Hydrodynamic Electron Flow and Hall Viscosity

Abstract

In metallic samples of small enough size and sufficiently strong momentum-conserving scattering, the viscosity of the electron gas can become the dominant process governing transport. In this regime, momentum is a long-lived quantity whose evolution is described by an emergent hydrodynamical theory. Furthermore, breaking time-reversal symmetry leads to the appearance of an odd component to the viscosity called the Hall viscosity, which has attracted considerable attention recently due to its quantized nature in gapped systems but still eludes experimental confirmation. Based on microscopic calculations, we discuss how to measure the effects of both the even and odd components of the viscosity using hydrodynamic electronic transport in mesoscopic samples under applied magnetic fields.The semiclassical theory of electronic conduction, based on relaxation of total momentum by impurities, phonons and umklapp scattering, occupies a central place in condensed matter physics. It is therefore of particular interest to study the cases for which it fails. One case that has attracted much interest is the possibility of a hydrodynamic regime, where transport is dominated by viscous effects . One needs a large separation of scales between momentum-relaxing and momentum-conserving scattering in order to see these effects. This was recently achieved in graphene [23,24] and PdCoO 2 [25,26].Interest in such a hydrodynamic regime also emanated from a conjectured bound on diffusion constants for the hydrodynamics of strongly interacting quantum systems [27,28]. Even though the physics described in this work is semiclassical and probably still quite far from these quantum-mechanical bounds, the observations that we hope to stimulate would constitute an important first step towards the understanding of emergent hydrodynamical regimes in electronic systems.A further motivation for the work is that reaching a viscous regime for a charged fluid enables one to break time-reversal symmetry by adding a magnetic field and hence to study a non-dissipative component to the viscosity tensor called the Hall viscosity. The recent interest in this Hall viscosity emanates from the fact that it is topologically quantized in gapped systems [29]. In order to study this effect experimentally, in analogy with the Hall conductivity, the first step would obviously be to measure the classical Hall viscosity. We show in this letter how this measurement could be done by describing specific size effects from Hall viscosity in transport in restricted 2D channels under transverse magnetic fields. This paper is organized as follows. We start by assuming a perfect hydrodynamic regime and calculate ρ xx and ρ xy . We show that the 1/W 2 component of ρ xy is proportional to the Hall viscosity, thereby providing a way of measuring it. In order to have realistic predictions to compare with experiments, one should also take into account other, non-viscous effects that can lead to a sizedependent resistivity. We thus perform a kinetic Boltzmann calculation in which t...

Hall Viscosity in Quantum Systems with Discrete Symmetry: Point Group and Lattice Anisotropy

“…This model still has internal degrees of freedom that need to be taken into account when considering rotations of the system, meaning that the generator of rotations is the full angular momentum operator Eq. (17), and the proper rotation operator is thuŝ…”

Spin–vorticity coupling in viscous electron fluids

“…, which, after Fourier transforming, yields K(q→0;iω n )∝ω n χ s (iω n ), where χ s (iω n ) is the imaginary-time spin-spin response function at zero wavevector (see equation (19) below), and where the zero-wavevector limit has to be taken because equation (15) can only be used in this limit. Moreover, ω n =2πn/(ÿβ) is a bosonic Matsubara frequency.…”

“…The spectral functions A δ (k, ω) are labeled by the Rashba spin-orbit-split band index δ=±. We incorporate electronelectron interactions into the spectral function by taking them equal to Lorentzians broadened by the electron collision time τ ee (this corresponds to dressing bare propagator lines in the spin bubble in equation (19) by selfenergy insertions), i.e.…”

“…The finite electron viscosity leads to several physical consequences, such as a negative nonlocal resistance [4] and super-ballistic transport through point contacts (PCs) [7,18]. These developments have spurred on a great deal of research, including proposals for measuring the Hall viscosity [19][20][21] and connections to strong-coupling predictions from string theory [22].…”

Quantum oscillations in the type-II Dirac semi-metal candidate PtSe2

“…A likely explanation is weak localization, however no such behavior was observed in previous studies [17] of the low-temperature resistivity of PtSe 2 . Thus it may also be possible that electronic interactions conspire with the lateral finite size confinement in the microstructure to yield positive corrections to the apparent device resistance due to viscous effects in a hydrodynamic picture of electron transport [25][26][27]. Clearly future experiments quantifying the resistance variation among samples and microstructure dimensions are required to address this question.…”