Physics 127c: Statistical Mechanics. Weakly Interacting Fermi Gas. The Electron Gas

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Raymond Merritt
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1 Physics 7c: Statistical Mechanics Wealy Interacting Fermi Gas Unlie the Boson case, there is usually no ualitative change in behavior going from the noninteracting to the wealy interacting Fermi gas for repulsive interactions. For example the excitation spectrum for adding a particle of momentum h with > F or for removing a particle with < F creating a hole has the same ualitative form ε µ = hv F F () with µ the chemical potential or Fermi energy and F the Fermi wave vector (which can be shown to unchanged by the interactions Luttinger s theorem). Here v F is called the Fermi velocity v F = ε () =F and defines the spectrum. The value of v F 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 ualitative change of behavior, unlie for the Boson case (where the canonical transformation was necessary) uantitative results can be calculated using perturbation theory. When are the interactions wea? 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 wea compared with the inetic terms at high density The Electron Gas We write the interaction potential as u(r) = e r ; ũ() =4πe, () where e is the electron charge in CGS units, and e = e /4πε with e the electron charge in SI units. An estimate of the interaction energy per particle is V N e, (4) r with r a measure of the interelectron spacing given by 4 πr = N, (5) i.e. a sphere of radius r contains on average one electron. Since F is related to N by the usual counting of states 4. (π) π F = N (6) we have F r = ( ) 9π / (7) 4

2 It is conventional to write r in terms of the Bohr radius a = h /me r = r s a (8) and r s the interelectron spacing in units of a is the dimensionless measure of the density of the electron gas. The estimate of the potential energy becomes The total inetic energy is V N e a r s. (9) E in N = 5 ε F = h F 5 m = e a 5 ( ) 9π / 4 rs e.. () a r s Thus the potential energy is relatively small for small r s, i.e. large density. For typical metals r s 6 (e.g. for sodium r s 4) and so the interactions are not wea. 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) ψ N (r,...r N ; σ...σ N ) = A,, N χ i (r i ; σ i ), () 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 χ i would be found self consistently by minimizing the resulting energy expression. In a homogeneous fermion gas the χ i remain spin up or down plane wave states, and the wavefunction reduces to the noninteracting one. Thus E E = ũ()a + +,σ a+,σ a,σ a,σ. () In the noninteracting ground state the four particle expectation value factorizes into the product of the nonzero pairwise averages, and as you proved in Homewor a + +,σ a+,σ a,σ a,σ = a + +,σ a,σ a +,σ a,σ a ++,σ a,σ a +,σ a,σ (a) i= = n,σ n,σ δ, n +,σ n,σ δ,+δ σσ (b) 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 n,σ is the noninteracting ground state occupation number { for <F n,σ =. (4) for > F

3 This gives ( ) E E = ũ() n.σ ũ()n +,σ n,σ. (5),σ,,σ The first term in E. (5) can be written as N ũ()/ and is just the mean potential interaction between all the particles. In this context it is nown 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 nown 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) V = N u(r)g(r)d r = N ũ()g( ). (6) 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 ) =. Also, remember the convention for Fourier transforms I am using f(r) = f()e i r (7) f() = f(r)e i r (8) where at the end of the calculation we evaluate the wavevector sum as... d... (9) (π) () Comparing with E. () we can introduce the spin dependent radial distribution function g σσ and write g() = g σσ () 4 σσ with g σσ () = a + (N/),σ a+ +,σ a,σ a,σ,, () or Fourier transforming g σσ (R) = e i R a + (N/),σ a+ +,σ a,σ a,σ,. () Using the same factorization procedure gives (writing the last term in terms of, rather than, ) ( )( ) g σσ (R) = 4 n N,σ n,σ δ σσ e i( ) R n,σ n,σ. (),

4 This can be written in terms of the real function φ(r) φ(r) = e i R n σ. (4) N as g σσ = δ σσ φ (R). (5) The function φ is easily evaluated, since n σ is for < F and for > F as F φ(r) = 4π d d(cos θ)e ir cos θ (6) n (π) = F r dy y sin y (7) ( F r) (writing the density n in terms of F and introducing y = F r), or φ(r) = ( F r) [sin( F r) ( F r)cos( F r)]. (8) g F r Figure : Radial distribution function for parallel spin fermions plotted from Es. (5) and (8). These results show that g = g = as might be expected for the noninteracting wave function. However g = g go to zero as r, and there is a correlation hole of radius F because of the antisymmetrization of the wavefunction, the nonlocal ramification of the Pauli exclusion principle, see Fig.. Note that the recovery to g = g = for r is oscillatory and with a power law tail g (r ) cos( F r). (9) r 4 These slowly decaying oscillations, appearing here and in other uantities, and resulting from the sharp discontinuity at the Fermi surface are nown as Friedel oscillations. 4

