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arXiv:1406.6337 · Physical Review B

Shear viscosity in a non-Fermi-liquid phase of a quadratic semimetal

Philipp T. Dumitrescu

Published Sep 2, 2015• 12 citations
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Abstract

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.

DOI
10.1103/physrevb.92.121102

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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 \ensuremathη\ensuremath{\eta}\ensuremathη. The ratio of shear viscosity to entropy density \ensuremathη/s\ensuremath{\eta}/s\ensuremathη/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, \ensuremathη/s\ensuremath{\eta}/s\ensuremathη/s is a universal number and we find it to be consistent with a bound proposed from gauge-gravity duality.
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.
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.
The ratio between the shear viscosity and the entropy η/s\eta/sη/s is considered a universal measure of the strength of interactions in quantum systems. In relativistic systems, this quantity was conjectured to have a universal lower bound (1/4π)ℏ/kB(1/4\pi)\hbar/k_{B}(1/4π)ℏ/kB​, 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 η/s∝T\eta/s\propto Tη/s∝T is \emph{unbounded}, scaling towards zero with decreasing temperature TTT in the perturbative limit. We find that the hydrodynamic scattering time between collisions τ∼ℏ/α2vkF\tau\sim\hbar/\alpha^{2}vk_{F}τ∼ℏ/α2vkF​, with vkFvk_{F}vkF​ the energy scale set by the radius of the ring and α\alphaα the fine structure constant. We suggest that the lower bound criteria should be modified to account for unscreened relativistic systems with a Fermi surface.
In an isotropic strongly interacting quantum liquid without quasiparticles, general scaling arguments imply that the dimensionless ratio $({k}_{B}/\ensuremath{\hbar})\phantom{\rule{0.16em}{0ex}}\ensuremath{\eta}/s$, where $\ensuremath{\eta}$ is the shear viscosity and sss is the entropy density, is a universal number. We compute the shear viscosity of the Ising-nematic critical point of metals in spatial dimension d=2d=2d=2 by an expansion below d=5/2d=5/2d=5/2. The anisotropy associated with directions parallel and normal to the Fermi surface leads to a violation of the scaling expectations: $\ensuremath{\eta}$ scales in the same manner as a chiral conductivity, and the ratio $\ensuremath{\eta}/s$ diverges at low temperature (TTT) as ${T}^{\ensuremath{-}2/z}$, where zzz is the dynamic critical exponent for fermionic excitations dispersing normal to the Fermi surface.
The thermodynamic and transport properties of the unitary Fermi gas at finite temperature TTT are governed by a quantum critical point at T=0T=0T=0 and zero density. We compute the universal shear viscosity to entropy ratio $\ensuremath{\eta}/s$ in the high-temperature quantum critical regime $T\ensuremath{\gg}|\ensuremath{\mu}|$ and find that this strongly coupled quantum fluid comes close to perfect fluidity $\ensuremath{\eta}/s=\ensuremath{\hbar}/(4\ensuremath{\pi}{k}_{B})$. Using a controlled large-NNN expansion, we show that already at the first nontrivial order the equation of state and the Tan contact density CCC agree well with the most recent experimental measurements and theoretical Luttinger-Ward and bold diagrammatic Monte Carlo calculations.