Introduction to Landau s Fermi Liquid Theory

Transcription

1 Introduction to Landau s Fermi Liquid Theory Erkki Thuneberg Deartment of hysical sciences University of Oulu 29

2 1. Introduction The rincial roblem of hysics is to determine how bodies behave when they interact. Most basic courses of classical and quantum mechanics treat the roblem of one or two articles or bodies. (An external otential can be considered as one very heavy body.) The roblem gets more difficult when the number of bodies involved is larger. In articular, in condensed matter we are dealing with a macroscoic number N 1 2 of articles, and tyically hundreds of them directly interact with each other. This roblem is commonly known as the many-body roblem. There is no general solution to the many-body roblem. Instead there is an great number of aroximations that successfully exlain various limiting cases. Here we discuss one of them, the Fermi-liquid theory. This tye of aroximation for a fermion many-body roblem was invented by Landau (1957). It was originally roosed for liquid He at very low temeratures. Soon it was realized that a similar aroach could be used to other fermion systems, most notably to the conduction electrons of metals. The Fermiliquid theory allows to understand very many roerties of metals. A generalization of the Fermi-liquid theory also allows to understand the suerconducting state, which occurs in many metals at low temeratures. Even when Landau s theory is not valid, it forms the standard against which to comare more sohisticated theories. Thus Fermiliquid theory is a aradigm of many-body theories, and it is resented in detail in many books discussing the manybody roblem. L.D. Landau and E.M. Lifshitz, Statistical Physics, Part 2 (Pergamon, Oxford, 198). P. Nozieres, Theory of interacting Fermi systems (Bejamin, New York 1964). D. Pines and P. Nozieres, The theory of quantum liquids, Vol. 1 Normal Fermi liquids (Bejamin, New York 1966). G. Baym and C. Pethick, Landau Fermi-liquid theory (Wiley, Ney York 1991). In this lecture I resent an introduction to the Landau theory. I resent the theory as if it could have logically develoed, but this not necessarily reflects historical facts. (If you know historical facts that contradict or suort this view, lease, let me know.) I will start with the calculation of secific heat in an ideal gas, and comare the result with the measurement in liquid He. This leads to the concet of quasiarticles with an effective mass differing from the atomic mass. It is then show that in order to make a consistent theory, one has to allow an interaction between the quasiarticles. Several alications of the Fermi-liquid theory are illustrated. The generalizations of the theory to metals and to the suerconducting state are discussed. 2. Preliminary toics Many-body roblem The many-body roblem for indentical articles can be formulated as follows. Consider articles of mass m labeled by index i = 1, 2,..., N. Their locations and momenta are written as r k and k. The Hamiltonian is H = N k=1 2 k 2m + V (r 1, r 2,...). (1) Here V describes interactions between any articles, and it could in many cases be written as a sum of airwise interactions V = V 12 + V V The classical many-body roblem is to solve the Newton s equations. In quantum mechanics the locations and momenta become oerators. In the Schrödinger icture k i h k, where k = ˆx x k + ŷ y k + ẑ z k. (2) An additional feature is that generally articles have sin. This is described by an additional index σ that takes values s, s + 1,..., s 1, s for a article of sin s. Thus the Hamiltonian oerator is H = N k=1 h 2 2m 2 k + V (r 1, σ 1, r 2, σ 2,...). () and the state of the system is described by a wave function Ψ(r 1, σ 1, r 2, σ 2,... ; t). Particles having integral sin are called bosons. Their wave function has to be symmetric in the exchange of any airs of arguments. For examle Ψ(r 1, σ 1, r 2, σ 2, r, σ, r 4, σ 4,...) = +Ψ(r, σ, r 2, σ 2, r 1, σ 1, r 4, σ 4,...) (4) where coordinates 1 and have been exchanged. Particles having half-integral sin are called fermions. Their wave function has to be anti-symmetric in the exchange of any airs of arguments. For examle Ψ(r 1, σ 1, r 2, σ 2, r, σ, r 4, σ 4,...) = Ψ(r, σ, r 2, σ 2, r 1, σ 1, r 4, σ 4,...) (5) The quantum many body roblem is to solve timedeendent Schrödinger equation i h Ψ t (r 1, σ 1,..., r N, σ N, t) = HΨ(r 1, σ 1,..., r N, σ N, t) or to solve energy eigenvalues and eigenstates. Ideal Fermi gas As stated in the introduction, there is no general solution of the many-body roblem. What one can do is to study some limiting cases. One articularly simle case is ideal (6) 1

