In interaction-dominated two-dimensional electron gases at intermediate temperatures, electron transport is not diffusive as in the conventional Drude picture but instead hydrodynamic. The relevant transport coefficient in this regime is the shear viscosity. Here, we develop a numerically exact basis expansion to solve the Fermi liquid equation, and apply it to compute the shear viscosity of the electron gas with screened Coulomb interactions. Our calculations are valid at all temperatures and in particular describe the response beyond the asymptotic low-temperature limit, where perturbative approaches exist. We show that even in this low-temperature limit, there is a nonanalytic exchange contribution to the shear viscosity, highlighting the need for a full nonperturbative solution of the Fermi liquid equation. We hope that the techniques developed in this work will serve as a platform to determine the response of interacting Fermi liquids.
We analyze how the logarithmic renormalizations in the Cooper channel affect the nonanalytic temperature dependence of the specific heat coefficient $\ensuremath{\gamma}(T)\ensuremath{-}\ensuremath{\gamma}(0)=A(T)T$ in a two-dimensional Fermi liquid. We show that
A(T) is expressed exactly in terms of the fully renormalized backscattering amplitude, which includes the renormalization in the Cooper channel. In contrast to the one-dimensional case, both charge and spin components of the backscattering amplitudes are subject to this renormalization. We show that the logarithmic renormalization of the charge amplitude vanishes for a flat Fermi surface when the system becomes effectively one dimensional.
The hydrodynamic regime of electron transport has been recently realized in conductors with ultralow densities of defects. Although relaxation processes in two-dimensional (2D) fluids have been studied in many theoretical works, the viscosity of the realistic Fermi gas of 2D electrons having a quadratic energy spectrum and interacting by Coulomb's law has not been reliably determined either in theory or in experiment. Here, we construct a theory of viscosity and thermal conductivity in such a system. We compare the calculated viscosity of the 2D electron Fermi gas and the previously known viscosity of a 2D Fermi liquid with available experimental data extracted from the hydrodynamic negative magnetoresistance of the best-quality GaAs quantum wells. Based on this comparison, we argue that measurements of the temperature dependence of the viscosity can allow us to trace the transition between an electron Fermi liquid and a Fermi gas.
Eighty years ago Eyring proposed that the shear viscosity of a liquid,
η, has a quantum limit
η≳nℏ where
n is the density of the fluid. Using holographic duality and the AdS/CFT correspondence in string theory Kovtun, Son, and Starinets (KSS) conjectured a universal bound
sη≥4πkBℏ for the ratio between the shear viscosity and the entropy density,
s. Using Dynamical Mean-Field Theory (DMFT) we calculate the shear viscosity and entropy density for a fermionic fluid described by a single band Hubbard model at half filling. Our calculated shear viscosity as a function of temperature is compared with experimental data for liquid
3He. At low temperature the shear viscosity is found to be well above the quantum limit and is proportional to the characteristic Fermi liquid
1/T2 dependence, where
T is the temperature. With increasing temperature and interaction strength
U there is significant deviation from the Fermi liquid form. Also, the shear viscosity violates the quantum limit near the crossover from coherent quasi-particle based transport to incoherent transport (the bad metal regime). Finally, the ratio of the shear viscosity to the entropy density is found to be comparable to the KSS bound for parameters appropriate to liquid
3He. However, this bound is found to be strongly violated in the bad metal regime for parameters appropriate to lattice electronic systems such as organic charge transfer salts.
We investigate the temperature dependence of the shear viscosity and spin diffusion in a two-dimensional Fermi gas with contact interactions, as realized in ultracold atomic gases. We describe the transport coefficients in terms of a Boltzmann equation and present a full numerical solution for the degenerate gas. In contrast to previous works we take the medium effects due to finite density fully into account. This effect reduces the viscosity-to-entropy ratio $\ensuremath{\eta}/s$ by a factor of 3, and similarly for spin diffusion. The trap-averaged viscosity agrees well with recent measurements by Vogt et al. [Phys. Rev. Lett. 108, 070404 (2012)].
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
D 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
D 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
D for second-layer
3He films on graphite, and thin $^{3}\mathrm{He}\ensuremath{-}^{4}\mathrm{He}$ superfluid mixtures. For
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
D a similar large increase is predicted from zero polarization to the polarization where
D is a maximum $(\ensuremath{\sim}0.74)$. We discuss the applicability of
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.