Physics 127c: Statistical Mechanics. Fermi Liquid Theory: Collective Modes. Boltzmann Equation. The quasiparticle energy including interactions

Transcription

1 Physics 27c: Statistica Mechanics Fermi Liquid Theory: Coective Modes Botzmann Equation The quasipartice energy incuding interactions ε p,σ = ε p + f(p, p ; σ, σ )δn p,σ, () p,σ with ε p ε F + v F (p p F ), acts as an effective Hamitonian for the quasipartices. Since δn p,σ can depend on space and time, through the spatia gradient of ε p,σ this gives a force acting on the quasipartices, and can ead to osciations or coective modes. Pictoriay, we can think of the modes as osciations of the Fermi sea, athough since the Fermi sea is a momentum space construction, and perturbations ead to veocities, the modes must aso invove spatia derivatives. For waveengths that are ong compared to the interpartice spacing, corresponding to mode wave vectors q with q k F, and frequencies sma compared with the Fermi energy ω ε F, the modes can be cacuated using a kinetic theory (Botzmann equation) for the quasipartices. This is a semicassica approximation in which the phase space point r, p of a quasipartice evoves driven by the effective Hamitonian ε p,σ.for q k F,ω ε F the uncertainty principe restrictions on the accuracy of a prescription of both of r, p is not important. To set up the equation suppose there is an constant equiibrium distribution n () p,σ and a perturbation δn p,σ (r,t) n p,σ (r,t)= n () p,σ + δn p,σ (r,t). (2) We wi study zero temperature so that n () p,σ is the Fermi sea. The Botzmann equation is n p,σ + ε p,σ p n p,σ ε p,σ n p,σ p = I({ δn p,σ ), (3) where I is the coision term coming from the scattering of quasipartices. The Botzmann equation (3) simpifies consideraby if we ony keep terms inear in δn p,σ. The spatia dependence necessariy invoves δn p,σ, and so the mutipying terms in (3) can be repaced by their zeroth order vaues, to give δn p,σ + v p δn p,σ n() p,σ p with v p = ε p / p v F. At zero temperature f(p,σ; p,σ ) δnp,σ p,σ = I( { δn p,σ ), (4) Thus we finay get δn p,σ + v p δn p,σ δn () p,σ p v p n () p,σ ε p = δ(ε p ε F )v p. (5) + δ(ε p ε F )v p f(p,σ; p,σ ) δnp,σ p,σ = I( { δn p,σ ). (6) Note that athough we postuated an unperturbed distribution of quasipartices n () p,σ as if it were we defined for a p, ony its properties for p p F where the concept makes sense were invoved in the derivation, and an expicit δ(ε p ε F ) occurs in the equation for δn p,σ.

2 Hydrodynamic Equations As in the Botzmann equation for the cassica gas, the coision term conserves the tota partice number, momentum, and energy. Taking appropriate moments of the Botzmann equation we coud write down conservation equations for the mass density, momentum density, and energy density. These conservation equations woud invove the divergence of correspond currents or fuxes, with expressions for these quantities given in terms of moments of δn p,σ. In equiibrium these currents are zero. In the ow frequency imit, the system is cose to equiibrium, and we can approximatey sove for δn p,σ by baancing the coision term with the driving terms coming from the gradients of n () p,σ (T, µ, v) where T,µ and the veocity v are space and time dependent. This gives δn p,σ proportiona to ωτ or qv F τ with τ the coision time characterizing the coision integra. The currents can be characterized by kinetic coefficients such as the therma conductivity, viscosity etc. This is the hydrodynamic imit vaid for ωτ, qv F τ, when the system is everywhere cose to a (oca) thermodynamic equiibrium. These cacuations are very simiar to the ones for the cassica gas, discussed in??, except that the coision integra must take into account the Fermi properties of the quasipartices and the excusion principe etc. Coisioness Limit A more nove imit is the coisioness imit ωτ, when the dynamics is dominated by the evoution of the state in phase space, and the coisions redistributing the quasipartices amongst the different states can be ignored. This is where thinking of the modes as osciations of the Fermi sea becomes usefu. Indeed, if we imagine a dispacement u,σ of the Fermi surface of the spin σ component at direction we can write δn p,σ = δ(ε p ε F )v F u,σ (7) The interaction term in the Botzmann equation (6) can be evauated as f(p,σ; p,σ ) δn p,σ d F,σσ = N() dε p δ(ε p ε F ) 4π 2N() P ( u,σ )v F (8) p,σ,σ d = v F F,σσ P ( ) u,σ, (9),σ for the moment expanding the interaction parameter in Legendre poynomias but not using the spin symmetric and antisymmetric notation F,σσ = F (s) + σσ F (a). Supposing a singe mode disturbance so that u,σ e i(q r ωt) The Botzmann equation becomes d (v F q ω)u,σ + v F q F,σσ P ( )u,σ =. (),σ Introducing spin symmetric and antisymmetric dispacements u, = u (s) u, = u (s) + u(a) () u(a) (2) the two components decoupe. Then dividing through by qv F and writing λ for the dimensioness speed of the coective mode and θ for the ange between and q, gives (for either s or a) (cos θ λ)u (s,a) + cos θ d F (s,a) P ( )u (s,a) =. (3) 2

