003•2 citations

Universal behavior of the condensation energy of Superconducting BCS Bose gases

Juan José Valencia Acevedo, M. Atala, Patricia Salas Casales +1 more

Using the Boson-Fermion formalism of superconductivity we calculate the condensation energy for several superconductors ranging from conventional to unconventional, or high temperature superconductors. It is calculated as the difference between the Helmholtz free energies of the superconducting and the normal state, which is a gas of N attractive electron gas, while the superconducting state is formed by the condensed Cooper pairs taken as composite bosons, coming from a fraction of the electrons inside the Debye shell, plus those electrons inside and outside the Debye shell that are unable to pair. After giving the analytic expressions for the internal energy U and the entropy S we obtain the Helmholtz free energy F = U -TS for both the superconducting and the normal states as functions of temperature. In the search for universalities, we calculate the ratio of the condensation energy at T=0 to the Sommerfeld constant (the normal state electronic specific heat over the temperature when T it's almost zero) using two different methods: the Boson-Fermion formalism developed here, as well as an analytical expression deduced from a combination of the BCS and Ginzburg-Landau theories. We find for the Boson-Fermion formalism $E_{cond}/γ_0= 0.252\,T_{c}^{1.997}$, which is the same behavior described by the experimental fit of Kim et al $E_{cond}/γ_0= 0.2\,T_{c}^{2.06}$ and by the one recently reported by Tallon et al for overdoped cuprate superconductors; while for the Ginzburg-Landau-BCS we get the expression $E_{cond}/γ_0= 0.236\,T_{c}^{2}$, also in very good agreement with the partifirst method.

arXiv:2606.25316•Jun 24, 2026•0 citations

Correlated bosons on a lattice: Dynamical mean-field theory for Bose-Einstein condensed and normal phases

Krzysztof Byczuk, D. Vollhardt

We formulate a bosonic dynamical mean-field theory (B-DMFT) which provides a comprehensive, thermodynamically consistent framework for the theoretical investigation of correlated lattice bosons. The B-DMFT is applicable for arbitrary values of the coupling parameters and temperature and becomes exact in the limit of high spatial dimensions $d$ or coordination number $Z$ of the lattice. In contrast to its fermionic counterpart, the construction of the B-DMFT requires different scalings of the hopping amplitudes with $Z$ depending on whether the bosons are in their normal state or in the Bose-Einstein condensate (BEC). A detailed discussion of how this conceptual problem can be overcome by performing the scaling in the action rather than in the Hamiltonian itself is presented. The B-DMFT treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling. It reproduces all previously investigated limits in parameter space, such as the Beliaev-Popov and Hartree-Fock-Bogoliubov approximations, and generalizes the existing mean-field theories of interacting bosons. The self-consistency equations of the B-DMFT are those of a bosonic single-impurity coupled to two reservoirs corresponding to bosons in the condensate and in the normal state, respectively. We employ the B-DMFT to solve a model of itinerant and localized, interacting bosons analytically. The local correlations are found to enhance the condensate density and the BEC transition temperature ${T}_{\text{BEC}}$. This effect may be used experimentally to increase ${T}_{\text{BEC}}$ of bosonic atoms in optical lattices.

