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Introduction to Landau’s Fermi Theory

Erkki Thuneberg

Department of physical sciences University of Oulu 2009 1. Introduction 2. Preliminary topics The principal problem of is to determine how bodies behave when they interact. Most basic courses of Many-body problem classical and quantum mechanics treat the problem of one The many-body problem for indentical particles can be or two particles or bodies. (An external potential can be formulated as follows. Consider particles of mass m labeled considered as one very heavy body.) The problem gets mo- by index i = 1, 2,...,N. Their locations and momenta are re difficult when the number of bodies involved is larger. written as r and p . The Hamiltonian is In particular, in condensed matter we are dealing with a k k macroscopic number N 1023 of particles, and typically N X p2 hundreds of them directly∼ interact with each other. This H = k + V (r , r ,...). (1) 2m 1 2 problem is commonly known as the many-body problem. k=1 There is no general solution to the many-body problem. Here V describes interactions between any particles, and Instead there is an great number of approximations that it could in many cases be written as a sum of pairwise successfully explain various limiting cases. Here we discuss interactions V = V12 + V13 + ... + V23 + .... The classical one of them, the Fermi-liquid theory. This type of approxi- many-body problem is to solve the Newton’s equations. mation for a many-body problem was invented by Landau (1957). It was originally proposed for liquid 3He In quantum mechanics the locations and momenta beco- me operators. In the Schr¨odinger picture pk i¯h∇k, at very low . Soon it was realized that a simi- → − lar approach could be used to other fermion systems, most where notably to the conduction of . The Fermi- ∂ ∂ ∂ liquid theory allows to understand very many properties ∇k = xˆ + yˆ + zˆ . (2) ∂xk ∂yk ∂zk of metals. A generalization of the Fermi-liquid theory al- so allows to understand the superconducting state, which An additional feature is that generally particles have . occurs in many metals at low temperatures. Even when This is described by an additional index σ that takes values Landau’s theory is not valid, it forms the standard against s, s + 1, . . . , s 1, s for a particle of spin s. Thus the − − − which to compare more sophisticated theories. Thus Fermi- Hamiltonian operator is liquid theory is a paradigm of many-body theories, and it N 2 is presented in detail in many books discussing the many- X ¯h 2 H = + V (r1, σ1, r2, σ2,...). (3) body problem. − 2m∇k k=1

L.D. Landau and E.M. Lifshitz, Statistical Physics, and the state of the system is described by a wave func- • Part 2 (Pergamon, Oxford, 1980). tion Ψ(r1, σ1, r2, σ2,... ; t). Particles having integral spin are called . Their wave function has to be symmet- P. Nozieres, Theory of interacting Fermi systems (Be- ric in the exchange of any pairs of arguments. For example • jamin, New York 1964). Ψ(r1, σ1, r2, σ2, r3, σ3, r4, σ4,...) D. Pines and P. Nozieres, The theory of quantum = +Ψ(r , σ , r , σ , r , σ , r , σ ,...) (4) • , Vol. 1 Normal Fermi liquids (Bejamin, New 3 3 2 2 1 1 4 4 York 1966). where coordinates 1 and 3 have been exchanged. Particles G. Baym and C. Pethick, Landau Fermi-liquid theory having half-integral spin are called . Their wave • (Wiley, Ney York 1991). function has to be anti-symmetric in the exchange of any pairs of arguments. For example In this lecture I present an introduction to the Landau Ψ(r1, σ1, r2, σ2, r3, σ3, r4, σ4,...) theory. I present the theory as if it could have logically = Ψ(r3, σ3, r2, σ2, r1, σ1, r4, σ4,...) (5) developed, but this not necessarily reflects historical facts. − (If you know historical facts that contradict or support this view, please, let me know.) I will start with the calculation The quantum many body problem is to solve time- of specific heat in an ideal , and compare the result with dependent Schr¨odinger equation the measurement in liquid 3He. This leads to the concept of ∂Ψ with an effective mass differing from the ato- i¯h (r , σ ,..., r , σ , t) = HΨ(r , σ ,..., r , σ , t) ∂t 1 1 N N 1 1 N N mic mass. It is then show that in order to make a consistent (6) theory, one has to allow an interaction between the qua- or to solve eigenvalues and eigenstates. siparticles. Several applications of the Fermi-liquid theory are illustrated. The generalizations of the theory to metals and to the superconducting state are discussed. Ideal As stated in the introduction, there is no general solution of the many-body problem. What one can do is to study some limiting cases. One particularly simple case is ideal

