Lecture 3: Fermi-liquid theory. 1 General considerations concerning condensed matter

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1 Phys 769 Selected Topics in Condensed Matter Physics Summer 200 Lecture 3: Fermi-liquid theory Lecturer: Anthony J. Leggett TA: Bill Coish General considerations concerning condensed matter (NB: Ultracold atomic gasses need separate discussion) Assume for simplicity a single atomic species. Then we have a collection of N (typically 0 23 ) nuclei (denoted α, β,...) and (usually) ZN electrons (denoted i, j,...) interacting via a Hamiltonian Ĥ. To a first approximation, Ĥ is the nonrelativistic limit of the full Dirac Hamiltonian, namely Ĥ NR = 2 2m i 2 i 2 2M 2 α + e 2 2 4πǫ 0 α ij + (Ze) 2 2 4πǫ 0 αβ r i r j R α R β Ze 2 2 4πǫ 0 For an isolated atom, the relevant energy scale is the Rydberg (R) Z 2 R. iα r i R α. () In addition, there are some relativistic effects which may need to be considered. Most important is the spin-orbit interaction: Ĥ SO = µ B c 2 σ i (v i V (r i )) (2) i (µ B is the Bohr magneton, v i is the velocity, and V (r i ) is the electrostatic potential at r i as obtained from ĤNR). In an isolated atom this term is o(α 2 R) for H and o(z 3 α 2 R) for a heavy atom (inner-shell electrons) (produces fine structure). The (electron-electron) magnetic dipole interaction is of the same order as ĤSO. The (electron-nucleus) hyperfine interaction is down relative to ĤSO by a factor µ n /µ B 0 3, and the nuclear dipole-dipole interaction by a factor (µ n /µ B ) In addition to the above intrinsic terms, there may be terms due to external magnetic and electric fields; with currently available fields (E 0 7 V/m, B 60 T) the maximum value of ea 0 E is 0.5 mev (i.e. 0 4 R) and the maximum value of µ B B is a little larger, R. For the moment, ignore externally applied fields

2 Symmetries of Ĥ NR : T,P,T (translation), SO(3) orb, SU(2) spin. Relativistic corrections break the last two, but not T, P, or T (but external fields do). When atoms combine to form a liquid or solid, the interatomic spacing is of the order of the atomic size. Then it is usually a good approximation to regard the closed-shell ( valence ) 2 electrons as rigidly tied to nuclei (so, in particular, we can regard rare-gas atoms as fixed units even in the liquid/solid state). However, open-shell ( conduction ) electrons (Z c /atom) may be distributed over the whole space. Then, in principle, we can implement the Born-Oppenheimer approximation: that is, we solve the time-independent Schrödinger equation (TISE) for the conduction electrons for fixed ionic positions, then feed back the resultant energies into the ionic Hamiltonian. Since the time scale of ionic motion is much greater than that of the conduction electrons, this generally gives good results (but use caution for metals). In solving the TISE for the conduction electrons, we must in principle use the Coulomb potential of the ions and require orthogonalization with core states; ( pseudo-potential method). However, in practice it is often adequate to replace the core electrons by a modification of the Coulomb potential to some phenomenological potential U(r (possibly also spin-dependent). Thus, if for the moment we ignore ionic motion, the new problem is that of NZ c conduction electrons described to lowest order by the Hamiltonian 3 Ĥ NR = 2 2m 2 i i i U(r i ) + e 2 2 4πǫ 0 r i r j where U(r) is the phenomenological potential of the (supposed fixed) ionic cores; this may be either periodic (crystal) or aperiodic (liquid or glass). Note that in general Ĥ NR is not invariant under either T or SO(3) orb (but is still invariant under SU(2) spin, and more importantly, under P and T ). The relativistic terms break the SU(2) spin invariance but (in zero external field) still preserve P and T. Energy scales: Since U(r) is at least of the order of magnitude of the Coulomb potential of the ions, and the atomic size ( interatomic spacing) is determined by the competition of kinetic and potential energies, the order of magnitude of both the first two terms in Eq. (3) is R (or perhaps better, Z c R). The last term might also be expected to be of the same order of magnitude; thus, prima facie, the conduction electrons in any (non-rare-gas) liquid or solid are strongly interacting (More technically, U V ǫ F ). 2 I am using the terms valence and conduction somewhat differently from the standard ways, e.g., in semiconductor physics. 3 Strictly speaking, this ignores the possible effect of polarization of the core shells by the conduction electrons. This may be handled phenomenologically by e 2 e 2 /ǫ c. ij (3) 2

3 Let s now take into account the possibility of ionic motion. Then we must include in the Hamiltonian an ionic kinetic energy, a direct ion-ion interaction (which will not in general be pure Coulomb, because of, e.g., exclusion-principle effects) and an interaction between the (displaced) ions and the conduction electrons. Most of the last can be eliminated by the Born-Oppenheimer technique in favor of an extra effective ion-ion interaction. Thus we get extra terms in H NR of the form H ion = 2 2 α + V ion (R α,r β,r γ,...) + H ion el (4) 2M α αβγ where M is the ionic mass ( nuclear mass) and V ion (R α,r β,r γ,...) is in general very complicated; however, a crucial point is that its order of magnitude will still be R or Z c R. The last term is any part of the ion-electron interaction which we have been unable to eliminate by the Born-Oppenheimer technique. Since the order of magnitude of V ion is the same as that of U or V in Eq. (3), while the mass in the kinetic-energy term is much larger, we see that the characteristic frequencies (vibrational energies) of the ionic system, which are (V /M) /2, are down by a factor (m/m) / relative to the characteristic electronic energies. This is a quite general conclusion, and independent of whether the system is crystalline, amorphous, or even liquid. (In particular, in a crystalline solid Debye energies are typically room temperature, which is about /300 of R). 2 Sommerfeld model Free fermions of spin /2 (electrons) moving in volume Ω. (No periodic/other potential, no interaction). Plane wave states are specified by k, σ: (periodic boundary conditions) ψ kσ = Ω e ik r σ ; σ = ±. (5) ǫ k = 2 k 2 2m In thermal equilibrium at temperature T (H = 0): n kσ = e (ǫ k µ)/k B T + µ(t) = µ(0) + o((k B T) 2 /µ(0)). Hence at zero temperature (6) (7) n kσ = θ(µ(0) ǫ k ) putµ(0) ǫ F. (8) so fills sphere of radius k F = (3π 2 n) /3 (n N/Ω) ǫ F = 2 k 2 F/2m = ( 2 /2m)(3π 2 n) 2/3 (9) 3

4 T F ǫ F /k B typically K at all solid/liquid temperatures k B T ǫ F µ(t) µ(0), and all the action is close to the Fermi surface. The density of states (DOS) (both spins) is dn/dǫ = 3n/2ǫ F [N(0) 2 (dn/dǫ)], we also define p F k F, v F p F /m = Fermi velocity. As such, the model implies that σ (no scattering). If we introduce a phenomenological mean-free path l, we have a relaxation time τ l/v F, then the d.c. conductivity σ is (Drude): σ = ne2 τ m = dn 3 v2 F dǫ τ (0) (the second form refers only to the Fermi surface). More generally, σ(ω) is complex and is given by σ(ω) = (ne2 /m)τ + iωτ ǫ 0ω 2 pτ + iωτ, ω2 p ne2 mǫ 0 () If we combine this result with Maxwell s equations, we find that electromagnetic radiation is absorbed (a) for ωτ ( Drude peak ), and also (b) in a δ-function peak at ω p ( plasmon ). In most textbook metals, ω p ǫ F (visible/uv). Most predictions agree well with experiments on conventional (textbook) metals. 3 Bloch model This model takes into account the periodic crystalline potential (but still no interactions). Now ψ k Ω /2 u kn expik r. ǫ( k ) ǫ n (k). If the number of electrons per unit cell is even, then (usually) filled bands imply an insulator. If odd, then the Fermi surface intersects one (or more) band(s). Properties are qualitatively similar to the Sommerfeld model, except for σ(ω) (interband absorption) (also, σ σ ij ) (anisotropic crystal)) 4 Landau Fermi-liquid theory Originally done for (normal) liquid 3 He (no crystalline potential, interactions short-ranged), later generalized to electrons in metals (crystalline potential, long-ranged Coulomb interaction). Recap: Sommerfeld model at T = 0, Fermi sea filled up to k F = (3π 2 n) /3. Formally, n(p, σ) = θ(ǫ F ǫ(p)). Excited states (N-conserving): take a particle from below the Fermi surface with momentum 4