3 Equation (3) is the dynamica equation for spin symmetric and antisymmetric modes. The paremter λ = is determined as the eigenvaues of the equation. We may expand u (s,a) on spherica harmonics Y,m (θ, φ). The different m modes decoupe. The equations coupe different however, eading in genera to compicated mode equations that can ony be soved numericay. We can gain intuition about the modes by making simpifying assumptions for the interaction parameters F (s,a). Zero Sound Zero sound First sound F (s) 5 Figure : Speed of zero sound and first sound as a function of F (s) in the mode where ony this Fermi iquid parameter is nonzero. Zero sound is a spin symmetric mode that invoves = m = distortions, amongst others, and so coupes to the tota density. A simpe discussion of zero sound is given by assuming ony F (s) is nonzero. Then in Eq. (3) for the distortion of the Fermi sea F (s) P ( ) reduces to F (s) and the equation becomes (cos θ λ)u (s) + F (s) cos θ d 2 4π u(s) =. (4) Ceary and then substituting this form gives For λ>the integra is easiy done d cos θ 4π cos θ λ = 2 to give the impicit equation for λ u (s) = C cos θ cos θ λ (5) + d 2 F (s) cos θ =. (6) 4π cos θ λ F (s) x x λ dx = + λ ( ) λ 2 n λ + (7) = (λ) = λ ( ) λ + 2 n. (8) λ 3

4 ω > < Partice-hoe excitations 2k F q Figure 2: Partice-hoe excitation spectrum. A coective mode with < (dashed ine) is strongy damped by Landau damping. For λ>the function (λ) ranges over a positive vaues, and so for any positive F (s) Eq. (8) can be inverted to find λ(f (s) ).For <F(s) < there are no rea soutions to Eq. (8) for λ. Indeed returning to the integra expression Eq. (6) shows that λ must be compex. Redoing the integras yieds soutions for λ with rea and imaginary parts comparabe, yieding a strongy damped coective mode (decay time comparabe to frequency). This damping coming not from coisions but from a resonant interaction of the coective mode with partice-hoe excitations is known as Landau damping. What is going on can be understood by first considering the range of energies for exciting a partice and hoe with tota momentum q hω q = ε p+q ε p (9) for a p with p >p F and p + q <p F. This is sketched in Fig.. The upper boundary of the region of partice-hoe energies for sma q is ω = qv F. Now consider the coective mode with = Re λ. For Re λ>the mode frequency is outside of the partice-hoe band. However for < Re λ<, the coective mode is immersed in the excitation sea, can resonanty excite partice hoe pairs, and becomes strongy damped. For F (s) < the soution for λ corresponds to an exponentiay growing time dependence. This signas the instabiity of the system. The mode we have cacuated has nonzero vaues of p,σ δn p,σ and p,σ pδn p,σ (i.e. tota number and momentum), as we as other moments, and so experimentay woud be detected as sound. Since the mode resuts from the coherent interaction of the quasipartices in the coisioness imit, rather than the near equiibrium behavior at ow frequencies, it is known as zero sound, and the hydrodynamic sound is caed first sound. The speed for first sound in the mode where ony F (s) is nonzero can be expressed as ω + F (s) =. (2) qv F 3 This sound exists as a propagating mode for a F (s) >, whereas zero sound ony propagates for F (s) >. (Note the instabiity F (s) < aso appears as a negative compressibiity.) For our simpe mode of ony 4

5 F (s) important, the speed of zero sound (where it exists) is aways greater than the speed of first sound, with the two speeds becoming equa for F (s). First Sound and Zero Sound First sound occurs in the ow frequency imit ωτ, and zero sound in the high frequency imit ωτ. For first sound, the coisions restore the quasipartice distribution to equiibrium, and the damping is proportiona to the deviation from equiibrium. Thus the damping reative to the propagation (e.g. Im q/re q for an experiment with driving at some frequency ω) is proportiona to ωτ. On the other hand zero sound is a coective motion of the quasipartices, and is disrupted by coisions. The reative dissipation from coisions is proportiona to (ωτ). If the drive frequency of the experiment is increased, the mode wi cross over from first to zero sound at a frequency ω τ, and there wi be a dissipation peak at this frequency. Aternativey, the experimentaist might ower the temperature (aways with k B T ε F ). Since the coision rate τ is proportiona to T 2 by the usua phase space arguments, the product ωτ increases as the temperature is owered. Experiments in iquid He 3 for exampe [see Abe, Anderson, and Wheatey Phys. Rev. Lett. 7, 74 (966) ] show a crossover from first sound at high temperatures to zero sound at ow temperatures, signaed by an increase in the speed of propagation and a dissipation peak. Further Reading The Theory of Quantum Liquids, o. I by Nozieres and Pines.4 and.7-. and Statistica Mechanics, part 2 of by Lifshitz and Pitaevskii 4 discuss the dynamics of Fermi iquids. May 4, 24 5