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Physical review. B, Condensed matter

Critical and tricritical microscopic exponents of a Fermi-Bose mixture

Partha Goswami

Published May 1, 1986• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.33.6094

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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.
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.
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.
Inverse Fourier transforms of dynamic correlation functions of order-parameter fluctuations in the singlet-spin-pairing and equal-spin-pairing states of an interacting Fermi system are calculated in a mean-field approximation. The behavior of these functions in the corresponding critical regions, i.e., in the neighborhood of a critical point for an anisotropic, singlet superfluid (and for a superconductor) and for an equal-spin-pairing-type, triplet superfluid, are examined. It is found that these functions do not conform to certain generalized homogeneity assumptions. The static counterparts of these functions are found to be nonsingular at corresponding critical points. A comparison of these findings with corresponding findings for a Bose system, reported in a previous work, highlights the role of Fermi statistics in the critical regions under consideration.
We determine the dynamical critical exponent z appearing at the Bose glass to superfluid transition in two dimensions by performing large scale numerical studies of two microscopically different quantum models within the universality class: The hard-core boson model and the quantum rotor (soft core) model, both subject to strong on-site disorder. By performing many simulations at different system size L and inverse temperature β close to the quantum critical point, the position of the critical point and the critical exponents, z, ν, and η can be determined independently of any implicit assumptions of the numerical value of z, in contrast to most prior studies. This is done by a careful scaling analysis close to the critical point with a particular focus on the temperature dependence of the scaling functions. For the hard-core boson model we find z=1.88(8), ν=0.99(3), and η=-0.16(8) with a critical field of h(c)=4.79(3), while for the quantum rotor model we find z=1.99(5), ν=1.00(2), and η=-0.3(1) with a critical hopping parameter of t(c)=0.0760(5). In both cases do we find a correlation length exponent consistent with ν=1, saturating the bound ν≥2/d as well as a value of z significantly larger than previous studies, and for the quantum rotor model consistent with z=d.
Renormalization techniques of field theory are used to prove (order by order to all orders) dynamical scaling laws on a Ginzburg-Landau stochastic model studied by Halperin, Hohenberg, and Ma. The dynamical exponent is calculated to order ${\ensuremath{\epsilon}}^{3}$ and so is the new exponent ${\ensuremath{\omega}}_{b}$, which governs the vanishing of the imaginary part of the (renormalized) kinetic coefficient, and appears among the corrections to scaling. Difficulties of previous calculations taking a microscopic approach to the critical dynamics of a Bose system are commented upon.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc{d}_{c}dc​ which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He^{4}\mathrm{He}4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.