5 p, place it above the Fermi surface at a state with momentum p. δe = ǫ(p) ǫ(p ). More generally, E E 0 = pσ ǫ(p)δn(pσ), S z = σ σδn(pσ), P = σ pδn(pσ), with { δn(pσ) = 0 or, p > pf, δn(pσ) = 0 or, p < p F. (2) It is convenient to measure ǫ p from ǫ F, and consider not E but E µn (recall that at low T µ(t) const. = ǫ F ) With this definition of E and ǫ p E = pσ ǫ(p)δn(pσ) (3) independent of whether pσ δn(pσ) = 0. Energy eigenstates are completely specified by {δn(pσ)}. Now, turn on the interaction adiabatically: assume the interaction is of the form 2 ij V ( r i r j ) (true to a good approximation in 3 He) conserves linear and angular momentum as well as spin. ** Fundamental assumption: Adiabatic evolution of low-lying states. If true, we can label each (low-lying) state of the interacting system by the set {δn(pσ)} which described the original free system by construction the Fermi momentum is unchanged. Terminology: δn(pσ) = + ( p > p F ) quasiparticles in state pσ, δn(pσ) = ( p < p F ) quasihole in state pσ (Idea of dressing ). Note that since [V,S] = [V,P] = 0, S z = σδn(pσ) pσ ( P = pσ and, trivially, δn = pσ pδn(pσ) (4) δn(pσ) ) (5) However, note that while for the noninteracting system, the spin current J spin is given by J spin = σ σpδn(pσ), (6) this is not true for Landau Fermi-liquid theory (since in general [J spin, V ] 0). (in particular, J spin is not even diagonal in the quasiparticle basis). Taylor expansion of energy E E{δn(pσ)}: E E 0 = pσ δe δn(pσ) δn(pσ) + 2 p,p,σ,σ δ2 E δn(pσ)δn(p σ ) δn(pσ)δn(p σ ) + (7) 5

6 Definition: δe δn(pσ) δ 2 E δn(pσ)δn(p σ) ǫ(pσ) (8) f(pp, σσ ) Ω (9) Why can we stop at the second term? Suppose N ex pσ δn(pσ) N. Then the second term is Nexf 2 Nex/Ω 2 N ex (N ex /N), while the third term is NexΩ 3 2 N ex (N ex /N) 2, so second term. On the other hand, we often (in fact, almost always) find that in the first term, the contribution linear in N ex vanishes from symmetry, so the second and first are of the same order of magnitude. Thus, terms explicitly kept in Eq. (7) are enough. Symmetry: For a rotationally invariant system, we must have (a) ǫ(pσ) = ǫ( p ). Expand around ǫ F (recall ǫ measured relative to ǫ F µ): ( ) dǫ ǫ ǫ F = (p p F ) + o ( (p p F ) 2) (20) dp p F Definition: ( ) dǫ v F, p F /v F m ( effective mass ) (2) dp p=p F so the single-particle energy spectrum is completely parametrized by m. In particular, the density of states of the Fermi liquid is given by ( )( ) dn dǫ = p2 F π 2 2 = m p F m dn v F π 2 2 m dǫ free gas (b) Landau interaction function: rotational invariance (22) f(pp σσ ) = f( p, p, ˆp ˆp, σ σ ) (23) (σ σ is a generalization of σσ ) but can set p p p F, so: f(pp σσ ) = f(cos θ, σ σ ) = f s (cos θ) + f a (cos θ)σ σ θ ˆp ˆp (24) To get a volume-independent quantity, multiply by the total density of states Ω(dn/dǫ), thus F s (cos θ) Ω(dn/dǫ)f s (cos θ), F a (cos θ) Ω(dn/dǫ)f a (cos θ) (25) F s, F a dimensionless. Finally, expand in Legendre polynomials: F s (cos θ) = l F s l P l(cos θ) (etc.) (26) 6