1 gas, where we assume no interactions, V 0. Below we where npσ = 1 for an occupied state and is zero othewise. concentrate on ideal spin-half (s = 1/2) Fermi≡ gas. The of a system with N particles has N In the absence of interactions we can assume a facto- lowest energy single-particle states occupied and others rizable form empty. Here the maximal kinetic energy of an occupied state is called F . We also define the Fermi Ψ (r , σ , r , σ ,... ; t) = φ (r , σ )φ (r , σ ) .... (7) 0 1 1 2 2 a 1 1 b 2 2 kF and Fermi pF =hk ¯ F so that

2 2 2 consisting of a product of single-particle wave functions pF ¯h kF F = = . (14) φα(r, σ). This does not yet satisfy the antisymmetry requi- 2m 2m rement (5) but permuting the arguments in Ψ and sum- 0 In momentum space this defines the (p = ming them all together multiplied by ( 1)nP , where n is P p ). All states inside the Fermi surface (p < p ) are occu- the number of pairwise permutations in− a permutation P , F F pied, and the ones outside are empty. one can generate a proper wave function 1 py X nP × ×××××××××××××× Ψ(r1, σ1, r2, σ2,...) = ( 1) √N − × ×××××××××××××× P × ×××××××××××××× × ×××××××××××××× φa(rP (1), σP (1))φb(rP (2), σP (2))) .... (8) × ××××××××××××××pF × × ×××××××××××××× × ×××××××××××××× px This is known as Slater determinant since it can also be ×× ××××××××××××× × ×××××××××××××× presented as a determinant × ×××××××××××××× × ×××××××××××××× 2π- × ×××××××××××××× h φa(r1, σ1) φb(r1, σ1) ... 1 × ×××××××××××××× L Ψ = φa(r2, σ2) φb(r2, σ2) ... . (9) × ×××××××××××××× √N . . . × ×××××××××××××× ...... The number of particles can be calculated as

4 3 We can now see if any two of the single-particle wave X πpF N = 2 1 = 2 3 , (15) functions φ , φ , . . . are identical, the resulting wave func- (2π¯h/L)3 a b p

2 Next we calculate the specific heat of the ideal Fermi gas (For detailed derivation see Ashcroft-Mermin, state at low temperatures. The average energy is given by physics.) Z X X 2 3 E =  f( ) =  f( )d p 3 p p (2π¯h/L)3 p p Liquid He σ p 4 3 8π Z ∞ has two stable , He and He. The former = p2 f( )dp (19) is by far more common in naturally occurring helium. It is (2π¯h/L)3 p p 0 a since in the ground state both the two electrons Changing  = p2/2m as the integration variable we get have total spin zero, and also the nuclear spin is zero. It has a lot of interesting properties that could be discussed, but 3 Z ∞ 3 E √2m 3/2 here we concentrate on the other . He is a fermion =  f()d 3 2 3 because the nuclear spin is one half, s = 1/2. Studies of He V π ¯h 0 Z ∞ were started as it became available in larger quantities in = g()f()d (20) the nuclear age after world war II as the decay product of 0 tritium. The two isotopes of helium are the only substances In the second line we have expressed the same result by that remain liquid even at the of temperature. defining a g() = m√2m/π2¯h3. The specific heat is now obtained as the derivative of energy ∂E(T,V,N) C = (21) ∂T In order eliminate µ appearing in the distribution function (17) one has to simultaneously satisfy N Z ∞ = g()f()d (22) V 0 The result calculated with Mathematica is shown below 2 C 3 N kB Figure: the specific hear of liquid 3He at two different 1.0 densities (Greywall 1983). 0.8 We see that at low temperatures, the specific heat is li- near in temperature. This resembles the discus- 0.6 sed above, but is quite puzzling since the in a liquid

