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Physical review. B, Condensed matter

Nonuniversality of tricritical behavior of Fermi-Bose mixtures

K. K. Singh

Published Jan 1, 1987• 3 citations
Publisher

Abstract

Tricritical amplitude ratios have been calculated for a mixture of weakly interacting fermions and bosons. The ratios are characterized by a quantum parameter \ensuremath{\alpha}\ensuremath{\propto}(${b}^{3}$/${\ensuremath{\lambda}}_{B}^{4}$${K}_{F}$${)}^{1/2}$ where ${K}_{F}$, ${\ensuremath{\lambda}}_{B}$, and b denote, respectively, the Fermi momentum of the fermion component, the boson thermal wavelength, and the scattering length associated with the fermion-boson interaction. To first order in \ensuremath{\alpha}, the results are formally the same as those obtained by Fisher and Sarbach for a classical n-component spin model in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}; to order ${\ensuremath{\alpha}}^{2}$ they differ. It is argued that nonobservance of nonuniversal behavior in $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures is not inconsistent with the theoretical estimate following from \ensuremath{\alpha}. Better experimental accuracy in the determination of the amplitude ratios may exhibit nonuniversal behavior.

DOI
10.1103/physrevb.35.377

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Tricritical amplitude ratios have been calculated for a mixture of weakly interacting fermions and bosons. The ratios are characterized by a quantum parameter \ensuremath{\alpha}\ensuremath{\propto}(b3{b}^{3}b3/{\ensuremath{\lambda}}_{B}^{4}$${K}_{F}$${)}^{1/2} where KF{K}_{F}KF​, \ensuremathλB{\ensuremath{\lambda}}_{B}\ensuremathλB​, and b denote, respectively, the Fermi momentum of the fermion component, the boson thermal wavelength, and the scattering length associated with the fermion-boson interaction. To first order in \ensuremath{\alpha}, the results are formally the same as those obtained by Fisher and Sarbach for a classical n-component spin model in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}; to order \ensuremathα2{\ensuremath{\alpha}}^{2}\ensuremathα2 they differ. It is argued that nonobservance of nonuniversal behavior in 3\ensuremath−4^{3}\mathrm{\ensuremath{-}}^{4}3\ensuremath−4He mixtures is not inconsistent with the theoretical estimate following from \ensuremath{\alpha}. Better experimental accuracy in the determination of the amplitude ratios may exhibit nonuniversal behavior.
Thermodynamic behavior of a mixture of weakly interacting fermions and bosons is investigated in 4-\ensuremath{\epsilon} dimensions by the renormalization-group method with the purpose of studying scaling and crossover properties of the system in the tricritical region. Conventional tricritical scaling, first found to break down for a classical infinite-component model, is seen to do so more spectacularly in the case of the mixture. Whereas in the infinite-component model, conventional scaling holds in the ordered and disordered phases separately (i.e., with different tricritical exponents), it is impossible in either of the phases of the mixture. The breakdown of scaling in the mixture is associated with the dimensionless strength v6{v}_{6}v6​ of the six-point interaction in the effective Hamiltonian that causes the parameters of the renormalized Hamiltonian to depend on two combinations of scaling fields rather than one. The strength v6{v}_{6}v6​ is a quantum-mechanical parameter, being proportional in three dimensions to b3{b}^{3}b3/${\ensuremath{\lambda}}_{T}^{4}$KF{K}_{F}KF​, where ${\ensuremath{\lambda}}_{T}$, KF{K}_{F}KF​, and b denote, respectively, the boson thermal wavelength, the Fermi momentum of the fermion component, and the scattering length associated with the fermion-boson interaction. The square root of this quantity agrees with the nonuniversality parameter, which was found to characterize tricritical amplitude ratios in three dimensions in an earlier work.
Deviations from the classical phenomenological theory of tricriticality are analyzed on the basis of the exact solution (in four field variables) of the spherical-model limit ($n\ensuremath{\rightarrow}\ensuremath{\infty}$) in three dimensions. The scaling functions are nonuniversal but are parametrized by a single variable, which vanishes for infinite-range interactions. Special relations between critical amplitude ratios correlate well with observations on antiferromagnets and 3He^{3}\mathrm{He}3He-4He^{4}\mathrm{He}4He mixtures. Novel qualitative predictions are made for the threephase monohedron below Tt{T}_{t}Tt​.
