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Physical Review

Theory of Dilute Mixtures of He3 in Liquid He4 at Low Temperatures

Victor J. Emery

Published Aug 5, 1966• 31 citations
Publisher

Abstract

Dilute mixtures of ${\mathrm{He}}^{3}$ in liquid ${\mathrm{He}}^{4}$ are regarded as a set of ${\mathrm{He}}^{3}$ atoms intereacting with elementary excitations of the ${\mathrm{He}}^{4}$. The coupling between the ${\mathrm{He}}^{3}$ and the ${\mathrm{He}}^{4}$ is chosen phenomenologically and it is not weak but, for low temperatures, its main effects may be eliminated by means of two canonical transformations. The equations which determine the transformations describe the motion of a ${\mathrm{He}}^{4}$ excitation in the presence of one or two ${\mathrm{He}}^{3}$ atoms and, in the latter case, they resemble Faddeev's equations in the three-body problem. At low temperatures, there are few ${\mathrm{He}}^{4}$ excitations and many of the properties of the mixture are determined by the effective ${\mathrm{He}}^{3}$ Hamiltonian produced by the transformation. In this respect, the system behaves like a Fermi liquid. The ${\mathrm{He}}^{3}$-${\mathrm{He}}^{3}$ potential is essentially momentum-independent and consists of the Van der Waals interaction together with an effectively repulsive potential induced by the ${\mathrm{He}}^{4}$. It is suggested that a Fermi-liquid analysis is the best way of obtaining information about this potential, and the possible existence of a fermion superfluid phase transition is discussed from that point of view.

DOI
10.1103/physrev.148.138

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He4 effective interaction for dilute solutions of He4 in liquid He3 at low temperatures

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Interactions Between He3 Atoms in Dilute Solutions of He3 in Superfluid He4

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The magnitude, sign, and momentum dependence of the effective interaction between ${\mathrm{He}}^{3}$ atoms in superfluid ${\mathrm{He}}^{4}$ are derived from the experimental data. Calculations of the Fermi-liquid parameters for the normal state of ${\mathrm{He}}^{3}$ in solution yield results in agreement with experiment; the temperature for a superfluid transition associated with the ${\mathrm{He}}^{3}$ is estimated to be \ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ \ifmmode^\circ\else\textdegree\fi{}K.

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Dilute helium mixtures at low temperatures : properties and cooling methods

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This thesis describes experimental work on dilute mixtures of ³He in 4He, mainly at millikelvin temperatures. The isotopic helium mixture has the unique property of remaining a miscible liquid down to the absolute zero temperature. In the mK regime, it consists of two very different components: perfectly superfluid 4He, and a weakly interacting degenerate Fermi liquid of ³He, predicted by theory to undergo transition to the superfluid state at an extremely low temperature. To discover that transition, new ways of cooling helium mixtures need to be developed, as it is not likely that the conventional method of nuclear demagnetization of copper can be improved to reach helium temperatures notably below the temperatures of order 0.1 mK attained so far. \n Adiabatic melting of 4He in the presence of liquid ³He is probably the most promising method of cooling helium mixtures to microkelvin temperatures. It produces helium mixture colder than the initial temperature as a direct consequence of the mixing of the isotopes. The starting configuration is attainable by pressurizing liquid helium mixture to its solidification pressure at a temperature below some tens of mK, as only 4He then enters the solid phase. This thesis describes an experiment in which the method of adiabatic melting was, for the first time, realized at sub-millikelvin temperatures, where the superfluidity of the pure ³He phase enables, in principle, a drastic decrease of temperature. In the experiments, cooling from initial temperatures between 0.3 and 0.9 mK was detected, temperature reduction remaining below a factor of two for recognized reasons of a technical rather than fundamental nature. \n A capacitive differential pressure transducer, constructed for the experiment, was used for high accuracy measurements of the temperature dependence of the melting pressure of helium mixtures with several ³He concentrations. The melting pressure is suitable for thermometry and carries information on the interactions of ³He particles in the mixture. Also, the response of a quartz tuning fork immersed in helium was studied. Its sensitivity to the properties of the surrounding fluid was utilized to determine the saturation concentration of dilute ³He across the entire accessible pressure range. The tuning fork was found to exhibit a complex pattern of anomalies attributed to resonant modes of second sound, or concentration waves, inside its cylindrical container.