3 gas, where we assume no interactions, V. Below we concentrate on ideal sin-half (s = 1/2) Fermi gas. In the absence of interactions we can assume a factorizable form Ψ (r 1, σ 1, r 2, σ 2,... ; t) = φ a (r 1, σ 1 )φ b (r 2, σ 2 ).... (7) consisting of a roduct of single-article wave functions φ α (r, σ). This does not yet satisfy the antisymmetry requirement (5) but ermuting the arguments in Ψ and summing them all together multilied by ( 1) n P, where n P is the number of airwise ermutations in a ermutation P, one can generate a roer wave function Ψ(r 1, σ 1, r 2, σ 2,...) = 1 ( 1) n P N φ a (r P (1), σ P (1) )φ b (r P (2), σ P (2) )).... (8) This is known as Slater determinant since it can also be resented as a determinant Ψ = 1 φ a (r 1, σ 1 ) φ b (r 1, σ 1 )... φ a (r 2, σ 2 ) φ b (r 2, σ 2 ).... (9) N..... We can now see if any two of the single-article wave functions φ a, φ b,... are identical, the resulting wave function vanishes identically. (Verify this in the case of two fermions.) This is the Pauli exclusion rincile, which states that a single state can be occuied by one fermion only. The natural choice for wave functions of a single free article are lane wave states. In order to incororate the sin, we have sin-u states { 1 φ (r, σ) = V e i r/ h if σ = 1 2 if σ = 1. (1) 2 and sin-down states { if σ = 1 2 φ (r, σ) = 1 V e i r/ h if σ = 1 2. (11) Here the wave vector k or the momentum = hk aears as a arameter. In order to count the states, it is most simle to require that the wave functions are eriodic in a cube of volume V = L, which allows the momenta (n x, n y and n z integers) x = 2π hn x L, y = 2π hn y L, z = 2π hn z L. (12) We suose that the volume V is very large. Then we can take the limit V in quantities that do not essentially deend on V. The energy of a single-article states is ɛ = 2 /2m. The total energy is this summed over all occuied states E(n σ ) = σ P 2 2m n σ (1) where n σ = 1 for an occuied state and is zero othewise. The ground state of a system with N articles has N lowest energy single-article states occuied and others emty. Here the maximal kinetic energy of an occuied state is called Fermi energy ɛ F. We also define the Fermi wave vector k F and Fermi momentum F = hk F so that ɛ F = 2 F 2m = h2 k 2 F 2m. (14) In momentum sace this defines the Fermi surface ( = F ). All states inside the Fermi surface ( < F ) are occuied, and the ones outside are emty. y F x 2πh- L The number of articles can be calculated as N = = 2 π F (2π h/l), (15) < F where the factor 2 comes from sin. From this we get a relation between the Fermi wave vector and the article density, N V = F. (16) π2 h The excited states of the system consist of states where one or more fermions is excited to higher energy states. The average occuations of the states at a given temerature T is given by the Fermi-Dirac distribution f(ɛ) = 1 e β(ɛ µ) + 1. (17) Here β = 1/k B T and k B = J/K is Boltzmann s constant, which is needed to exress the temerature T in Kelvins, and µ is the chemical otential. At T = the system is in its ground state, { 1 for ɛ < µ f(ɛ) = (18) for ɛ > µ. with µ = ɛ F. When T >, the occuation f(ɛ) gets rounded so that the change from f 1 to f takes lace in the energy interval k B T. k f B T T = ε F T > ε 2

