Landau Theory of the Fermi Liquid

Transcription

1 Chater 5 Landau Theory of the Fermi Liquid 5. Adiabatic Continuity The results of the revious lectures, which are based on the hysics of noninteracting systems lus lowest orders in erturbation theory, motivates the following descrition of the behavior of a Fermi liquid at very low temeratures. These assumtions (and results) are usually called the Landau Theory of the Fermi Liquid, or more commonly Fermi Liquid Theory (FLT). n( k) Z F k Figure 5.: Discontinuity in the occuation number at the Fermi surface in a a free and in an interacting system. The hysical icture is the following. In the non-interacting system, the sectrum consists of articles and holes with a certain disersion (the free We will follow closely the treatment in Baym and Pethick.

2 2 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID one-article sectrum) ε 0 (). These excitations are infinitely long-lived since they cannot decay in the absence of matrix elements (interactions). The ground state is characterized by a distribution function n 0 ( ) which at T = 0 has the form shown in Fig. (5.). Landau reasoned that what interactions do is to roduce (virtual) articlehole airs. Thus, the distribution function must change: n 0 ( ) n( ). He further assumed that this change is a smooth function (i.e. analytic) of the interaction. In other words, as the interactions are slowly turned on, the non-interacting states are assumed to smoothly and continuously evolve into interacting states, and that this evolution takes lace without hitting any singular behavior. Such a singularity would signal an instability of the ground state and should be viewed as a hase transition. Thus, if there are no hase transitions, there should be a smooth connection between noninteracting and interacting states. In articular the quantum numbers used to label the non-interacting states should also be good quantum numbers in the resence of interactions. The one-electron (article) state becomes a quasiarticle which carries the same charge ( e) and sin (±/2) of the bare electron. Non-interacting electron (article) > P F hole ( < P F ) Interacting quasiarticle > F quasihole < F charge ±e, sin 2 always stable charge ±e, sin 2 stable only at low energies (ω 0) The state of the interacting systems can be arameterized by the actual distribution function n( ). Let δn( ) n( ) n 0 ( ). For the system to be stable δn() must be non-zero only for F and the ground state energy is determined by δn(). If n 0 () n() = n 0 () + δn() with δn/n (for all close to the fermi surface), the total energy ise = E 0 + δe and we can exand the excitation energy δe in owers of the change of the distribution function δn( ) as δe = ε δn() + (5.) The excitation energy δe should also tell us how much energy does it cost to add an excitation of momentum close to the Fermi surface. Thus,

3 5.. ADIABATIC CONTINUITY 3 emty occuied F (a) Non-interacting system emty occuied F (b) Interacting system Λ Figure 5.2: Fermi surface in a) a non-interacting system and b) in an interacting system. Only the states in the narrow region Λ F contribute in the resence of interactions. the quasiarticle energy ε() should be given by ε() = δe δn() (5.2) The difference between the energy of the ground state with N + articles and N articles is (by definition) the chemical otential µ of the system, Hence, we see that or, what is the same, E(N + ) E(N) = µ (5.3) µ = ε( ), with, = F (5.4) µ = E N = ε( F) (5.5) The higher order corrections to δe in owers of δn( ) know about the interactions among quasiarticles. This is the Landau exansion. Is ε() indeendent of the existence of other quasiarticles? If so ε() would be indeendent of δn(). In that case ε() = ε 0 ()! This is obviously not true. Thus, to low orders in δn( ) we must have terms of the form E(δn) = ε()δn( ) + 2, f(, )δn( )δn( ) + O((δn( )) 3 ) (5.6) where we have introduced the Landau arameters, the symmetric function f(, ) = f(, ).

