Physics 127c: Statistical Mechanics
Weakly Interacting Fermi Gas
Unlike the Boson case, there is usually no qualitative change in behavior going from the noninteracting to the weakly interacting Fermi gas for repulsive interactions. For example the excitation spectrum for adding a particle of momentum h¯k with |k| >kF or for removing a particle with |k|
∂ε v = k F ∂k (2) k=kF and defines the spectrum. The value of vF is altered by interactions. The familiar spectrum leads to the usual temperature dependences at low temperature, e.g. specific heat C ∝ T , susceptibility χ independent of temperature. Furthermore, since there is no qualitative change of behavior, unlike for the Boson case (where the canonical transformation was necessary) quantitative results can be calculated using perturbation theory. When are the interactions weak? For a short range pair interaction potential this is small potential or low density. On the other hand for a Coulomb interaction, as for the electron gas, interactions are weak compared with the kinetic terms at high density
The Electron Gas We write the interaction potential as
e2 4πe2 u(r) = ;˜u(q) = , (3) r q2
2 2 where e is the electron charge in CGS units, and e = qe /4πε0 with qe the electron charge in SI units. An estimate of the interaction energy per particle is
V e2 ∼ , (4) N 2r0 with r0 a measure of the interelectron spacing given by 4 πr3 = , (5) 3 0 N i.e. a sphere of radius r0 contains on average one electron. Since kF is related to N by the usual counting of states 4 3 2. πkF = N (6) (2π)3 3 we have / 9π 1 3 kF r = ' 1.9192 .... (7) 0 4
1 2 2 It is conventional to write r0 in terms of the Bohr radius a0 = h¯ /me
r0 = rsa0 (8) and rs the interelectron spacing in units of a0 is the dimensionless measure of the density of the electron gas. The estimate of the potential energy becomes
V e2 1 ∼ . (9) N 2a0 rs The total kinetic energy is
/ E 3 3 h¯ 2k2 e2 3 9π 1 3 1 e2 2.21 kin = ε2 = F = ' . (10) F 2 2 N 5 5 2m 2a0 5 4 rs 2a0 rs
Thus the potential energy is relatively small for small rs, i.e. large density. For typical metals 2 . rs . 6 (e.g. for sodium rs ' 4) and so the interactions are not weak.
Perturbation Theory Potential Energy The ground state energy correction to lowest order in the interactions is given by the expectation value of the potential energy in the noninteracting ground state. In conventional wavefunction notation we would just write the wavefunction as an antisymmetrized product (Slater determinant) of single particle wavefunctions (functions of coordinate and spin) YN ψN (r1,...rN ; σ1 ...σN ) = A χi(ri; σi), (11) i=1 with A the antisymmetrization operator (giving the Slater determinant form). In chemistry, where we are dealing with electrons in an external potential as in the trapped Bose gases, the best single particle states χi would be found self consistently by minimizing the resulting energy expression. In a homogeneous fermion gas the χi remain spin up or down plane wave states, and the wavefunction reduces to the noninteracting one. Thus * + X 1 + + E − E = u(˜ q)a a 0 0 a 0,σ 0 a ,σ . (12) 0 k+q,σ k −q,σ k k 2 , 0, k k q 0 In the noninteracting ground state the four particle expectation value factorizes into the product of the nonzero pairwise averages, and as you proved in Homework 1 D E D E D E D E D E + + + + + + a a a 0 0 a = a a a a 0 0 − a a 0 0 a a k+q,σ k0−q,σ 0 k ,σ k,σ k+q,σ k,σ k0−q,σ 0 k ,σ k+q,σ k ,σ k0−q,σ 0 k,σ (13a) 0 0 0 0 0 = nk,σ nk0,σ 0 δq,0 − nk+q,σ nk,σ δk0,k+qδσσ0 (13b) where the minus sign for the second term comes because we have changed the order of the middle pair of Fermi operators (remember the anticommutation rules!), and nk,σ is the noninteracting ground state occupation number 1 for k
2 This gives ! X 2 X 1 E − E = u(˜ 0) n .σ − u(˜ q)n + ,σ n ,σ . (15) 0 2 k k q k k,σ k,q,σ 1 N 2u(˜ )/ The first term in Eq. (15) can be written as 2 0 and is just the mean potential interaction between all the particles. In this context it is known as the Hartree term. You might have expected this to be the complete result since we are just using the noninteracting wavefunction, and indeed this would be the case for distinguishable particles. The second term arises from the antisymmetrization of the wavefunction, and is known as the exchange or Foch term. Together we have Hartree-Foch theory.