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Theory of a Superfluid Fermi Liquid. I. General Formalism and Static Properties

A. J. Leggett

The microscopic theory of a superfluid Fermi liquid at finite temperature is developed for the case of a pure system with $S$-wave pairing, and applied to the calculation of the static properties. As a function of $\ensuremath{\theta}\ensuremath{\equiv}\frac{T}{{T}_{c}}$ these properties are determined entirely by the Landau parameters ${F}_{0}$, ${F}_{1}$, ${Z}_{0}$, etc., characterizing quasiparticle interactions in the normal phase. In particular the spin susceptibility $\ensuremath{\chi}$ and the density of the normal component ${\ensuremath{\rho}}_{n}$ are given by $\frac{\ensuremath{\chi}(\ensuremath{\theta})}{\ensuremath{\chi}(1)}=(1+\frac{1}{4}{Z}_{0})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{4}{Z}_{0}f(\ensuremath{\theta})]},$ $\frac{{\ensuremath{\rho}}_{n}}{\ensuremath{\rho}}=(1+\frac{1}{3}{F}_{1})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{3}{F}_{1}f(\ensuremath{\theta})]},$ where the universal function $f(\ensuremath{\theta})\ensuremath{\equiv}\ensuremath{-}{[\ensuremath{\nu}(0)]}^{\ensuremath{-}1}{\ensuremath{\Sigma}}_{\mathrm{p}}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is the "effective density of states near the Fermi surface" relative to its value $\ensuremath{\nu}(0)$ in the normal phase. Thus the often-quoted expression ${\ensuremath{\rho}}_{n}=\frac{1}{3}{\ensuremath{\Sigma}}_{\mathrm{p}}{\mathrm{p}}^{2}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is valid for an interacting system only in the limit $T\ensuremath{\rightarrow}0$. In the latter part of the paper a simple phenomenological theory of "Fermi-liquid" effects on $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ is developed for arbitrary conditions (including the presence of impurities and pairing with $l\ensuremath{\ne}0$); it is found that under most circumstances explicit expressions for $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ may be obtained which involve only the Landau parameters and a suitably generalized effective density of states. The theory should apply to the possible superfluid phase of ${\mathrm{He}}^{3}$ and to most superconductors. It is suggested that the Knight shift in nontransition-metal superconductors should display some "Fermi-liquid" effects. The weak-field dc penetration depth $\ensuremath{\lambda}(T)$ is shown to be insensitive to such effects both in the Pippard limit and near ${T}_{c}$; however, in a London superconductor at lower temperatures the correction to $\ensuremath{\lambda}(T)$ should be observable and yield a direct estimate of ${F}_{1}$.

Physical Review•Dec 13, 1965•284 citations

The role of pair correlations in the formation of the ground state and the elementary excitation spectrum in a superfluid Bose liquid (A Review)