4 Next we calculate the secific heat of the ideal Fermi gas at low temeratures. The average energy is given by E = 2 ɛ f(ɛ ) = (2π h/l) ɛ f(ɛ )d σ 8π = (2π h/l) 2 ɛ f(ɛ )d (19) Changing ɛ = 2 /2m as the integration variable we get E 2m V = π 2 h ɛ /2 f(ɛ)dɛ = g(ɛ)ɛf(ɛ)dɛ (2) In the second line we have exressed the same result by defining a density of states g(ɛ) = m 2mɛ/π 2 h. The secific heat is now obtained as the derivative of energy C = E(T, V, N) T (21) In order eliminate µ aearing in the distribution function (17) one has to simultaneously satisfy (For detailed derivation see Ashcroft-Mermin, Solid state hysics.) Liquid He Helium has two stable isotoes, 4 He and He. The former is by far more common in naturally occurring helium. It is a boson since in the ground state both the two electrons have total sin zero, and also the nuclear sin is zero. It has a lot of interesting roerties that could be discussed, but here we concentrate on the other isotoe. He is a fermion because the nuclear sin is one half, s = 1/2. Studies of He were started as it became available in larger quantities in the nuclear age after world war II as the decay roduct of tritium. The two isotoes of helium are the only substances that remain liquid even at the absolute zero of temerature. N V = g(ɛ)f(ɛ)dɛ (22) The result calculated with Mathematica is shown below 2 C N k B At high temeratures T T F the Fermi gas behaves like classical gas, where the average energy er article is k B T/2, known as the equiartition theorem, and this gives the secific heat k B /2 er article. We see that at lower temeratures T < T F the secific heat is reduced. This can be understood that only the articles with energies close to the Fermi surface can be excited. Those further than energy k B T from the Fermi surface cannot be excited, and thus do not contribute to secific heat. At temeratures T T F the secific heat is linear in T : F Figure: the secific hear of liquid He at two different densities (Greywall 198). We see that at low temeratures, the secific heat is linear in temerature. This resembles the ideal gas discussed above, but is quite uzzling since the atoms in a liquid are more like hard balls continuously touching each other! Also, quantitative comarison of the sloe with the ideal Fermi gas gives that the measured sloe is by factor 2.7 larger. C = π2 g(ɛ F )k 2 BT + O(T 2 ). (2) We see that the linear term is determined by the density of states at the Fermi surface g(ɛ F ) = m F. (24) π2 h

5 . Construction of the theory Landau s idea The exeriment above raises the following idea. Could it be ossible that low temerature liquid He would effectively be like an ideal gas? This was the roblem Landau started thinking. He had to answer the following questions How could dense helium atoms behave like an ideal gas? If there is exlanation to the first question, how one can understand the difference by 2.7 in the density of states? If revious questions have ositive answers, are any other modifications needed comared to the ideal gas? An obvious roblem with the ideal gas wave function (8) is that there is too little correlation between the locations r k of the articles. The Pauli rincile rohibits for two articles with the same sin to occuy the same location, but there is no such restriction for articles with oosite sins. Thus it is equally likely to find two oosite-sin articles just at the same lace than at any other laces in the ideal gas wave function (8). Weak interactions As a first attemt to answer the questions, consider oint-like articles (instead of real He atoms). In an ideal gas the articles fly straight trajectories without ever colliding. If we now allow some small size for the articles, they will collide with each other. U(r) c d a b Figure: illustration of various article-article otentials U(r): (a) the otential between two He atoms, (b) ideal gas otential U, (c) otential used in scattering aroach, (d) otential used in erturbation theory. We have to consider two articles with momenta 1 and 2 colliding and leaving with momenta 1 and 2. In such a rocess the momentum and energy has to be conserved, r = (25) ɛ 1 + ɛ 2 = ɛ 1 + ɛ 2 (26) At least qualitatively the collision rate can be calculated using the golden rule Γ = 2π h f H int i 2 δ(e f E i ) (27) f We see that the rate is roortional to the number of available final states f. Consider secifically the case of filled Fermi shere lus one article at energy ɛ 1 > ɛ F. We wish to estimate the allowed final states when article 1 collides with any article inside the Fermi shere, ɛ 2 < ɛ F. The final state has to have two articles outside the Fermi shere (ɛ 1 > ɛ F, ɛ 2 > ɛ F ) since the Pauli rincile forbids all states inside. We see that the number final states gets very small when the initial article is close to the Fermi energy, namely both ɛ 2 and ɛ 1 have to be chosen in an energy shell of thickness ɛ 1 ɛ F. This means that the final states are limited by factor (ɛ 1 ɛ F ) 2. Thus the scattering of low energy articles is indeed suressed and thus resembles the one in ideal gas. But He atoms are not oint articles, rather they touch each other continuously. Thus for one article to move, the others must give the way. This is one of the hardest roblems in many body theory even today, but one can get some idea of what haens with a model: instead of true He- He interaction otential, one assumes a weak otential, whose effect can be calculated using quantummechanical erturbation theory. We will ski this calculation here (see Landau-Lifshitz). The result is that the excitation sectrum remains qualitatively similar as in free Fermi gas but there is a shift in energies. Consider secifically the case, already mentioned above, of a filled Fermi shere lus one article at momentum, with > F. This excited state of the ideal gas corresonds to the excitation energy ɛ ɛ F = 2 2m ɛ F F m ( F ) (28) where the aroximation is good if is nof far from the Fermi surface ( F F ). The effect of the weak interactions is now that the excitation energy still is linear in F, but the coefficient is no more F /m. It is customary to write the new excitation energy in the form ɛ ɛ F = F m ( F ) (29) where we have defined the effective mass m. Note that the Fermi momentum F is not changed, equation (16) still remains valid. With the new disersion relation (29) we get a new density of states g(ɛ F ) = m F. () π2 h This is determined by the effective mass m, not the bare article mass m as for ideal gas (24). We now see that weak interactions can exlain that the secific heat coefficient (2) differs form its ideal gas value. However, the theory is valid for small erturbations, say 1%, and thus is insufficient to exlain the factor 2.7. Quasiarticles Landau now made the following assumtions. i) Even for strong interactions, the excitation sectrum remains as in (29). Such excitations are called quasiarticles: they develo continuously from single-article excitations when the interactions are turned on, but they consist of correlated 4

6 motion of the whole liquid. ii) The quasiarticles have long life time at low energy, like in the scattering aroximation above. It should be noticed that Landau s theory is henomenological. At this state it has one arameter, m, whose value is unknown theoretically, but can be obtained from exeriments. Although the detailed structure of the quasiarticle remained undetermined, we can develo a qualitative icture with a model. Consider a sherical object moving in otherwise stationary liquid. The details of this model are discussed in the aendix. The main result is that associated with the moving object, there is momentum in the fluid in the same direction. In the literature this is sometimes called back flow, but I find this name misleading. Rather it should be called forward flow or that the moving object drags with itself art of the surrounding fluid. Thinking now that the total momentum of the quasiarticle is fixed, this means that switching on the interactions slows the original fermion down, since art of the momentum goes into the surrounding fluid and less is left for the original fermion. The same icture is obtained by quantum mechanical analysis. In order to get the velocity of the quasiarticle, we have to form a localized wave acket. This travels with the grou velocity. Based on the disersion relation (29) the grou velocity is v grou = de d = F m. (1) This means that the momentum of the original fermion in the interacting system is mv grou ˆ = (m/m ), i.e. the original fermion carries fraction m/m of the momentum and the fraction 