Generalized Electron Hydrodynamics, Vorticity Coupling, and Hall Viscosity in Crystals
Georgios Varnavides, Adam S. Jermyn, Polina Anikeeva, Claudia Felser, Prineha Narang
GGeneralized Electron Hydrodynamics, Vorticity Coupling, and Hall Viscosity in Crystals
Georgios Varnavides,
1, 2, 3, ∗ Adam S. Jermyn, ∗ Polina Anikeeva,
2, 3
Claudia Felser, and Prineha Narang † Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA,USA Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA Max-Planck-Institut für Chemische Physik fester Stoffe, Dresden, Germany (Dated: February 24, 2020)
Theoretical and experimental studies have revealed thatelectrons in condensed matter can behave hydrodynami-cally, exhibiting fluid phenomena such as Stokes flow andvortices [1–9]. Unlike classical fluids, preferred directionsinside crystals lift isotropic restrictions, necessitatinga generalized treatment of electron hydrodynamics.We explore electron fluid behaviors arising from themost general viscosity tensors in two and three dimen-sions, constrained only by thermodynamics and crystalsymmetries. Hexagonal 2D materials such as graphenesupport flows indistinguishable from those of an isotropicfluid. By contrast 3D materials including Weyl semimet-als [10, 11], exhibit significant deviations from isotropy.Breaking time-reversal symmetry, for example in mag-netic topological materials, introduces a non-dissipativeHall component to the viscosity tensor [7, 12–15]. Whilethis vanishes by isotropy in 3D, anisotropic materialscan exhibit nonzero Hall viscosity components. Weshow that in 3D anisotropic materials the electronicfluid stress can couple to the vorticity without breakingtime-reversal symmetry. Our work demonstrates theanomalous landscape for electron hydrodynamics insystems beyond graphene, and presents experimentalgeometries to quantify the effects of electronic viscosity.
Electron hydrodynamics is observed when microscopicscattering processes conserve momentum over time- andlength-scales which are large compared to those of theexperimental probe. However, even as momentum is con-served, free energy may be dissipated from the electronicsystem, giving rise to a measurable viscosity in the electronflow [10, 16–21]. When momentum is conserved, a fluidobeys Cauchy’s laws of motion [22]: ρ (cid:219) u i = ∂ j τ ji + ρ f i (1) ρ (cid:219) σ i = ∂ j m ji + ρ l i + (cid:15) ijk τ jk , (2)where u and ρ are the fluid velocity and density, f and l are body forces and couples, τ and m are the fluid stressand couple stress, and σ is the intrinsic angular momen- ∗ These authors contributed equally to this work † Electronic address: [email protected] tum density (internal spin). The superscript dot denotes thematerial derivative, (cid:219) x = ∂ t x + u j ∂ j x , and (cid:15) is the rank-3 alter-nating tensor. We assume couple stresses and body couplesto be zero. In steady state and at experimentally accessibleReynolds numbers [20, 23], this implies that the stress ten-sor is symmetric [22]. In this limit, electron fluids obey theNavier Stokes equation ρ u j ∂ j u i = − ∂ i p + ∂ j τ ji , (3)where τ is symmetric. Note that in electron fluids, currentdensity is analogous to the fluid velocity and voltage dropsare analogous to changes in pressure. Assuming the fluidvelocity is much smaller than the electronic speed of sound, u (cid:28) c s , the electron fluids are nearly incompressible, thus ∂ i u i = . (4)In this limit ρ is a constant, which we take to be unity. Sincethe fluid stress appears in a divergence, it is defined only upto a constant, which we choose to make τ vanish when u is uniformly zero [24, 25]. We further assume that the fluidstress vanishes for uniform flow, so that it is only a functionof the velocity gradient. Without further loss of generality,the constitutive relation is written to first order as [24] τ ij = A ijkl ∂ l u k , (5)where A is the fluid viscosity, a rank-4 tensor relating the fluidvelocity gradient ( ∂ j u i ) and the fluid stress. Since we take τ to be symmetric, A is invariant under permutation of its firsttwo indices, i.e. A ijkl = A jikl [24, 25]. Viscosity is repre-sented as the sum of three rank-4 tensors basis elements [15],summarized in Table I A ( ij ) kl = α (( ij )( kl )) + β [( ij )( kl )] + γ ( ij )[ kl ] . (6)Tensor α describes dissipative behavior respecting both stresssymmetry and objectivity, i.e. α ijkl = α jikl = α klij . Tensor β on the other hand, describes non-dissipative Hall viscos-ity [7, 12–15], i.e. β ijkl = − β klij , and is non-zero only whentime-reversal symmetry is broken. Finally, γ breaks stressobjectivity, i.e. γ ijkl = − γ ijlk , coupling fluid stress to the vor-ticity. The fifth column in Table I specifies whether the tensoris defined according to a handedness convention. a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Tensor Tensor Symmetries Indep. Comp. i ↔ j k ↔ l i j ↔ kl type 3D 2D α (( ij )( kl )) + + + proper 21 6 β [( ij )( kl )] + + - pseudo 15 3 γ ( ij )[ kl ] + - N/A pseudo 18 3 TABLE I . Rank-4 tensors used as orthogonal basis elements forthe viscosity tensor. Even and odd symmetries are represented us-ing parentheses and square brackets respectively. The fifth columnspecifies whether the tensor changes sign under mirror operations.
