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Physical review. B, Condensed matter

Scaling and crossover in a fermion-boson mixture

K. K. Singh

Published Jan 1, 1988• 5 citations
Publisher

Abstract

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 ${v}_{6}$ 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 ${v}_{6}$ is a quantum-mechanical parameter, being proportional in three dimensions to ${b}^{3}$/${\ensuremath{\lambda}}_{T}^{4}$${K}_{F}$, where ${\ensuremath{\lambda}}_{T}$, ${K}_{F}$, 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.

DOI
10.1103/physrevb.37.419

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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}$${K}_{F}, where \ensuremathλT{\ensuremath{\lambda}}_{T}\ensuremathλT​, 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.
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/${\ensuremath{\lambda}}_{B}^{4}$KF{K}_{F}KF​)1/2{)}^{1/2})1/2 where KF{K}_{F}KF​, ${\ensuremath{\lambda}}_{B}$, 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.
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 tricritical behavior of a classical three-well-potential model for two-component systems (such as He3{\mathrm{He}}^{3}He3-He4{\mathrm{He}}^{4}He4 mixtures) is discussed by using renormalization-group techniques. The tricritical exponents and scaling fields are calculated for three dimensions.
We present a detailed investigation of the momentum-dependent self-energy $\ensuremath{\Sigma}(k)$ at zero frequency of weakly interacting bosons at the critical temperature Tc{T}_{c}Tc​ of Bose-Einstein condensation in dimensions $3\ensuremath{\leqslant}D<4$. Applying the functional renormalization group, we calculate the universal scaling function for the self-energy at zero frequency but at all wave vectors within an approximation which truncates the flow equations of the irreducible vertices at the four-point level. The self-energy interpolates between the critical regime k⪡kck⪡{k}_{c}k⪡kc​ and the short-wavelength regime k⪢kck⪢{k}_{c}k⪢kc​, where kc{k}_{c}kc​ is the crossover scale. In the critical regime, the self-energy correctly approaches the asymptotic behavior $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}$, and in the short-wavelength regime the behavior is $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2(D\ensuremath{-}3)}$ in D>3D>3D>3. In D=3D=3D=3, we recover the logarithmic divergence $\ensuremath{\Sigma}(k)\ensuremath{\propto}\mathrm{ln}(k∕{k}_{c})$ encountered in perturbation theory. Our approach yields the crossover scale kc{k}_{c}kc​ as well as a reasonable estimate for the critical exponent $\ensuremath{\eta}$ in D=3D=3D=3. From our scaling function we find for the interaction-induced shift in Tc{T}_{c}Tc​ in three dimensions, $\ensuremath{\Delta}{T}_{c}∕{T}_{c}=1.23a{n}^{1∕3}$, where aaa is the sss-wave scattering length and nnn is the density, in excellent agreement with other approaches. We also discuss the flow of marginal parameters in D=3D=3D=3 and extend our truncation scheme of the renormalization group equations by including the six- and eight-point vertex, which yields an improved estimate for the anomalous dimension $\ensuremath{\eta}\ensuremath{\approx}0.0513$. We further calculate the constant ${\mathrm{lim}}_{k\ensuremath{\rightarrow}0}\phantom{\rule{0.2em}{0ex}}\ensuremath{\Sigma}(k)∕{k}^{2\ensuremath{-}\ensuremath{\eta}}$ and find good agreement with recent Monte Carlo data.
Renormalisation of a field theory model for tricritical behaviour in which the operator phi 6 is essential for thermodynamic stability is discussed in d=4- epsilon dimensions. The crossover scaling form of the equation of state is obtained explicitly to first order in epsilon for both positive and negative values of the four-spin coupling constant. Orthodox scaling, involving Gaussian tricritical exponents is obtained, but is shown to be physically inappropriate in the ordered region of the symmetry plane. A reformulation of scaling, using classical tricritical exponents is possible, but involves an additional parameter p with its own scaling exponent phi p =- 1 / 2 (1- epsilon ). This parameter, introduced by Sarbach and Fisher for the many-component, spherical model limit, is related to the coefficient of (nabla phi ) 2 in the field-theoretic Hamiltonian.
The physics of interacting Fermi systems is extremely sensitive to the energy scale. Of particular interest is the low energy regime where correlation induced collective behavior emerges. The theory of interacting Fermi systems is confronted with the occurrence of very different phenomena along a continuum of scales calling for methods capable of computing physical observables as a function of energy scale. In this thesis, we perform a comprehensive renormalization group analysis of two and three-dimensional Fermi systems at low and zero temperature. We examine systems with spontaneous symmetry-breaking and quantum critical behavior by deriving and solving flow equations within the functional renormalization group framework. We extend the Hertz-Millis theory of quantumphase transitions in itinerant fermion systems to phases with discrete and continuous symmetry-breaking, and to quantum critical points where the zero temperature theory is associated with a non-Gaussian fixed point. The order parameter is implemented by a bosonic Hubbard-Stratonovich field, which - for continuous symmetry-breaking - splits into two components corresponding to longitudinal and transversal Goldstone fluctuations. We compute the finite temperature phase boundary near the quantum critical point explicitly including non-Gaussian fluctuations. We then set up a coupled fermion-boson renormalization group theory that captures the mutual interplay of gapless fermions with massless order parameter fluctuations when approaching a quantum critical point. As a first application, we compute the complete set of quantum critical exponents at the semimetal-to-superfluid quantum phase transition of attractively interacting Dirac fermions in two dimensions. Both, the order parameter propagator and the fermion propagator become nonanalytic functions of momenta destroying the Fermi liquid behavior. We finally compute the effects of quantum fluctuations in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The flow equations capture the influence of longitudinal and Goldstone order parameter fluctuations on non-universal quantities such as the fermionic gap and the fermion-boson vertex, as well as the exact universal infrared asymptotics present in every fermionic superfluid.