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arXiv:2310.08556 · Physical review. B./Physical review. B

Linear-in-temperature conductance in two-dimensional electron fluids

Serhii Kryhin, Qiantan Hong, Leonid S Levitov

Published Feb 7, 2025• 10 citations
arXivHTMLPDF

Abstract

A hallmark of electron hydrodynamics---a transport regime dominated by electron collisions---is that collisions enhance conduction and reduce dissipation, causing resistance to drop and conductance to rise with temperature ($T$). Here, the authors present a microscopic study of two-dimensional electron fluids, predicting linear-in-$T$ scaling of hydrodynamic conductance. This behavior, contrasting with the ${T}^{2}$ scaling in three-dimensional Fermi liquids, arises from a cascade of multiple long-lived viscous modes unique to two-dimensional systems. The number of modes relevant for transport grows as $T$ decreases, resulting in a non-Fermi-liquid linear-in-$T$ behavior.

DOI
10.1103/physrevb.111.l081403

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A hallmark of electron hydrodynamics---a transport regime dominated by electron collisions---is that collisions enhance conduction and reduce dissipation, causing resistance to drop and conductance to rise with temperature (TTT). Here, the authors present a microscopic study of two-dimensional electron fluids, predicting linear-in-TTT scaling of hydrodynamic conductance. This behavior, contrasting with the T2{T}^{2}T2 scaling in three-dimensional Fermi liquids, arises from a cascade of multiple long-lived viscous modes unique to two-dimensional systems. The number of modes relevant for transport grows as TTT decreases, resulting in a non-Fermi-liquid linear-in-TTT behavior.
Two-dimensional electron systems offer an appealing platform to explore long-lived excitations arising due to collinear carrier scattering enabled by phase-space constraints at the Fermi surface. Recently it was found that these effects can boost excitation lifetimes over the fundamental bound set by Landau’s Fermi-liquid theory by a factor as large as (TF/T)α with α≈2. Long-lived degrees of freedom possess the capability to amplify the response to weak perturbations, producing lasting collective memory effects. This leads to non-Newtonian hydrodynamics in 2D electron fluids driven by multiple viscous modes with scale-dependent viscosity. We describe these modes as Fermi surface modulations of odd parity evolving in space and time, and discuss their implications for experimental studies of electron hydrodynamics.
We study thermal conductivity in one-dimensional electronic fluids combining kinetic [R. Samanta, I. V. Protopopov, A. D. Mirlin, and D. B. Gutman, Thermal transport in one-dimensional electronic fluid, Phys. Rev. Lett. 122, 206801 (2019)] and hydrodynamic [I. V. Protopopov, R. Samanta, A. D. Mirlin, and D. B. Gutman, Anomalous hydrodynamics in one-dimensional electronic fluid, Phys. Rev. Lett. 126, 256801 (2021)] theories. The kinetic approach is developed by partitioning the Hilbert space into bosonic and fermionic sectors. We focus on the regime where the long-living thermal excitations are fermions and compute thermal conductivity. From the kinetic theory standpoint, the fermionic part of thermal conductivity is normal, while the bosonic one is anomalous, that scales as ω–1/3 and thus dominates in the infrared limit. The multi-mode hydrodynamic theory is obtained by projecting the fermionic kinetic equation on the zero modes of its collision integral. On a bare level, both theories agree and the thermal conductivity computed in hydrodynamic theory matches the result of the kinetic equation. The interaction between hydrodynamic modes leads to renormalization and consequently to anomalous scaling of the transport coefficients. In a four-mode regime, all modes are ballistic and the anomaly manifests itself in Kardar-Parisi-Zhang-like broadening with asymmetric power-law tails. “Heads” and “tails” of the pulses contribute equally to thermal conductivity, leading to ω–1/3 scaling of heat conductivity. In the three-mode regime, the system is in the universality class of a classical viscous fluid [Herbert Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154, 1191 (2014); O. Narayan and S. Ramaswamy, Anomalous heat conduction in one-dimensional momentum-conserving systems, Phys. Rev. Lett. 89, 200601 (2002)].
A precise characterization of the recently discovered crossover to hydrodynamic transport in electron liquids, and in particular of a conjectured exotic odd-parity transport regime, requires a full solution of the Fermi-liquid collision integral at all temperatures beyond state-of-the-art analytic leading-logarithmic approximations. We develop a general basis expansion of the linearized collision integral applicable at all temperatures, and employ this to determine the decay rates of Fermi surface perturbations in two-dimensional electron liquids. In particular, we provide exhaustive data on the decay rates for a new odd-parity regime, in which parity-even Fermi surface deformations decay rapidly but odd-parity modes are anomalously longlived. We find that this transport regime exists at fairly large temperatures $T\ensuremath{\lesssim}0.1\phantom{\rule{0.16em}{0ex}}{T}_{F}$, putting this regime within the reach of current experiments.
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.
Clean two-dimensional Fermi liquids are now known to exhibit an intermediate \emph{tomographic} regime, between ballistic and Navier--Stokes transport, caused by the anomalously slow relaxation of parity-odd multipolar deformations of the Fermi surface. Here we show that this anomaly extends to the dynamical realm. Starting from a microscopic numerical evaluation of the linearized electron--electron collision operator, we find that the finite-frequency nonlocal conductivity is controlled at low frequency by a single hydrodynamic pole, σ(q,ω)=D(q)/(iω+η⋆qz)σ(q,ω)=\mathcal{D}(q)/(iω+η_\star q^z)σ(q,ω)=D(q)/(iω+η⋆​qz), with dynamical exponent z=4/3z=4/3z=4/3 and superdiffusive viscosity η⋆η_\starη⋆​. Remarkably, the pole residue itself is scale dependent and obeys D(q)∼q−α\mathcal{D}(q)\sim q^{-α}D(q)∼q−α with α=1/3α=1/3α=1/3, so the dynamical properties are described by two separate exponents rather than one. We interpret the residue suppression using a Krylov-chain description of current relaxation: as qqq increases, the longest-lived quasinormal mode ceases to be a nearly pure current excitation and spreads over higher odd angular harmonics. Finally, we show that AC transport in narrow channels provides a direct experimental probe of these phenomena.
We construct multimode viscous hydrodynamics for one-dimensional spinless electrons. Depending on the scale, the fluid has six (shortest lengths), four (intermediate, exponentially broad regime), or three (asymptotically long scales) hydrodynamic modes. Interaction between hydrodynamic modes leads to anomalous scaling of physical observables and waves propagating in the fluid. In the four-mode regime, all modes are ballistic and acquire Kardar-Parisi-Zhang (KPZ)-like broadening with asymmetric power-law tails. "Heads" and "tails" of the waves contribute equally to thermal conductivity, leading to ω^{-1/3} scaling of its real part. In the three-mode regime, the system is in the universality class of a classical viscous fluid [O. Narayan and S. Ramaswamy, Anomalous Heat Conduction in One-Dimensional Momentum-Conserving Systems, Phys. Rev. Lett. 89, 200601 (2002).PRLTAO0031-900710.1103/PhysRevLett.89.200601, H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154, 1191 (2014).JSTPBS0022-471510.1007/s10955-014-0933-y]. Self-interaction of the sound modes results in a KPZ-like shape, while the interaction with the heat mode results in asymmetric tails. The heat mode is governed by Levy flight distribution, whose power-law tails give rise to ω^{-1/3} scaling of heat conductivity.