Radial Distribution Function The Foch term can be understood better by remembering the result for the potential energy from our study of the classical interacting gas. For a pairwise interaction the potential energy just depends on the pair correlation function, conveniently expressed in terms of the radial distribution function g(r) Z 1 N 2 1 N 2 X hV i = u(r)g(r)d3r = u(˜ q)g(−q). (16) 2 2 2 q The radial distribution function g(r) is proportional to the probability of finding a particle at r given that there is one at the origin. It is normalized so that g(r →∞) = 1. Also, remember the convention for Fourier transforms I am using X f(r) = f(˜ q)eiq·r (17) q Z 1 f(˜ ) = f( )e−iq·r q r (18) where at the end of the calculation we evaluate the wavevector sum as Z X ...→ d3q.... (19) (2π)3 q
Comparing with Eq. () we can introduce the spin dependent radial distribution function gσσ0 and write 1 X g(˜ q) = g˜σσ0 (q) (20) 4 σσ0 with * + X + + g˜σσ0 (q) = a a 0 0 a 0,σ 0 a ,σ , (21) (N/ )2 k−q,σ k +q,σ k k 2 , 0 k k 0 or Fourier transforming * + X X 1 iq·R + + gσσ0 (R) = e a a 0 0 a 0,σ 0 a ,σ . (22) (N/ )2 k−q,σ k +q,σ k k 2 , 0 q k k 0 Using the same factorization procedure gives (writing the last term in terms of k, k0 rather than k, q) ! ! X X X 4 i(k−k0)·R gσσ0 (R) = n ,σ n 0,σ 0 − δσσ0 e n 0,σ n ,σ . (23) N 2 k k k k k k0 k,k0
3 This can be written in terms of the real function φ(R) 2 X φ( ) = eik·Rn . R N kσ (24) k as 2 gσσ0 = 1 − δσσ0 φ (R). (25)
The function φ is easily evaluated, since nkσ is 1 for k
(writing the density n in terms of kF and introducing y = kF r), or 3 φ(R) = [sin(kF r) − (kF r)cos(kF r)] . (28) (kF r)3
1
g↑↑
10 20 kFr
Figure 1: Radial distribution function for parallel spin fermions plotted from Eqs. (25) and (28).
These results show that g↑↓ = g ↓↑= 1 as might be expected for the noninteracting wave function. −1 However g↑↑ = g↓↓ go to zero as r → 0, and there is a “correlation hole” of radius ∼ kF because of the antisymmetrization of the wavefunction, the nonlocal ramification of the Pauli exclusion principle, see Fig. 1. Note that the recovery to g↑↑ = g↓↓ = 1 for r →∞is oscillatory and with a power law tail
cos(2kF r) 1 − g↑↑(r →∞) ∝ . (29) r4 These slowly decaying oscillations, appearing here and in other quantities, and resulting from the sharp discontinuity at the Fermi surface are known as Friedel oscillations.
4 k k+q q
Figure 2: Shaded volume gives the integral in the Foch term of the ground state energy.
Potential Energy for the Electron Gas For the electron gas with a uniform positive background (the jellium model) the first term in Eq. (15)is cancelled by the interaction with the positive background. This leaves the Foch term. There is a lowering of the energy because as we have seen from g↑↑ like spins tend to stay apart, reducing the repulsive interaction. To evaluate Eq. (15)or(??) we need to calculate X nk+q,σ nk,σ . (30) k
This is easily evaluated geometrically, since the product nk+qnk is unity inside the intersection of the Fermi sea and a Fermi sea displaced by −q (shaded in Fig. (2)) and zero outside. The shaded volume is Z " # 1 4 3 q 1 q 3 k3 π 2 θd( θ) = πk3 − + 2 F sin cos F 1 k k (31) q/2kF 3 2 2 F 2 2 F for q<2kF and zero otherwise. Thus " # X q q 3 4 3 3 1 nk+q,σ nk,σ = πkF 1 − + 2(2kF − 2kF ) (32) (2π)3 3 2 2kF 2 2kF k " # N 3 q 1 q 3 = 1 − + 2(2kF − q). (33) 2 2 2kF 2 2kF
The Foch term to the perturbation of the energy is then Z " # −1 2kF 4πe2 N 3 q 1 q 3 E − E = 4πq2 1 − + dq. (34) 0 3 2 2 8π 0 q 2 2 2kF 2 2kF Z 2 1 2 Ne 4kF 3 1 −Ne 3kF =− dx[1 − x + x3] − (35) 2 π 0 2 2 2 2π or in terms of rs / Ne2 3 9π 1 3 1 E − E0 =− . (36) 2α0 2π 4 rs
5 This gives for the total energy e2 2.21 0.916 E = − +··· (37) 2 2a0 rs rs where the kinetic energy is given by summing over the Fermi sphere in the usual way. Evaluating the higher order terms was a major focus of many in the early days of many body physics, using diagrammatic perturbation theory or other approaches. We can also calculate the excitation energy ε¯k,σ —the energy required to add an additional particle— within the same approximation: X X ∂E 1 1 0 ε¯k,σ = ' εk +˜u(0) nk0,σ 0 − u(˜ k − k )nk0,σ (38) ∂nk,σ k0,σ 0 k0 with kinetic, Hartree and Foch terms. (Note in the later the interaction is only with the same spin particles— there is no spin sum in this term.). It turns out the for the Coulomb interaction, the last term gives a logarithmic singularity at k = kF , so that vF = ∂εk/∂k is infinite. This is an unphysical result, and results from the diverging potential at small q. In this limit we must take screening into account—an electron repels its neighbors via the Coulomb interaction giving an effective positive screening cloud, so that the inter-electron interaction is reduced. Using the Thomas-Fermi approximation, the same method we used in the calculation of white dwarf stars (see the Appendix below), leads to the replacement
4πe2 4πe2 → (39) 2 2 2 q q + qTF with the Thomas-Fermi wavevector given by
πne2 2 6 4 qTF = = kF /a0. (40) εF π
We can think of this as replacing the screening the 1/q2 potential with a dielectric constant
4πe2 u˜eff (q) = (41) q2ε(q) with the Thomas-Fermi approximation to the dielectric constant
2 qTF εTF(q) = 1 + . (42) q2
In real space this corresponds to a Yukawa potential
e2 u (r) = e−qTFr eff r (43) with an exponential decay at large separations. This screened potential leads to a finite Fermi velocity. The corresponding corrections to the ground state energy are in the +··· in Eq. (37).