1 m/m is carried by other fermions surrounding the original one. Quasiarticle interactions Thus far we have arrived at the icture that the low energy roerties of a Fermi liquid can be understood as an ideal gas with the difference that the effective mass m aears instead of the article mass m. In the following we show that this cannot be the whole story, and one more ingredient has to be added in order to arrive at a consistent theory. A general requirement of any hysical theory is that the redictions of the theory should be indeendent of the coordinate system chosen. In the resent case, one has to ay attention to Galilean invariance. That means that the hysics should be the same in two coordinate frames that move at constant velocity with resect to each other. To be secific consider a coordinate system O, and a second coordinate system O that moves with velocity u as seen in the frame O. We assume to study a system of N articles (interacting or not) of mass m. If the total momentum of this system in O is P, then the momentum seen in frame O has to be P = P + Nmu. Now the Galilean invariance requires that if one determines the state of the system in O at fixed total momentum P, it is the same as one would do in O with momentum P. The ideal gas obviously has to satisfy Galilean invariance. However, when relace the article mass m by m in (29), the Galilean invariance is broken. The cure for this roblem is that we have to allow interactions between the quasiarticles. Thus we rewrite (29) into the form ɛ ɛ F = F m ( F )+ 1 V σ f(, )(n n () ). (2) Here n is the distribution of the quasiarticles, n () = Θ( F ) is the distribution function in the ground state, where Θ(x) is the ste function (Θ(x) = for x < and Θ(x) = 1 for x > ). The function f(, ) describes the interaction energy between two quasiarticles having momenta and. The first thing to notice is that when only one or a few quasiarticles are excited, the interaction term in (2) is negligible since n n (). In this case the excitation energy ɛ (2) reduces to its the revious exression (29). Consider next an uniformly dislaced Fermi shere, n = n () mu = Θ( F mu ). This is the stationary ground state in the O frame. In order to the theory to be Galilean invariant, the excitation energy in the must be changed from (29) to the ideal gas value (28) at the dislaced Fermi surface. As a formula F m m ˆ u = F m m ˆ u + 1 V σ f(, )(n () mu n () ). () We see that this could not be satisfied without the interaction term. In order to work () further, we need to study f(, ). Because of sherical symmetry, it can deend only on the relative directions of and, i.e. f( ˆ ˆ,, ). Futher since we are interested only in quasiarticles close to the Fermi surface, we aroximate f( ˆ ˆ,, ) f( ˆ ˆ, F, F ). This means that f deends only on the angle between and, i.e. f( ˆ ˆ ). In addition, it is conventional to define F ( ˆ ˆ ) = g(ɛ F )f( ˆ ˆ ). Such a function can be exanded in Legendre olynomials F ( ˆ ˆ ) = l= F s l P l ( ˆ ˆ ) (4) where P (x) = 1, P 1 (x) = 1, P 2 (x) = (x 2 1)/2, etc. Using these we can now reduce the requirement () to m m = 1 + F s 1. (5) We have arrived at the result that in order to have m m, we also should include an interaction between the quasiarticles of the form of the F1 s term in (4). The other interaction term wiht coefficients Fl s are not required for internal consistency of the theory but some of them aear as a result of erturbation theory. In order to make general henomenology, they all should be retained. 5