4 4 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID Near = F the Fermi velocity is given by v F ( ) = ε( ). Hence, we can define an effective mass m = F v F ( F ) (5.7) which is isotroic only if the Fermi surface is isotroic. Hence we conclude that ε(), which is defined by ε() = δe δn() (5.8) has the form ε() = ε 0 () + f(, )δn( ) + (5.9) The correction term gives a measure of the change of the quasiarticle energy due to the resence of other quasiarticles. The function f(, ) measures the strength of quasiarticle-quasiarticle interactions. Hence f(, ) is an effective interaction for excitations arbitrarily close to the Fermi surface. What about sin effects? If the system is isotroic and there are no magnetic fields resent, the quasiarticle with u sin ( ) has the same energy as the quasiarticle with down sin ( ). Hence ε ( ) = ε ( ) (Note: This relation is changed in the resence of an external magnetic field by the Zeeman effect). Likewise, the interactions between quasiarticles deends only on the relative orientation of the sins and. The Landau interaction term is modified by sin effects as f(, )δn( ) δn( ) f (, )δn ( )δn ( ) (5.0),,, By symmetry considerations we exect that f =f f (S) + f (A) f =f f (S) f (A) (5.) Thus we can also write the quasiarticle interaction term as the sum of a

5 5.. ADIABATIC CONTINUITY 5 symmetric and an antisymmetric (or exchange) term f (, )δn () δn ( ) =,, symmetric, f (S) (, )(δn () + δn ())(δn ( ) + δn ( )) + antisymmetric, f (A) (, )(δn () δn ())(δn ( ) δn ( )) The density of quasiarticle states at Fermi surface, N(0), is, (5.2) N(0) = δ(ε 0 V µ) = n 0 (5.3) V ε where we have used the fact that n 0, is the fermi function at T = 0 (a ste function). Hence, we find N(0) = m F (5.4) π 2 3 where m is the effective mass. To avoid confusion, note that ε 0 reresents energy of quasiarticles at the Fermi-surface, while ε 0 () reresents the energy of the non-interacting system, as indicated reviously. In a general scenario, where all of the quasiarticle sins are not quantized along the same axis, the sin olarization of the Fermi liquid is (in the following equations τ reresents Pauli matrices, and α, α reresents the matrix indices) i = (τ i ) αα [n( )] αα (5.5) Equivalently, and Consequently δ 2 E = V 2 αα n( ) = (n ) αα (5.6) 2 α ( ) = ( τ) αα [n( )] αα (5.7) 2 αα ααα α f αα, α α (δn()) αα(δn( )) αα (5.8)

6 6 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID where f αα, α α = fs δ ααδ α α + fa τ αα τ α α (5.9) which can also be written more comactly as f f (S) + f (A) τ τ (no trace) (5.20) must de- For a rotationally invariant system, the interaction functions f S,A, end only on the angle θ defined by cosθ(, ) = 2 F (5.2) Hence we should be able to use an exansion of the form f S,A, = f S,A l P l (cosθ) (5.22) where l=0 f S,A l = 2l + dxp l (x) (f, ± f, ) (5.23) which leads to the definition of the Landau arameters in terms of angular momentum channels F S,A l 5.2 Equilibrium Proerties 5.2. Secific Heat: N(0)f S,A l (5.24) The low temerature secific heat of a Fermi liquid, just as in the case of non-interacting fermions, is linear in T with a coefficient determined by the effective mass m of the quasiarticles at F. Let s comute the low temerature entroy, or rather the variation of the quasiarticle entroy (er unit volume) as T T + δt S = k B V [n ( ) ln n ( ) + ( n ( )) ln( n ( ))] (5.25) where n ( ) is the Fermi-Dirac distribution n ( ) = e (ε( ) µ)/k BT + (5.26)

7 5.2. EQUILIBRIUM PROPERTIES 7 and ε ( ) is the quasiarticle excitation energy, i.e., where F(δn( ) is the free energy. Thus, the variation of the entroy is where δs = TV δf δn ( ) = ε ( ) (5.27) (ε ( ) µ)δn ( ) (5.28) δn ( ) = n [ ( ) (ε ] ( ) µ) δt + δε ( ) δµ ε ( ) T }{{} (5.29) where the term in braces is due to the quasiarticle interactions. Here, contribution from the first term is δs = V n ( ) ε ( ) (ε ( ) µ) δt (5.30) T 2 Since n( ) ε ( ) is non-zero only within k BT of the Fermi energy, we find δs = 2d 4π [ ]( ) 2 ε µ dε (2π ) 3dε δt ε e (ε µ)/k BT + T + kb 2 N(0) dx ( ) x 2 δt x e x + (5.3) Hence we find that the low temerature contribution of the quasiarticles is (to leading order) given by and that the secific heat is ( ) S C v = T T V S = π2 3 N(0)k2 B T (5.32) = S = π2 3 N(0)k2 B T = m F 3 3 k2 B T (5.33)

