Landau Fermi Liquid Theory Research Articles

Landau theory of relativistic Fermi liquids

A relativistic extension of the Landau Fermi liquid theory, applicable to the study of high density matter, is developed. Consequences of Lorentz invariance in the theory are explored. The formalism is illustrated by a study of relativistic Fermi systems weakly interacting via scalar and vector meson exchange. Second order exchange energies for both massless scalar and massless vector interactions are calculated in terms of Landau parameters on the Fermi surface. Zero sound and “color-plasma oscillations” are studied in quark matter with SU(3) color gluon coupling.

The logarithmic temperature dependence of the susceptibility of paramagnetic Fermi liquids or metals

For the logarithmic temperature dependence of the magnetic susceptibility of normal paramagnetic Fermi liquids or metals, the discrepancy between predictions from the Landau Fermi-liquid theory and those from microscopic theory is discussed. The coefficient of the T2 ln T term in the temperature dependence of the susceptibility is not triply (or quadruply) enhanced, as approximate microscopic theory predicts, but doubly enhanced with respect to the enhancement factor. Similarly the term next to T2 ln T is not of the form T3 ln T but of the form T4 ln T.

Specific heat of a normal Fermi liquid. II. Microscopic approach

We consider the thermodynamic properties of a normal Fermi liquid, starting from the expression for the thermodynamic potential in terms of the fully renormalized single-particle propagator. It is shown that the entropy may be expressed as the dynamical quasiparticle contribution, given by the usual quasiparticle result for the entropy evaluated using as quasiparticle energies the poles of the single-particle propagator, plus a correction term. The correction term is shown to come from terms in perturbation theory which have at least two vanishing energy denominators. Using a reduced-graph technique similar to that used for analyzing analytic properties of Feynman amplitudes in field theory we investigate in detail the terms which have two vanishing energy denominators. The results are then applied to calculate the contribution from long-wavelength fluctuations to the ${T}^{3}$ $\mathrm{ln}T$ term in the specific heat of a normal Fermi liquid; agreement is obtained with earlier calculations based on Landau Fermi-liquid theory. The differences between various definitions of the quasiparticle energy, and the effect of quasiparticle lifetimes on thermodynamic properties are also considered.

Experimental determination of the Landau Fermi-liquid-theory parameters: Spin waves in sodium and potassium

This paper presents the results of a study made on spin waves in high-purity sodium and potassium foils at cryogenic temperatures (1.4-10\ifmmode^\circ\else\textdegree\fi{}K). The measurements were made using the microwave transmission technique at $X$-band microwave frequencies. Spin waves in an alkali metal are a mode of excitation of the conduction electrons that exists solely due to the interactions, or many-body effects, among the electrons. We present a description of the general properties of the spin waves, such as their dependence upon sample purity, temperature, magnetic field orientation, thickness, and other relevant parameters. By comparing the experimental data with detailed theoretical line-shape formulas derived from the Landau Fermi-liquid theory, we are able to deduce values for the first three Legendre coefficients of the spin-dependent part of the Landau interaction function. The values are ${B}_{0}=\ensuremath{-}0.215\ifmmode\pm\else\textpm\fi{}0.03$, ${B}_{1}=\ensuremath{-}0.005\ifmmode\pm\else\textpm\fi{}0.04$, and ${B}_{2}=0.0\ifmmode\pm\else\textpm\fi{}0.05$ for sodium; and ${B}_{0}=\ensuremath{-}0.285\ifmmode\pm\else\textpm\fi{}0.02$, ${B}_{1}=\ensuremath{-}0.06\ifmmode\pm\else\textpm\fi{}0.03$, and ${B}_{2}=0.0\ifmmode\pm\else\textpm\fi{}0.05$ for potassium. The numbers we obtained are compared with results from other methods, both experimental and theoretical, where applicable. A discussion of the difficulties encountered in determining the coefficients is given along with a description of the experimental techniques we have used. Finally, we consider how these measurements might be extended in the future to yield further information on interacting conduction-electron systems.