7 Until now we have not considered the fermion sin excet that they roduced factors of two. In magnetic field the system becomes sin olarized, and this is no more sufficient. All the revious analysis can be generalized to include sin deendence. This means that the quasiarticle energy ɛ() becomes a 2 2 sin matrix. Equation (2) has to be generalized to where + 1 V ɛ ην () ɛ F δ ην = F m ( F )δ ην f ηα,νβ (, )[n αβ () n () ()δ αβ ]. (6) f ηα,νβ (, ) g(ɛ F ) = (Fl s δ ηα δ νβ +Fl a σ ηα σ νβ )P l ( ˆ ˆ ) (7) l= We see that including the sin deendence the theory has two sets of arameters, Fl s and Fl a with l =, 1,.... (Here s denotes symmetric and a antisymmetric.) One of these arameters, F1 s is related to m by (5). 4. Alications In normal fermi liquid: Secific heat (see above) C = π2 g(ɛ F )k 2 BT + O(T 2 ). (8) Magnetic suscetibility χ = µ µ 2 g(ɛ F ) m 1 + F a, (9) where µ m is the magnetic moment of a article. Sound velocity Zero sound Sin waves 1 c = v F (1 + F s)(1 + 1 F 1 s ). (4) Acoustic imedance Generalizations to suerfluid state: almost all quantities affected by Fermi-liquid interactions. 4.1 Vibrating wire The figure shows the frequency (horizontal) and daming (vertical) of a vibrating arametrically as the mean free ath gets bigger with lowering temerature (from small to large daming). The exeriment is done in mixtures of He and 4 He at three different He concentrations x. Source: Martikainen et al 22. One articular roblem is to exlain why the frequency at in the ballistic limit (lowest temeratures) gets larger than in the high temerature limit. This has to do with quasiarticle interactions. b quasiarticle beam quasiarticle trajectory s 6

8 In order to understand what haened for the roagating quasiarticle, let us consider an arbitrary quasiarticle trajectory crossing a quasiarticle beam. In a Fermi gas, a quasiarticle on the trajectory would not react to the beam at all. In a Fermi liquid it exeriences the otential change δɛ caused by the beam. In the simle case that F is the only relevant Fermi-liquid arameter, the otential δɛ = [F /2N()]n is determined by the article density in the beam. µ ε F ε F δε ε The basic assumtion of the Fermi-liquid theory is that the otential is small comared to the Fermi energy ɛ F, i.e. δɛ ɛ F. This means that a quasiarticle is slightly decelerated when it enters the beam and it is accelerated back when it leaves the crossing region. In the energy oint of view (figure above), the quasiarticle flies at constant energy ɛ µ and the otential is effectively comensated by deletion of fermions with the same momentum direction in the crossing region. If there were a second quasiarticle beam along the trajectory, there aears to be no interaction between the two stationary beams. Let us now consider the case that the intensity of the (original) quasiarticle beam is varying in time. This means that the otential seen on the crossing trajectory changes. This changes the number of articles stored in the crossing region, and thus leads to emission of article or hole like quasiarticles from the crossing region. This takes lace on all crossing trajectories. In the case of a vibrating wire, the motion of the wire generates a beam of quasiarticles. This beam interacts with the quasiarticles that are coming to the wire. In case of negative F this increases the restoring force of the wire and thus also the frequency. In order to determine this quantitatively, one has to solve the Landau-Boltzmann equation. ε s Aendix: hydrodynamic model of a quasiarticle The starting oint is Euler s equation and the equation of continuity v t + v v = 1 ρ ρ + (ρv) t =, (44) where is now the ressure. We assume a small velocity so that the nonlinear term can be droed and assume incomressible fluid, ρ = constant. The boundary condition on the surface of a shere of radius a is n v = n u, where u is the velocity of the shere and n the surface normal. We assume that far from the shere v. The velocity can be reresented using the otential v = χ where χ = a u r/2r. By the Euler equation the ressure = ρ χ, and the force exerted by the shere on the fluid F = da = 2πa ρ u (45) is roortional to the acceleration u. Therefore, associated with a moving shere, there is momentum in the fluid = 2πa ρ u (46) corresonding to half of the fluid dislaced by the shere and moving in the same direction and at the same velocity as the shere. Note that this may differ from the total momentum of the fluid, which is not essential here, and is undetermined in the resent limit of unlimited fluid. n(x,, t) n(x,, t) + t x n(x,, t) ɛ(x,, t) x ɛ(x,, t) The linearized kinetic equation simlifies to = I(n(x,, t)) (41) n l t + v F ˆ (n l + δ(ɛ () µ)δɛ) = I, (42) φ t + v F ˆ (φ + δɛ) = I. (4) 7