8 8 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID We now introduce the Fermi temerature T F = 2 F 2m k B ε F k B (5.34) as the Fermi energy in temerature units, in terms of which the secific heat becomes C V = π2 2 nk T B = π2 T CV 0 (5.35) T F 3 T F where C 0 V = 3 2 nk B is the secific heat of a classical ideal gas, and n is the article density. Using these results we find that the low temerature correction to the Free Energy, F = E T S, is δf SδT (to lowest order), i.e., T 2 F E 0 π2 4 nk B (5.36) T F where E 0 is the ground state energy. The chemical otential µ is (note that T F is a function of m and hence a function of n) ( ) F µ(n, T) = n T Let us now comute the comressibility: ( = µ(n, 0) π2 4 k B 3 + n ) m T 2 (5.37) m n T F κ = V V P = n 2 n µ (5.38) where P is the ressure. At T = 0, δn ( ) = n ( ) ε ( ) (δε ( ) δµ) (5.39) The quasiarticle energy ε ( ) deends on µ only through its deendence on δn ( ) (i.e., quasiarticle interactions, see Eq. 5.9). As T 0 both n and ε δn () vanish unless all momenta are at the Fermi-surface. δε () = f S 0 V δn ( ) f0 S δn (5.40),

9 5.3. SPIN SUSCEPTIBILITY 9 where f S 0 and Similarly, Thus, and is Landau arameter with with l = 0. Hence, we have δn ( ) = n ( ) ε ( ) (fs 0 δn δµ) (5.4) δn = δn ( ) = n ( ) V V ε ( ) (fs 0 δn δµ) (5.42),, n ( ) ε ( ) δ( F) for T 0 (5.43) δn = N(0)(f0 S δn δµ) (5.44) δn[ + N(0)f0 S ] = N(0)δµ (5.45) using the exression for the (s-wave) symmetric (singlet) Landau arameter, F S 0 = N(0)fS 0 (5.46) we can write n µ = N(0) + F0 S which leads to an exression for the comressibility κ: κ = n 2 N(0) + F S 0 (5.47) (5.48) which includes the Fermi liquid correction exressed in terms of the Landau arameter F S Sin Suscetibility We will now determine the (sin) magnetic suscetibility of a Fermi liquid. Thus we need to consider its resonse to an external magnetic field. Here we will be interested in the effect of the Zeeman couling, which causes the quasiarticle energy to change by an amount that deends on the sin olarization: 2 γ zh (5.49)

10 0 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID where γ is the gyromagnetic ratio, z is the diagonal Pauli matrix, and H is the external (uniform) magnetic field. By taking into account also the change caused to the distribution functions we find δε, = 2 γ zh + f, (, )δn V, (5.50), where, as before, δn, = n, (δε, δµ) (5.5) ε, The chemical otential is a scalar (and time reversal invariant ) quantity and as such it cannot have a linear variation with the magnetic field. Hence the only ossible deendence of µ with H must be an even ower and (at least) of order H 2. Hence it does not contribute to the magnetic suscetibility (within linear resonse). We will neglect this contribution. Hence, δn, δε,, are indeendent of the direction of the momentum, and have oosite sign for and quasiarticles. Since δn, 0 only is is on the Fermi surface (which we will assume to be isotroic), we find V, f, (, )δn, = 2fA 0 δn = z f A 0 (δn δn ) (5.52) where δ is the change in the total number of articles (er unit volume) with sin. Hence, δn = 2 N(0) ( 2 γ zh 2f A 0 δn ) (5.53) The net sin olarization is and the total magnetization M is δn δn = 2 γn(0)h + F A 0 M = γ 2 We can thus identify the sin suscetibility χ with χ = 2 4 (5.54) ( γ 2 2) N(0) H (5.55) + F0 A γ 2 N(0) + F S 0 (5.56) which is the (Pauli) sin suscetibility of a free Fermi gas with mass m, with the Fermi liquid correction.