Phase Transition in Bose Liquid. II

We apply the method proposed in a previous paper to the calculation of thermodynamical functions. This method resembles the Landau Fermi liquid theory in the sense that the energy, excitation energy, etc., are expressed by the functionals of occupation numbers of quasi-particles. It is shown that if the excitation energy of a quasi-particle Ep is proportional to pr with 2>r>3/2, then the specific heat cp behaves as log(Tc/IT-T.J). We discuss also the thermal behaviour of condensation density. It is shown that the :fluctuation of condensa­ tion density becomes infinite on the transition line. § I. Introduction In a previous paper referred to as I, we have shown that the interacting Bose gas can be worked out in a way similar to the Landau Fermi liquid theory1> by considering r>3/2, then the specific heat cp behaves as log(T.//T-T./), and if epocp313, then Cp-'log[log(T.//T-T./)]. These results are different from those of Patashinskii and Pokrovskii 2> (hereafter referred to as PP).

Effects of Anisotropy on Magnetoplasma Waves in a Charged Fermi-Liquid System

The propagation of the plasma modes in the presence of a static magnetic field and in an electron liquid is studied in terms of the Landau Fermi-liquid theory for a system with an anisotropic Fermi surface. Two types of anisotropy are studied: (a) Weak anisotropy; the resonant frequency of the normal modes is calculated in the long-wavelength limit, $q\ensuremath{\rightarrow}0$, for a system with a nearly spherical Fermi surface. It is found that the deviation from the isotropic case is of the linear order of the anisotropy. Application to the alkali metals is briefly discussed. Effects of weak anisotropy of the Landau $F$ function is also studied. (b) Strong anisotropy; the dispersion relations of the modes propagating parallel to the magnetic field near the cyclotron frequency ${\ensuremath{\omega}}_{c}$ and its harmonics $m{\ensuremath{\omega}}_{c}$ are calculated for a system with a simplified Fermi surface which satisfies the Gor'kov-Dzyaloshinskii condition. It is found that, except for the $m=1$ modes, all higher-$m$ modes have large finite (or infinite) cutoff wave vectors even in the weak coupling limit; while in the isotropic case, all cutoff wave vectors approach zero in the same limit. It is suggested that this simplified model may be relevant to the noble metals.

Electron Spin Waves in Nonmagnetic Conductors: Self-Consistent-Field Theory

A general derivation of the electrodynamic response of a quantum many-electron gas in a nonmagnetic conducting solid immersed in an applied magnetic field is given. Self-consistent-field (SCF) theory of the equation of motion of the one-electron density matrix is used in such a way as to include, from the outset, one-electron effects such as complex energy band structure, spin-orbit coupling, and spin paramagnetism. This treatment specifically omits exchange effects such as those encountered in an extended random-phase approximation or Landau-Fermi liquid theory. The aim is to study the properties of wave propagation in the gas, looking for spin waves and/or characteristic effects which uniquely involve the spin degree of freedom and the paramagnetism of the equilibrium state. The derived results contain terms which have been neglected previously and terms which do not evolve from a simple generalization of previous treatments of the quantum dielectric theory of a Fermi gas. There are interesting spin effects in the plasma wave properties both with and without spin-orbit mixing of the one-electron states. In an effective mass approximation for the one-electron states, it is shown that there are resonances and cutoffs associated with electron spin resonance in the transverse wave propagation (both perpendicular and parallel to the magnetic field). For spin-orbit mixed states, one finds zeros of the longitudinal dielectric constant (for long wavelength) near the electron spin-flip frequency. The mechanism for the spin wave associated with this zero is a correlation of the motion of electrons with "opposite spins" by the long-range Coulomb field through the spin-orbit coupling of the crystalline eigenstates.

What is a Quasi-Particle?

The concept of a quasi-particle excitation in an interacting many-body system will be discussed from both the physical and the mathematical points of view. The physical origin of mass enhancement, wave function renormalization, interactions between quasi-particles, etc. will be presented. Landau's Fermi liquid theory, including the quasi-particle kinetic equation, will be reviewed. Finally, the domain of validity of the quasi-particle approximation will be discussed.

