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Lecture 3: Fermi-Liquid Theory 1 General Considerations Concerning Condensed Matter

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Lecture 3: Fermi-Liquid Theory 1 General Considerations Concerning Condensed Matter Phys 769 Selected Topics in Condensed Matter Physics Summer 2010 Lecture 3: Fermi-liquid theory Lecturer: Anthony J. Leggett TA: Bill Coish 1 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 1023) nuclei (denoted α,β,...) and (usually) ZN electrons (denoted i,j,...) interacting ∼ via a Hamiltonian Hˆ . To a first approximation, Hˆ is the nonrelativistic limit of the full Dirac Hamiltonian, namely1 ~2 ~2 1 e2 1 Hˆ = 2 2 + NR −2m ∇i − 2M ∇α 2 4πǫ r r α 0 i j Xi X Xij | − | 1 (Ze)2 1 1 Ze2 1 + . (1) 2 4πǫ0 Rα Rβ − 2 4πǫ0 ri Rα Xαβ | − | Xiα | − | For an isolated atom, the relevant energy scale is the Rydberg (R) – Z2R. In addition, there are some relativistic effects which may need to be considered. Most important is the spin-orbit interaction: µ Hˆ = B σ (v V (r )) (2) SO − c2 i · i × ∇ i Xi (µB is the Bohr magneton, vi is the velocity, and V (ri) is the electrostatic potential at 2 3 2 ri as obtained from HˆNR). In an isolated atom this term is o(α R) for H and o(Z α R) for a heavy atom (inner-shell electrons) (produces fine structure). The (electron-electron) magnetic dipole interaction is of the same order as HˆSO. The (electron-nucleus) hyperfine interaction is down relative to Hˆ by a factor µ /µ 10−3, and the nuclear dipole-dipole SO n B ∼ interaction by a factor (µ /µ )2 10−6. In addition to the above “intrinsic” terms, there n B ∼ may be terms due to external magnetic and electric fields; with currently available fields (E . 107 V/m, B . 60 T) the maximum value of ea E is 0.5meV (i.e. 10−4 R) and 0 ∼ ∼ the maximum value of µ B is a little larger, 4 10−4 R. B ∼ × 1For the moment, ignore externally applied fields 1 Symmetries of Hˆ : ,P ,T (translation), SO(3) , SU(2) . Relativistic corrections NR T orb spin break the last two, but not , P , or T (but external fields do). T 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 (Zc/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 Schr¨odinger 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 NZc conduction electrons described to lowest order by the Hamiltonian3 ~2 2 ˆ ′ 2 1 e 1 HNR = i U(ri)+ (3) −2m ∇ − 2 4πǫ0 ri rj Xi Xi Xij | − | where U(r) is the phenomenological potential of the (supposed fixed) ionic cores; this may ˆ ′ be either periodic (crystal) or aperiodic (liquid or glass). Note that in general HNR is not invariant under either T or SO(3)orb (but is still invariant under SU(2)spin, and more importantly, under P and ). The relativistic terms break the SU(2) invariance but (in T spin 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, ZcR). 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 2I am using the terms “valence” and “conduction” somewhat differently from the standard ways, e.g., in semiconductor physics. 3Strictly speaking, this ignores the possible effect of polarization of the core shells by the conduction 2 2 electrons. This may be handled phenomenologically by e → e /ǫc. 2 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 HNR of the form ~2 2 ′ H = + V (R , R , R ,...)+ H − (4) ion −2M ∇α ion α β γ ion el α ··· X αβγX where M is the ionic mass ( nuclear mass) and V (R , R , R ,...) is in general very ≃ ion α β γ complicated; however, a crucial point is that its order of magnitude will still be R or ∼ ZcR. 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 Vion 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)1/2, are down by a factor (m/M)1/2 10−2 10−3 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 1/300 of R). ∼ 2 Sommerfeld model Free fermions of spin 1/2 (electrons) moving in volume Ω. (No periodic/other potential, no interaction). Plane wave states are specified by k,σ: (periodic boundary conditions) 1 ik·r ψkσ = e σ ; σ = 1. (5) √Ω | i ± ~2k2 ǫ = (6) k 2m In thermal equilibrium at temperature T ( = 0): H 1 nkσ = (7) e(ǫk−µ)/kBT + 1 2 µ(T )= µ(0) + o((kBT ) /µ(0)). Hence at zero temperature nk = θ(µ(0) ǫ ) put µ(0) ǫ . (8) σ − k ≡ F so fills sphere of radius k = (3π2n)1/3 (n N/Ω) ǫ = ~2k2 /2m = (~2/2m)(3π2n)2/3 (9) F ≡ F F 3 T ǫ /k typically 104 105 K at all solid/liquid temperatures k T ǫ ⇒ F ≡ F B ∼ − ⇒ B ≪ 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 [N(0) 1 (dn/dǫ)], we also ≡ F ≡ 2 define p ~k , v p /m = Fermi velocity. F ≡ F F ≡ F 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 , then the d.c. conductivity σ is ≡ F (Drude): ne2τ 1 dn σ = = v2 τ (10) m 3 F dǫ (the second form refers only to the Fermi surface). More generally, σ(ω) is complex and is given by 2 2 2 (ne /m)τ ǫ0ωpτ 2 ne σ(ω)= , ωp (11) 1+ iωτ ≡ 1+ iωτ ≡ mǫ0 If we combine this result with Maxwell’s equations, we find that electromagnetic radiation is absorbed (a) for ωτ . 1 (“Drude peak”), and also (b) in a δ-function peak at ωp (“plas- mon”). In most “textbook” metals, ωp & ǫF (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). −1/2 Now ψk Ω uk exp ik r. ǫ( k ) ǫ (k). If the number of electrons per unit cell is ∼ n · | | → n 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 ) (anisotropic crystal)) → ij 4 Landau Fermi-liquid theory Originally done for (normal) liquid 3He (no crystalline potential, interactions short-ranged), later generalized to electrons in metals (crystalline potential, long-ranged Coulomb interac- tion). 2 1/3 Recap: Sommerfeld model at T = 0, Fermi sea filled up to kF = (3π n) . Formally, n(p,σ)= θ(ǫ ǫ(p)). F − Excited states (N-conserving): take a particle from below the Fermi surface with momentum 4 p, place it above the Fermi surface at a state with momentum p′. δE = ǫ(p) ǫ(p′). More − generally, E E0 = pσ ǫ(p)δn(pσ), − δn(pσ) = 0 or 1,p>pF , p with (12) Sz = σ σδnP ( σ), ( δn(pσ)=0or 1,p<pF .
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