Gerry and Fermi Liquid Theory. Thomas Schaefer North Carolina State
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1 Ë Ë Ë³ Gerry and Fermi Liquid Theory Thomas Schaefer North Carolina State
2 Introduction I learned about Fermi liquid theory (FLT from Gerry. I was under the imression that the theory amounted to the oeration m m I only later realized that Gerry had a very sohisticated understanding of FLT (see his many-body lectures. One of Gerry s legacies is his insistence that the nuclear many body roblem should be understood in terms of quasi-articles and effective interactions. Some of the best work along these lines was done in Stony Brook (and Darmstadt since the late 90s, based on V low k and the renormalization grou.
3 Many imrovements could have been made, esecially in Chater XIII on effective forces in nuclei, but time is short, and I shall make them in later editions, when I am too old to ski. Of course, nobody will be interested in the subject by then. Unified Theory of Nuclear Models and Forces 3rd edition, 1970.
4 Fermi liquid theory a la Landau In a cold Fermi system the low energy excitations are sin 1/2 quasi-articles. Define a distribution function f = f 0 + δf. Then E = E 0 + δe δf δf δ 2 E δf δf E = δe δf t = δ2 E δf δf δf δf +... The distribution function satisfies a Boltzmann equation ( t + v r + F f (r, t = C[f ] with v = E and F = r E.
5 Fermi liquid theory a la Polchinski-Shankar Free non-relativistic quasi-articles near Fermi surface d 3 S = dt (2π 3 ψ( (i t (ǫ( ǫ F ψ( Exand momenta around Fermi momentum = k + l ǫ( ǫ F = v F (k l + O(l 2 Study scaling behavior l s l. Scaling dimensions [k] = 0, [l] = 1, [ t ] = 1, [d 3 ] = 1, [ψ] = 1 2 Interaction S int = dt [ 4 i=1 d 3 i (2π 3 ] ψ ( 4 ψ ( 3 ψ( 2 ψ( 1 δ 3 ( tot U( i
6 Marginal Interactions BCS Landau BCS : U( ˆ 3, ˆ 3, ˆ 1, ˆ 1 = V (ˆ 1 ˆ 3 = l V l P l (ˆ 1 ˆ 3, LFL : U(ˆ 4, ˆ 3, ˆ 2, ˆ 1 ˆ1 ˆ 2 =ˆ 3 ˆ 4 = F(ˆ 1 ˆ 2, φ 12,34
7 Fermi surface RG in nuclear hysics Achim and Bengt: Evolve V low k towards Fermi surface Γ 1 2 d dλ 1 Γ = 2 1 Γ Γ Γ Γ 2 + Γ + 1 Γ Fermi liquid arameter t = [F l + G l (σ σ ]P l (cos θ F full direct [fm ] k F F direct [fm ] k F full
8 Non-FLT in QCD: High Density Effective Theory QCD lagrangian L = ψ (id/ + µγ 0 m ψ 1 4 Ga µνg a µν Quasi-articles (holes ε E ± = µ ± 2 + m 2 µ ± holes articles Λ Effective field theory on v-atches ψ v± = e iµv x ( 1 ± α v 2 ψ v F v = µv +l + l
9 Effective Theory for l < m L = ψ v ( iv D D2 ψ v + L 4f 1 2µ 4 Ga µνg a µν + L HDL L HDL = m2 2 Transverse gauge boson roagator v G a µα v α v β (v D 2 Gb µβ D ij (k = δ ij ˆk iˆkj k 2 0 k 2 + i π 2 m2 k 0 k, Scaling of gluon momenta k k 1/3 0 m 2/3 k 0 gluons are very sacelike
10 Non-Fermi Liquid Effective Theory Gluons very sacelike k k 0. Quark kinematics? Scaling relations k 0 k + k2 2µ k k k 0 k k2 2µ k m 2/3 k 1/3 0, k m 4/3 k 2/3 0 /µ Proagators S αβ = iδ αβ + 2 2µ iǫsgn( 0 D ij = iδ ij k 2 i π 2 m2 k 0, k
11 Non-Fermi Liquid Exansion Scale momenta (k 0, k, k (sk 0, s 2/3 k, s 1/3 k [ψ] = 5/6 [A i ] = 5/6 [S] = [D] = 0 Scaling behavior of vertices s 1/6 s 1/2 s 5/6 s Systematic exansion in ǫ 1/3 (ω/m 1/3
12 Non-Fermi liquid effects Quark self energy near Fermi surface ( ( Σ( = g2 2 5/2 m 9π 2 0 log π 0 + i π O (ǫ 5/3 Luttinger: G 1 FS = 0 n = Vol(FSh Quasi-article velocity vanishes, IR freedom near Fermi surface d log α v dλ = + 4α2 v 9π α = g2 v F 4π Enhanced color suerconducting ga [ = µex ( 3π2 5 log(g + 4 log(128π π2 + 4 ] 2g 8 Unusual transort: η µ 4 m 2/3 /(g 4 T 5/3 µ 4 /(g 10/3 T +...
13 Ê Ê Ê Ê Ê Ä Ê Remnants of Fermi liquid theory Mass corrections to µ Å Å Ý and scattering amlitudes Ê Ä Å Ê Ä Å Ä Ä LL Instanton vertex: χsb in CFL hase L R = L R RR
14 The return of the master: FLT a la Landau The discovery of nearly erfect fluidity in the QGP and in ultracold gases has led to (renewed interest in transort roerties as a measure of the quasi-article interaction. ( t + v r + F f (r, t = C[f ] Require consistency between transort and thermodynamics: E = δe δf t = δ2 E δf δf
15 Examle I: Bulk viscosity in a dilute Fermi gas Conformal symmetry breaking (thermodynamics 1 2E 3P = C 12πmaP 1 λ 6π nλ3 a How does this translate into ζ 0? Momentum deendent m (. Im Σ(k zt Bulk viscosity ζ = π λ 3 ( zλ a Re Σ(k zt λ a T ǫ k Erf T ǫ k F D 2 ζ ( 1 2E 3P ( ǫk T ( ǫk 2 η T T
16 Examle II: Kinetics of the chiral magnetic effect Chiral fermions modify measure n = d 3 (1 + B Ω n π = d 3 (1 + B Ω n E = d 3 (1 + B Ω ǫ n where Ω = A = ±ˆ/(2µ 2 is the Berry curvature. Get energy/momentum conservation 0 E + π = E j 0 π + ˆΠ = n E + j B and the anomaly 0 n + j = ± 1 4π 2 E B Son & Yamamoto, Zahed, Basar et al.
17 Coda: Non-FLT engineering in AdS/CMT Consider (deformations of AdS/Reissner-Nordstrom black holes. Solve for sinor eigenmodes G R (ω, k = h 1 ω v F (k k F + h 2 ω 2ν k F (2ν kf 1 regular/marginal/singular Fermi liquid Study transort roerties σ(0 T 2ν k σ(ω = σ(0 1 iτω (2ν K > 1 QFT: Non-interacting FS interacting with strongly couled CFT. Faulkner, Polchinski, Iqbal, McGreevy.
18 Summary Contrary to Gerry s 1970 rediction, there is a renaissance of interest in effective forces. Much of this interest is driven by develoments initiated or encouraged by Gerry: chiral forces, soft otentials, free sace and Fermi surface RG. Landau Fermi liquid theory is alive and well. Indeed we have come to (reareciate Landau s insight of basing the theory on kinetics rather than effective lagrangians. I am looking forward to the next 45 years of nuclear theory at Stony Brook!
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