In classical fluids the added consideration of rotational in-variance requires A to be isotropic, reducing it to the form A ijkl = λδ ij δ kl + µ (cid:0) δ il δ jk + δ jl δ ik (cid:1) + B (cid:0) (cid:15) ik δ jl + δ ik (cid:15) jl (cid:1) + Γ δ ij (cid:15) kl , (7)where δ is the Kronecker delta, (cid:15) is the rank-two alternatingtensor, and the Lamé parameters λ and µ can be identifiedas the two independent components of the proper tensor α . In the incompressible case λ does not contribute to thestress [24]. B and Γ are constants parametrizing termswith the symmetry of β and γ respectively. Since β and γ are pseudo-tensors, the last three terms in eq. (7) are onlynon-zero in two dimensions [12, 15].In crystals, however, there exist preferred directionsand we cannot assume rotational invariance. Instead wemust consider the effect of the crystal symmetry given byNeumann’s Principle [26, 27], which requires that physicalproperties described by rank-4 tensors, such as viscosity,remain invariant under the transformation law: A (cid:48) ijkl = | s | η s im s jn s ko s lp A mnop , (8)where s is the space-representation of any given point groupsymmetry of the crystal, | s | = ± η = η = u in d -dimensions ∫ u i ∂ j ( A ijkl ∂ k u l ) d d r ≤ . (9)Letting the Fourier transform of u be˜ u ( q ) = ∫ e i q · r u ( r ) d d r (10)in d -dimensions, we find ∫ q j q k ˜ u ∗ i ( q ) ˜ u l ( q ) A ijkl d d q ≥ . (11) This is satisfied when A ijkl has a positive definite biquadraticform in il and j k , so we impose this constraint in addition to i j symmetry and crystal symmetry.Viscosity tensors are then generated to satisfy theaforementioned constraints [28]. The viscosity tensor isassumed to be spatially uniform in all cases. To demonstratethe differences between these general viscosity tensors andthose more strongly constrained by symmetry we solve forthe velocity and pressure of low Reynolds number flowsin several geometries. The parametrization of the viscositytensor in eq. (6) allows us to explore the effects of breakingstress objectivity and time-reversal symmetry. We highlightthe effects of symmetry in the last two indices ( kl ) becauseit implies that the stress only couples to the strain rate( ∂ k u l + ∂ l u k ) and not to the vorticity ( ∂ k u l − ∂ l u k ). This is aproperty of classical fluids, which means that rigid-rotationalflows are stress-free, and hence are only sensitive to rotationvia weaker effects like the Coriolis force. Below we demon-strate that with more general viscosity tensors this is not thecase, and that the resulting rotational stresses can be probedin experimentally accessible geometries.We first consider rotational flow in an annulus withinner radius R inner = R outer = ω = ≡ ω R | A | = . , (12)where | A | = A ijkl A ijkl . (13)The zero-pressure point is fixed at the bottom of the annu-lus. Experimentally, such rotational flows can be achieved bythreading a time-varying magnetic flux through a Corbinodisk geometry [29], shown in Figure 1(a). For a fluid withan isotropic viscosity, the steady-state velocity field rotatesrigidly with the angular velocity set by the inner boundarycondition (Fig. 1(b)).To investigate the effects of anisotropy in two-dimensionalmaterials, we considered materials with D (hexagonal) and D (square) symmetry. Notably, D materials do not deviatefrom isotropic behavior (Fig. 1(c)), consistent with experi-mental observations for graphene [9, 20]. We note that 2Dmaterials with C (three-fold), C (six-fold), and D (trian-gular) symmetry also exhibit isotropic viscosity tensors (seeSupplementary Material). By contrast, the flow deviates con-siderably from isotropic behavior in D materials (Fig. 1(d)).Figures 1(e,f) illustrate these points further, by showing thesteady-state velocity flow difference between the isotropiccase and D and D materials respectively. In the latter, weobserve steady-state vortices emerging at ∼