11 5.4. EFFECTIVE MASS AND GALILEAN INVARIANCE 5.4 Effective mass and Galilean Invariance In Galilean invariant systems, there is a simle relation between m, the bare mass m and the Landau Fermi liquid arameter F S, given by m m = + 3 F (S) (5.57) To see how this comes about we will consider a Galilean transformation to a frame at seed v. The Hamiltonian of the system transforms as follows H H = H P v + 2 M v2 (5.58) where P is the total momentum oerator in the laboratory frame and M = Nm is the total mass of the system. Hence the transformed total energy and total momentum are E = E P v + 2 M v2 P = P M v (5.59) Consider the change in energy due to adding a quasiarticle of momentum in the lab frame. The total mass changes as M M + m where m is the bare mass. The addition of one quasiarticle involves the addition of one bare article. In lab frame the momentum increases by and the energy by ε ( ). In the moving frame the momentum increases by and the energy increases by m v (5.60) ε v + 2 m v2 (5.6) Therefore the quasiarticle energy in the moving frame is given by ε m v = ε v + 2 m v2 ε = ε +m v v 2 m v2 (5.62)

12 2 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID which is a consequence of Galilean invariance. Exanding to order v, and using the definition of the effective mass m, we have: ( ) m m ε m v ε + v (5.63) From the moving frame the ground state looks like a Fermi surface centered at = m v, hence the occuation numbers change as follows m n = n 0 +m v = n 0 + m v n 0 + (5.64) where n 0 +m v refers to the lab frame. The quasiarticle energy in the moving frame is ε = ε {n } = ε {n 0 +m v } (5.65) Note that this is valid for one-comonent systems only. We have n n 0 = ε ε ε = ε + V = ε F S 3 f S For = F, the Fermi momentum, we have m m n 0 m v m ε m m v (5.66) m = m m F S 3 m m = + 3 F (S) (5.67) This imlies that the relative deviation of m from m is determined by F (S). 5.5 Thermodynamic Stability The ground state should be a minimum of the (Gibbs) free energy which imlies that there should be restrictions of the Landau arameters. Consider

13 5.5. THERMODYNAMIC STABILITY 3 a distortion of the Fermi Surface characterized by a direction deendent Fermi momentum F (θ). 2 n ( ) = θ( F (θ) ) (5.68) For a stable system, the thermodynamic otential G = E µn must be a minimum. Therefore the change in the (Gibbs) free energy due to the distortion is (E µn) (E µn) 0 = (ε 0 µ)δn () + V 2V 2 where, δn () = n () n 0 () = δ F δ( F ) 2 (δ F) 2 δ( F ) f, δn ()δn ( ) (5.70) where the change of the Fermi momentum is δ F = F (θ) 0 F. The first term in Eq is where V ε (ε 0 µ)δn () = 4 N(0)v2 F v F δ F (θ) The second term in Eq is 8 (N(0)v F) 2 d cosθ 2 l=0 d(cosθ) 2 (δ F(θ, )) 2 (5.7) v l, P l (cosθ) (5.72) d cosθ f, 2 δ F(θ, )δ F (θ, ) (5.73) which imlies that ( ) N(0) δe µδn = [(v l + v l ) 2 + F (S) l 8(2l + ) 2l + l=0 ( )] (5.74) +(v l v l ) 2 + F (A) l 2l + 2 For simlicity we assume that the distorted Fermi surface has azimuthal symmetry. (5.69)

