An imbalanced Fermi gas in 1 + ɛ dimensions. Andrew J. A. James A. Lamacraft

Transcription

1 An imbalanced Fermi gas in 1 + ɛ dimensions Andrew J. A. James A. Lamacraft 2009

2 Quantum Liquids Interactions and statistics (indistinguishability) Some examples: 4 He 3 He Electrons in a metal Ultracold atomic gas

3 Fermi Liquids e.g. electrons in metals, 3 He V = 0 Response ε Q ω Occupation prob E F ω

4 Fermi Liquids e.g. electrons in metals, 3 He Response ε Q ω V = 0 Occupation prob E F ω Propagator G(ω, Q) a ω ξ Q + iη ξ KF = 0 V 0, Q K F Response ε Q a QP ω Occupation prob E F a QP ω η 1 τ ImG(ω, K F ) = aδ(ω)

5 Fermion pairing BCS superconductivity K K' 2Λ K F -K' -K Effective Hamiltonian H = kσ ξ k ψ kσ ψ kσ + V ψ k +Q ψ k ψ k+q ψ k k,k,q

6 Fermion pairing BCS diagrams Cooper instability Γ = V V 2 log(λ/µ) +... Γ = V 1 + V log(λ/µ)

7 Fermion pairing BCS diagrams Cooper instability Γ = V V 2 log(λ/µ) +... Γ = V 1 + V log(λ/µ) RG flow dγ d log Λ = 0 dv d log Λ = V 2 BCS 0 FL V

8 Inhomogenous Superconductors What happens when the two spin populations are unequal? Layered superconductor in a magnetic field...

9 Inhomogenous Superconductors What happens when the two spin populations are unequal? Layered superconductor in a magnetic field... 1st order h 0 Δ Superfluid Normal 1st order transition when h = / 2 at T = 0.

10 Cooper s Problem Two fermions interacting above their Fermi seas [ ] 2 a 2 b + V δ(r a r b ) E ψ(r a, r b ) = 0 2m a 2m b

11 Cooper s Problem Two fermions interacting above their Fermi seas [ ] 2 a 2 b + V δ(r a r b ) E ψ(r a, r b ) = 0 2m a 2m b 1 V = Q p >K F,a p >K F,b d d p 1 (2π) 2 ɛ a (Q p) + ɛ b (p) E Bound state: E = µ a + µ b + E B, E B < 0

12 Larkin Ovchinnikov Fulde Ferrell A possibility for (K F,a > K F,b ) Q = K F,a K F,b Q K F,a (x) exp(iq x) K F,b

13 Larkin Ovchinnikov Fulde Ferrell A possibility for (K F,a > K F,b ) Q = K F,a K F,b Q K F,a (x) exp(iq x) K F,b LOFF 0 1 Superfluid Normal Seen in CeCoIn 5, heavy fermion superconductor? h

14 Polarised Fermi gases RF pulse through ultracold Fermi gas, alters relative spin populations (don t have to work with spins though) Mean field and experiment suggest that P LOFF Normal 0 PS 0 V Y. Sin et al. PRL 2006 P = n a n b n a + n b

15 The model Effective Hamiltonian H = K,s (ɛ K µ s )ψ K,s ψ K,s + V ψ K +Q,a ψ K,b ψ K+Q,bψ K,a K,K,Q ɛ K µ s v F,s k and k = K K F,s One loop RG (for continuum path integral formulation) Γ = PP PH

16 One dimension (balanced) Cancellation = PP PH Flow vanishes Γ = V dγ d log Λ = dv d log Λ

17 One dimension (balanced) Cancellation = PP PH Flow vanishes Γ = V dγ d log Λ = 0 dγ d log Λ = dv d log Λ

18 One dimension (balanced) Cancellation = PP PH Flow vanishes Γ = V dγ d log Λ = dv d log Λ dγ d log Λ = 0 β(v ) = 0

19 One dimension (balanced) Cancellation = PP PH Flow vanishes Γ = V dγ d log Λ = dv d log Λ dγ d log Λ = 0 β(v ) = 0 Cancellation occurs to all orders, (Ward identities, bosonization, exact solution of XXZ spin chain) Luttinger liquid, power law behaviour with interaction dependent exponents

20 One + ɛ dimensions Expect flow to appear as we turn on new dimension

21 One + ɛ dimensions Expect flow to appear as we turn on new dimension Angular measure π d d k S d 1 dkk d 1 dθ(sin θ) d 2 0 PP θ K b Q K a Λ

22 One + ɛ dimensions Angular cut off S d 1 dkk d 1 θ M 0 dθ(sin θ) d 2 2KF θ M,a k QK F,b Λ θ

23 Γ in 1 + ɛ Combine PP and PH diagrams Γ = V V 2 2(2K F,bK F,a ) ɛ/2 ( Λ ) ɛ/2 F π v F ɛ K F,a F = ( 1 K F,b K F,a ) ɛ/2 ( 1 + K F,b K F,a ) ɛ/2 1st term in F comes from PP, 2nd from PH