15% of the bulkflow rate overlaid onto the isotropic velocity field. While the e SO(2)-D Streamplot f SO(2)-D Streamplot h D Pressure plot - - - - g D Pressure plot a Corbino DiskGeometry �� �� t � c D Streamplot d D Streamplot b SO(2)Streamplot
Figure 1 . Effect of viscosity tensor anisotropy on rotational flow in an annulus. (a)
Corbino disk geometry schematic. The time-varyingmagnetic flux gives rise to a Lorentz force, inducing rotational electron flow. Steady state (b) streamplot plot using an isotropic ( SO ( ) ) viscositytensor. Streamplots using (c) hexagonal ( D ), and (d) square ( D ) viscosity tensors. Difference in steady state streamplot between isotropicand (e) D , and (f) D viscosity tensors, highlighting the emergence of steady-state vortices. Steady state pressure plot using (g) D and (h) D viscosity tensors, illustrating the breaking of azimuthal symmetry in the latter. steady-state pressure field in D materials mirrors that of anisotropic fluid (Fig. 1(g)), the pressure field in D materialsalso exhibits four vortices (Fig. 1(h)), with orientation set bythe underlying crystal axes.We next examine the importance of symmetry in the last twoindices of the viscosity tensor. We calculate the flow profilefor the annulus in Figure 1 scaled by a factor of two, equippedwith a pressure gauge, as shown in Figure 2(a). The pressuregauge is a channel with no-slip boundary conditions, allowingus to measure the difference between the flow and a nearly-stationary fluid. To isolate the effects of B and Γ in eq. (7),Figures 2(a,b) show the flow and pressure fields in the annulusfor a material with isotropic viscosity tensor where B and Γ have both been set to zero ( SO ( ){ α } ). These are nearlyunchanged inside the annulus as compared to Figures 1(b,g),with a constant pressure in the gauge. Allowing for non-zero stress-breaking components, i.e. using a material withisotropic viscosity for B = Γ = .
25 ( SO ( ){ γ } ), weobserve a significant pressure build-up near the gauge. Thisis due to the shear stress between the rotating and stationaryfluids, while the pressure within the gauge itself is nearlyuniform, as shown in Figure 2(c).To quantify the pressure difference between SO ( ){ α } and SO ( ){ γ } , note that the pressure is fixed to zero at a point p ,the bottom of the annulus domain. The pressure in the gaugemay be written as the path integral p gauge = ∫ gp ∇ p · d s , (14)where g is a point in the gauge. At low Reynolds numbers wemay neglect u j ∂ j u i in equation eq. (3), to find in steady state ∇ p = A ijkl ∂ i ∂ k u l . (15) p gauge = ∫ gp A ijkl ∂ i ∂ k u l ds j . (16)Taking into account eq. (7) and noting that the changes influid flow are negligible, we find: ∆ p gauge = ∫ gp ∆ A ijkl ∂ i ∂ k u l ds j = Γ ∫ gp ∂ i ω ds i = Γ ∆ ω, (17)where ω i = (cid:15) ijk ∂ j u k is the vorticity of the flow. For the ge-ometry used, we find ∆ p gauge = .
15, vorticity in the gauge iszero and that in the annulus is 0.6, so ∆ ω = . - - - - - - - SO(2){α}
Pressure plot
SO(2){α}
Streamplot
SO(2){γ}
Pressure plot D -C Pressure plot ba c d
Figure 2 . Proposed setup to quantify effect of viscosity tensor asymmetry and Hall coefficient.
Steady state (a) streamplot and (b) pressure plot using a viscosity tensor. (c)
Difference in steady state pressure between the viscosity tensor and the same with additional stressobjectivity breaking terms. The asymmetry introduces an additional pressure-like contribution, which can be directly measured. (d)
Differencein steady state pressure between the D h and C v viscosity tensors along the ab plane. Since we chose Γ = .
25, we see that in this setup the pres-sure gauge ( ∆ p gauge = Γ ∆ ω = . × . = .