14 4 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID The Fermi liquid is stable under the deformation if δe µδn > 0 (5.75) which requires that + F S,A l 2l + 0 F S,A l (2l + ) (Pormeranchuk) (5.76) Notice that for l = ( N(0) + ) 3 F (S) l 0 (5.77) is always satisfied for a Galilean invariant one-comonent system. What haens if a Pomeranchuk inequality is violated? Clearly if this haens the system will gain energy by distorting the Fermi surface. Thus, the ground state of a system that violates the Pomeranchuk bound has a broken rotational invariance. The simlest examle of such a state in the nematic Fermi fluid in which the symmetry is broken in the quadruolar (l = 2 or d-wave) channel Non-Equilibrium Proerties In ractice we are also interested in dynamical effects, involving the roagation of excitations. Thus we need to consider systems slightly away from thermal equilibrium and slightly inhomogeneous. We wish to generalize the revious discussion to this case and to define osition and time deendent distributions n ( r, t). Clearly there is a roblem with the uncertainty rincile since we cannot define both and r with arbitrary recision. At temerature T the momentum fluctuates with a characteristic value k BT v F. If we wish to define localized quasiarticles a tyical length λ, we must have λ for the classical icture to work, which imlies that λ v F k B T (5.78) 3 To stabilize this state one needs to consider contributions to the Gibbs free energy at orders higher than (δ F (θ)) 2.

15 5.6. NON-EQUILIBRIUM PROPERTIES 5 q Figure 5.3: A article-air with relative momentum q and sin olarizations and, on the FS at. As T 0 only macroscoic excitations can be described by a classical icture (λ as ). In general we will have to use a Wigner distribution function T W( r, r 2 2 ; t), i.e., the amlitude for removing a article at r with sin at time t and at the same time to add a article at r 2 with sin 2. The Wigner function is defined as d 3 d 3 2 W( r, r 2 2 ; t) = (2π ) 3 (2π ) e i 3 ( r 2 r 2 ) a 2 2 (t)a (t) (5.79) Define [n ( r, t)] = ) d 3 r e i r W ( r + r r, ; r 2 2, ; t (5.80) d 3 q i (2π ) 3 e q r a + q, (t)a q,(t) 2 2 where FS is the Filled Fermi Sea or the ground state against which the exectation values are evaluated, and a + q (t)a 2, q,(t) FS is a articlehole air with relative momentum q localized at on the Fermi Surface. 2 Clearly a smooth distortion of the Fermi Sea requires a large number of such airs leading to coherent states of article-hole airs. (We ll come back to this later). The quasi-article density is d 3 (2π ) 3[n ( r, t)] = W( r, r) (5.8)

16 6 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID and the number of quasi-articles with momentum is d 3 r[n ( r, t)] = d 3 r d 3 r 2 e i r r 2 W( r, r 2 ). (5.82) For λ v F, note that [n k B T (r, t)] becomes a classical distribution. If the system is inhomogeneous, the total energy E(t) may vary with time. We can still define ε ( r, t) as the quasiarticle energy at osition r δe(t) = d 3 rδe( r, t) = d 3 d 3 r (2π ) E ( r, t)δn 3 (r, t) (5.83) where δe( r, t) is the local energy density. We have δe ( r, t) = d 3 r d 3 (2π ) f 3, ( r, r, t)δn ( r, t) (5.84) If the system is neutral the interactions are tyically (assumed to be!) short-ranged and vary only over microscoic scales of the order of F. Therefore, we can relace f, ( r, r, t) by a local form. If there are Coulomb forces f, ( r, r, t) e2 r r δ(t t ) + f, δ(r r ) (5.85) However many of these assumtions (concerning the existence and stability of quasiarticles) fails for transverse interactions (mediated by gauge fields) and in one-dimension. We will come back to this roblem later. 5.7 Kinetic Equation We now turn to the roblem of the evolution (i.e. dynamics) of the quasiarticle disturbances. We will use Landau s quantum kinetic theory. We begin by looking at the regime in which δn ( r, t) can be regarded as a classical distribution where it should obey a kinetic equation, i.e. the Boltzmann equation. As usual this equation is simly the continuity equation for δ n, ( r, t) and embodies the condition of local charge conservation in the fluid. In the absence of collisions between quasiarticles their number must be constant. Hence, d dt δn ( r, t) = 0 (5.86)

