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.
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.
We carried out experiments on a spin polarized 3He- 4He mixture with 3He concentration x 3 = 6.26 × 10−4, and on pure 3He liquid. Spin polarization affects the transport properties of these Fermi systems. The effect on momentum transport was studied by using a vibrating-wire viscometer to measure viscosity of the 3He-4He mixture over the temperature range 6.09 mK–100 mK in 7.96 T and 1.00 T magnetic fields. A large viscosity increase was observed upon application of the 7.96 T magnetic field for temperature T < TF(TF = 19.5 mK is the Fermi temperature). The observed viscosity is in very good agreement with theoretical calculations for a dilute Fermi gas by Jeon and Mullin [1988, 1989] and Mullin and Jeon [1992]. The polarization effect on spin transport was investigated by measuring the transverse diffusion coefficient D ⊥ in pure 3He liquid at saturated vapor pressure and at 15.85 bar over the temperature range 4.5 mK–159 mK in a 7.96 T magnetic field. We used a pulsed NMR spin echo technique in a field gradient of 16.0 G/cm to do the measurements and fits to the Leggett equations [1970] to obtain D⊥. For T < 20 mK, we found D⊥ is less than measured in earlier experiments at lower magnetic fields. D⊥ does not increase with decreasing temperature as 1/T2, but appears to approach a constant as T → 0 while it is expected that the longitudinal spin diffusion coefficient D∥ ∝ 1/ T2. This is called spin diffusion anisotropy and it was observed for the first time in our 3He liquid experiments. The anisotropy temperature we determined for 3He liquid was Ta = 16.4 ± 2.2 mK at saturated vapor pressure and in a 7.96 T magnetic field. The transverse spin diffusion in 3 He results agree qualitatively with theories proposed by Meyerovich and Musaeflan [1992, 1994]. They also agree qualitatively with theories proposed by Golosov and Ruckenstein [1995, 1998] by extrapolation of the dilute theory to a strongly interacting system.
We study the thermodynamic response, effective masses, and collective excitations for thin (submonolayer) 3He films as a function of density and polarization using Fermi liquid theory. The Landau parameters ${f}^{\ensuremath{\uparrow}\ensuremath{\uparrow}},{f}^{\ensuremath{\uparrow}\ensuremath{\downarrow}},{f}^{\ensuremath{\downarrow}\ensuremath{\downarrow}}$ are obtained to quadratic order in the low-density s-wave and p-wave T-matrix interaction parameters. Values for the effective interaction components are determined by fitting zero-polarization experimental data for the cases of thin 3He films on graphite and also thin 3He films in surface states of thin superfluid 4He films. By fitting the interaction parameters, we can calculate the Landau parameters to all orders, and we show results for the behavior of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{\ensuremath{\ell}}^{\ensuremath{\uparrow}\ensuremath{\uparrow}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\ensuremath{\ell}\ensuremath{\leqslant}30$. With knowledge of the Landau parameters we calculate the polarization dependence of the state-dependent effective masses, compressibility, spin susceptibility, and zero-sound spectra for the mobile submonolayer range of areal densities. Our results predict a dramatic decrease in the effective masses for 3He on graphite as a function of polarization at higher coverages. At fixed density, the compressibility is shown to decrease monotonically with increasing polarization. At small polarization the inverse spin susceptibility is largest for small density whereas at large polarization it is largest for large density. We show that zero sound will propagate at all densities and polarizations whereas spin zero sound does not propagate in these systems. The condition for thermodynamic stability in an arbitrarily polarized Fermi liquid film is derived and discussed. We show explicitly the Fermi surface distortion due to the presence of a zero-sound mode. A table of predicted values for the zero polarization Fermi liquid parameters F0sandF1a is provided.
Landau (1958) gave a rigorous microscopic derivation of his phenomenological Fermi -liquid theory starting from the first principles Hamiltonian of strongly interacting Fermions. The derivation is based on an asymptotic expansion in small parameters such as q/kF, bc.>/EF, kT/EF, where q and c.> represent the wave numbers and frequencies of real excitations of the system. Landau showed that the dynamics of these low-lying excitations (quasiparticles and collective modes) is governed by classical transport equations with all quantum many-body effects absorbed into effective interaction parameters. Landau's approach has been generalized to superfluid Fermi liquids and superconductors by Leggett (1965) and, going beyond linear response, by Eliashberg (1971). Their ideas merged into a recent review on the quasiclassical approach to superfluid 3 He by Serene and Rainer (1983). I will use the notation of this review throughout this paper. Landau's transport theory together with its offsprings will be called 'quasiclassical theory in the following. In this context it is important to mention that Eilenberger (1968) and Larkin and Ovchinnikov (1968) have shown that the complete theory of superconductors can be formulated as a quasiclassical transport theory. Surprisingly, this theory permits calculating microscopic properties, e.g. the quasiparticle excitation spectrum, which are predominantly governed by quantum effects.
Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.
We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ4 field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large. We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.
Abstract. I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant Kρ, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and phonons, and list important open problems. WHAT IS A LUTTINGER LIQUID ANYWAY? Ordinary, three-dimensional metals are described by Fermi liquid theory. Fermi liquid theory is about the importance of electron-electron interactions in metals. It states that there is a 1:1-correspondence between the low-energy excitations of a free Fermi gas, and those of an interacting electron liquid which are termed “quasiparticles” [1]. Roughly speaking, the combination of the Pauli principle with low excitation energy (e.g. T ≪ EF) and the large phase space available in 3D, produces