0.4 are more like hard balls continuously touching each other! Also, quantitative comparison of the slope with the ideal 0.2 Fermi gas gives that the measured slope is by factor 2.7 T larger. 0.0 0.5 1.0 1.5 2.0 TF

At high temperatures T TF the Fermi gas behaves like classical gas, where the average energy per particle is 3kBT/2, known as the , and this gi- ves the specific heat 3kB/2 per particle. We see that at lower temperatures T < TF the specific heat is reduced. This can be understood that only the particles with ener- gies close to the Fermi surface can be excited. Those further than energy kBT from the Fermi surface cannot be excited, and thus do not contribute to specific heat. At temperatu- res T TF the specific heat is linear in T :  π2 C = g( )k2 T + O(T 2). (23) 3 F B We see that the linear term is determined by the density of states at the Fermi surface

mpF g(F ) = . (24) π2¯h3

3 3. Construction of the theory Fermi sphere plus one particle at energy 1 > F . We wish to estimate the allowed final states when particle 1 collides Landau’s idea with any particle inside the Fermi sphere, 2 < F . The final state has to have two particles outside the Fermi sp- The experiment above raises the following idea. Could it here (0 >  , 0 >  ) since the Pauli principle forbids all 3 1 F 2 F be possible that low temperature liquid He would effec- states inside. We see that the number final states gets very tively be like an ideal gas? This was the problem Landau small when the initial particle is close to the Fermi energy, started thinking. He had to answer the following questions 0 namely both 2 and 1 have to be chosen in an energy shell of thickness 1 F . This means that the final states are How could dense helium atoms behave like an ideal ∝ − 2 • limited by factor (1 F ) . Thus the scattering of low gas? energy particles is∝ indeed− suppressed and thus resembles If there is explanation to the first question, how one the one in ideal gas. • can understand the difference by 2.7 in the density of But 3He atoms are not point particles, rather they touch states? each other continuously. Thus for one particle to move, If previous questions have positive answers, are any the others must give the way. This is one of the hardest • other modifications needed compared to the ideal gas? problems in many body theory even today, but one can get some idea of what happens with a model: instead of 3 3 An obvious problem with the ideal gas wave function (8) true He- He interaction potential, one assumes a weak is that there is too little correlation between the locations potential, whose effect can be calculated using quantum- mechanical . We will skip this calcula- rk of the particles. The Pauli principle prohibits for two particles with the same spin to occupy the same location, tion here (see Landau-Lifshitz). The result is that the exci- but there is no such restriction for particles with opposite tation spectrum remains qualitatively similar as in free Fer- spins. Thus it is equally likely to find two opposite-spin mi gas but there is a shift in . Consider specifically particles just at the same place than at any other places in the case, already mentioned above, of a filled Fermi sphe- the ideal gas wave function (8). re plus one particle at momentum p, with p > pF . This excited state of the ideal gas corresponds to the excitation energy Weak interactions 2 As a first attempt to answer the questions, consi- p pF p F = F (p pF ) (28) der point-like particles (instead of real 3He atoms). In − 2m − ≈ m − an ideal gas the particles fly straight trajectories wit- where the approximation is good if p is nof far from the hout ever colliding. If we now allow some small size Fermi surface (p pF pF ). The effect of the weak inte- for the particles, they will collide with each other. ractions is now that− the excitation energy still is linear in c a p pF , but the coefficient is no more pF /m. It is customary U(r) to− write the new excitation energy in the form

