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arXiv:1612.09239

Stokes Paradox, Back Reflections and Interaction-Enhanced Conduction

Haoyu Guo, Ekin Ilseven, Gregory Falkovich, Leonid S Levitov

Published Dec 29, 2016• 11 citations
arXivHTMLPDF

Abstract

Interactions in electron systems can lead to viscous flows in which correlations allow electrons to avoid disorder scattering, reducing momentum loss and dissipation. We illustrate this behavior in a viscous pinball model, describing electrons moving in the presence of dilute point-like defects. Conductivity is found to obey an additive relation $σ=σ_0+Δσ$, with a non-interacting Drude contribution $σ_0$ and a contribution $Δσ>0$ describing conductivity enhancement due to interactions. The quantity $Δσ$ is enhanced by a logarithmically large factor originating from the Stokes paradox at the hydrodynamic lengthscales and, in addition, from an effect of repeated returns to the same scatterer due to backreflection in the carrier-carrier collisions occurring at the ballistic lengthscales. The interplay between these effects is essential at the ballistic-to-viscous crossover.

DOI
10.48550/arxiv.1612.09239

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Interactions in electron systems can lead to viscous flows in which correlations allow electrons to avoid disorder scattering, reducing momentum loss and dissipation. We illustrate this behavior in a viscous pinball model, describing electrons moving in the presence of dilute point-like defects. Conductivity is found to obey an additive relation σ=σ0+Δσσ=σ_0+Δσσ=σ0​+Δσ, with a non-interacting Drude contribution σ0σ_0σ0​ and a contribution Δσ>0Δσ>0Δσ>0 describing conductivity enhancement due to interactions. The quantity ΔσΔσΔσ is enhanced by a logarithmically large factor originating from the Stokes paradox at the hydrodynamic lengthscales and, in addition, from an effect of repeated returns to the same scatterer due to backreflection in the carrier-carrier collisions occurring at the ballistic lengthscales. The interplay between these effects is essential at the ballistic-to-viscous crossover.
Strongly interacting electrons can move in a neatly coordinated way, reminiscent of the movement of viscous fluids. Here, we show that in viscous flows, interactions facilitate transport, allowing conductance to exceed the fundamental Landauer's ballistic limit [Formula: see text] The effect is particularly striking for the flow through a viscous point contact, a constriction exhibiting the quantum mechanical ballistic transport at [Formula: see text] but governed by electron hydrodynamics at elevated temperatures. We develop a theory of the ballistic-to-viscous crossover using an approach based on quasi-hydrodynamic variables. Conductance is found to obey an additive relation [Formula: see text], where the viscous contribution [Formula: see text] dominates over [Formula: see text] in the hydrodynamic limit. The superballistic, low-dissipation transport is a generic feature of viscous electronics.
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.
Recent progress in experimental techniques has made it possible to extract detailed information on dynamics of carriers in a correlated electron material from its optical conductivity, [Formula: see text]. This review consists of three parts, addressing the following three aspects of optical response: (1) the role of momentum relaxation; (2) [Formula: see text] scaling of the optical conductivity of a Fermi-liquid metal, and (3) the optical conductivity of non-Fermi-liquid metals. In the first part (section 2), we analyze the interplay between the contributions to the conductivity from normal and umklapp electron-electron scattering. As a concrete example, we consider a two-band metal and show that although its optical conductivity is finite it does not obey the Drude formula. In the second part (sections 3 and 4), we re-visit the Gurzhi formula for the optical scattering rate, [Formula: see text], and show that a factor of [Formula: see text] is the manifestation of the 'first-Matsubara-frequency rule' for boson response, which states that [Formula: see text] must vanish upon analytic continuation to the first boson Matsubara frequency. However, recent experiments show that the coefficient b in the Gurzhi-like form, [Formula: see text], differs significantly from b = 4 in most of the cases. We suggest that the deviations from Gurzhi scaling may be due to the presence of elastic but energy-dependent scattering, which decreases the value of b below 4, with b = 1 corresponding to purely elastic scattering. In the third part (section 5), we consider the optical conductivity of metals near quantum phase transitions to nematic and spin-density-wave states. In the last case, we focus on 'composite' scattering processes, which give rise to a non-Fermi-liquid behavior of the optical conductivity at T = 0: [Formula: see text] at low frequencies and [Formula: see text] at higher frequencies. We also discuss [Formula: see text] scaling of the conductivity and show that [Formula: see text] in the same model scales in a non-Fermi-liquid way, as [Formula: see text].
Recent experiments performed on weakly pinched quantum point contacts have shown a resistance that tends to decrease at low source drain voltage. We show that enhanced Coulomb interactions, prompted by the presence of the point contact, may lead to anomalously large multiple-particle scattering at finite bias voltage. These processes tend to decrease at low voltage, and thus may account for the observed reduction of the resistance. We concentrate on the case of a normal point contact, and model it by a spinful interacting Tomonaga-Luttinger liquid, with a single impurity, connected to noninteracting leads. We find that sufficiently strong Coulomb interactions enhance two-electron scatterings, which dominate the conductance. Our calculation shows that the effective charge, probed by the shot noise of such a system, approaches a value proportional to e∗=2e{e}^{*}=2ee∗=2e at a sufficiently large backscattering current. This distinctive hallmark may be tested experimentally. We discuss possible applications of this model to experiments conducted on Hall bars.
In order to extend the Landauer formulation of quantum transport to correlated fermions, we consider a spinless system in which charge carriers interact, connected to two reservoirs by non-interacting one-dimensional leads. We show that the mapping of the embedded many-body scatterer onto an effective one-body scatterer with interaction-dependent parameters requires to include parts of the attached leads where the interacting region induces power law correlations. Physically, this gives a dependence of the conductance of a mesoscopic scatterer upon the nature of the used leads which is due to electron interactions inside the scatterer. To show this, we consider two identical correlated systems connected by a non-interacting lead of length LC. We demonstrate that the effective one-body transmission of the ensemble deviates by an amount A/LC from the behavior obtained assuming an effective one-body description for each element and the combination law of scatterers in series. A is maximum for the interaction strength U around which the Luttinger liquid becomes a Mott insulator in the used model, and vanishes when U ↦0 and U ↦∞. Analogies with the Kondo problem are pointed out.
Transport in strongly correlated fermions cannot be understood by fermionic quasiparticles alone. We present a theoretical framework for quantum transport that incorporates strong local correlations of fermion pairs. These contact correlations add essential contributions to viscous, thermal and sound transport coefficients. The bulk viscosity, in particular, receives its dominant contribution from pair excitations. Moreover, it can be measured elegantly by observing the response to a time-dependent scattering length even when the fluid is not moving. Rapid changes of the scattering length drive the system far out of local equilibrium, and we show how it relaxes back to equilibrium following a hydrodynamic attractor before a Navier–Stokes description becomes valid. This paper summarizes a talk given at the Symposium “Open questions in the quantum many-body problem” at the Institut Henry Poincaré, Paris, in July 2024.