17 5.7. KINETIC EQUATION 7 This imlies that t δn ( r, t) + r ( v δn ( r, t)) + ( f ( r, t)δn (r, t)) = 0 (5.87) The quasiarticle grou velocity in sace is v ( nt) = ε ( r, t) (5.88) The rate of change of quasiarticle momentum (the force) is f ( r, t) = r ε ( r, t) d dt Hence we obtain Landau s kinetic equation (5.89) t δn {ε, δn } PB = I[δn ] (5.90) where I[δn ] is the collision integral, and PB denotes the Poisson bracket {ε, δn } PB = r ε δn ε δn r (5.9) Landau s kinetic equation differs from the Boltzmann equation in that. ε can be a function of r, t 2. r ε includes effective field contributions. For examle, if the system interacts with an external robe of the form of a otential U( r, t), then the total energy is increased by d r U( r, t)δn( r, t). Therefore r ε ( r, t) = d 3 U( r, t) + r (2π ) f 3 r δn (r, t). (5.92) The first term on the right hand side in the above equation is resent in dilute gases, wheras the second term arises from self-consistency condition as effects of other quasiarticles. In the quantum case one needs to use Wigner functions, giving rise to the quantum mechanical version of the kinetic equation. In Landau s aroach

18 8 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID one relaces the Poisson Brackets by multilied by commutators, roducing the quantum mechanical equations of motion (see Baym and Pethick, i ages 9-20). Near equilibrium we can linearize the transort equation, which then becomes the Landau-Silin Equation δn + v ( ) δn n0 t r ε δε = I[n ] (5.93) where v and n0 ε are equilibrium functions. The Fourier transformed equation is (ω q v )δn ( q, ω) q v n 0 ε δε ( q, ω) = ii[n ] (5.94) 5.8 Conservation Laws Let us first define n(r, t) d 3 (2π ) n ( r, t) (5.95) 3 By integrating the Landau equation over (and ) we get ( ) ( ) n + t r v n + ( f n ) = I [n ] (5.96) The number of quasiarticles is conserved uon collisions, therefore I[n ] = 0 [f n ] = 0 Let us define the current j(r, t) which is just charge conservation, is where j = n( r, t) t v n = (5.97) v n. The continuity equation, + j( r, t) = 0 (5.98) ε (r, t)n (r, t) (5.99)

19 5.8. CONSERVATION LAWS 9 On linearizing around equilibrium we get an exression for the current density (suerscrit zero denotes equilibrium quantities) j = ( ε 0 δn + δε n 0 ) (5.00) Since δn = δn n0 ε δε (5.0) which allows us to write the current as j = [ ε 0 δn ] n 0 δε Using that the current now becomes Similarly we can write δε δn (5.02) n 0 = n0 ε ε 0 (5.03) j = ( ε 0 )δn (5.04) = f, δn (r, t) (5.05) = δn n0 f, ε δn (5.06) I ll kee only the Fermi surface contribution, and find that the current takes the form j(r, t) = ( ( ε 0 )δn ( r, t) + ) 3 F (S) ( + 3 = F S ) (5.07) δn m (r, t) Galilean invariance imlies + 3 F (S) j = m = m m. Hence we have δn = P m (5.08)

20 20 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID where the last ratio is momentum density over the bare mass. Otherwise, the general result is ( + 3 j = F S ) P (5.09) m 5.8. Momentum Conservation The local momentum density g( r, t) is given by g(r, t) = n (r, t) (5.0) It obeys the local conservation law g i t + jt ij + ε r i n = 0 (5.) where T ij = i ε j n (5.2) is (almost!) the stress tensor of the Fermi fluid. Using that ε r i n = r i (ε n ) ε n r i (5.3) and ε n r i = r i E n( r, t) i U( r, t) (5.4) we can define the stress tensor Π ij which imlies that Π ij = T ij + δ ij ( ) ε n E (5.5) g i ( r, t) t + j Π ij ( r, t) + n( r, t) i U( r, t) = 0 (5.6)