5 + Figure : 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 bacground (the jellium model) the first term in E. (5) is cancelled by the interaction with the positive bacground. This leaves the Foch term. There is a lowering of the energy because as we have seen from g lie spins tend to stay apart, reducing the repulsive interaction. To evaluate E. (5)or(??) we need to calculate n +,σ n,σ. () This is easily evaluated geometrically, since the product n + n is unity inside the intersection of the Fermi sea and a Fermi sea displaced by (shaded in Fig. ()) and zero outside. The shaded volume is [ F π sin θd(cos θ) = 4 / F π F + ( ) ] () F F for < F and zero otherwise. Thus [ n +,σ n,σ = 4 (π) π F + ( ) ] ( F F ) () F F [ = N + ( ) ] ( F ). () F F The Foch term to the perturbation of the energy is then E E = ( ) [ F 4π 4πe N + ( ) ] 8π F F = Ne 4 F π dx[ x + x ] Ne F π d. (4) (5) or in terms of r s E E = Ne α π ( ) 9π /. (6) 4 r s 5

6 This gives for the total energy [ E = e..96 ] + (7) a rs r s where the inetic 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,σ the energy reuired to add an additional particle within the same approximation: ε,σ = E n,σ ε +ũ() n,σ ũ( )n,σ (8),σ with inetic, 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 = F, so that v F = ε / is infinite. This is an unphysical result, and results from the diverging potential at small. In this limit we must tae 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 4πe 4πe + TF (9) with the Thomas-Fermi wavevector given by TF = 6πne ε F = 4 π F /a. (4) We can thin of this as replacing the screening the / potential with a dielectric constant ũ eff () = 4πe ε() (4) with the Thomas-Fermi approximation to the dielectric constant In real space this corresponds to a Yuawa potential ε TF () = + TF. (4) u eff (r) = e r e TFr (4) 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 E. (7). Diagrammatic Language The approximation we have done can be expressed (if you lie such things!) in diagrammatic language as shown in Fig. (). The diagrams for the screening of the interaction (the last line of the figure) can be 6

7 = + Σ Self energy Σ = + Hartree Foch Screened interaction = + Π Figure : 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 u eff () = 4πe + 4πe ()u eff () (44) 4πe u eff () = (45) + 4πe [ ()] where () 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 is freuency dependent ( represents,ω), and the full dielectric constant screening the interaction ε() = 4πe () (46) is also freuency 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. This gives the Lindhard approximation to the dielectric constant. Using this interaction to evaluate the properties of the electron gas is nown 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 7 of Ph7a 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 inetic terms V(r) + ε F (n(r)) = const, (47) 7

8 where n is the local electron density, ε F F n/ is the local Fermi energy, and we are considering low temperatures B T ε F. To calculate the dielectric constant we assume a small external potential, and may then linearize E. (47) δv (r) + ε F δn =, (48) n with n,ε F the values without the perturbation. The potential derives from the sum of external charges and eδn from the electrons Euations (48) and (49) are easily solved by Fourier transforming with the Thomas-Fermi wave vector given by δv = 4π(ρ ext eδn). (49) δṽ() = 4π ρ ext + TF, (5) TF = 6πn e ε F. (5) The potential without the screening would be δṽ ext = 4πρ ext / and so we may introduce the dielectric constant δṽ = δṽ ext /ε() with ε() = + TF. (5) This is singular at small 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 ũ() = 4πe + TF, u(r) = e r e TFr, (5) a screened potential finite as in Fourier representation, and a Yuawa 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 suare root of the spacing TF = 4 π ( F a ) /.56 r / s a. (54) April 5, 4 8

Preface Introduction to the electron liquid

Preface Introduction to the electron liquid Table of Preface page xvii 1 Introduction to the electron liquid 1 1.1 A tale of many electrons 1 1.2 Where the electrons roam: physical realizations of the electron liquid 5 1.2.1 Three dimensions 5 1.2.2