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Physical Review B

Fermi-liquid theory for thin arbitrarily polarized 3 He films

David Z. Li, R. H. Anderson, Michael David Miller

Published Jun 8, 2012• 5 citations
Publisher

Abstract

We study the thermodynamic response, effective masses, and collective excitations for thin (submonolayer) ${}^{3}$He 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 ${}^{3}$He films on graphite and also thin ${}^{3}$He films in surface states of thin superfluid ${}^{4}$He 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 ${}^{3}$He 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 ${F}_{0}^{\text{s}}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{F}_{1}^{\text{a}}$ is provided.

DOI
10.1103/physrevb.85.224511

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We study the thermodynamic response, effective masses, and collective excitations for thin (submonolayer) 3{}^{3}3He films as a function of density and polarization using Fermi liquid theory. The Landau parameters f\ensuremath↑\ensuremath↑,f\ensuremath↑\ensuremath↓,f\ensuremath↓\ensuremath↓{f}^{\ensuremath{\uparrow}\ensuremath{\uparrow}},{f}^{\ensuremath{\uparrow}\ensuremath{\downarrow}},{f}^{\ensuremath{\downarrow}\ensuremath{\downarrow}}f\ensuremath↑\ensuremath↑,f\ensuremath↑\ensuremath↓,f\ensuremath↓\ensuremath↓ are obtained to quadratic order in the low-density sss-wave and ppp-wave TTT-matrix interaction parameters. Values for the effective interaction components are determined by fitting zero-polarization experimental data for the cases of thin 3{}^{3}3He films on graphite and also thin 3{}^{3}3He films in surface states of thin superfluid 4{}^{4}4He films. By fitting the interaction parameters, we can calculate the Landau parameters to all orders, and we show results for the behavior of F\ifmmode~\else˜\fi\ensuremathℓ\ensuremath↑\ensuremath↑for\ensuremathℓ\ensuremath⩽30{\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}30F\ifmmode~\else˜\fi​\ensuremathℓ\ensuremath↑\ensuremath↑​for\ensuremathℓ\ensuremath⩽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 3{}^{3}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{F}_{0}^{\text{s}}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{F}_{1}^{\text{a}}F0s​andF1a​ is provided.
We examine the thermodynamics of a two-dimensional Fermi gas model describing adsorbed 3{}^{3}3Hein a thin superfluid 4{}^{4}4Hefilm. The 3{}^{3}3Hesystem is characterized by both a hydrodynamic effective mass and a set of discrete transverse single-particle states that model the effects of the interaction with the substrate that supports the film. We show that the magnetization steps seen in experiment are a simple manifestation of the T=0KT=0\mathrm{K}T=0K equation of state. We prove that perfectly horizontal magnetization steps rigorously disappear at any finite temperature. We show that the thermal stability of the steps is determined by the larger of $\ensuremath{\Delta}\ensuremath{\epsilon}/2,$ one-half of the level spacing, and ${\ensuremath{\mu}}_{m}{\mathcal{H}}_{0},$ the magnetic energy. We derive the conditions under which there exist points in the phase space (termed invariant points) through which all low-temperature isotherms pass exponentially close. The invariant points appear for magnetization versus 3{}^{3}3Hecoverage, chemical potential versus magnetization, magnetic susceptibility versus coverage, and speed of sound squared versus coverage. We compare our calculated invariant points for the magnetization versus coverage with experiment and find good agreement. We show that there exist small regions of thermodynamic phase space in which the temperature derivative of the pressure is negative. We explain these anomalous regions as the result of 3{}^{3}3Heatoms ``spilling over'' from a full Fermi sea to an empty Fermi sea upon the application of a small increase in temperature. Through a Maxwell relation, this behavior can also be seen as in the appearance of a local peak in low-temperature entropy isotherms versus coverage. In the limit of a two state model, we calculate the specific heat and show that a Schottky peak develops in the low 3{}^{3}3Hecoverage limit. We calculate the magnetic susceptibility and predict that it should exhibit a steplike structure versus 3{}^{3}3Hecoverage similar to that of the magnetization. We calculate the speed of sound and show that it should exhibit zero-temperature discontinuities at the points where new Fermi seas begin to be occupied. Both the steplike structure in the magnetic susceptibility and the discontinuity-type structure in the speed of sound persist to temperatures on the order of 100 mK, and are analyzed in terms of invariant points in their respective phase diagrams. We show explicitly that, in the Fermi gas limit, the zero-field magnetic susceptibility is simply proportional to the isothermal compressibility.
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We calculate expressions for the state-dependent quasiparticle lifetime ${\ensuremath{\tau}}_{\ensuremath{\sigma}}$, the thermal conductivity $\ensuremath{\kappa}$, the shear viscosity $\ensuremath{\eta}$, and discuss the spin diffusion coefficient DDD for Fermi-liquid films in two dimensions. The expressions are valid for low temperatures and arbitrary polarization. In two dimensions, as in three dimensions, the integrals over the transition rates factor into energy and angular parts. However, the angular integrations contain a weak divergence. This problem is addressed using the method of K. Miyake and W. J. Mullin [Phys. Rev. Lett. 50, 197 (1983); J. Low Temp. Phys. 56, 499 (1984)]. The low-temperature expressions for the transport coefficients are essentially exact. We find that ${\ensuremath{\kappa}}^{\ensuremath{-}1}\ensuremath{\sim}TlnT$, and ${\ensuremath{\eta}}^{\ensuremath{-}1}\ensuremath{\sim}{T}^{2}$ for arbitrary polarizations $0\ensuremath{\le}\mathcal{P}\ensuremath{\le}1$. These results are in agreement with earlier zero-polarization results of H. H. Fu and C. Ebner [Phys. Rev. A 10, 338 (1974).], but differ from the temperature dependence of the shear viscosity found by D. S. Novikov (arXiv:cond-mat/0603184). They also differ from the discontinuous change of temperature dependence in DDD from zero to nonzero polarization that was discovered by Miyake and Mullin. We note that in two dimensions the shear viscosity requires a unique analysis. We obtain predictions for the density, temperature, and polarization dependence of $\ensuremath{\kappa},\phantom{\rule{0.16em}{0ex}}\ensuremath{\eta}$, and DDD for second-layer 3He^{3}\mathrm{He}3He films on graphite, and thin $^{3}\mathrm{He}\ensuremath{-}^{4}\mathrm{He}$ superfluid mixtures. For 3He^{3}\mathrm{He}3He on graphite, we find roughly an order of magnitude increase in magnitude for $\ensuremath{\kappa}$ and $\ensuremath{\eta}$ as the polarization is increased from 0 to 1. For DDD a similar large increase is predicted from zero polarization to the polarization where DDD is a maximum $(\ensuremath{\sim}0.74)$. We discuss the applicability of 3He^{3}\mathrm{He}3He thin films to the question of the existence of a universal lower bound for the ratio of the shear viscosity to the entropy density.
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