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arXiv:1911.06431 · Physical review. B./Physical review. B

Hydrodynamic transport in the Luttinger-Abrikosov-Beneslavskii non-Fermi liquid

Julia M. Link, Igor F. Herbut

Published Mar 26, 2020• 12 citations
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Abstract

We determine the shear viscosity and the dc electrical conductivity of interacting three-dimensional Luttinger semimetals, which have a quadratic band touching point in the energy spectrum, in the hydrodynamic regime. It is well known that when the chemical potential is right at the band touching point, the long-range Coulomb interaction induces the Luttinger-Abrikosov-Beneslavskii (LAB) phase at $T=0$, which is an interacting, scale-invariant, non-Fermi-liquid state of electrons. Upon combining the renormalization-group (RG) analysis near the upper critical spatial dimension of 4 with the Boltzmann kinetic equation, we determine the universal ratio of viscosity over entropy and the electrical dc conductivity of the system at the interacting LAB fixed point of the RG flow. The projection of the Coulomb interaction on the eigenstates of the system is found to play an important quantitative role for the scattering amplitude in the collision integral, and the so-called Auger processes make a large numerical contribution to the inverse scattering time in the transport quantities. The obtained leading-order result suggests that the universal ratio of the viscosity over entropy, when extrapolated to the physical three spatial dimensions, is above, but could be rather close to, the Kovtun-Son-Starinets lower bound.

DOI
10.1103/physrevb.101.125128

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We determine the shear viscosity and the dc electrical conductivity of interacting three-dimensional Luttinger semimetals, which have a quadratic band touching point in the energy spectrum, in the hydrodynamic regime. It is well known that when the chemical potential is right at the band touching point, the long-range Coulomb interaction induces the Luttinger-Abrikosov-Beneslavskii (LAB) phase at T=0T=0T=0, which is an interacting, scale-invariant, non-Fermi-liquid state of electrons. Upon combining the renormalization-group (RG) analysis near the upper critical spatial dimension of 4 with the Boltzmann kinetic equation, we determine the universal ratio of viscosity over entropy and the electrical dc conductivity of the system at the interacting LAB fixed point of the RG flow. The projection of the Coulomb interaction on the eigenstates of the system is found to play an important quantitative role for the scattering amplitude in the collision integral, and the so-called Auger processes make a large numerical contribution to the inverse scattering time in the transport quantities. The obtained leading-order result suggests that the universal ratio of the viscosity over entropy, when extrapolated to the physical three spatial dimensions, is above, but could be rather close to, the Kovtun-Son-Starinets lower bound.
We study finite temperature transport in the Luttinger-Abrikosov-Beneslavskii phase---an interacting, scale invariant, non-Fermi-liquid phase found in quadratic semimetals. We develop a kinetic equation formalism to describe the dc transport properties, which are dominated by collisions, and compute the shear viscosity $\ensuremath{\eta}$. The ratio of shear viscosity to entropy density $\ensuremath{\eta}/s$ is a measure of the strength of interaction between the excitations of a quantum fluid. As a consequence of the quantum critical nature of the system, $\ensuremath{\eta}/s$ is a universal number and we find it to be consistent with a bound proposed from gauge-gravity duality.
Eighty years ago Eyring proposed that the shear viscosity of a liquid, ηηη, has a quantum limit η≳nℏη\gtrsim n\hbarη≳nℏ where nnn 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 ηs≥ℏ4πkB\fracη{s}\geq \frac{\hbar}{4πk_{B}}sη​≥4πkB​ℏ​ for the ratio between the shear viscosity and the entropy density, sss. 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 3^{3}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/T21/T^{2}1/T2 dependence, where TTT is the temperature. With increasing temperature and interaction strength UUU 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 3^{3}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.
The ratio between the shear viscosity and the entropy $\ensuremath{\eta}/s$ is considered a universal measure of the strength of interactions in quantum systems. This quantity was conjectured to have a universal lower bound $(1/4\ensuremath{\pi})\ensuremath{\hbar}/{k}_{B}$, which indicates a very strongly correlated quantum fluid. By solving the quantum kinetic theory for a nodal-line semimetal in the hydrodynamic regime, we show that $\ensuremath{\eta}/s\ensuremath{\propto}T$ violates the universal lower bound, scaling toward zero with decreasing temperature TTT in the perturbative limit. We find that the hydrodynamic scattering time between collisions is nearly temperature independent, up to logarithmic scaling corrections, and can be extremely short for large nodal lines, near the Mott-Ragel-Ioffe limit. Our finding suggests that nodal-line semimetals can be very strongly correlated quantum systems.
We study the quantum electrodynamics of Luttinger fermions with quadratic band-crossing dispersion in three dimensions. The model can be viewed as the low-energy effective theory of a putative U(1) quantum spin liquid with fermionic Luttinger spinons, or as an extension of the Luttinger-Abrikosov-Beneslavskii (LAB) model that accounts for transverse gauge fluctuations with finite photon velocity. Aided by a renormalization group analysis below four dimensions, we elucidate the presence and stability of quantum critical phenomena in this model. We find that the non-Fermi liquid LAB phase is stable against gauge fluctuations and can thus also be viewed as a U(1) spin liquid with gapless Luttinger spinons. We discover a multicritical point with Lifshitz scaling that corresponds to a time-reversal symmetry-breaking quantum phase transition from the LAB state to a chiral spin liquid with spinon Landau levels and birefringent emergent photons. This multicritical point is characterized by a finite fermion-photon coupling in the infrared and can be viewed as a fermionic analog of the Rokhsar-Kivelson point in three-dimensional quantum dimer models.
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.
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.