pF p F = (p pF ) (29) d − m∗ − b ∗ 0 r where we have defined the effective mass m . Note that 0 the Fermi momentum pF is not changed, equation (16) still remains valid. With the new (29) Figure: illustration of various particle-particle potentials we get a new density of states U(r): (a) the potential between two 3He atoms, (b) ideal ∗ gas potential U 0, (c) potential used in scattering ap- m pF ≡ g(F ) = . (30) proach, (d) potential used in perturbation theory. π2¯h3 We have to consider two particles with momenta p1 and This is determined by the effective mass m∗, not the ba- 0 0 p2 colliding and leaving with momenta p1 and p2. In such re particle mass m as for ideal gas (24). We now see that a process the momentum and energy has to be conserved, weak interactions can explain that the specific heat coef- 0 0 ficient (23) differs form its ideal gas value. However, the p1 + p2 = p1 + p2 (25) 0 0 theory is valid for small perturbations, say 10%, and thus 1 + 2 = 1 + 2 (26) is insufficient to explain the factor 2.7. At least qualitatively the collision rate can be calculated using the golden rule Quasiparticles

2π X 2 Landau now made the following assumptions. i) Even for Γ = f Hint i δ(Ef Ei) (27) ¯h |h | | i| − strong interactions, the excitation spectrum remains as in f (29). Such excitations are called quasiparticles: they deve- We see that the rate is proportional to the number of avai- lop continuously from single-particle excitations when the lable final states f. Consider specifically the case of filled interactions are ”turned on”, but they consist of correlated

4 motion of the whole liquid. ii) The quasiparticles have long O’ at fixed total momentum P 0, it is the same as one would life time at low energy, like in the scattering approximation do in O with momentum P . above. The ideal gas obviously has to satisfy Galilean inva- It should be noticed that Landau’s theory is phenome- riance. However, when replace the particle mass m by m∗ nological. At this state it has one parameter, m∗, whose in (29), the Galilean invariance is broken. The cure for this value is unknown theoretically, but can be obtained from problem is that we have to allow interactions between the experiments. quasiparticles. Thus we rewrite (29) into the form Although the detailed structure of the re- pF 1 X X 0 (0) mained undetermined, we can develop a qualitative pic- p F = (p pF )+ f(p, p )(np np ). (32) − m∗ − V − ture with a model. Consider a spherical object moving in σ p otherwise stationary liquid. The details of this model are (0) discussed in the appendix. The main result is that associa- Here np is the distribution of the quasiparticles, np = Θ(pF p) is the distribution function in the ground sta- ted with the moving object, there is momentum in the fluid − in the same direction. In the literature this is sometimes te, where Θ(x) is the step function (Θ(x) = 0 for x < 0 0 called ”back flow”, but I find this name misleading. Rat- and Θ(x) = 1 for x > 0). The function f(p, p ) describes her it should be called ”forward flow”or that the moving the interaction energy between two quasiparticles having 0 object drags with itself part of the surrounding fluid. Thin- momenta p and p . king now that the total momentum of the quasiparticle is The first thing to notice is that when only one or a few fixed, this means that switching on the interactions slows quasiparticles are excited, the interaction term in (32) is the original fermion down, since part of the momentum (0) negligible since np np 0. In this case the excitation goes into the surrounding fluid and less is left for the ori- − ≈ energy p (32) reduces to its the previous expression (29). ginal fermion. Consider next an uniformly displaced Fermi sphere, np = The same picture is obtained by quantum mechanical n(0) = Θ(p p mu ). This is the stationary ground analysis. In order to get the velocity of the quasiparticle, p−mu F state in the O’ frame.− | − In order| to the theory to be Galilean we have to form a localized wave packet. This travels with invariant, the excitation energy in the must be changed the group velocity. Based on the dispersion relation (29) from (29) to the ideal gas value (28) at the displaced Fermi the group velocity is surface. As a formula dEp pF v = = . (31) pF pF 1 X X (0) group dp m∗ mpˆ u = mpˆ u + f(p, p0)(n n(0)). m m∗ V p−mu p · · σ p − This means that the momentum of the original fermion in (33) ∗ the interacting system is mvgrouppˆ = (m/m )p, i.e. the We see that this could not be satisfied without the interac- ∗ original fermion carries fraction m/m of the momentum tion term. In order to work (33) further, we need to study p and the fraction 1 m/m∗ is carried by other fermions 0 − f(p, p ). Because of spherical , it can depend on- surrounding the original one. ly on the relative directions of p and p0, i.e. f(pˆ pˆ0, p, p0). Futher since we are interested only in quasiparticles· clo- Quasiparticle interactions se to the Fermi surface, we approximate f(pˆ pˆ0, p, p0) ˆ ˆ0 · ≈ Thus far we have arrived at the picture that the low f(p p , pF , pF ). This means that f depends only on the angle· between p and p0, i.e. f(pˆ pˆ0). In addition, it is energy properties of a Fermi liquid can be understood as 0 · 0 ∗ conventional to define F (pˆ pˆ ) = g(F )f(pˆ pˆ ). Such a an ideal gas with the difference that the effective mass m · · appears instead of the particle mass m. In the following function can be expanded in Legendre polynomials we show that this cannot be the whole story, and one more ∞ 0 X s 0 ingredient has to be added in order to arrive at a consistent F (pˆ pˆ ) = F Pl(pˆ pˆ ) (34) · l · theory. l=0