21 5.9. COLLECTIVE MODES: ZERO SOUND Energy Conservation Multilying by ε and ε n t we obtain + ( ε ) ε n = 0 (5.7) We can now define the energy current density j E = ( ε )(ε U)n (5.8) which obeys the energy conservation equation t (E Un) + j E = j U (5.9) 5.9 Collective modes: Zero sound We will now look at the solutions of the Landau-Silin kinetic equation, Eq.(5.9), in the limit T 0 in which the collision integral can be neglected. By linearizing this equation we obtain ( ) t + v δn (r, t) n0 v ε δε (r, t) = 0 (5.20) with δε (r, t) = U(r, t) + f δn (r, t) (5.2) We now go to its Fourier transform and assume that the external otential is monocromatic U(r, t) Ue i( q r ωt) (5.22) and we obtain δn (r, t) = δn (q, ω)e i( q r ωt) (5.23) (ω q v )δn + n0 ε q v (U + f δn ) = 0 (5.24)

22 22 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID θ q Figure 5.4: A article-hole fluctuation with momentum roagating with q near a oint on the Fermi surface. Let us write δn in terms of ν defined by δn n0 ε ν (5.25) where we have assumed that only the Fermi surface matters. Within this notation we find that the linearized kinetic equation takes the form ν + q v f(, ) n0 ν = ω q v ε q v ω q v U (5.26) with on the Fermi surface. We will now make use of the azimuthal symmetry to exand the fluctuation in artial waves ν = ν l P l (cos θ) (5.27) and write f(, ) n0 ε l=0 ν = l=0 2l + F (S) l P l (cosθ)ν l (5.28) Let us now define the dimensionless arameter s s = ω qv F (5.29)

23 5.9. COLLECTIVE MODES: ZERO SOUND 23 and Ω ll (s) = Ω l l(s) = 2 The Landau-Silin Equation now takes the form ν l 2l + + l =0 x dxp l (x) x s P l (x) (5.30) Ω ll (s)f s l ν l 2l + = Ω l0(s)u (5.3) This equation has solutions of the form ν l (s). Let us consider first the s-wave channel, l = 0. In this channel we find Ω 00 (s) = + s ( s ) 2 ln s + = + s 2 ln s + i π s θ( s ) (5.32) s + 2 and similar exressions in the other channels. For examle, if we assume that the only non-vanishing Landau arameter is in the s-wave channel, F0 s 0 and F (s) l = 0(l ), we find the simle equation ν 0 (s) + Ω 00 (s)f (s) 0 ν 0 (s) = Ω 00 (s)u (5.33) whose solution is ν 0 (s) = Ω 00(s)U + F (s) 0 Ω (5.34) 00 The equation for the other angular momentum modes, with l, is ν l 2l + + Ω l0(s)f (s) 0 ν 0 = Ω l0 (s)u (5.35) Hence, The equation ν l 2l + Ω l0f 0 (s)ω (s) 00 U + F s 0Ω 00 (s) = Ω l0(s)u (5.36) ν l 2l + = Ω(s) l0 U[ F (s) 0 Ω 00 (s) F (s) 0 Ω 00 (s)] + F (s) 0 Ω 00(s) (5.37) ν l 2l + = Ω l0(s) Ω 00 (s) ν 0(s) (5.38) ν 0 (s) = Ω 00(s)U + F0 sω 00(s) (5.39)

24 24 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID ω zero sound qv F article-hole continuum q has oles at the zeros, s 0, of We have the following regimes Figure 5.5: Sectrum of collective modes + F s 0 Ω 00(s 0 ) = 0 (5.40) For 0 s <, Ω 00 (s) is comlex. This solution corresonds to the article-hole continuum. For s <, Ω 00 (s) is a real and monotonically increasing function of s. In articular, in the latter case we find that if F (S) 0 > 0, then there is a simle ole with s 0 > + e F 0, for F 0 s 0 = ω q v F = F (s) 0 3, for F 0 (5.4) This solution corresonds to an (undamed) sound mode with disersion ω = q v F s 0 and a sound velocity c 0 = v F s 0. This collective mode is known as zero sound. Notice that the edge of the article-hole continuum is at ω = qv F. (See Fig.5.5). 5.0 The Quasiarticle Lifetime The Landau interactions can in rincile give rise to a finite lifetime τ for the quasiarticles. However, if the lifetime remains finite as ω µ = E F