Ernst A. Pashitskii

The paradoxes and disparities in the contemporary microscopic theory of superfluid helium (He–II) are discussed along with possible ways of resolving them by taking pair correlations of He4 atoms into consideration. It is shown that most paradoxes are associated with the commonly accepted initial assumption concerning the dominating role of single-particle Bose condensate (SPBC) in the quantum microstructure of the superfluid component ρs. The existence of intensive SPBC leads to a strong hybridization of the elementary excitation branches and to a common dispersion law for all boson branches, which is identified with the quasiparticle spectrum E(p) observed experimentally from slow neutron scattering in liquid helium. However, the stability of this spectrum during a transition through the λ-point and the large value of the gap in the vicinity of the “rotonic” minimum contradict both the Landau theoretical criterion of superfluidity and the small value of experimentally measured critical velocity. At the same time, a strong interaction between particles in the Bose liquid He4 strongly suppresses the SPBC which amounts to less than 1% of all He4 atoms and hence cannot be the main constituent of the superfluid component, unlike the case of a weakly nonideal Bose gas. Moreover, for a quite strong attraction between particles in a certain region of the momentum space, bound pairs of bosons can be formed in the superfluid Bose liquid, and a coherent pair condensate (CPC) analogous to the Cooper pair condensate in superconductors may appear. Such a strong CPC may completely suppress the weak SPBC. In this case, the one-particle spectrum ε(p) of elementary excitations does not hybridize with the collective (two-particle) spectrum and does not appear in the structure of the dynamic form factor S(p,ε), i.e., does not coincide with the spectrum measured from neutron scattering. The dispersion of one-particle spectrum is defined by the momentum dependence of the pair order parameter Ψ̃(p) and may have a minimum or a point of inflection at p≠0. This peculiarity in the one-particle spectrum of a Bose liquid with CPC but without SPBC vanishes together with Ψ̃(p) at the temperature Tc=Tλ of the phase transition from the superfluid to the normal state (unlike the rotonic minimum in the collective spectrum), while the corresponding critical velocity vc=min[ε(p)/p] vanishes at the λ-point in accordance with the Landau criterion and the experimental data. The assumption that the strong “Cooper-like” CPC is responsible for the quantum structure of the superfluid component ρs is confirmed indirectly by the successful application of the Justrow approximation (based on strong pair correlations) for describing the properties of liquid He4 and quantum liquid mixtures He3–4He on one hand, and by an anomalously large effective mass of He3 impurity atoms in He4, which is approximately equal to total mass of He3 and He4 atoms, thus pointing to the existence of helium atoms in superfluid liquid He–II. The value of the superfluid velocity circulation quantum in the Onsager–Feynman vortices in a Bose liquid with CPC but without SPBC is discussed as well as the critical velocities of superfluid He4 in ultrathin films and channels in which the creation and motion of quantum vortices are ruled out, and the quasiparticle spectrum undergoes dimensional quantization.

Low Temperature Physics•Feb 1, 1999•14 citations

Topics in low temperature Fermi liquid theory

Daryl W. Hess

Several topics in quantum liquids are discussed including the elementary excitation spectrum of 3 He under pressure, spin polarized 3 He, and an early attempt to formulate a Fermi liquid theory to describe the low temperature thermodynamic and transport properties of the heavy electron system UPt3. The elementary excitation spectrum of ordinary liquid 3 He is calculated at several pressures using the polarization potential theory of Aldrich and Pines together with a simple model to describe the effect of multipair excitations. The calculated elementary excitation spectra, with no adjustable parameters, agree well with the recent experiment of Scherm et al. The effective interactions between quasi particles in fully spin polarized 3 He are obtained from physical arguments and sum rules. The interactions between two down spin impurities and that between an up and down spin are also deduced. The volume occupied by a down spin impurity is found to be 17% to 7% less than that occupied by an up· spin. Nonlocal scalar pseudopotentials of the Aldrich-Pines form are developed along with two extreme models for the momentum dependent backflow potential. The transport properties are calculated and found to be two to three orders of magnitude larger than those of the unpolarized liquid at the same density. The elementary excitation spectrum is found to be quite sensitive to the momentum dependence of the backflow potential. The regime of small polarization is considered next. Using the phenomenological model of Bedell and Sanchez-Castro together with an ansatz form for the spin flip interaction, a large increase in the singlet scattering rate as a function of polarization is obtained. The transport properties are calculated as a function of polarization and the viscosity obtained is found to be in good agreement with the recent experiment of P. Kopietz et al. Finally, an early attempt at describing the low temperature thermodynamic and transport properties of the heavy electron system UPt3 by an extension of Landau Fermi liquid theory is described. Although the Fermi surface geometry is treated naively, the transport calculations underscore the importance of electron-electron scattering in these systems.

