Physics 127c: Statistical Mechanics
Fermi Liquid Theory: Thermodynamics
Energy Expansion For a small number of excited quasiparticles the energy expanded about the ground state is X 2 E = E0 + εpδnp,σ + O(δn ) (1) p,σ where δnp,σ is plus one for every excited quasiparticle for p>pF and minus one for every quasihole (empty state in the noninteracting limit) for p
εp = εF + vF (p − pF ) +··· (3) where εF ,vF are not the same as in the noninteracting system. The single quasiparticle energy does not ∗ depend on spin in a spin invariant system. Since pF is unchanged, an effective mass m is often defined as ∗ vF = pF /m . (4) Using these expressions we can calculate the density of states of the single quasiparticle excitations. An energy band 1ε corresponds to a momentum shell near pF of width 1ε/vF . In this band there are V 1ε 2 1Nσ = 4πpF (5) (2πh)¯ 3 vF states of one spin type, with V the volume of the system. The density of states (per energy per volume) for each spin is written N(0), where the 0 refers to “at zero energy relative to the Fermi surface”. Thus ∗ m pF N(0) = . (6) 2π 2h¯ 3 This is changed from the noninteracting gas by the factor m∗/m. It is usually convenient to look at the free energy F = E − µN (this is not the Helmholtz free energy A = E − TS), and subtracting off the ground state value F0 since we are interested in the changes from the ground state due to the excitations X F − F0 = (εp − µ)δnp,σ +··· . (7) p,σ
Since at zero temperature µ = εF , and εp − εF is small for p ' pF where quasiparticles are well defined objects, this term is “second order small”. Landau had the significant insight to realize that for consistent answers the O(δn2) terms must also be retained, so that X X 1 0 0 F − F ' (εp − µ)δn ,σ + f(p,σ; p σ )δn ,σ δn 0,σ 0 . (8) 0 p 2V p p p,σ p,p0 σ,σ 0
1 Equation (8) introduces the effective interaction f(p,σ; p0σ 0) between the quasiparticle. The prefactor 1/2V is chosen in comparison with the usual expression for the interaction between particles. We know very little about the effective interaction, except the properties deriving from symmetries, and that it should vary smoothly with its arguments. Since p ' pF for quasiparticles to be well defined, we can ignore the 0 dependence on p, p0, and by rotational invariance this means f can only depend on pˆ · pˆ i.e. the angle between p and p0. By spin rotation symmetry there is just a spin-parallel, and spin-antiparallel interaction. This means we can write X∞ 0 0 (s) 0 (a) 0 f(p,σ; p σ ) = fl + σσ fl Pl(pˆ · pˆ ) (9) l=0 where the angular dependence is expanded in Legendre polynomials, σσ0 = 1 for parallel spins and −1 for (s,a) antiparallel spins, and the fl are constants, that depend of course on the nature of the physical system, (s,a) pair potential etc. Since fl has the dimensions of energy times volume, it is conventional to introduce (s,a) dimensionless Fermi liquid parameters Fl through
(s,a) (s,a) Fl = 2N(0)fl (10) with 2N(0) the total density of states per energy per volume at the Fermi surface. The quasiparticle interaction is still parameterized in terms of a discrete infinity of constant that must be found by other means (experiment or microscopic theory) but in practice interesting physical properties often just depend on small l angular components. Fermi liquid theory then provides a phenomenological (s,a) theory in terms of the small l interaction parameters Fl . Some examples are given in the next few subsections. In all the calculations it should be remembered that only excitations for p near pF can be considered, and we can usually approximate p ' pF pˆ with pˆ the unit vector in the direction of p, so that the integration over the radial |p| direction is usually trivial, and only the angular integration is left. The calculations can often be formulated in terms of infinitesimal perturbations of the Fermi sea—indeed we will see in the next lecture that collective modes such as sound can be understood in terms of the “ringing” of the Fermi sea. For a displacement of the Fermi surface by δpF at momentum pˆpF we then have
dn0 δnp,σ =− vF δpF (pˆ) (11) dεp with n0(εp) the Fermi function. At zero temperature the Fermi function is a step function, and so
δnp,σ = δ(εp − εF )vF δpF (pˆ). (12)
Since thisP is a singular function, I prefer to formulate the calculation directly in terms of the relevant sums, δn such as p,σ p,σ to give the change in the number of particles. However you will often see the calculations in text books done in terms of Eq. (12). At some stage when the sum over p is done, it can be replaced by an angular average and an integration over |p| or over εp using the density of states 2N(0) (if the two spins are behaving in the same way) Z Z X V d ···→ 2N(0) dεpδ(ε − εF ) vF δpF (pˆ) ··· (13) (2π)3 p 4π p,σ
2 Specific Heat
Because of the quasiparticle interaction, the energy cost of adding an excitation at p is no longer εp but is X δF 1 0 0 ε˜p,σ − µ = = (εp − µ) + f(p, p ; σ, σ )δnp0,σ 0 , (14) δn ,σ V p p,σ and depends on the other quasiparticles present. In general it may depend on p and σ not only on |p|. Since the counting of states is exactly the same as in the noninteracting system, the entropy S( δnp,σ ) for a given quasiparticle distribution is the same as in the noninteracting system X S =−kB δnp,σ ln δnp,σ + (1 − δnp,σ ) ln(1 − δnp,σ ) . (15) p,σ