15) is directlysensitive to the asymmetry in the viscosity, which couples therotation directly to the pressure field and the stress. We notethat the same setup is sensitive to Hall viscosity coefficientsfor time-reversal broken systems, i.e. for the case where both B and Γ are non-zero the pressure gauge generalizes to ∆ p gauge = ∫ gp ∆ A ijkl ∂ i ∂ k u l ds j = (B + Γ ) ∆ ω. (18)While time reversal and stress objectivity breaking termspersist in two-dimensional isotropic materials, the handed-ness of the pseudo-tensor implies that mirror operations setthem to zero in 3D. This can be directly observed in com-paring low- and high-symmetry three-dimensional crystals.We consider the same rotational flow along the ab crystalplane of orthorhombic materials, such as the hydrodynam-ically reported Weyl semi-metal WP [10, 11]. Along thisplane, the difference between the two viscosity tensors canbe parametrized as follows: A C ( ) v ijkl = A D ( ) h ijkl + Γ δ ij (cid:15) kl + Γ σ zij (cid:15) kl + B (cid:0) δ li (cid:15) jk − (cid:15) li δ jk (cid:1) + B (cid:16) δ ij σ xkl − σ xij δ kl (cid:17) , (19)where B , B , Γ , and Γ are constants parametrizing termswith the symmetry of β and γ respectively, σ x and σ z arePauli matrices. Figure 2(d) shows the pressure difference be-tween a material with D h symmetry and one with C v sym-metry (for B = B = Γ = Γ = . T d (tetrahedron) and O h (cubic) point groups. In particu-lar, we consider flows along the polar { } , nonpolar { } ,and semipolar { } family of planes (Figure 3(b)): A T ( ) d ijkl = A O ( ) h ijkl + B (cid:16) σ xij σ zkl − σ zij σ xkl (cid:17) + Γ δ ij (cid:15) kl (20a) A T ( ) d ijkl = A O ( ) h ijkl (20b) A T ( ) d ijkl = A O ( ) h ijkl + B (cid:16) σ zij δ zkl − δ ij σ zkl (cid:17) + Γ σ xij (cid:15) kl . (20c)Along these planes, the difference between the two viscositytensors can be parametrized according to eqs. (20a) to (20c).We impose fully-developed (parabolic) inlet and outlet flowswith constant discharge, and solve for the steady state flowat low Reynolds number. Figure 3(c) shows the differencebetween the flow in an isotropic material and the flow ina cubic material along a { } close-packed plane, whichexhibits rotational invariance. Along the nonpolar { } planes, terms with β and γ symmetry vanish. However, A O ( ) h is anisotropic along this plane, with Figure 3(d)showing the difference in flow between the isotropic case.Finally, along the semipolar { } family of planes, the vis-cosity tensor is both anisotropic (Fig. 3(e)), and asymmetric.Figure 3(f) quantifies the additional vortices generated bythe asymmetry at ∼ B = Γ = . eba cd f SO(3)Streamplot SO(3)-O h{111}
StreamplotSO(3)-O h{110}
Streamplot SO(3)-T d{001}
Streamplot O h{001} -T d{001} Streamplot{111}Cubic Planes(Polar) {001}Cubic Planes(Semipolar){110}Cubic Planes(Nonpolar)
Figure 3 . Three dimensional projected flows through an expanding geometry. (a)
Steady state streamplot using an isotropic ( SO ( ) )viscosity tensor. (b) High-symmetry family of planes in cubic crystals. In crystals with T d (tetrahedral) symmetry, these are further identifiedas polar { } , nonpolar { } , and semipolar { } . Difference in steady state streamplot between using an isotropic viscosity tensor andusing (c) (111)-projected, (d) (110)-projected, and (e) (001)-projected viscosity tensors of cubic crystals with T d symmetry. Note that (c) showsno difference. (f) Effect of viscosity tensor asymmetry difference between O h and T d crystals along a semipolar { } plane. We found that electron fluids in crystals with anisotropicand asymmetric viscosity tensors can exhibit steady-statefluid behaviors not observed in classical fluids. Even aminor deviation from isotropy allows the fluid stress tocouple to the fluid vorticity with or without breakingtime-reversal symmetry, for the case of Hall viscosityand objectivity-breaking viscosity respectively. Recentmeasurements of spatially-resolved flows [9, 20, 30], suggestthat these effects can be directly observed in systems beyondgraphene. Our findings further hint at potential applications.For instance, the pressure gauge in Figure 2 could be usedas a magnetometer, converting a time-varying magneticflux through a modified corbino disk geometry into currentin the annulus, and ultimately into a voltage drop betweenit and the gauge. Our work highlights the importance ofcrystal symmetry on electronic flow, and invites furtherexploration of time-dependent flows in systems with in-ternal spin degrees of freedom and asymmetric stress tensors.
Acknowledgments.
The authors thank Prof. AndrewLucas of the University of Colorado Boulder for fruitfuldiscussions. The authors acknowledge funding from theDefense Advanced Research Projects Agency (DARPA)Defense Sciences Office (DSO) Driven and NonequilibriumQuantum Systems program. ASJ is supported by theFlatiron Institute of the Simons Foundation. PN is a MooreInventor Fellow supported by the Gordon and Betty MooreFoundation.
Author Contributions.
GV and ASJ jointly conceivedthe ideas and developed the framework. GV, CF, and PNidentified material systems and link with transport mea-surements of topological systems. GV, PA, and PN jointlyworked on introducing the effects of crystal symmetries. Allauthors discussed the findings and contributed to the writingof the manuscript.
Competing Financial Interests.
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