Electron hydrodynamics in anisotropic materials

Rotational invariance strongly constrains the viscosity tensor of classical fluids. When this symmetry is broken in anisotropic materials a wide array of novel phenomena become possible. We explore electron fluid behaviors arising from the most general viscosity tensors in two and three dimensions, constrained only thermodynamics and crystal symmetries. We find nontrivial behaviors in both two- and three-dimensional materials, including imprints of the crystal symmetry on the large-scale flow pattern. Breaking time-reversal symmetry introduces a non-dissipative Hall component to the viscosity tensor, and while this vanishes for 3D isotropic systems we show it need not for anisotropic materials. Further, for such systems we find that the electronic fluid stress can couple to the vorticity without breaking time-reversal symmetry. Our work demonstrates the anomalous landscape for electron hydrodynamics in systems beyond graphene, and presents experimental geometries to quantify the effects of electronic viscosity.


Supplementary Methods
Stress Tensor Symmetries. The viscosity tensors used in the manuscript are all invariant under permutation of the first two indices, i.e.
= . This is a direct consequence of the stress tensor being symmetric under the assumptions and conditions used, which we elaborate on in this section. The anti-symmetric part of the stress tensor, is constrained by Cauchy's laws of motion [1] = where the symbols have the same meaning as in the manuscript.
In the absence of internal spin degrees of freedom σ, body couples l, and stress couples m, eq. (2) necessitates that the anti-symmetric part of the stress tensor, given by the pseudovector A = = 0, must vanish. If we allow for nonzero internal spin, then eq. (2) instead specifies: Recall that the superscript dot denotes the material derivative, = + , which for low Reynolds number flows can be approximated as simply the partial time derivative, ≈ , which by definition must vanish at steadystate. Therefore, at the experimentally accessible conditions we consider in the paper, the anti-symmetric part of the stress tensor again must vanish at steady-state.
In the most general case where we consider stress couples (and/or body couples), the anti-symmetric part of the stress tensor does not vanish. To account for it, we need to augment our constitutive relations to read [1,2]: where B is a rank-4 tensor, and ξ and κ are rank-3 tensors anti-symmetric with respect to their first two indices.
Breaking stress symmetry, therefore allows one to express the viscosity tensor in an expanded basis set [2]: Finally, we note the interesting observation that the stress tensor always appears in a divergence in eq. (1). This suggests that there is an ambiguity in the choice of stress tensor, and in-fact one can always write a physically indistinguishable symmetrized tensor [3][4][5]: where χ is a rank-3 tensor anti-symmetric in and , i.e. = − . While it's straight-forward to show the divergence of the additional term in eq. (7) vanishes, we need to establish this symmetrizes˜ . Consider the following choice: where φ is anti-symmetric, and chosen as the solution to Poisson's equation with the anti-symmetric part of the stress tensor as a source: Substituting eqs. (8) and (9) in eq. (7), we obtain [5]: which is symmetric as desired.
Irreversible Thermodynamics. Viscosity is a non-equilibrium transport tensor which we analyze using irreversible thermodynamics. The entropy production for a system evolving irreversibly from a non-equilibrium state is given by [6][7][8]: where = / are generalized (intensive) forces, and = / are the thermodynamic fluxes of the conjugate generalized (extensive) displacements . The thermo-dynamic fluxes can be expressed as linear combinations of the generalized forces via Onsager's phenomenological equations: J = X, where L is a matrix of kinetic coefficients. The Onsager reciprocal relations postulate L is a symmetric matrix, concluding that entropy production is given by: We note that Onsager derived that the kinetic coefficients matrix is symmetric, due to the microscopic reversibilty of equilibrium states, but that the symmetry is not necessitated by eq. (13) [9,10]. Consider a general kinetic coefficients matrix given by the sum of a symmetric and antisymmetric component L = L S + L A . The antisymmetric component does not contripute to dissipation, i.e. it describes isentropic processes [10]: We identify the generalized force as the fluid stress τ and the generalized displacement as the velocity gradient . We split the viscosity tensor as in the manuscript in the three tensors: Equation (13) then imposes additional 'major' symmetries on α, and β, i.e. under permutation of ↔ , = and = − . The 'odd' Hall viscosity tensor, identified as β, which breaks time-reversal symmetry, is dissipationless. Tensor γ which breaks stress objectivity is not symmetric under the permutation of ↔ , but rather spans a twodimensional subspace with both 'even' and 'odd' components [2].
Onsager's regression hypothesis can be put on firmer footing by deriving Green-Kubo relations using time correlation functions of the generalized flux variables [2,11]. The procedure is analogous, with the resulting transport equations derived in Epstein and Mandadapu These reduce to the ones used in the manuscript, e.g. the 2D case with no internal spins = 3 = 5 = 0: Momentum Relaxing Body Force. Strictly speaking, Cauchy's laws of motion, given by eqs. (1) and (2), arise as a consequence of momentum conservation. Electrons in condensed matter however, especially at elevated temperatures, can undergo multiple momentum-relaxing scattering events, e.g. against impurities or phonons. We incorporate such momentum-relaxation in the manuscript by adding a body force term of the form: where R is a rank-two, positive-semidefinite tensor which is inversely proportional to a microscopic momentum-relaxing lifetime. In order to highlight the viscous effects of crystal anisotropy, the steady-state solutions shown in the manuscript (with the exception of Fig.1(e) justifying this choice) assumed R → 0. In this section, we investigate the effects of this term.
First, we establish a baseline effect for the isotropic (SO (2)) case in our annulus geometry, where the momentum relaxing term is given by = .
Since both the viscous and momentum-relaxing terms are isotropic, the velocity profile is angularly symmetric (Supplementary Fig. 1(a)). As the relative magnitude of the momentum relaxing term increases, the flow dissipates quicker, resulting in a significant part of the annulus exhibiting stationary flow for |R|/|A| > 1. Next, we investigate the case where the viscous and momentumrelaxing terms have different symmetries. According to Neumann's principle, this is allowed by the difference in rank in the two material property tensors. For example, in the square ( 4 ) point group, while the viscous tensor exhibits anisotropy, the momentum relaxing term is still given by = .
As such, the underlying four-fold symmetry of the steady-state vortices remains the same (Supplementary Fig.1(b-e)), with the momentum-relaxing term smearing out these vortices for | |/| | > 1 (Supplementary Fig.1(f)). The magnitude of the vortices however, is largely unchanged at ∼ 12.5%, indicating the effect to be as observable.
Finally, we investigate the effects of momentum-relaxation on a purely hydrodynamic phenomenon we proposed, namely the pressure gauge. Supplementary Figure 2 shows that the effect of momentum-relaxation on the pressure drop used to quantify objectivity-breaking terms in the manuscript. While for values of | |/| | < 0.1 the momentum-relaxing term doesn't alter the results significantly, the pressure drop is no longer directly proportional to the objectivity-breaking terms for | |/| | > 0.1.  Neumann's Principle. In this section, we demonstrate the application of Neumann's principle on the rank-4 viscosity tensor. For convenience, we restrict ourselves in a two-dimensional projection ( -plane) of the most general symbolic viscosity tensor in three dimensions, with -symmetry, given by: and take into account the effects of four-fold ( 4 ) and square ( 4 ) symmetry elements given below: In general, we only need to consider the minimal set of symmetry generators, underlined for each point group above. Applying the 4 operation to eq. (18) according to the transformation law, we obtain: The remaining three coefficients can be recovered using the following parametrization: