Shear viscosity and spin diffusion coefficient of a two-dimensional Fermi gas
SShear viscosity and spin diffusion coefficient of a two-dimensional Fermi gas
G. M. Bruun Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, 8000 Aarhus C, Denmark
Using kinetic theory, we calculate the shear viscosity and the spin diffusion coefficient as well as theassociated relaxation times for a two-component Fermi gas in two dimensions, as a function of tem-perature, coupling strength, polarization, and mass ratio of the two components. It is demonstratedthat the minimum value of the viscosity decreases with the mass ratio, since Fermi blocking becomesless efficient. We furthermore analyze recent experimental results for the quadrupole mode of a 2Dgas in terms of viscous damping obtaining a qualitative agreement using no fitting parameters.
INTRODUCTION
The properties of 2-dimensional (2D) Fermi systemsare fundamental for our understanding of a wide rangeof phenomena including organic and high- T c supercon-ductors, 2D nano-structures, and He films. A new gen-eration of experiments are now probing the many-bodyproperties of atomic Fermi gases in 2D traps [1–3]. Thisprovides a unique possibility to systematically explorethe physics of 2D systems using the high experimen-tal control characterizing atomic gases. Recently, therehas been a lot of interest in the transport properties ofatomic gases. One reason is that transport coefficientsprovide excellent probes for strong correlations, sincethey can change by orders of magnitude due to interac-tions. It has been shown experimentally that 3D atomicgases may form a perfect fluid with a shear viscosity η having the least possible value consistent with quantummechanics [4]. This has inspired a lot work investigat-ing the connections between the physics of atomic gases,and other strong coupling systems including quark-gluonplasmas, and liquid Helium [5]. Also, recent experimentsdemonstrate that the spin diffusion coefficient approachesa scale set by quantum mechanics for a 3D resonantly in-teracting atomic gas [6]. The first experiments probingthe collective mode spectrum of a strongly interacting 2DFermi gas were recently reported [7]. The frequency ofthe breathing mode was shown to be provide evidence ofa classical dynamical scaling symmetry [8], whereas thedamping of the quadrupole mode was used as a measurefor the shear viscosity of a 2D Fermi gas.We calculate the shear viscosity and the spin diffusioncoefficient as well as the associated relaxation times fora two-component Fermi gas in 2D using kinetic theory.The dependence of the viscosity on the mass ratio of thetwo components is analyzed, and we show that the mini-mum value is reduced for systems with a mass imbalance.We furthermore analyze the recent experimental resultsfor the quadrupole mode in terms of viscous damping [7]obtaining a qualitative agreement. However, our analy-sis shows that further work is needed to understand theexperiments quantitatively. FORMALISM
Consider a 2D gas of two fermionic species σ = 1 , m σ and density n σ = k F σ / π so that thetotal density is n = n + n . The range of the interac-tion is taken to be much shorter than the interparticlespacing, and there is therefore no interaction betweenidentical fermions. We shall focus on two steady statenon-equilibrium situations: one with a spatially varyinglocal mean velocity u ( r ), and one with a spatially vary-ing magnetization M ( r ) = n ( r ) − n ( r ), which we forconcreteness take to have the forms u ( r ) = [ u x ( y ) ,
0] and M ( r ) = M ( x ). As a result of the velocity field u x ( y ),there is a net current Π xy in the y -direction of momen-tum along the x -direction, and likewise M ( x ) induces anet magnetization current j M in the x -direction. Withinlinear response, we can writeΠ xy = − η∂ y u x and j M = − D∂ x M (1)which defines the shear viscosity η and the spin diffusioncoefficient D .We briefly outline a variational method to calculate theshear viscosity and the spin diffusion coefficient withinkinetic theory. Further details are given in Refs. [9–12]. In kinetic theory, both coefficients are obtainedfrom a steady state solution to the Boltzmann equa-tion. In the hydrodynamic limit, the distribution func-tions f σ ( r , p ) are close to the local equilibrium form f le σ =1 / [exp( βξ le σ )+1] with ξ le σ = p / m σ − u ( r ) · p − µ σ for thecase of a local velocity field u and ξ le σ = p / m σ − µ σ ( r )appropriate for a local magnetization M . Here µ ( r ) and µ ( r ) are the spatially varying chemical potentials corre-sponding to the magnetization and β = 1 /T (we useunits where k B = (cid:126) = 1). When these local equilibriumfunctions are plugged into the left side of the linearizedBoltzmann equation, it can be written as ∂f σ ∂(cid:15) Φ ησ ∂u x ∂y = I σ and ∂f σ ∂(cid:15) Φ Dσ ∂µ σ ∂x = I σ (2)where I σ is the collision operator for component σ , and f σ = 1 / [exp β ( (cid:15) σ − µ σ ) + 1] is the equilibrium functionwith (cid:15) σ = p / m σ . Here Φ ησ = p x p y /m σ and Φ Dσ = p x /m σ for shear viscosity and spin diffusion respectively. a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n The momentum and spin currents are given byΠ xy = (cid:90) d ˇ k (Φ η f + Φ η f ) j M = (cid:90) d ˇ k (Φ D f − Φ D f ) (3)with d ˇ k = d k/ (2 π ) . To proceed, we need an approxi-mate solution to the Boltzmann equation (2). In 3D, theansatz δf σ ∝ Φ σ f σ (1 − f σ ) for the deviation of f σ ( r , p )away from equilibrium is known to yield results within2% of the exact result for the viscosity [9, 13]. We there-fore use this ansatz for the 2D case which yields η = 2 β (cid:104) Φ η (cid:105) (cid:104) Φ η H [Φ η ] (cid:105) and D = βχ (cid:104) Φ D (cid:105) (cid:104) Φ D H [Φ D ] (cid:105) . (4)as variational expressions for the viscosity and spin dif-fusion coefficient. We have defined the average (cid:104) Φ (cid:105) ≡ − (cid:80) σ (cid:82) d ˇ kf σ (1 − f σ )Φ σ and χ = ∂ ( n − n ) /∂ ( µ − µ )is the magnetic susceptibility. The linearized collision in-tegral can after symmetrization be written as (cid:104) Φ H [Φ] (cid:105) = 14 (cid:90) d ˇ k d ˇ k p r m r (cid:90) π dθ (cid:48) dσdθ (cid:48) × (∆Φ) f f (1 − f )(1 − f ) (5)where p r = ( m p − m p ) /M is the relative momen-tum of the incoming scattering particles, θ (cid:48) is the anglebetween the outgoing and incoming relative momenta,and dσ/dθ is the differential cross section. The to-tal mass is M = m + m , and m − r = m − + m − is the reduced mass. We have defined the function∆Φ ≡ Φ η + Φ η − Φ η − Φ η for shear viscosity and∆Φ ≡ Φ D − Φ D − Φ D + Φ D for spin diffusion. It de-termines the contribution of a given collision to the mo-mentum and spin transport respectively. The reason forthe factor 2 difference in the expressions for η and D in(4) is that the two components contribute with the samesign to the momentum current and with opposite signs tothe magnetic current, see (3). This also causes the signdifferences in the expressions for ∆Φ in the two cases. Relaxation times
The viscous and spin relaxation times τ η and τ D whichgive the typical time between collisions for the two typesof motion are useful for estimating whether a system isin the hydrodynamic regime. Suitable definitions can beobtained by writing the collision integral as I σ (cid:39) δf σ /τ .which gives η = 2 τ η β (cid:104) (Φ η ) (cid:105) and D = τ D β (cid:104) (Φ D ) (cid:105) /χ .Performing the integrals yields ηn = τ η (cid:82) ∞ d(cid:15)(cid:15)f (cid:82) ∞ d(cid:15)f and Dn = τ D mχ (6) where we have taken m = m and n = n = n/ η = nτ η (cid:15) F / D = (cid:15) F τ D /m in the degenerate limit, whereas η = nτ η T and D = τ D T /m in the classical limit. We have used χ = m/ π for T (cid:28) T F and χ = n/ T for T (cid:29) T F . Scattering cross section
When the range of the interaction is much shorter thanthe typical interparticle spacing, the scattering betweenthe σ = 1 and σ = 2 fermions is predominantly s -wave.The 2D cross section for relative momentum p r is σ = m r |T ( p r / m r ) | /p r with the T -matrix given by [14–16] T ( (cid:15) ) = 2 πm r | E b | /(cid:15) ) + iπ (7)which has a pole at a 2-body bound state with energy E b = − / m r a . CLASSICAL LIMIT
Consider the classical limit T (cid:29) T F σ = k F σ / m σ . Inthis limit f σ (cid:28)
1, and the integrals in (4) are straight-forward to perform. We obtain for the viscosity η cl = m r π ( n + n ) n n TI η ( T /E b ) (8)with I η ( T /E b ) = (cid:90) ∞ dte − t t ln( | E b | /T t ) + π (cid:39) | E b | /T ) − . + π . (9)The shear viscosity depends only on the reduced massin the classical limit. This is because the scatteringonly depends on the relative coordinates in this limit,since there is no Fermi-blocking. For fixed total density n + n , the viscosity is minimum for n = n as expected,since the scattering becomes less frequent with increas-ing population imbalance. The viscosity in the classicallimit for m = m and n = n was reported while thismanuscript was being written [17] and the result agreeswith (8)-(9) for that case.Likewise, (4) yields for the spin diffusion coefficient inthe classical limit D cl = T π n I D ( T /E b ) (10)with I D ( T /E b ) = (cid:90) ∞ dte − t t ln( | E b | /T t ) + π (cid:39) | E b | /T ) − . + π . (11)Here, we have for simplicity taken m = m and n = n = n/ χ = n/ k B T in the classical limit.Equations (8)-(9) and (10)-(11) should be comparedwith the analogous expressions obtained for the 3Dcase: η = 15( mk B T ) / / √ π [11, 18] and D =3 √ m ( k B T ) / / √ π [12] for a classical gas in the uni-tarity regime. The reason for the more complicated T -dependence in 2D is the intrinsic energy dependence ofthe T matrix (7), which means one never recovers thesimple power law predictions for an energy independentcross section: η ∼ nvml mf ∝ √ T and D ∼ vl mf ∝ √ T where l mf ∼ /nσ is the mean free path. NUMERICAL RESULTS
In this section we present numerical results for the vis-cosity and spin diffusion coefficient obtained from (4).
Viscosity
In Fig. 1 (a), we plot the viscosity as a function of T for the mass ratios m /m = 1 and m /m = 6 /
40. Thelatter corresponds to a mixture of K and Li atoms.We have taken the density of the two components to beequal and ln( | E b | /T F ) = 1. For high temperatures, theviscosity approaches the classical result (8), whereas itincreases strongly for low T due to Fermi blocking [19].This results in a minimum of the viscosity at T (cid:39) . T F for m /m = 1, whereas the minimum is located at T (cid:39) . T F for m /m = 6 /
40 due to the larger Fermitemperature for the light component σ = 1.An important result is that the minimum viscosityof the mass imbalanced mixture is significantly smallerthan for the mass balanced mixture; for the mass ra-tio m /m = 6 /
40 it is a factor 0 . m /m = 1 clearly does not reduce the minimumvalue of η/n , since this simply amounts to rescaling T F ;however, reducing m while keeping m fixed makes theFermi blocking less efficient on the scale of T F and theminimum value of η/n is reduced essentially since theclassical result (8) holds for lower T /T F . The mini-mum value of the viscosity is subject to intense inter-est due to a conjecture inspired by results for a certainclass of strong coupling theories [20], which states thatthe ratio of the viscosity over the entropy of any sys-tem obeys the universal bound η/s > / π [21]. In theinset of Fig. 1 (a), we therefore plot η/s for the sameparameters as in the main plot. The entropy density s = s + s is obtained from the ideal gas expression s σ = − (cid:82) d ˇ k [ f σ ln f σ − (1 − f σ ) ln(1 − f σ )]. Again, wesee that the minimum value of η/s is significantly smallerfor the mass ratio m /m = 6 /
40. Intriguingly, it seemsto follow from kinetic theory that a two-component sys-
T [T F2 ] (cid:100) [ nh / (cid:47) ] (cid:100) / s [ h / (cid:47) k B ] (a) Classical limitm /m =1m /m =6/40 (cid:239) log(|E b |/ (cid:161) F2 ) (cid:100) [ nh / (cid:47) ] m /m =1m /m =6/40 (b) FIG. 1: (color on-line)(a) The viscosity as a function of tem-perature for ln( | E b | /T F ) = 1 and m /m = 1 (blue line), m /m = 6 /
40 (cyan dash-dot line). The classical limitsare plotted as red dashed and purple dotted lines. The insetshows η/s as a function of T . (b) The viscosity as a functionof ln( | E b | /T F ) for T = T F and m /m = 1 (blue line), and m /m = 6 /
40 (cyan dash-dot line). tems with a sufficiently large mass ratio can break theconjectured bound η/s > / π . A similar effect is in factpresent for 3D systems.In Fig. 1 (b), we plot η/n as a function of ln( | E b | /T F )for T = T F . The viscosity is minimum in the strongcoupling regime ln( | E b | /T F ) ∼ O (1) as expected. In theclassical limit, it follows from (8) that the minimum is lo-cated at ln( | E b | /T ) ∼ .
92. With decreasing m /m , theminimum moves to larger values of ln( | E b | /T F ) because T F increases. T [T F ] D [ h / (cid:47) m ] (cid:239) b |/ (cid:161) F2 ) FIG. 2: (color on-line) The spin diffusion coefficient as a func-tion of temperature for ln( | E b | /T F ) = 1 and m /m = 1.The classical limit is plotted as a red dashed line. The insetshows D as a function of the interaction strength for T = T F . Spin diffusion coefficient
The spin diffusion coefficient is plotted in Fig. 2as a function of temperature for m /m = 1 andln( | E b | /T F ) = 1. For high T , it approaches the clas-sical value (10) whereas Fermi blocking makes it in-crease strongly for low T , leading to a minimum valueat T = 0 . T F . The inset shows D as a function ofln( | E b | /T F ) for T = T F . Again, the minimum valueis for ln( | E b | /T F ) ∼ O (1). For high T , (10) predictsln( | E b | /T ) ∼ .
42 to be the minimum value.
Validity of kinetic theory
Let us briefly discuss the range of validity of kinetictheory. For weak coupling | ln( | E b | /(cid:15) F | ) | (cid:29)
1, the kineticapproach is accurate except for extremely low tempera-ture, as has been shown for the viscosity in 3D [18]. Forstrong coupling, one must expect the Boltzmann equa-tion (2) to break down for low temperature where thereare no well-defined quasi-particles. In 3D, calculations ofthe viscosity based on the Kubo-formalism show that thekinetic approach is accurate down to temperatures signif-icantly below T F , even for strong coupling [13, 18, 22, 23].We expect a similar result to hold in 2D. In particular,kinetic theory is likely to be reliable at the tempera-tures where we predict η and D to be minimum. Theoccupation of the closed channel molecule can further-more have significant effects on thermodynamic proper-ties in 2D [24]. Similarly, corrections to the single channelapproximation for the T -matrix (7) could influence thetransport properties considered here, although one would expect small effects for a broad resonance. EXPERIMENTS
The frequency and damping of the quadrupole modeof a 2D Fermi gas of K atoms with equal populationsin two hyperfine states were recently measured [7]. Theresults were interpreted in terms of viscous damping ap-propriate for the hydrodynamic regime. The amplitudedamping of a collective mode can be calculated from [25]Γ = |(cid:104) ˙ E mech (cid:105) t | (cid:104) E mech (cid:105) t (12)where (cid:104) E mech (cid:105) t is the time averaged mechanical energyof the mode. Taking the velocity field of the quadrupolemode to have the form u ( r ) = ( x, − y ) cos ωt , we get (cid:104) E mech (cid:105) t = m (cid:90) d r n ( r ) u ( r ) , (13)where we have used that the potential energy of the modeis equal to the kinetic energy, and we have neglected anyinteraction energy. The viscous damping is for this ve-locity field, following [25, 26], given by (cid:104) ˙ E mech (cid:105) = − b (cid:90) d r η ω Q τ η ( r ) . (14)Here we have used the real part of the complex dynamicalviscosity η ( ω ) = η/ [1 − iωτ η ( r )] [27] evaluated at thequadrupole frequency ω Q to obtain a cut-off in the outerclassical regions of the cloud, where the viscosity is givenby (8) with m = m and n = n = n/
2, and thereforeis independent of density. In the classical regime, (12)becomes using (8) and (6)Γ cl = 2 ωπ √ N I η (cid:90) ∞ du
11 + ω Q τ η (15)where N = N + N is the total number of particlestrapped and ω is the 2D trapping frequency.In Fig. 3 (a), we plot the damping of the quadrupolemode taking T /T F = 0 .
47 and N = ( E F / (cid:126) ω ⊥ ) = 4300particles trapped which are the experimental parametersappropriate for Fig. 1 in Ref. [7]. We have calculatedthe damping as a function of ln( k F a ) and the × ’s arethe experimental results reported in Ref. [7]. We seethat the theory agrees qualitatively with the data. InFig. 3 (b), we plot the damping as a function of T forvarious coupling strengths taking N = 3500 particlestrapped to model the experimental situation of Fig. 3in Ref. [7]. Again, the theory accounts qualitatively forthe experimental results which are plotted as × ’s. Notethat we have no fitting parameters. For both sets of data,the agreement between theory and experiment is best for log(k F a ) D a m p i ng r a t e (cid:75) [ (cid:116) (cid:140) (cid:239) ] ExperimentEquation (12) (a)
T [T F ] (cid:75) Q [ (cid:116) (cid:140) (cid:239) ] log(kFa)=2.7log(kFa)=5.3log(kFa)=9.7log(kFa)=18 (cid:116) (cid:140) (cid:111) (cid:100) (b) FIG. 3: (color on-line) (a) The damping of the quadrupolemode as a function of interaction strength. The × ’s arethe experimental results of Ref. [7]. (b) The damping ofthe quadrupole mode as a function of T for various couplingstrengths with the × ’s the experimental results of Ref. [7].The inset shows the viscous collision rate ω ⊥ τ η . large ln( k F a ) and large T /T F whereas there are signif-icant quantitative discrepancies in the strong couplingregime of small ln( k F a ). Similar results were reportedin Ref. [17] for the spatial average of the viscosity usingthe classical limit approximation.It is perhaps surprising that the agreement is best in the weak coupling regime where the system is collisionlessrather than hydrodynamic. This is illustrated in the insetin Fig. 3 (b) which shows ω ⊥ τ η : the hydrodynamic condi-tion ω ⊥ τ η < k F a ) = 2 .
7, whereas ω ⊥ τ η > k F a ) indicating collisionlessdynamics. However, despite being based on hydrody-namics, the viscous damping approach turns out to workrather well in the collisionless regime. In 3D it has in factbeen shown to yield exact results in the collisionless limitprovided one uses the complex dynamical viscosity eval-uated at the collisionless collective mode frequency [26].The reason for the discrepancy between theory andexperiment for small ln( k F a ) where the system is hy-drodynamic, can be strong coupling effects making thekinetic approach quantitatively inaccurate as discussedabove. Better agreement could also be obtained by solv-ing the Boltzmann equation approximately by taking mo-ments with basis functions for δf σ [23, 26, 28]. Such anapproach has indeed been successful in describing the fre-quency and damping of collective modes in 3D.To summarize, we have, using kinetic theory, calcu-lated the shear viscosity and the spin diffusion coeffi-cient for a two-component Fermi gas in 2D. Both trans-port coefficients have a minimum value somewhat belowthe Fermi temperature. We showed that the minimumvalue of the viscosity can be reduced significantly withincreasing mass ratio of the two components. Using aviscous damping approach, we qualitatively accountedfor recent experimental results for the damping of thequadrupole mode of a 2D Fermi gas. However, our re-sults also showed that further work is needed to obtain aquantitative understanding of the results.I am grateful to M. K¨ohl for discussing the experimen-tal results in Ref. [7] with me. [1] K. Martiyanov, V. Makhalov, and A. Turlapov, Phys.Rev. Lett. , 030404 (2010);[2] M. Feld et al ., Nature , 75 (2011); B. Fr¨ohlich et al .,Phys. Rev. Lett. , 105301 (2011).[3] A.Sommer et al ., arXiv:1110.3058.[4] C. Cao et al. , Science , 28 (2011).[5] T. Sch¨afer and D. Teaney, Rep. Prog. Phys. ,126001(2009).[6] A. Sommer et al. , Nature , 201 (2011); A. Sommer,M. Ku, and M. W. Zwierlein, New J. Phys. , 055009(2011).[7] E. Vogt, M. Feld, B. Fr¨ohlich, D. Pertot, M. Koshorreck,and M. K¨ohl, arXiv:1111.1173.[8] J. Hofmann, arXiv:1112.1384.[9] H. Smith and H. Højgaard Jensen, Transport Phenomena (Oxford University Press, 1989).[10] G. Baym and C. J. Pethick,
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