•Jan 1, 1987•0 citations
Dilute mixtures of He3{\mathrm{He}}^{3}He3 in liquid He4{\mathrm{He}}^{4}He4 are regarded as a set of He3{\mathrm{He}}^{3}He3 atoms intereacting with elementary excitations of the He4{\mathrm{He}}^{4}He4. The coupling between the He3{\mathrm{He}}^{3}He3 and the He4{\mathrm{He}}^{4}He4 is chosen phenomenologically and it is not weak but, for low temperatures, its main effects may be eliminated by means of two canonical transformations. The equations which determine the transformations describe the motion of a He4{\mathrm{He}}^{4}He4 excitation in the presence of one or two He3{\mathrm{He}}^{3}He3 atoms and, in the latter case, they resemble Faddeev's equations in the three-body problem. At low temperatures, there are few He4{\mathrm{He}}^{4}He4 excitations and many of the properties of the mixture are determined by the effective He3{\mathrm{He}}^{3}He3 Hamiltonian produced by the transformation. In this respect, the system behaves like a Fermi liquid. The He3{\mathrm{He}}^{3}He3-He3{\mathrm{He}}^{3}He3 potential is essentially momentum-independent and consists of the Van der Waals interaction together with an effectively repulsive potential induced by the He4{\mathrm{He}}^{4}He4. It is suggested that a Fermi-liquid analysis is the best way of obtaining information about this potential, and the possible existence of a fermion superfluid phase transition is discussed from that point of view.
The authors present for the first time an effective interaction between two 4He^{4}\mathrm{He}4He atoms in a bath of liquid 3He^{3}\mathrm{He}3He at temperatures below 0.1 K. This interaction turns out to be, in its long-range limit, an oscillatory function of the 4He^{4}\mathrm{He}4He interatomic separation, the oscillations arising essentially from the sharp discontinuity in the momentum distribution at the Fermi surface. In addition, an expression is formulated for the oscillations in the density of the 3He^{3}\mathrm{He}3He bath induced by one 4He^{4}\mathrm{He}4He impurity, as well as a modified Friedel sum rule, which relates the relative amount of matter displaced by this impurity to the phase shifts pertaining to the impurity-3He^{3}\mathrm{He}3He quasiparticle scattering at the Fermi surface. Finally, the dependence of the results on temperature and on the 3He^{3}\mathrm{He}3He superfluidity gap is discussed.
The magnitude, sign, and momentum dependence of the effective interaction between He3{\mathrm{He}}^{3}He3 atoms in superfluid He4{\mathrm{He}}^{4}He4 are derived from the experimental data. Calculations of the Fermi-liquid parameters for the normal state of He3{\mathrm{He}}^{3}He3 in solution yield results in agreement with experiment; the temperature for a superfluid transition associated with the He3{\mathrm{He}}^{3}He3 is estimated to be \ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ \ifmmode^\circ\else\textdegree\fi{}K.
This thesis describes experimental work on dilute mixtures of ³He in 4He, mainly at millikelvin temperatures. The isotopic helium mixture has the unique property of remaining a miscible liquid down to the absolute zero temperature. In the mK regime, it consists of two very different components: perfectly superfluid 4He, and a weakly interacting degenerate Fermi liquid of ³He, predicted by theory to undergo transition to the superfluid state at an extremely low temperature. To discover that transition, new ways of cooling helium mixtures need to be developed, as it is not likely that the conventional method of nuclear demagnetization of copper can be improved to reach helium temperatures notably below the temperatures of order 0.1 mK attained so far. \n Adiabatic melting of 4He in the presence of liquid ³He is probably the most promising method of cooling helium mixtures to microkelvin temperatures. It produces helium mixture colder than the initial temperature as a direct consequence of the mixing of the isotopes. The starting configuration is attainable by pressurizing liquid helium mixture to its solidification pressure at a temperature below some tens of mK, as only 4He then enters the solid phase. This thesis describes an experiment in which the method of adiabatic melting was, for the first time, realized at sub-millikelvin temperatures, where the superfluidity of the pure ³He phase enables, in principle, a drastic decrease of temperature. In the experiments, cooling from initial temperatures between 0.3 and 0.9 mK was detected, temperature reduction remaining below a factor of two