7 The interquasiparticle interaction is completely parametrized by an infinite set of dimensionless numbers Fl s, F l a. (but for most purposes, we only need l 2). NB: In the most general case, δn(pσ) has to be a matrix. (hence σ σ not σσ ). 5 Generalized molecular fields ex. only F a 0 nonzero, i.e. (for δn pσ diagonal in the S z -basis) equivalent molecular field: f(pp σσ ) = Ω (dn/dǫ) F0 a σσ (27) δe (2) = Ω (dn/dǫ) F0 a σσ δn pσ δn p σ (28) pσ,p σ = 2 Ω (dn/dǫ) F a 0 S 2 (29) H mol = ( ) dn F0 a S (30) dǫ so calculate the response of the spin density to an external field H ext by S(kω) = χ 0 s(kω)h tot (kω) [response of free Fermi gas with effective mass m ] H tot (kω) = H ext (kω) + H mol (kω) (3) H mol (kω) = (dn/dǫ) F0 as(kω) the true susceptibility χ s (kω) δs(kω)δh ext (kω) is given by χ s (kω) = χ 0 s(kω) + (dn/dǫ) F a 0 χ0 s(kω) e.g. for ω 0 then k 0, χ 0 s is the static response = dn/dǫ, so (32) χ s /χ 0 s = ( + F0 a ) ( Wilson ratio ) (33) General principle: for no net polarization of the Fermi surface, all molecular field effects vanish. Ex.: no effect on specific heat, de Haas van Alphen,... (When do molecular fields have an effect? e.g., zero sound, high-field NMR...). Lifetime of quasiparticles: due to e.g. qp qp + qp + qh (34) Result of calculation: τ (ǫ) ǫ F ( ǫ 2 + π 2 k 2 BT 2) /ǫ 2 F (35) so for ǫ, k B T ǫ F, τ (ǫ) ǫ. (Justifies the original Landau argument). 7

8 6 Generalization to metals. Crystalline lattice: can handle by adiabatic technique, but [P, V ] 0. So we can label the Bloch states k by the original states from which they evolved, but k is no longer momentum of quasiparticle with wavevector k : (rather, quasimomentum). Also, ǫ(k) has gaps at the Brillouin zone boundaries. K = kσ kn kσ not conserved because of Umklapp processes Lack of rotational invariance we must classify ǫ(k), f(k,k ) by symmetry operations of the crystal group (actually sometimes simplifies things). 2. Coulomb interaction: must handle as a special type of molecular field. Result: χ(q, ω) = χ 0 (q, ω) + (e 2 /ǫ 0 q 2 )χ 0 (q, ω) (36) where χ 0 (qω) is the bare Fermi-liquid density response. 3. Impurity scattering: (quasi)-momentum not conserved. However, we can reformulate Fermi-liquid theory in terms of exact s,p eigenstates of the noninteracting gas in the presence of impurities (so we no longer have plane waves). NB: Subtle interaction of impurity and interaction effects (Altshuler, Aronov). 4. Electron-phonon interactions two different Fermi-liquid theories (ω ω D : phonons ineffective, ω ω D : all parameters renormalized). Topological interpretation of Fermi-liquid theory. 7 Some references D. Pines and P. Nozières, Theory of Quantum Liquids, Addison-Wesley, 989 W. Harrison, Solid State Theory, McGraw-Hill 970 (concentrates on band theory) P. Nozières, Theory of Interacting Fermi Systems, Benjamin 964 (very technical) A. J. Leggett, Quantum Liquids, Oxford 2006, appendix 5A. (closest to notes) Ashcroft and Mermin, Solid state physics, Hold Reinehart and Winston; 976 (good on Sommerfeld model, less detail on Fermi-liquid theory) 8