This means that the equilibrium distribution (minimize F − TS with respect to δnp,σ ) remains the Fermi distribution δn (T ) = 1 , p,σ β(ε˜ −µ) (16) e p,σ + 1 with the difference that ε˜p,σ now depends self consistently on δn through Eq. (14). Actually,R in calculating the specific heat all these complications drop out. This is because summing over 2 2 4 |p|, dp p δnp,σ is O(T ). This means the interaction term in the energy is O(T ), whereas the single 2 2 particle terms are O(T ). In addition µ = εF + O(T ), and this change can again be neglected. The calculation proceeds just as for the noninteracting gas [see Lecture 17 of Ph127a, especially Eqs. (50-53)] but with the density of states Eq. (6), giving CV ∝ T , and
∗ CV m = . (17) CV free m
Compressibility
pF δ pF
Figure 1: Construction to calculate the bulk modulus
The isothermal bulk modulus (the inverse of the compressibility), is ∂P KT = n . (18) ∂n T,V
3 with n the density (number of particles per volume). Using the Gibbs-Duhem expression
dµ = n−1dP − sdT (19) this can be written ∂µ 2 KT = n . (20) ∂n T,V The speed of sound s is given by KT n ∂µ s2 = = . (21) nm m ∂n T,V δµ/δn δn = p To calculate we start from a reference groundP state, and put p,σ 1 in a shell at F of width δpF so that the change in the total number of particles is δnp = Vδn(see Fig. 1). The extra energy of a adding a quasiparticle at pF + δpF is then δµ which from Eq. (14)is X 1 (s) δµ = vF δpF + f 0 (δn 0,↑ + δn 0,↓) (22) V p,p p p p,σ shaded
Obviously the spin antisymmetric part of fp,p0 does not contribute. Similarly only the l = 0 component contributes, and so (s) F 1 X δµ = v δp + 0 (δn + δn ) . F F N( ) V p,↑ p,↓ (23) 2 0 p,σ shaded
The term in [ ] is just δn, which is also 2N(0)vF δpF ,so δn (s) δµ = 1 + F . (24) 2N(0) 0 Thus (s) ∂µ ∂µ 1 + F = 0 (25) ∂n ∂n m∗/m free 2 3 with (∂µ/∂n)free = π h¯ /mpF the value in the noninteracting gas of the same density. The compressibility s2 ( + F (s))/(m∗/m) and are therefore changed from the values of the noninteracting system by the factor 1 0 . The speed of sound is actually given by
(s) 1 pF 2 1 + F s2 = 0 . (26) 3 m m∗/m
Magnetic susceptibility
Similar arguments give the renormalization ofP the magneticP susceptibility. Now spins are flipped, so that to B δn =− δn linear order in the magnetic field we have p p,↑ p p,↓. The Fermi sea of up spins with lower energy in the field increases at the expense of the down spins, so that the chemical potentials are equal. The final results for the susceptibility (you will drive this in the homework) is
2N(0) m∗/m χ = µ2 = χ . (27) B + F (a) free + F (a) 1 0 1 0
4 δn↓ δ n↑
pF δ pF
Figure 2: Construction to calculate the Fermi liquid corrections to the magnetic susceptibility.
δn=-1 δn=1
mvG
Figure 3: Change δnp due to Galilean boost.
Effective Mass For a translationally invariant system the requirement of Galilean invariance leads to the expression for the effective mass ∗ m 1 (s) = 1 + F . (28) m 3 1 Consider the change in energy of a quasiparticle at momentum p due to a Galilean boost vG. We can evaluate this in two ways. The first way is a direct boost for an excitation of momentum p
δEp = p · vG. (29)
The second way is to construct the change in energy in the lab frame, from the change in the single particle and interaction terms coming from shifting all momentum by mvG X 1 (s) δE = vF pˆ · (mvG) + f 0 δn 0,↑ + δn 0,↓ (30) p V p,p p p p0 where δnp,σ is the spin-symmetric change from the shift of the Fermi, see Fig. 3. Since δnp,σ ∝ cos θ with θ ˆ l = f (s) the angle between p and vG only the 1 component of p,p0 will contribute, and we can make the
5 replacement F (s) (s) 1 0 f 0 → pˆ ·ˆp . (31) p,p 2N(0)
Equating the two expressions for δEp then gives
F (s) X 1 1 0 p · vG = vF pˆ · (mvG) + pˆ · pˆ δn 0,↑ + δn 0,↓ (32) 2N(0) V p p p0
We can calculate the sum X 1 0 ˆ δn 0 + δn 0 = 2 θ N( )v mv V p p ,↑ p ,↓ cos 2 0 F G (33) p0
2 ∗ and cos θ = 1/3, so that dividing through by p ' pF and using vF = pF /m gives m 1 (s) 1 = 1 + F , (34) m∗ 3 1 which is Eq. (28).
Parameters The correction factors are often not small. For example for liquid He3 the values are
m∗ F (s) F (a) F (s) s2 χ m 0 0 1 s2 χ free free melting pressure 6.2 94. −0.74 15.7 15.3 24. zero pressure 3.0 10.1 −0.52 6.0 2.3 6.3
Particularly at the melting pressure, where the density is largest, some of the correction factors are very large, F (s) e.g. 0 approaching 100. Nevertheless Fermi liquid theory is found to work very well: the qualitative behavior is the same as the noninteracting system (CV ∝ T , χ → const), with changed coefficients given by these parameters.
Further Reading The Theory of Quantum Liquids, Vol. I by Nozieres and Pines is a readable reference on phenomenological Fermi liquid theory. Statistical Mechanics, part 2 of the Landau and Lifshitz series is also good, but less readable. Pathria has a brief discussion in §10.8.
April 30, 2004
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