A general requirement of any physical theory is that the 2 where P0(x) = 1, P1(x) = 1, P2(x) = (3x 1)/2, etc. predictions of the theory should be independent of the coor- Using these we can now reduce the requirement− (33) to dinate system chosen. In the present case, one has to pay attention to Galilean invariance. That means that the phy- m∗ F s = 1 + 1 . (35) sics should be the same in two coordinate frames that mo- m 3 ve at constant velocity with respect to each other. To be specific consider a coordinate system O, and a second coor- We have arrived at the result that in order to have dinate system O’ that moves with velocity u as seen in the m∗ = m, we also should include an interaction between 6 s frame O. We assume to study a system of N particles (in- the quasiparticles of the form of the F1 term in (34). The s teracting or not) of mass m. If the total momentum of this other interaction term wiht coefficients Fl are not requi- system in O’ is P 0, then the momentum seen in frame O red for internal consistency of the theory but some of them has to be P = P 0 + Nmu. Now the Galilean invariance appear as a result of perturbation theory. In order to make requires that if one determines the state of the system in general phenomenology, they all should be retained.

5 Until now we have not considered the fermion spin except 4. Applications that they produced factors of two. In magnetic field the sys- In normal fermi liquid: tem becomes spin polarized, and this is no more sufficient. All the previous analysis can be generalized to include spin Specific heat (see above) dependence. This means that the quasiparticle energy (p) 2 π 2 2 becomes a 2 2 spin matrix. Equation (32) has to be ge- C = g(F )kBT + O(T ). (38) neralized to × 3

pF Magnetic susceptibility  (p)  δ = (p p )δ ην F ην m∗ F ην − − g( ) 1 X χ = µ µ2 F , (39) + f (p, p0)[n (p) n(0)(p)δ ]. (36) 0 m a V ηα,νβ αβ αβ 1 + F0 p − where µm is the of a particle. where Sound velocity 0 ∞ fηα,νβ(p, p ) X s a 0 r = (Fl δηαδνβ +Fl σηα σνβ)Pl(pˆ pˆ ) (37) 1 s 1 s g(F ) · · c = vF (1 + F0 )(1 + F1 ). (40) l=0 3 3