for recognized reasons of a technical rather than fundamental nature. \n A capacitive differential pressure transducer, constructed for the experiment, was used for high accuracy measurements of the temperature dependence of the melting pressure of helium mixtures with several ³He concentrations. The melting pressure is suitable for thermometry and carries information on the interactions of ³He particles in the mixture. Also, the response of a quartz tuning fork immersed in helium was studied. Its sensitivity to the properties of the surrounding fluid was utilized to determine the saturation concentration of dilute ³He across the entire accessible pressure range. The tuning fork was found to exhibit a complex pattern of anomalies attributed to resonant modes of second sound, or concentration waves, inside its cylindrical container.
The microscopic theory of a superfluid Fermi liquid at finite temperature is developed for the case of a pure system with SSS-wave pairing, and applied to the calculation of the static properties. As a function of $\ensuremath{\theta}\ensuremath{\equiv}\frac{T}{{T}_{c}}$ these properties are determined entirely by the Landau parameters F0{F}_{0}F0​, F1{F}_{1}F1​, Z0{Z}_{0}Z0​, etc., characterizing quasiparticle interactions in the normal phase. In particular the spin susceptibility $\ensuremath{\chi}$ and the density of the normal component ${\ensuremath{\rho}}_{n}$ are given by $\frac{\ensuremath{\chi}(\ensuremath{\theta})}{\ensuremath{\chi}(1)}=(1+\frac{1}{4}{Z}_{0})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{4}{Z}_{0}f(\ensuremath{\theta})]},$ $\frac{{\ensuremath{\rho}}_{n}}{\ensuremath{\rho}}=(1+\frac{1}{3}{F}_{1})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{3}{F}_{1}f(\ensuremath{\theta})]},$ where the universal function $f(\ensuremath{\theta})\ensuremath{\equiv}\ensuremath{-}{[\ensuremath{\nu}(0)]}^{\ensuremath{-}1}{\ensuremath{\Sigma}}_{\mathrm{p}}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is the "effective density of states near the Fermi surface" relative to its value $\ensuremath{\nu}(0)$ in the normal phase. Thus the often-quoted expression ${\ensuremath{\rho}}_{n}=\frac{1}{3}{\ensuremath{\Sigma}}_{\mathrm{p}}{\mathrm{p}}^{2}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is valid for an interacting system only in the limit $T\ensuremath{\rightarrow}0$. In the latter part of the paper a simple phenomenological theory of "Fermi-liquid" effects on $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ is developed for arbitrary conditions (including the presence of impurities and pairing with $l\ensuremath{\ne}0$); it is found that under most circumstances explicit expressions for $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ may be obtained which involve only the Landau parameters and a suitably generalized effective density of states. The theory should apply to the possible superfluid phase of He3{\mathrm{He}}^{3}He3 and to most superconductors. It is suggested that the Knight shift in nontransition-metal superconductors should display some "Fermi-liquid" effects. The weak-field dc penetration depth $\ensuremath{\lambda}(T)$ is shown to be insensitive to such effects both in the Pippard limit and near Tc{T}_{c}Tc​; however, in a London superconductor at lower temperatures the correction to $\ensuremath{\lambda}(T)$ should be observable and yield a direct estimate of F1{F}_{1}F1​.
The paradoxes and disparities in the contemporary microscopic theory of superfluid helium (He–II) are discussed along with possible ways of resolving them by taking pair correlations of He4 atoms into consideration. It is shown that most paradoxes are associated with the commonly accepted initial assumption concerning the dominating role of single-particle Bose condensate (SPBC) in the quantum microstructure of the superfluid component ρs. The existence of intensive SPBC leads to a strong hybridization of the elementary excitation branches and to a common dispersion law for all boson branches, which is identified with the quasiparticle spectrum E(p) observed experimentally from slow neutron scattering in liquid helium. However, the stability of this spectrum during a transition through the λ-point and the large value of the gap in the vicinity of the “rotonic” minimum contradict both the Landau theoretical criterion of superfluidity and the small value of experimentally measured critical velocity. At the same time, a strong interaction between particles in the Bose liquid He4 strongly