Transmission Electron Spin Resonance in Dilute Copper-Chromium Alloys

Utilizing the transmission conduction-electron spin-resonance technique, we have observed the spin resonance associated with the combined local-moment-conduction-electron system in Cu-Cr over the temperature range 1.4-40\ifmmode^\circ\else\textdegree\fi{}K and concentration range 4-36 ppm. The measured $g$ values, linewidths, and lineshape parameters are interpreted utilizing the Hasegawa model extended to include electron magnetization diffusion and additional local-moment relaxation modes. The predictions of this phenomenological model successfully account for the behavior of the system and we are able to deduce the following parameters: the intrinsic $g$ value for Cr in Cu (2.006\ifmmode\pm\else\textpm\fi{}0.002), the spin-flip scattering rate of the conduction electrons in copper by the Cr impurities ($\frac{1}{{T}_{\mathrm{sl}}}=4.8\ifmmode\times\else\texttimes\fi{}{10}^{+7}$ ${\mathrm{sec}}^{\ensuremath{-}1}$/ppm), the intrinsic (phenomenological) relaxation rate of the Cr to the copper lattice ($\frac{1}{{T}_{\mathrm{dl}}}=2.1\ifmmode\times\else\texttimes\fi{}{10}^{9}$ ${\mathrm{sec}}^{\ensuremath{-}1}$), a lower limit to the effective $s\ensuremath{-}d$ exchange coupling ($|J|\ensuremath{\ge}0.3$ eV), and the ratio of the local moment to conduction-electron spin susceptibilities. Combining this ratio with a separate measurement of the local-moment susceptibility, we are able to determine the conduction-electron spin susceptibility of pure copper. This value, (1.08\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}7}$ emu/g, is compared with the free-electron theory from which ${F}_{0}$, the first spin-dependent parameter of the Landau-Fermi liquid theory, is deduced.

Ground-state and low-excited properties of liquid 3He calculated with a two-body potential

Although scattering phase shifts reveal that the interaction of two free 3He atoms with the momenta and spins of quasi-particles in liquid 3He is attractive, the experimental spin-averaged effective quasi-particle interaction ( F s) defined in Landau's Fermi-liquid theory is repulsive in liquid 3He. To understand the role of many-body effects in the repulsive interaction, we have calculated the bulk properties of liquid 3He using the Bethe-Goldstone method and using Goldstone perturbation theory. The compressibility yields a value for F 0 s, the angular average of F s. The potential used in our calculations is a hard-core square-well potential adjusted to approximate the phase shifts of the de Boer 6–12 potential. The properties we computed with the Bethe-Goldstone method give good agreement with experiment for effective masses m B ∗ between 1.0 and 1.4 times the true mass. We were unable to determine m B ∗ self-consistently. In liquid 3He three-body correlations are expected to be important. Using techniques developed for nuclear matter by Bethe and his co-workers, we estimated all two and three-body Goldstone diagrams. The properties we computed are in very good agreement with experiment. Since the net three-body energy is obtained in our calculation by adding repulsive and attractive contributions much larger in absolute value than their sum, these numbers should not be taken too seriously. Three-body diagrams contribute most of the binding energy in our calculation. The three-body repulsive energy is about 60% of the two-body repulsive energy. We describe qualitatively how the Pauli principle helps suppress the contributions of higher clusters. As particles are added to a self-bound many-body system or a system under pressure, the binding energy per particle decreases. Those particles originally present are rearranged into an “excited” state. In both the Bethe-Goldstone and Goldstone-perturbation-theory calculations, the rearrangement contributions are large enough to make the calculated effective quasi-particle interaction repulsive.

A quarks-on-springs model of the baryons: J. A. Shapiro. Department of Physics, University of California, Berkeley, California

The Fermi-liquid properties of the model nuclear system described by a relativistic quantum field theory are examined in terms of a relativistic extension of Landau's Fermi-liquid theory. The relativistic Landau parameters are derived microscopically from the ground state energy in the mean-field approximation, and are used to describe the thermodynamic properties of the system, such as the compressibility, the symmetry energy and the hydrodynamic sound velocities. We reproduce the previous results at nuclear saturation density (n0 = 0.19 fm−3) and extrapolate to all density regions. It is shown that the system exhibits instability against the long wavelength density fluctuations in the low density region (nB <0.70 n0) and becomes stable at and above the nuclear saturation density due to the relativistic reduction of the attractive scalar meson component in the quasiparticle interaction. In the extreme high-density region, we reproduce the correct causal results for sound velocities. The existence of collisionless zero-sound oscillation is also examined.

