Sign of viscous magnetoresistance in electron fluids
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Sign of viscous magnetoresistance in electron fluids
Ipsita Mandal a,b and Andrew Lucas c,d a Laboratory of Atomic And Solid State Physics, Cornell University, Ithaca, NY 14853, USA b Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway c Department of Physics, Stanford University, Stanford, CA 94305, USA d Department of Physics, University of Colorado, Boulder, CO 80309, USA [email protected], [email protected]
January 22, 2020
Abstract:
In sufficiently clean metals, it is possible for electrons to collectively flow as a viscous fluidat finite temperature. These viscous effects have been predicted to give a notable magne-toresistance, but whether the magnetoresistance is positive or negative has been debated.We argue that regardless of the strength of inhomogeneity, bulk magnetoresistance is alwayspositive in the hydrodynamic regime. We also compute transport in weakly inhomogeneousmetals across the ballistic-to-hydrodynamic crossover, where we also find positive magne-toresistance. The non-monotonic temperature dependence of resistivity in this regime (abulk Gurzhi effect) rapidly disappears upon turning on any finite magnetic field, suggestingthat magnetotransport is a simple test for viscous effects in bulk transport, including at theonset of the hydrodynamic regime. a r X i v : . [ c ond - m a t . s t r- e l ] J a n ntroduction1 In an ultrapure solid-state device, it may be the case that momentum-conserving electron-electron scat-tering is the fastest process which can scatter an electronic quasiparticle [1]. Historically such metals didnot exist: in a Fermi liquid, the electron-impurity scattering rate is always faster as temperature T → T usually an umklapp process (off electrons or phonons) is sufficient to relax the electronicmomentum. Nevertheless, experiments have increasingly discovered evidence for this hydrodynamic flowregime in clean samples of graphene, GaAs, and other compounds, in recent years [2, 3, 4, 5, 6, 7, 8, 9];see [10] for a review.While there are possible applications for hydrodynamic electron flow ranging from high conductancemesoscopic devices [11] to terahertz radiation generation [12], this paper is inspired by a simpler question:is it possible that hydrodynamic effects have already been seen in experiments via the unusual behavior ofelectrical resistivity as a function of temperature, etc.? A number of clear predictions have already beenmade for the resistivity of a Fermi liquid of viscous electrons [13, 14, 15, 16], but there is no compellingexperimental observation thus far.It was recently suggested [17] that viscous electron flow through an inhomogeneous device wouldlead to negative magnetoresistance in a (quasi-)two-dimensional metal, where the dissipative resistivitydecreases as one turns on a small magnetic field: ∂ρ/∂ ( B ) <
0. This work was inspired by an experimenton relatively large samples of GaAs [18], where negative magnetoresistance has been observed, albeit notnecessarily in a hydrodynamic regime. More recently, negative magnetoresistance was seen in very narrowchannels of graphene at moderately high temperatures (where hydrodynamic effects are observed) [19].Later, [20, 21] pointed out that in bulk crystals, the magnetoresistance would always be positive; however,their argument is perturbative in the strength of the inhomogeneity.In this paper, we will argue that the conclusion of [20, 21] is valid more generally, and thus thatnegative magnetoresistance is not a signature of viscous flow in a bulk crystal. We do so from twoperspectives. First, we will argue that even when disorder and inhomogeneity are arbitrarily strong,a generic Fermi liquid will exhibit positive magnetoresistance. Our conclusion is based on an exactanalysis of the hydrodynamic transport problem in systems which are homogeneous in one out