24 Flow and Fixed Point Flow d d log Λ Γ = 0

25 Flow and Fixed Point Flow ( ) d d log Λ Γ = 0 d d log Λ V = (2K F,bK F,a ) ɛ/2 Λ ɛ/2fv 2π v F K 2 F,a +...

26 Flow and Fixed Point Flow ( ) d d log Λ Γ = 0 d d log Λ V = (2K F,bK F,a ) ɛ/2 Λ ɛ/2fv 2π v F K 2 F,a +... ( ) Define a dimensionless coupling, g = 2(2K F,bK F,a ) ɛ/2 Λ ɛ/2v 2π v F K F,a d β(g) = d log Λ g = ɛ 2 g Fg

27 Flow and Fixed Point Flow ( ) d d log Λ Γ = 0 d d log Λ V = (2K F,bK F,a ) ɛ/2 Λ ɛ/2fv 2π v F K 2 F,a +... ( ) Define a dimensionless coupling, g = 2(2K F,bK F,a ) ɛ/2 Λ ɛ/2v 2π v F K F,a d β(g) = d log Λ g = ɛ 2 g Fg Non trivial fixed point g = ɛ 2F LOFF g * FL 0 g

28 Fixed Point Properties Limits P 0 g = ɛ 2F ɛ ( KF,a K ) F,b ɛ/2 2 K F,a P 1 g = K F,a /2K F,b 1

29 Fixed Point Properties Limits P 0 g = ɛ 2F ɛ ( KF,a K ) F,b ɛ/2 2 K F,a P 1 g = K F,a /2K F,b 1 Integrated flow for weak imbalance g g g = g 0 (Λ/Λ 0 ) ɛ/2 V independent of Λ Expected result for couplings when P = 0 and Q 0 In contrast g = g V Λ ɛ/2 LOFF g * FL 0 g

30 Self Energy Ω, K F,s + Retarded self energy Σ R s ( k = K F,s, Ω) = g 2 ( Ω Ω ɛ 4 2Λ v 1 F iπ Ω 1+ɛ) + O(g 3 ) ɛ Ω = Ω/2Λ v F

31 Propagator and Residue Greens function G s ( k = K F,s, Ω) = Z Ω Σ R s ( k = K F,s, Ω) d ln Z d ln Λ = g 2ɛ β(g) = g O(g 3 ),

32 Propagator and Residue Greens function G s ( k = K F,s, Ω) = Z Ω Σ R s ( k = K F,s, Ω) d ln Z d ln Λ = g 2ɛ β(g) = g O(g 3 ), At the fixed point No pole! Residue has vanished G s ( k = K F,s, Ω) ω η 1, η = ɛ 2 /16F 2 Response 0 ω η-1 ω

33 Residue Z flow Linearise about fixed point to get Z (g 0 g ) ɛ/8f 2 Dependence on start of flow Z g* 0 g

34 Residue Z flow Linearise about fixed point to get Z (g 0 g ) ɛ/8f 2 Dependence on start of flow Z (V-V *) 2η/ε Z g* 0 g V * 0

35 Two Species, with Spin Interaction H int = σ,σ [ (Ve δ σ,σ + V e δ σ, σ )ψ b,σ ψ a,σ ψ a,σ ψ b,σ +(V d δ σ,σ + V d δ σ, σ )ψ b,σ ψ a,σ ψ a,σ ψ b,σ ] Heisenberg Interaction Set V = V e + V d, V = V d, then J z = 4V and J xy = 2V e V e σ V d σ' σ σ' σ' σ σ σ'

36 Repulsive Fixed Point Flows β(g ) = ɛ 2 g g 2 e β(g d ) = ɛ 2 g d 1 2 g 2 e β(g e ) = ɛ 2 g e g e (g d g ) First two equations are equivalent if V = V d Repulsive fixed point (g, g e ) = (ɛ/4, ɛ/2) Spin density wave order J z = J xy = ɛπ v F /(K F,b Λ) ɛ/2

37 Hole Pockets e.g. Iron Pnictides 0 mev -20 mev -60 mev -40 mev M Γ (a) (b) (d) (c) LDA Schematic EF M (π, 0) Γ X Γ M Band Schematic (e) Γ π π X ( 2, 2) (f) M Y. Xia et al Spin density wave order? = πv /(K /2 Jz? = Jxy F F,b Λ) (g)

38 Summary RG flow for imbalanced Fermi gas in d = 1 + ɛ Spinless case Finite momentum pairing Vanishing residue at fixed point Imbalance dependent critical exponent Spinful cases, SDW order