Fermi-Liquid Effects on Plasma Wave Propagation in Alkali Metals

The dispersion of plasma-wave propagation in the alkali metals sodium and potassium has been observed to deviate significantly from that expected for a noninteracting electron gas. Modifications of the dispersion have been calculated using the Landau Fermi-liquid theory and a quantitative evaluation of the first two moments of the interaction function has been made for the case of potassium.

Transport Phenomena in the Simple Metals

The low-excitation-energy phenomena characteristic of the simple metals is discussed. The model used takes into account both the electron-phonon and electron-electron interactions. The result obtained is an extended form of Landau's Fermi-liquid theory. The parameters of the theory are related to the underlying interactions, and the relationships corresponding to the relation of the effective mass with the "scattering function" discovered by Landau are developed. The theory is used to classify the renormalization effects in the interacting electron-phonon model of the metal. The results are valid when excitation energies no greater in magnitude than the Debye energy are involved, with the exception that the usual differential form of the Landau-Boltzmann transport equation does not hold if time variations of frequency comparable to the Debye frequency are considered.

Fermi liquid theory and spectrum of excitations of the Pb 208 nucleus

The Pb 208 excitation spectrum is calculated in terms of Landau's Fermi liquid theory. It is shown that the region of decrease of nuclear density is essential in this problem, and the parameters of the effective forward scattering amplitude are found. The calculated excitation spectrum is in agreement with experiment.

Finite Temperature Equilibrium Properties of LiquidHe3in Fermi Liquid Theory

Landau's Fermi liquid theory is used to obtain expressions for the respective ${T}^{3}$ and ${T}^{2}$ terms in lowtemperature expansions of the specific heat and magnetic susceptibility of liquid ${\mathrm{He}}^{3}$ at a temperature $T$. Such a calculation is essential toward checking the internal consistency of the Fermi liquid theory above 0.05\ifmmode^\circ\else\textdegree\fi{}K; but previous authors have considered only the zero-temperature limit when working with the Landau parameter, $f$. Coefficients of the finite-temperature corrections derived here are specific functions of $f$ evaluated on the Fermi surface. Existing experimental data on the pressure dependence of the sound velocity, specific heat, and magnetic susceptibility may be used to assign numerical values to these derivatives under the assumption that $f$ depends only on|k-k\ensuremath{'}|. This method of estimation is believed to place an upper limit to the magnitude of the ${T}^{3}$ term in the specific heat; and its resulting value is more than sufficient to account for the specific heat's deviating from linearity below 0.05\ifmmode^\circ\else\textdegree\fi{}K. The strong curvature in the specific heat versus temperature curve above 0.05\ifmmode^\circ\else\textdegree\fi{}K is thus concluded to be consistent with the Fermi liquid theory. The ${T}^{2}$ term in the magnetic susceptibility contains the same function as occurs in the ${T}^{3}$ term for the specific heat plus an additional correction related to derivatives of the spin-dependent part of $f$. A reasonable amount of accuracy is expected in the numerical estimate of this latter term. Assuming this to be the case, this theory is in accord with the susceptibility data of Meyer et al. together with the specific-heat data of Brewer et al. or Fairbank et al., but it cannot reconcile the specific heat data of Wheatley et al. with any existing susceptibility measurements.

Complex Dielectric Constant of Electron Fluid

A solution of the kinetic equation in L. D. Landau's Fermi liquid theory (J. Exp. Theoret. Phys. U.S.S.R., 30: adopted as the crudest approximation for the correlation function of the Coulomb interaction, an expression giving only the plasma frequency with the exchange effect is obtained. The surface impedance in the region of extreme anomalous skin effect is obtained. (L.N.N.)