25 5.0. THE QUASIPARTICLE LIFETIME 25 hole f article article Figure 5.6: A rocess that leads to a finite quasiarticle lifetime; f denotes a Landau interaction. the whole theory breaks down. Thus, stability of the Fermi liquid requires that the lifetime to diverge, τ (at T = 0) as ω µ. Similarly, if T > 0 the lifetime must also diverge as T 0(ω = µ). In rincile the lifetime τ is a function of ω µ and T. The rate of decay is = Γ(ω µ, T). We can comute this function for τ ω µ µ and T µ(µ = ε F = k B T F ). In terms of the Green function the decay rate shows u as an imaginary art of the self energy, namely where G(, ω) = = Hence the rate is given by G 0 (, ω) Σ(, ω) = ω (ε 0 ( ) µ) Σ(, ω) (5.42) ω (ε 0 ( ) µ + ReΣ(, ω)) iimσ(, ω) sgn(iσ(, ω)) = sgn(ω µ) at = F (5.43) Γ(, ω) = ImΣ(, ω) = τ (5.44) and in general it is a function of both and ω. In terms of diagrams, the lifetime arises because there is a finite amlitude for a rocess with a quasiarticle at (, ω) in the initial state and a quasiarticle (with some momentum and frequency) and some article-hole airs in the finite state (see Fig.5.6). The amlitude is determined by the Landau interaction arameters f that we defined above. How does this rocess enter in the comutation of the self-energy? There is a term in erturbation theory of the form shown in Fig Alternately,

26 26 CHAPTER 5. LANDAU THEORY OF THE FERMI LIQUID Figure 5.7: Feynman diagram with a contribution to ImΣ. the imaginary art comes from effects of the collision integral (see Baym and Pethick. 87) and is given in terms of the t-matrix: τ = d 3 q d 3 (2π) 3 (2π) 3 2π q, + q t, 2 δ(ε() + ε( ) ε( q) ε( q)) [n 0 ( )( n 0 ( + q))( n 0 ( q)) + ( n 0 ( ))n 0 ( + q)n 0 ( q)] (5.45) which follows from using Fermi s Golden Rule. The first term in the exression in brackets in Eq.(5.43) reresents the rate at which quasiarticles are scattered into new unoccuied states while the second term reresents the blocking of such rocesses due to occuied states. The t-matrix amlitude q, + q t, is reresented by summing u article-article (or article-hole) ladder diagrams, scattering rocesses of the tye shown in Fig.5.8, given by the solution of the Bethe-saleter Equation (see Baym and Pethick,. 77): t (q, ω + iη) = f, f, q n0 ω + iη q v t (q, ω + iη) (5.46) The matrix elements of the t-matrix can also be slit into a singlet t S and trilet t A channels, and further be exanded in angular momentum comonents, t S,A l : t (q, 0) = l t l (q, 0)P l (cosθ) (5.47)

27 5.0. THE QUASIPARTICLE LIFETIME 27 whose coefficients are given by t S l = t A l = f S l + F s l 2l+ f A l + F A l 2l+ q F (5.48) q F (5.49) After some algebra on finds that the decay rate at finite temerature T and frequency ε at zero momentum transfer (i.e. on the Fermi surface) is given by π(n(0)t(0))2 τ 8πv F ( 2 F /q c) [(ε µ)2 + (πk B T) 2 ] + (5.50) (q c Λ is a momentum cutoff) which satisfies Landau s assumtions. The Landau icture we discussed works very well in neutral Fermi fluids (such as the normal hase of 3 He) and in most (simle) metals. However we will see that it fails in a number of imortant situations. In articular it fails in one dimension (for any value of the interaction couling constants) and also near a quantum hase transition. It also fails in systems with strong correlation. Figure 5.8: Feynman diagram that contributes to t (q, ω+iη) in the Bethe- Saleter Equation (5.44).