of thetwo spatial dimensions, along with a discussion of the density dependence of the local hydrodynamiccoefficients. Secondly, we will argue that near a low temperature transition between viscous electron flowand ballistic electron flow, there is an enormous positive magnetoresistance in weakly inhomogeneoussystems. Therefore, the conclusion of [20, 21] cannot be avoided by studying systems near the onsetof viscous flow. In fact, we argue that the magnetic field dependence of resistivity is so strong thatmagnetoresistance is an excellent test for whether resistance minima (where ρ ( T ) is a decreasing functionat low T ) are a consequence of viscous effects [15] or other effects, such as Kondo physics [22]. Hydrodynamics2
We begin by describing hydrodynamic transport in a Fermi liquid. For simplicity, we assume an isotropicFermi surface; new transport coefficients generically arise in the absence of rotational symmetry [23]. Inthe limit where temperature T is very small compared to Fermi energy E F , and in a background magneticfield, we will see that it is acceptable to neglect energy conservation and treat the only hydrodynamicdegrees of freedom as charge and energy [10].For simplicity, we focus on flows in media which are only inhomogeneous in a single direction y . Weassume an isotropic fluid, so that incoherent conductivities may be neglected at low temperatures. Weexpect that the resulting cartoon qualitatively captures the physics of flows in media which are inhomo-geneous in both directions. The hydrodynamic equations of charge, energy and momentum conservation up to sources) read respectively: ∂ y ( nv y ) = 0 , (1a) ∂ y ( T sv y − κ∂ y T ) = 0 , (1b) − ∂ y ( η∂ y v x + η H ∂ y v y ) = n ( E x + Bv y ) , (1c) n∂ y µ + s∂ y T − ∂ y (( ζ + η ) ∂ y v y − η H ∂ y v x ) = n ( E y − Bv x ) , (1d)where we have approximated that η and η H , the shear and Hall viscosity respectively, can be treatedas constants. Here T is a background temperature which is independent of y . For simplicity, we areneglecting the vorticity susceptibility [24] and bulk viscosity. We have also chosen units of charge so thatthe electron has charge +1, for convenience in what follows. We begin by briefly reviewing the scenario for negative magnetoresistance proposed in [17]. Consider along narrow channel of width w , inside of which is a completely homogeneous electron fluid. We choosecoordinates so that the channel is the region w ≤ | y | . We assume that the scattering at the boundaries islargely diffuse, in which case it is appropriate to assume no slip boundary conditions: v y = 0 at y = ± w .Suppose that we apply an electric field E x , oriented down the channel. We wish to solve (1) for µ , v x and v y , to linear order in E x , in order to calculate linear response transport coefficients. Theconstraint that no electric current flows through the boundary, together with charge conservation, impliesthat v y = 0. It is straightforward to then obtain the solution to (1) consistent with boundary conditions: v x = nE x η (cid:18) w − y (cid:19) , (2a) µ = − η H E x η y − B nE x η w y + B nE x η y . (2b)We have assumed for this subsection that n and η do not depend on position. The resistivity is definedas 1 ρ = 1 w w/ (cid:90) − w/ d y J x E x = n w η . (3)The last fact which we need to use is that in an isotropic two-dimensional Fermi liquid, η ( B ) = η ω c τ ee ) , (4a) η H ( B ) = 2 ω c τ ee η ( B ) , (4b)where 1 /r c = B/p F is the cyclotron radius, (cid:96) ee is (predominantly) the mean free path for momentum-conserving electron-electron collisions, and η ∼ np F (cid:96) ee (5)is an increasing function of the background density. In other words, in a two-dimensional Fermi liquidwith dispersion relation (cid:15) ∼ p z as p → (cid:96) ee ∼ T − n (2 z − / . (6) ombining (3) and (4a), we obtain that ρ ( B ) = 12 η n w (cid:32) − (cid:18) (cid:96) ee p F (cid:19) B + O (cid:0) B (cid:1)(cid:33) . (7)Hence we predict that there is a negative magnetoresistance. Since (cid:96) ee ∼ T − , the effect should be morepronounced at lower temperatures (if hydrodynamics is valid). This effect has been observed experimen-tally in narrow channels [19].A key point, however, is that (2b) implies the presence of a Hall voltage from y = − w/ y = w/ not immediately generalize into a continuous medium, where thefluctuating chemical potential µ must be continuous. This means that a separate theory is required tounderstand transport in inhomogeneous media, which we turn to next. We now consider the equations (1) in an infinite medium with local fluid density n ( y ). The local viscosity η and Hall viscosity η H will also generically depend on y . For simplicity, we suppose that all of thesefunctions are periodic with some large period L . Clearly, charge conservation implies that J y = n ( y ) v y ( y ) (8)is a constant.First, let us consider applying an electric field in the y -direction: E x = 0 and E y (cid:54) = 0. Integrating the x -momentum equation over the periodic direction and denoting (cid:104)· · · (cid:105) = 1 L (cid:90) d y · · · , (9)we conclude that 0 = (cid:104) n ( E x + Bv y ) (cid:105) = BJ y + (cid:104) n (cid:105) E x . (10)Hence J y = v y = 0, which implies that ∂ y ( η∂ y v x ) = 0 . (11)This is only satisfied for periodic functions by constant v x . The y -momentum equation is then onlysatisfied for constant µ and v x = E y B . (12)We conclude that σ xy = (cid:104) n (cid:105) B , (13a) σ yy = 0 . (13b)Now we assume E y = 0 and E x (cid:54) = 0. Then (10) implies that σ yx = J y E x = − (cid:104) n (cid:105) B . (14)For convenience in what follows, we define the periodic function Ψ by the integrable equation ∂ y Ψ = n − (cid:104) n (cid:105) . (15) long with the constraint (cid:104) Ψ (cid:105) = 0. The x -momentum equation becomes − ∂ y (cid:18) η∂ y v x + η H J y ∂ y n (cid:19) = E x ( n − (cid:104) n (cid:105) ) = E x ∂ y Ψ, (16)which is solved by v x = C − (cid:90) d y (cid:20) E x Ψη + η H J y η ∂ y n (cid:21) . (17)To determine the unknown constant C , note that − BC = (cid:28) ∂ y µ + sn ∂ y T − n ∂ y (( ζ + η ) ∂ y v y − η H ∂ y v x ) (cid:29) = (cid:28) sn ∂ y T + ∂ y n (cid:18) ( ζ + η ) ∂ y J y n − η H η ∂ y v x (cid:19)(cid:29) = (cid:42) sn ∂ y T − E x (cid:104) n (cid:105) B η ( η + ζ ) + η η (cid:18) ∂ y n (cid:19) + η H E x η Ψ ∂ y n (cid:43) . (18)To fix ∂ y T , we use the energy conservation equation, which can be straightforwardly integrated: ∂ y T = T sJ y nκ − (cid:104) κ − (cid:105) κ (cid:28) T sJ y nκ (cid:29) (19)(the constant is fixed by periodicity of T ). Lastly, to determine σ xx , observe that (cid:104) J x (cid:105) = (cid:104) ( (cid:104) n (cid:105) + ∂ y Ψ ) v x (cid:105) = C (cid:104) n (cid:105) − (cid:104) Ψ ∂ y v x (cid:105) , (20)which implies that σ xx = (cid:104) n (cid:105) B (cid:42) ( η + ζ ) (cid:18) ∂ y n (cid:19) (cid:43) + (cid:42) η (cid:18) Ψ − (cid:104) n (cid:105) B η H η ∂ y n (cid:19) (cid:43) + (cid:104) n (cid:105) T B (cid:18)(cid:28) s n κ (cid:29) − (cid:104) κ − (cid:105) (cid:68) snκ (cid:69) (cid:19) . (21)Clearly, we have found a positive semidefinite conductivity tensor as required on physical grounds. Itis straightforward to convert to a resistivity matrix: ρ xx = 0 , (22a) ρ xy = − ρ yx = − B (cid:104) n (cid:105) , (22b) ρ yy = (cid:42) ( η + ζ ) (cid:18) ∂ y n (cid:19) + 1 η (cid:18) B (cid:104) n (cid:105) Ψ − η H η ∂ y n (cid:19) (cid:43) + T (cid:18)(cid:28) s n κ (cid:29) − (cid:104) κ − (cid:105) (cid:68) snκ (cid:69) (cid:19) . (22c)It remains to check whether the magnetoresistance can be negative. We first explain that, consistentwith [20, 21], the thermal conductivity can be neglected. ( i ) At low temperature in a Fermi liquid, κ ∼ T − and s ∼ T , while η ∼ T − . Hence, the correction to the hydrodynamic resistivity arising fromthe thermal conductivity is suppressed by a power of T , which is quite small at low temperatures. ( ii ) Atvery long wavelengths, but finite temperature, the dominant term in ρ yy is the Ψ -dependent term, whichdiverges with the wavelength of the inhomogeneity in the charge density; in contrast, the contribution to ρ yy coming from the thermal conductivity does not diverge with the length scale of the inhomogeneity.For the rest of this section, we focus our study on the cartoon system shown in Figure 1. From thiscartoon, and (4b), we can estimate that (cid:42) η (cid:18) B (cid:104) n (cid:105) Ψ − η H η ∂ y n (cid:19) (cid:43) ∼ η ( n , B ) B n ( n w ) ∼ ( Bw ) η ( n , B ) , (23a) A sketch of n ( y ) in a periodic system. n ( y ) is approximately n in half of the channel(length w/
2) and approximately n in the other half. The transition region between the two“domains” is of length a . We assume a (cid:28) w and n (cid:28) n . (cid:42) η (cid:18) ∂ y n (cid:19) (cid:43) ∼ aw η ( n , B ) n (cid:16) n a (cid:17) . (23b)As B →
0, we can estimate that ∂ρ yy ∂B ∼ − n (cid:96) , n awp , η ( n ,
0) + w η ( n , . (24)Hence, magnetoresistance is negative when aw (cid:28) (cid:18) n n η ( n ) (cid:96) ee , n p F , (cid:19) ∼ (cid:18) n n (cid:96) , (cid:19) (25)Of course, hydrodynamics itself is only valid when w (cid:29) a (cid:29) (cid:96) ee , where the last inequality should(conservatively) hold for the maximal value which (cid:96) ee takes in the domain. Assuming (6), we find that(25) becomes n z − (cid:28) n z − , which implies z <
1. We do not know of any physical systems with z <
1, sothis argument suggests that even the cartoon model above, which is perhaps absurd for a realistic metal,is insufficient to lead to negative magnetoresistance at small B .At large B , using (4a), we instead have: ρ yy ∼ B w η ( n , (cid:96) , p , + n awn η ( n , p , (cid:96) , B , (26)which exhibits positive magnetoresistance whenever B (cid:38) p F , (cid:96) / , (cid:96) / , , (27)assuming a, w (cid:29) (cid:96) ee , . As (27) is a sufficiently small magnetic field to estimate the magnetoresistance bythe B -correction to resistivity, we conclude that for any value of B , magnetoresistance will generally bepositive, even in highly inhomogeneous metals.The one shortcoming in our argument is, of course, that the system was only inhomogeneous in one ofthe two directions. However, we do not expect that a fully two-dimensional calculation would qualitativelychange the physics described above. In the presence of a magnetic field, it is not possible to push the lectron fluid along contours of almost zero resistance due to local Hall effects. We leave a final resolutionof the two-dimensional transport problem to elsewhere.To summarize our findings thus far, we have seen that even in highly inhomogeneous metals withsharp “domain walls” between regimes of different density, the magnetoresistance remains positive. Thisis despite our attempt to engineer a flow through a narrow channel analogous to [17]. The differencebetween our calculation and that of [17] is that, as mentioned previously, in the narrow channel of[17] there is a Hall voltage between the two sides of the channel while no current flows between them( J y = 0), while in a continuous medium the voltage must be continuous and a current flows ( J y (cid:54) = 0).This qualitative change to the flow of current creates additional magnetic field dependent corrections totransport which cause a large positive magnetoresistance [20, 21] to the electrical resistivity governing bulktransport. It is, however, plausible that a negative magnetoresistance could be seen in narrow channels,such as in a recent experiment in graphene [19]. Kinetic Theory3
Next, we ask whether it is possible to have negative magnetoresistance at the onset of hydrodynamicbehavior at ultra low temperatures, below which the physics is described by essentially free quasiparticlesmoving through a random medium.
To begin, we briefly recall some known results from the theory of transport in weakly inhomogeneousmetals [25]. Such results can be derived from kinetic theory as well [15, 16], though we will not doso here. Consider an arbitrary quantum many-body system (not only a Fermi liquid with well-definedquasiparticles) with a conserved U(1) charge, whose (effective) Hamiltonian H is translation invariant in the continuum and hence momentum conserving. Suppose that the low energy theory is described byHamiltonian H = H − (cid:90) d d x µ ( x ) n ( x ) , (28)where n ( x ) is the charge density operator and µ ( x ) is a perturbatively small inhomogeneous coefficient.Then the electrical resistivity tensor ρ ij is ρ ij = 1 n (cid:90) d d k (2 π ) d k i k j | µ ( k ) | × A nn ( k ) + O( µ ) , (29)where n = (cid:104) n ( x ) (cid:105) H is the average charge density, µ ( k ) is the Fourier transform of µ ( x ), up to an overallcoefficient related to the volume of spacetime, and A nn ( k ) = lim ω → Im (cid:0) G R nn ( k , ω ) (cid:1) ω . (30)is the spectral weight of the charge density operator at wave number k . Details of this derivation can befound in [25]. This result can be understood as a generalization of the Born approximation for electron-impurity scattering to generic interacting systems scattering off of arbitrary kinds of weak disorder. All we need to do in order to calculate resistivity is to evaluate A nn ( k ), and towards this end we use kinetictheory as a toy model for the spectral weight of the density operator across the ballistic-to-hydrodynamiccrossover. We restrict our focus to a toy model of the kinetic theory of a two-dimensional Fermi liquid
2, 11, 15, 26, 27], whose properties have been extensively studied in these previous papers. Here wesimply review what the model is and how to solve it. Let f ( x , p ) = f eq ( p ) + δ f ( x , p ) (31)be the distribution function of the fermionic quasiparticles, with f eq the Fermi-Dirac distribution and δ f a perturbatively small correction due to the presence of the external electric field. The linearizedBoltzmann equation reads ∂ t δ f + v · ∂ x δ f + E · ∂ p f eq + ( v × B ) · ∂ p δ f = − W ⊗ δ f, (32)where W corresponds (abstractly) to the linearized collision operator. Note that in a background magneticfield, the global momentum is not a conserved quantity due to the Lorentz force; the fact that electron-electron collisions conserve momentum alone implies that the linearized collision operator W does notrelax momentum. As explained in [10], at very low temperatures in a Fermi liquid, we may approximatethat f eq = Θ( (cid:15) F − (cid:15) ( p )) and that δ f = δ ( (cid:15) F − (cid:15) ( p )) × Φ ( x , θ ) , (33)where the variable Φ is defined only on the Fermi surface. In a rotationally invariant model, the Fermisurface is easily parameterized by an angle θ .For simplicity, we will begin by assuming a relaxation time approximation for the linearized collisionintegral. Defining Φ = (cid:88) n ∈ Z a n e i nθ , (34a) P Φ = (cid:88) | n |≥ a n e i nθ , (34b)we approximate that ∂ t Φ + v · ∂ x Φ + E · v + ω c ∂ θ Φ = − τ ee P Φ, (35)where the cyclotron frequency is ω c = v F Bp F . (36)Despite the fact that the collision integral is not quantitatively accurate [28, 29, 30], this model is exactlysolvable for many purposes, including ours. As discussed in the following section, we expect that thequalitative physics described below remains relevant for realistic metals close to the hydrodynamic regime.It is useful for us to recast the cartoon Boltzmann equation above as follows. Denote Φ ( θ ) as | Φ (cid:105) = (cid:88) n ∈ Z a n | n (cid:105) , (37)where we define the inner product (cid:104) n | m (cid:105) = ν , δ nm , (38)with ν the density of states of the Fermi liquid. Defining the matrices L ( ∇ ) | n (cid:105) = v F ∂ x + i ∂ y ) | n − (cid:105) + v F ∂ x − i ∂ y ) | n + 1 (cid:105) + i nω c | n (cid:105) , (39a) W | n (cid:105) = 1 τ ee (1 − δ n, − δ n, − δ n, − ) | n (cid:105) , (39b)the spectral weight A nn ( k ) is computed in kinetic theory as [15]: A nn ( k ) = (cid:104) | ( W + L ( k )) − | (cid:105) . (40)As our model is rotationally invariant, we can set k = k ˆ x when evaluating (40), without loss of generality. .3 Spectral Weight In order to evaluate (40), we use a simple trick from [11]. Let G ( k ) = ( W + L ( k )) − . (41)Suppose we can exactly evaluate G ( k ) = (cid:18) τ ee + L ( k ) (cid:19) − , (42)where the first term in the inverted matrix above implicitly multiplies the identity matrix. Denoting X as the projection onto the three | n | ≤ G = (cid:18) G − − τ ee X (cid:19) − , (43)we obtain a simple formula: if ˜ G denotes the 3 × G corresponding to the | n | ≤ G : ˜ G = (cid:18) − τ ee ˜ G (cid:19) − ˜ G . (44)These results follow from block matrix inversion identities. We only need the 3 × A nn , which is given by (40) and thus (cid:104) | ˜ G | (cid:105) .Now, we evaluate ˜ G . To do so, we find the eigenvectors of L , and it is easiest to do so by temporarilyreverting back to the θ -basis. With k = k ˆ x , we find(i kv F cos θ − ω c ∂ θ ) Φ λ ( θ ) = λΦ λ ( θ ) (45)with λ an eigenvalue to be determined. A simple calculation [31] shows that Φ λ ( θ ) = exp (cid:20) i˜ nθ + i kv F ω c sin θ (cid:21) , λ = − i˜ nω c . (46)In what follows, we define (cid:96) B = v F /ω c and (cid:96) ee = v F τ ee for simplicity. Since˜ G = (cid:88) ˜ n ∈ Z | Φ ˜ n (cid:105)(cid:104) Φ ˜ n | τ − − i˜ nω c , (47)and (cid:104) Φ ˜ n | n (cid:105) ν = π (cid:90) d θ π e i( n − ˜ n ) θ − i k(cid:96) B sin θ = J n − ˜ n ( k(cid:96) B ) , (48)we conclude that (cid:104) n (cid:48) | ˜ G | n (cid:105) ν = v F (cid:88) ˜ n ∈ Z J n (cid:48) − ˜ n ( k(cid:96) B )J n − ˜ n ( k(cid:96) B ) (cid:96) ee − i˜ n(cid:96) B . (49)Combining (40) (44) and (49) we find an explicit formula for A nn . Then we may use (30), along witha physically sensible choice for the inhomogeneity µ ( k ), to numerically evaluate the resistivity. Note thatthe inhomogeneity µ ( k ) is not a property of the kinetic theory computation, but is a property of theenvironment in which the electrons are moving. Plots of resistivity as a function of ξ/(cid:96) ee , for fixed values of (cid:96) B /ξ . Figure 3:
Plots of resistivity as a function of ξ/(cid:96) B , for fixed values of (cid:96) ee /ξ .A physically sensible choice is to assume random point-like charged impurities, placed at a distance ξ “above the plane” in which the electrons flow. (These are analogous to impurities in the gates of aheterostructure). One finds that [15] | µ ( k ) | ∝ e − k ξ ( k + k TF ) , (50)where k TF is a Thomas-Fermi screening wave number.In Figure 4, we show the resistivity ρ as a function of ξ/(cid:96) ee . Using (6), we can interpret this as ρ as a function of T (up to logarithms) in a Fermi liquid. Observe that when B = 0 ( (cid:96) B = ∞ ) that ρ is a decreasing function of temperature . This is the manifestation of the Gurzhi effect seen in a bulkcrystal [15], and when we consider the limit (cid:96) B → ∞ , we exactly recover the results of [15]. Rathersurprisingly, we find that a cyclotron radius (cid:96) B > ξ essentially leads to ∂ρ/∂T > ρ as a function of B at various temperatures. It is clear that regardless of thechoice of parameters, ρ ( B ) is a rapidly increasing function.While we have not demonstrated that magnetoresistance is positive across the ballistic-to-hydrodynamiccrossover for large amplitude inhomogeneity, we see no reason for this to not be the case. .4 A More Sophisticated Model Lastly, let us revisit the simplified model of kinetic theory discussed in the previous section. It has beenpointed out in [30] that when the temperature T (cid:28) T F (the Fermi temperature) in a low