arXiv:0706.0839•Physical Review B•Jun 10, 2008•100 citations
A model of hybridized bosons and fermions is studied beyond the mean field approximation. The divergent boson self-energy at zero temperature makes the Cooper pairing of fermions impossible. The frequency and momentum dependence of the self- energy and the condensation temperature TcT_{c}Tc​ of initially localized bosons are calculated analytically. The value of the boson condensation temperature TcT_{c}Tc​ is below 1K1K1K which rules out the boson-fermion model with the initially localized bosons as a phenomenological explanation of high-temperature superconductivity. The intra-cell density-density fermion-boson interaction dominates in the fermion self-energy. The model represents a normal metal with strongly damped bosonic excitations. The latter play the role of normal impurities.
A model of hybridized bosons and fermions is studied beyond the mean-field approximation. The divergent boson self-energy at zero temperature makes the Cooper pairing of fermions impossible. The frequency and momentum dependence of the self-energy and the condensation temperature of initially localized bosons are calculated analytically. The value of the boson condensation temperature is below 1 K and the long-wave spectrum is strongly damped which rules out the boson - fermion model with the initially localized bosons as a phenomenological explanation of high-temperature superconductivity. The intra-cell density - density fermion - boson interaction dominates in the fermion self-energy. The model represents a dirty normal metal, in which localized bosons play the role of impurities.
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 have developed a semi-analytical framework formulated in the canonical fermion representation to investigate strongly correlated electron systems. We consider the U=∞\infty∞ Hubbard model and used the equation of motion method to calculate the fermion self-energy which has two parts: single and two-boson exchange processes. The emergent bosons here are self-generated local charge and spin-density fluctuations which become strongly time-dependent due to extreme correlations. The computed boson spectral density is a diffusive damped mode with a long tail. The electron self-energy at d=∞d=\inftyd=∞ is computed self-consistently. The corresponding fermionic spectral density displays a pronounced coherence peak at ω=0ω=0ω=0, while its frequency derivative develops a two-peak structure at finite ωωω. The resistivity shows a linear temperature dependence over a broad range, crossing over to coherent Fermi-liquid behavior at extremely low temperatures.
The two-dimensional boson-fermion model of high-temperature superconductors, numerically studied by Domanski (Phys. Rev. B v.66, 134512 (2002)) is not a superconductor. The critical temperature of the model is zero for any symmetry of the order parameter. The opposite conclusion advocated by Domanski and some other authors stems from an erroneous mean-field approximation neglecting the boson self-energy. The approximation is qualitatively incorrect even in three dimensions. Taking into account the boson self-energy changes the phase transition in 3D BFM, which is not a mean-field second-order phase transition. There are real bosons in the model at and below Tc of 3D BFM, but not only fermions and their Cooper pairs.
Using the Boson-Fermion formalism of superconductivity we calculate the condensation energy for several superconductors ranging from conventional to unconventional, or high temperature superconductors. It is calculated as the difference between the Helmholtz free energies of the superconducting and the normal state, which is a gas of N attractive electron gas, while the superconducting state is formed by the condensed Cooper pairs taken as composite bosons, coming from a fraction of the electrons inside the Debye shell, plus those electrons inside and outside the Debye shell that are unable to pair. After giving the analytic expressions for the internal energy U and the entropy S we obtain the Helmholtz free energy F = U -TS for both the superconducting and the normal states as functions of temperature. In the search for universalities, we calculate the ratio of the condensation energy at T=0 to the Sommerfeld constant (the normal state electronic specific heat over the temperature when T it's almost zero) using two different methods: the Boson-Fermion formalism developed here, as well as an analytical expression deduced from a combination of the BCS and Ginzburg-Landau theories. We find for the Boson-Fermion formalism Econd/γ0=0.252 Tc1.997E_{cond}/γ_0= 0.252\,T_{c}^{1.997}Econd​/γ0​=0.252Tc1.997​, which is the same behavior described by the experimental fit of Kim et al Econd/γ0=0.2 Tc2.06E_{cond}/γ_0= 0.2\,T_{c}^{2.06}Econd​/γ0​=0.2Tc2.06​ and by the one recently reported by Tallon et al for overdoped cuprate superconductors; while for the Ginzburg-Landau-BCS we get the expression Econd/γ0=0.236 Tc2E_{cond}/γ_0= 0.236\,T_{c}^{2}Econd​/γ0​=0.236Tc2​, also in very good agreement with the partifirst method.
We formulate a bosonic dynamical mean-field theory (B-DMFT) which provides a comprehensive, thermodynamically consistent framework for the theoretical investigation of correlated lattice bosons. The B-DMFT is applicable for arbitrary values of the coupling parameters and temperature and becomes exact in the limit of high spatial dimensions ddd or coordination number ZZZ of the lattice. In contrast to its fermionic counterpart, the construction of the B-DMFT requires different scalings of the hopping amplitudes with ZZZ depending on whether the bosons are in their normal state or in the Bose-Einstein condensate (BEC). A detailed discussion of how this conceptual problem can be overcome by performing the scaling in the action rather than in the Hamiltonian itself is presented. The B-DMFT treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling. It reproduces all previously investigated limits in parameter space, such as the Beliaev-Popov and Hartree-Fock-Bogoliubov approximations, and generalizes the existing mean-field theories of interacting bosons. The self-consistency equations of the B-DMFT are those of a bosonic single-impurity coupled to two reservoirs corresponding to bosons in the condensate and in the normal state, respectively. We employ the B-DMFT to solve a model of itinerant and localized, interacting bosons analytically. The local correlations are found to enhance the condensate density and the BEC transition temperature TBEC{T}_{\text{BEC}}TBEC​. This effect may be used experimentally to increase TBEC{T}_{\text{BEC}}TBEC​ of bosonic atoms in optical lattices.