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arXiv:1708.02376 · Physical Review Letters

Tomographic Dynamics and Scale-Dependent Viscosity in 2D Electron Systems

Patrick J. Ledwith, Haoyu Guo, A. V. Shytov, Leonid S Levitov

Published Sep 13, 2019• 64 citations
arXivHTMLPDF

Abstract

Fermi gases in two dimensions display collective dynamics originating from head-on collisions, a collinear carrier scattering process that dominates angular relaxation at not-too-high temperatures T≪T_{F}. In this regime, a large family of excitations emerges, with an odd-parity angular structure of momentum distribution and exceptionally long lifetimes. This leads to "tomographic" dynamics: fast 1D spatial diffusion along the unchanging velocity direction accompanied by a slow angular dynamics that gradually randomizes velocity orientation. The tomographic regime features an unusual hierarchy of timescales and scale-dependent transport coefficients with nontrivial fractional scaling dimensions, leading to fractional-power current flow profiles and unusual conductance scaling versus sample width.

DOI
10.1103/physrevlett.123.116601

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Fermi gases in two dimensions display collective dynamics originating from head-on collisions, a collinear carrier scattering process that dominates angular relaxation at not-too-high temperatures T≪T_{F}. In this regime, a large family of excitations emerges, with an odd-parity angular structure of momentum distribution and exceptionally long lifetimes. This leads to "tomographic" dynamics: fast 1D spatial diffusion along the unchanging velocity direction accompanied by a slow angular dynamics that gradually randomizes velocity orientation. The tomographic regime features an unusual hierarchy of timescales and scale-dependent transport coefficients with nontrivial fractional scaling dimensions, leading to fractional-power current flow profiles and unusual conductance scaling versus sample width.
Abstract Two-dimensional electron gases are predicted to possess an anomalous “tomographic” transport regime that is marked by an odd-even effect in the relaxation times, with odd-parity deformations of the Fermi surface becoming long-lived in comparison to even-parity ones. In this work, we establish that neutral two-component atomic Fermi gases also exhibit this tomographic effect. By diagonalizing the Fermi liquid collision integral, we identify odd-parity modes with anomalously long lifetimes below temperatures T≤0.15T F , which is within reach of cold atom experiments. Furthermore, in contrast to electron gases, we find that the odd-even effect in neutral gases is widely tuneable with interactions along the BCS-BEC crossover and is suppressed on the BEC side. We propose as an experimental signature of the odd-even effect the damping rate of quadrupole oscillations, which is anomalously enhanced due to the presence of long-lived odd-parity modes. Our findings suggest that the dynamics of two-dimensional Fermi gases is richer than previously thought and should include additional long-lived modes.
Two-dimensional Fermi liquids at low temperatures have been theoretically established to exhibit an odd-even effect in the collective quasiparticle relaxation rates where even-parity deformations of the Fermi surface decay at a much faster rate than odd-parity ones. A predicted consequence of this effect is a new tomographic transport regime that mixes hydrodynamic and collisionless transport. In the presence of a magnetic field, however, the tomographic regime is expected to evolve towards conventional transport regimes as soon as the cyclotron radius becomes smaller than the dominant odd-parity mean-free path. In this work, we examine this transition from the point of view of collective modes, using a numerically exact solution of the linearized Boltzmann equation within a generalized relaxation time approximation for the odd-parity and even-parity modes. In the absence of a magnetic field, the transverse conductivity exhibits two diffusive tomographic collective modes, and we find that at a critical magnetic field one of these two tomographic modes disappears. Which tomographic mode persists depends on the Landau parameters, and becomes increasingly dominated by hydrodynamic modes at high fields. We corroborate our analysis using a variational approach for the Fermi surface deformation that captures the angular structure of the deformation and the critical magnetic field strength. The collective modes discussed here can in principle be observed by examining the damping of longitudinal and transverse current responses in finite magnetic fields.
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.
Recent theoretical works predict a hierarchy of long-lived, nonhydrodynamic modes in two-dimensional Fermi liquids arising from the feature—supposedly unique to two dimensions—that relaxation by head-on scattering is not efficient in the presence of Pauli blocking. This leads to a parity-based separation of scattering rates, with odd-parity modes relaxing much more slowly than even-parity ones. In this work, we establish that a similar effect exists in isotropic three-dimensional (3D) Fermi liquids, even though relaxation does not proceed solely by head-on scattering. We show that while the relaxation rates of even and odd modes in 3D share the same leading-order ∼ T 2 low-temperature scaling typical of Fermi liquids, their magnitudes differ, with odd-parity modes relaxing more slowly than even ones for a broad class of interactions. We find a relative difference between odd-parity and even-parity relaxation rates as large as 40 % just by Pauli blocking alone, with a strong additional dependence on the scattering potential, such that the odd-even staggering is further enhanced by interactions that favor large-angle scattering. We identify signatures of these odd-parity relaxation rates in the static transverse conductivity as well as the transverse collective mode structure. Our results establish the unexpected existence of a tomographiclike regime in higher-dimensional Fermi liquids and suggest experimental probes via transport measurements.
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 develop an analytically solvable model for interacting two-dimensional Fermi liquids with separate collisional relaxation rates for parity-odd and parity-even Fermi surface deformations. Such a disparity of collisional lifetimes exists whenever scattering is restricted to inversion-symmetric Fermi surfaces, and should thus be a generic feature of two-dimensional Fermi liquids. It implies an additional unanticipated ``tomographic'' transport regime (in between the standard collisionless and hydrodynamic regimes) in which even-parity modes are overdamped while odd-parity modes are collisionless. We derive expressions for both the longitudinal and the transverse conductivity and discuss the collective mode spectrum along the collisionless-tomographic-hydrodynamic crossover. Longitudinal modes cross over from zero sound in the collisionless regime to hydrodynamic first sound in the tomographic and hydrodynamic regime, where odd-parity damping appears as a subleading correction to the lifetime. In charged Fermi liquids with long-range Coulomb coupling, these modes reduce to plasmons with a strongly suppressed odd-parity correction to the damping. The transverse response, by contrast, has a specific tomographic transport regime with two imaginary odd-parity modes, one of which requires a finite repulsive interaction, distinct from both the shear sound in the collisionless regime and an overdamped diffusive current mode in the hydrodynamic limit. Our work demonstrates that there are deep many-body aspects of interacting Fermi liquids, which are often thought to be well understood theoretically, remaining unexplored.