Dissipative and Hall Viscosity of a Disordered 2D Electron Gas

Abstract

Hydrodynamic charge transport is at the center of recent research efforts. Of particular interest is the nondissipative Hall viscosity, which conveys topological information in clean gapped systems. The prevalence of disorder in the real world calls for a study of its effect on viscosity. Here we address this question, both analytically and numerically, in the context of a disordered noninteracting 2D electrons. Analytically, we employ the self-consistent Born approximation, explicitly taking into account the modification of the single-particle density of states and the elastic transport time due to the Landau quantization. The reported results interpolate smoothly between the limiting cases of weak (strong) magnetic field and strong (weak) disorder. In the regime of weak magnetic field our results describes the quantum (Shubnikov-de Haas type) oscillations of the dissipative and Hall viscosity. For strong magnetic fields we characterize the effects of the disorder-induced broadening of the Landau levels on the viscosity coefficients. This is supplemented by numerical calculations for a few filled Landau levels. Our results show that the Hall viscosity is surprisingly robust to disorder.Introduction. -Ordinary fluid motion is described by the theory of hydrodynamics, one of whose cornerstones is viscosity, which serves as the source of dissipation. Under certain conditions, charge transport in an electronic system can also be dominated by hydrodynamic viscous flow [1,2]. The discovery of graphene stimulated renewed theoretical [3][4][5][6][7][8][9][10][11] and experimental [12][13][14][15][16][17][18] interest in the hydrodynamic description of charge conduction.In the absence of time-reversal symmetry the viscosity tensor has non-dissipative antisymmetric components. In the presence of a magnetic field B, this nondissipative Hall viscosity (η H ) was studied theoretically in the classical limit of high temperature plasmas [19][20][21][22][23], and for low temperature electron gas [24]. Later, interest in the Hall viscosity was rekindled in quantum systems with a gapped spectrum, due to the connection between η H and geometric response [25][26][27][28][29][30][31], and its expected quantization in the presence of translational and rotational symmetries [29]. It was understood that beyond the Hall conductivity and viscosity there are additional non-dissipative electro-magnetic and geometrical response functions in gapped quantum systems [32][33][34][35][36][37][38][39][40][41][42][43][44]. Within the hydrodynamic description of electron transport, non-zero η H influences significantly the structure of the electron flow [45][46][47][48][49][50], which allows one to access η H experimentally [51]. Also, it was argued that the dissipative and Hall viscosity affect the spectrum of edge magnetoplasmons [52][53][54].For noninteracting electrons in the absence of disorder each filled Landau level (LL) gives a contribution to the Hall viscosity equal (2n + 1)/(8πl 2 B ) [25], where n denotes the LL index and l B = c/(eB) ...

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

“…Furthermore, note that the internal angular momentum contribution to the momentum density originates solely from the time-derivative term in the action Eq. (25). As such, even when the Hamiltonian breaks rotational symmetry explicitly, the form of the momentum density operator is unchanged.…”

“…For instance, the low-energy Dirac theory of graphene arises as a k · p expansion in a highly anisotropic band structure for a system with no translational symmetry. In spite of previous works examining the Hall viscosity in models with broken translational symmetry [22][23][24][25] , the connection between microscopic, low energy descriptions and long-wavelength hydrodynamics relevant to experiment has not been directly addressed. Furthermore, a comprehensive framework for treating momentum transport in systems with broken time reversal symmetry and no external magnetic field (analogous to the formalism for the anomalous Hall conductance) is lacking.…”

Statistical physics of flux-carrying Brownian particles

“…In spite of recent progresses in the study of chiral fluids, the implications of chirality on the kinetics as well as the hydrodynamics remains largely unexplored under dominant dissipative and noise effects [2,94], which are ubiquitous in real life experiments and applications. In this context, the flux-carrying Brownian motion constitutes an ideal testing ground to get useful insight of the hydrodynamic transport properties and electromagnetic response of open quantum systems whose microscopic components violates both time-reversal and parity invariance.…”

Hall viscosity and conductivity of two-dimensional chiral superconductors

“…+ I qu (iω n )q (i u j) (q • u * ) + I qu * (iω n )q (i u * j) (q • u * ) + I qq (iω n )q i q j + O(q 2 ), (D. 29) where the integrals I α are functions of the external frequency iω n . No matter what are the values of the integrals I α , the projection of S i jkl onto σ xz vanishes and there is no contribution to η (1) o .…”

“…In addition to the Hall conductivity, a clean two-dimensional system is characterized by a supplementary non-dissipative Hall response, the Hall (or odd) viscosity tensor η i jkl o (ω, q), that fixes the (odd under time-reversal) response of the stress tensor to the strain rate [15,16], see [17] for a review. Recently, observable signatures of the Hall viscosity have been vigorously studied in classical and quantum fluids both theoretically [18][19][20][21][22][23][24][25][26][27][28][29][30][31] and experimentally [32,33]. In a two-dimensional isotropic system that is invariant under the combined P T symmetry the odd viscosity tensor reduces to two independent components [34], in this paper to be denoted η (1) o (ω, q 2 ) and η (2) o (ω, q 2 ), respectively.…”