We consider a dilute homogeneous mixture of bosons and spin-polarized fermions at zero temperature. We first construct the formal scheme for carrying out systematic perturbation theory in terms of single particle Green's functions. We especially focus on the description of the boson-fermion interaction. To do so we need to introduce the renormalized boson-fermion TTT matrix, which we determine to second order in the boson-fermion sss-wave scattering length. We also discuss how to incorporate the usual boson-boson TTT matrix in mean field approximation to obtain the total ground-state properties of the system. The next-order term beyond mean field stems from the boson-fermion interaction and is proportional to aBFkF.{a}_{\mathrm{BF}}{k}_{\mathrm{F}}.aBF​kF​. The total ground-state energy density to this order is the sum of the kinetic energy of the free fermions, the boson-boson mean-field interaction, the usual mean-field contribution to the boson-fermion interaction energy, and the first boson-fermion correction beyond mean field. We also compute the bosonic and the fermionic chemical potentials, the compressibilities, and the modification to the induced fermion-fermion interaction. We discuss the behavior of the total ground-state energy and the importance of the correction beyond mean field for various parameter regimes, in particular considering mixtures of 6Li{}^{6}\mathrm{Li}6Li and 7Li{}^{7}\mathrm{Li}7Li and of 3He{}^{3}\mathrm{He}3He and 4He.{}^{4}\mathrm{He}.4He. Moreover, we determine the modification of the induced fermion-fermion interaction due to the effects beyond mean field. We show that there is no effect on the depletion of the Bose condensate to first order in the boson-fermion scattering length aBF.{a}_{\mathrm{BF}}.aBF​.
We study static properties of attractively interacting Bose-Fermi mixtures of uniform atomic gases at zero temperature. Using the Green's function formalism, we calculate the boson-fermion scattering amplitude and the fermion self-energy in the medium to lowest order of the hole-line expansion. We study the ground-state energy and pressure as a function of the scattering length for a few values of the boson-fermion mass ratio mb∕mf{m}_{b}∕{m}_{f}mb​∕mf​ and the number ratio Nb∕Nf{N}_{b}∕{N}_{f}Nb​∕Nf​. We find that the attractive contribution to the energy is greatly enhanced for small values of the mass ratio. We study the role of Bose-Fermi pair correlations in the mixture by calculating the pole of the boson-fermion scattering amplitude in the medium. The pole shows a standard quasiparticle dispersion for a Bose-Fermi pair. In addition, we also study the fermion dispersion relation. We find two dispersion branches with the possibility of avoided crossings. This strongly depends on the number ratio Nb∕Nf{N}_{b}∕{N}_{f}Nb​∕Nf​.
Critical and tricritical ``microscopic'' exponents (\ensuremath{\eta},\ensuremath{\nu},z) of a weakly interacting Fermi-Bose mixture are calculated in an approximation scheme. The scheme is based on an approximate form of the effective, low-momentum boson Hamiltonian derived in a previous paper and the Landau-theory equation of state. The approximate form is quadratic in boson operators with nonzero momenta. The scheme predicts its own range of validity. In the critical region of the condensed phase, the range is given by a condition reminiscent of the well-known Ginzburg-Levanyuk criterion. In the tricritical region, however, the scheme is exact. An attempt has been made to extract some information concerning the underlying dynamical structure which leads to the static critical and tricritical behavior observed. It is found that the spatial Fourier transform of the dynamic correlation function of order-parameter fluctuations is periodic in time with time period large compared to the inverse thermal frequency in the critical and tricritical regions. The Fourier transform in space and time conforms to the prediction of Halperin and Hohenberg. A dynamic-scaling assumption at a tricritical point is also verified. The value of the dynamic scaling exponent z, calculated with the aid of this assumption, agrees with that expected on the basis of conventional (Van Hove) prediction.
The collective oscillation frequencies and structure factor in the tricritical region of a mixture of weakly interacting bosons and fermions have been calculated by use of effective boson Hamiltonian derived in a previous publication. The collective mode corresponds to a phonon-type dispersion relation with a damping constant proportional to the magnitude of the wave vector q. The structure factor exhibits divergence for q\ensuremath{\rightarrow}0. Experimental investigation at tricriticality of a $^{3}\mathrm{\ensuremath{-}}^{4}$He mixture is worth attempting to confirm the latter.