We see that including the spin dependence the theory has Zero sound s a two sets of parameters, Fl and Fl with l = 0, 1,.... (Here s denotes symmetric and a antisymmetric.) One of these Spin waves s ∗ parameters, F1 is related to m by (35). Acoustic impedance Generalizations to superfluid state: almost all quantities affected by Fermi-liquid interactions. 4.1 Vibrating wire

The figure shows the frequency (horizontal) and damping (vertical) of a vibrating parametrically as the mean free path gets bigger with lowering temperature (from small to large damping). The experiment is done in mixtures of 3He 4 3 and He at three different He concentrations x3. Source: Martikainen et al 2002. One particular problem is to explain why the frequency at in the ballistic limit (lowest temperatures) gets larger than in the high temperature limit. This has to do with quasiparticle interactions. p quasiparticleb beam

quasiparticle trajectory s p

6 In order to understand what happened for the propaga- Appendix: hydrodynamic model of a quasipar- ting quasiparticle, let us consider an arbitrary quasiparticle ticle trajectory crossing a quasiparticle beam. In a Fermi gas, The starting point is Euler’s equation and the equation a quasiparticle on the trajectory would not react to the of continuity beam at all. In a Fermi liquid it experiences the potential change δ caused by the beam. In the simple case that F0 ∂v 1 + v ∇v = ∇p is the only relevant Fermi-liquid parameter, the potential ∂t · −ρ 0 δ = [F0/2N(0)]n is determined by the particle density in ∂ρ the beam. + ∇ (ρv) = 0, (44) ∂t · ε ε ε µ pF where p is now the . We assume a small veloci- ty so that the nonlinear term can be dropped and assume incompressible fluid, ρ = constant. The boundary condi- ε tion on the surface of a sphere of radius a is n v = n u, F · · δε where u is the velocity of the sphere and n the surface normal. We assume that far from the sphere v 0. The → p p p s velocity can be represented using the potential v = χ where χ = a3u r/2r3. By the Euler equation the pres-∇ − · The basic assumption of the Fermi-liquid theory is that sure p = ρχ˙, and the force exerted by the sphere on the fluid − the potential is small compared to the Fermi energy F , i.e. δ F . This means that a quasiparticle is slightly dece- Z 2πa3ρ lerated when it enters the beam and it is accelerated back F = da p = u˙ (45) 3 when it leaves the crossing region. In the energy point of view (figure above), the quasiparticle flies at constant ener- is proportional to the acceleration u˙ . Therefore, associated gy  µ and the potential is effectively compensated by with a moving sphere, there is momentum in the fluid depletion≈ of fermions with the same momentum direction in the crossing region. If there were a second quasiparticle 2πa3ρ p = u (46) beam along the trajectory, there appears to be no interac- 3 tion between the two stationary beams. corresponding to half of the fluid displaced by the sphere Let us now consider the case that the intensity of the and moving in the same direction and at the same velocity (original) quasiparticle beam is varying in time. This means as the sphere. Note that this may differ from the total that the potential seen on the crossing trajectory changes. momentum of the fluid, which is not essential here, and is This changes the number of particles stored in the crossing undetermined in the present limit of unlimited fluid. region, and thus leads to emission of particle or hole like quasiparticles from the crossing region. This takes place on all crossing trajectories. In the case of a vibrating wire, the motion of the wire generates a beam of quasiparticles. This beam interacts with the quasiparticles that are coming to the wire. In case of negative F0 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.

∂n(x, p, t) ∂n(x, p, t) ∂(x, p, t) + ∂t ∂x · ∂p ∂n(x, p, t) ∂(x, p, t) = I(n(x, p, t)) (41) − ∂p · ∂x The linearized kinetic equation simplifies to

∂nl (0) + v pˆ ∇(nl + δ( µ)δ) = I, (42) ∂t F · p −

∂φ + v pˆ ∇(φ + δ) = I. (43) ∂t F ·

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