suppresses the SPBC which amounts to less than 1% of all He4 atoms and hence cannot be the main constituent of the superfluid component, unlike the case of a weakly nonideal Bose gas. Moreover, for a quite strong attraction between particles in a certain region of the momentum space, bound pairs of bosons can be formed in the superfluid Bose liquid, and a coherent pair condensate (CPC) analogous to the Cooper pair condensate in superconductors may appear. Such a strong CPC may completely suppress the weak SPBC. In this case, the one-particle spectrum ε(p) of elementary excitations does not hybridize with the collective (two-particle) spectrum and does not appear in the structure of the dynamic form factor S(p,ε), i.e., does not coincide with the spectrum measured from neutron scattering. The dispersion of one-particle spectrum is defined by the momentum dependence of the pair order parameter Ψ̃(p) and may have a minimum or a point of inflection at p≠0. This peculiarity in the one-particle spectrum of a Bose liquid with CPC but without SPBC vanishes together with Ψ̃(p) at the temperature Tc=Tλ of the phase transition from the superfluid to the normal state (unlike the rotonic minimum in the collective spectrum), while the corresponding critical velocity vc=min[ε(p)/p] vanishes at the λ-point in accordance with the Landau criterion and the experimental data. The assumption that the strong “Cooper-like” CPC is responsible for the quantum structure of the superfluid component ρs is confirmed indirectly by the successful application of the Justrow approximation (based on strong pair correlations) for describing the properties of liquid He4 and quantum liquid mixtures He3–4He on one hand, and by an anomalously large effective mass of He3 impurity atoms in He4, which is approximately equal to total mass of He3 and He4 atoms, thus pointing to the existence of helium atoms in superfluid liquid He–II. The value of the superfluid velocity circulation quantum in the Onsager–Feynman vortices in a Bose liquid with CPC but without SPBC is discussed as well as the critical velocities of superfluid He4 in ultrathin films and channels in which the creation and motion of quantum vortices are ruled out, and the quasiparticle spectrum undergoes dimensional quantization.
Several topics in quantum liquids are discussed including the elementary excitation spectrum of 3 He under pressure, spin polarized 3 He, and an early attempt to formulate a Fermi liquid theory to describe the low temperature thermodynamic and transport properties of the heavy electron system UPt3. The elementary excitation spectrum of ordinary liquid 3 He is calculated at several pressures using the polarization potential theory of Aldrich and Pines together with a simple model to describe the effect of multipair excitations. The calculated elementary excitation spectra, with no adjustable parameters, agree well with the recent experiment of Scherm et al. The effective interactions between quasi particles in fully spin polarized 3 He are obtained from physical arguments and sum rules. The interactions between two down spin impurities and that between an up and down spin are also deduced. The volume occupied by a down spin impurity is found to be 17% to 7% less than that occupied by an up· spin. Nonlocal scalar pseudopotentials of the Aldrich-Pines form are developed along with two extreme models for the momentum dependent backflow potential. The transport properties are calculated and found to be two to three orders of magnitude larger than those of the unpolarized liquid at the same density. The elementary excitation spectrum is found to be quite sensitive to the momentum dependence of the backflow potential. The regime of small polarization is considered next. Using the phenomenological model of Bedell and Sanchez-Castro together with an ansatz form for the spin flip interaction, a large increase in the singlet scattering rate as a function of polarization is obtained. The transport properties are calculated as a function of polarization and the viscosity obtained is found to be in good agreement with the recent experiment of P. Kopietz et al. Finally, an early attempt at describing the low temperature thermodynamic and transport properties of the heavy electron system UPt3 by an extension of Landau Fermi liquid theory is described. Although the Fermi surface geometry is treated naively, the transport calculations underscore the importance of electron-electron scattering in these systems.