Diagrammatic Language The approximation we have done can be expressed (if you like such things!) in diagrammatic language as shown in Fig. (3). The diagrams for the screening of the interaction (the last line of the figure) can be
6 = + Σ
Self energy
Σ = +
Hartree Foch
=+ Screened Π interaction
Figure 3: Diagrams for the Hartree-Foch calculation of the single particle propagator using the screened interaction, with the screening calculated to lowest order. expressed algebraically as 4πe2 4πe2 u (q) = + 5(q)u (q) (44) eff q2 q2 eff 4πe2 u (q) = (45) eff q2 + 4πe2 [−5(q)] where 5(q) is given (to lowest order) by the loop or “bubble” diagram, and corresponds to an approximate evaluation of the polarizability of the electron gas. Actually 5 is frequency dependent (q represents q,ω), and the full dielectric constant screening the interaction 4πe2 ε(q) = 1 − 5(q) (46) q2 is also frequency dependent. In the static and long wavelength limit, the expression reduces to the Thomas- Fermi result. More generally we could use the expression for the bubble to give a better approximation to 5. This gives the Lindhard approximation to the dielectric constant. Using this interaction to evaluate the properties of the electron gas is known as the Random Phase Approximation or RPA.
Appendix: Thomas-Fermi Calculation of Dielectric Constant The Thomas-Fermi approach balances the degeneracy pressure against the forces from the electrostatic potential in a local approximation: at each point the degeneracy pressure is calculated for a gas at the local density as for a homogeneous gas. We used this before in the physics of white dwarf stars in Lecture 17 of Ph127a and trapped-atom Bose systems in Lecture 7 this term. Suppose that there is a net potential V(r) acting on the electrons, coming from some fixed set of charges and the electrons themselves. The chemical potential must be constant, and is the sum of potential and kinetic terms V(r) + εF (n(r)) = const, (47)
7 2 2/3 where n is the local electron density, εF ∝ kF ∝ n is the local Fermi energy, and we are considering low temperatures kB T εF . To calculate the dielectric constant we assume a small external potential, and may then linearize Eq. (47) 2 εF δV (r) + 0 δn = 0, (48) 3 n0 with n0,εF 0 the values without the perturbation. The potential derives from the sum of external charges and −eδn from the electrons
2 ∇ δV = 4π(ρext − eδn). (49)
Equations (48) and (49) are easily solved by Fourier transforming
4πρ˜ext δV(˜ q) =− , (50) 2 2 q + qTF with the Thomas-Fermi wave vector given by
πn e2 2 6 0 qTF = . (51) εF 0
˜ 2 The potential without the screening would be δVext =−4πρext /q and so we may introduce the dielectric ˜ ˜ constant δV = δVext /ε(q) with q2 ε(q) = 1 + TF . (52) q2 This is singular at small q due to the efficient screening at long wavelengths. For the interaction between electrons, the “external” charge is provided by another electron. The effective interaction in the Thomas-Fermi approximation is
4πe2 e2 u(˜ q) = , u(r) = e−qTFr , (53) 2 2 q + qTF r a screened potential finite as q → 0 in Fourier representation, and aYukawa potential decaying exponentially at large separations in real space. The Thomas-Fermi wavevector is typically comparable to the inverse spacing for the electron gas in metals, although it actually scales as the inverse of the square root of the spacing . 4 −1 1/2 1 5632 1 qTF = kF a ' . 0 1/2 (54) π rs a0
April 25, 2004
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