temperatureFermi liquid with a circular Fermi surface, analogous to our model above, that (cid:104) n | W | n (cid:105) ∼ ( T /T F ) n when n is odd, while (cid:104) n | W | n (cid:105) ∼ ( T /T F ) when n is even. (We have ignored logarithmic factors in both T /T F and n , for convenience.)One simple model which retains some of the simplicity of the relaxation time approximation, but atthe same time includes the hierarchy of scattering times described above, is as follows: choose n max ≥ W to be (cid:104) n | W | n (cid:48) (cid:105) = δ nn (cid:48) τ ee × | n | ≤ n = ± , ± , . . . ± ( n max + 2) , ± ( n max + 4) , . . .n / ( n max + 2) n = ± , . . . , ± n max . . (51)We say that the model with n max = 1 is the relaxation time model from above, with W given by (35).The parameter n max can be thought of as a simple proxy for T /T F , with n max ∼ (cid:114) T F T . (52)Metals such as graphene exhibit signatures of hydrodynamic flow when
T /T F ∼ .
1, so we expect that n max = 3 or n max = 5 are reasonable models for this system, as well as other low density Fermi liquidswhich might be near the hydrodynamic regime in experimental devices.An even simpler model, which we present for illustrative purposes and to demonstrate that our resultsare not finely tuned, is to simply assume that all odd harmonics with | n | ≤ n max are conserved quantities: (cid:104) n | W | n (cid:48) (cid:105) = δ nn (cid:48) τ ee × | n | ≤ n = ± , ± , . . . ± ( n max + 2) , ± ( n max + 4) , . . . n = ± , . . . , ± n max . . (53)Regardless of the collision integral that we choose, we solve this problem the same way as before. Weevaluate (43) with X = 1 − τ ee W (54)denoting the components of the collision integral which deviate from the identity (up to a factor of τ ee .As in our earlier model, X is only non-vanishing in a finite number of rows and columns. Hence, whenwe evaluate A nn using (40), we need only evaluate a finite dimensional block of (43). This makes theproblem numerically tractable.In Figure 4, we show the resistivity as a function of ξ/(cid:96) ee (a proxy for T , as before). While thecurves are qualitatively similar regardless of n max or the precise collision integral used, there are notablequantitative differences between the curves. Figure 5 shows that after turning on even a very smallmagnetic field, the quantitative discrepancies between these different models largely disappear. Just asimportantly, the trend of positive magnetoresistance persists and is not sensitive to the details of themicroscopic collision operator. Hence, our conclusion that magnetoresistance in bulk crystals is positiveat the onset of the hydrodynamic regime is not an artifact of the relaxation time approximation; rather,it is a generic feature of interacting two-dimensional Fermi liquids. Plots of resistivity as a function of ξ/(cid:96) ee for B = 0, for models with odd harmonics upto | n | ≤ n max included. Left: the model (53); right: the model (51). Figure 5:
Plots of resistivity as a function of ξ/(cid:96) B , for fixed values of (cid:96) ee /ξ , for models with oddharmonics | n | ≤ n max included. Left: the model (53); right: the model (51). Conclusion4
In this paper, we have extended the arguments of [20, 21] and demonstrated that magnetoresistance isessentially always positive in simple electron liquids with strong interactions, regardless of the strengthof inhomogeneity, and even when interactions are not very strong. We expect that our most relevantobservation is the extreme sensitivity of the resistivity to an external magnetic field in or near a hydro-dynamic flow regime of the electrons. We expect that this large positive magnetoresistance can serve asan experimental test for the Gurzhi effect, when manifested as a resistance minimum, in bulk transportmeasurements.
Acknowledgements
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