Unitary Fermi Gas: Quarky Methods

Transcription

1 Unitary Fermi Gas: Quarky Methods Matthew Wingate DAMTP, U. of Cambridge

2 Outline Fermion Lagrangian Monte Carlo calculation of Tc Superfluid EFT Random matrix theory

3 Fermion L

4 Dilute Fermi gas, 2 spins Hamiltonian H = 1 2m σ ψ σ 2 ψ σ g2 2 ( ψ1 ψ2 ψ 2 ψ 1 ) Lagrangian, real time L t = ψ i t ψ + 1 2m ψ 2 ψ + g2 2 ( ψψ ) 2 Galilean invariance t t, x x + vt ψ(t, x) ψ (t, x) = e im v x ψ(t, x vt) SU(2) spin symmetry U(1) fermion number ψ e i β σ ψ, ψ e iα ψ, ψ ψe i β σ ψ ψe iα

5 Transformations Hubbard-Stratonovich, as in Chen & Kaplan, PRL (2003). L t = ψ i t ψ + 1 2m ψ 2 ψ + g ψφψ 1 2 φ2 Imaginary time for finite temperature formulation ( Z = Dψ D ψ Dφ exp β 0 dτ d 3 x L τ ) with L τ = ψ τ ψ 1 2m ψ 2 ψ g ψφψ φ2 ψkψ φ2

6 Gor kov basis L τ = 1 2 (ψt, ψ(iσ 2 )) ( 0 K (iσ 2 ) K(iσ 2 ) 0 )( ψ iσ 2 ψt ) φ2 1 2 ΨT KΨ+ 1 2 φ2 SU(2) spin symmetry U(1) fermion number γ k γ 5 ( ) Ψ V Ψ = Ψ + i σ 0 β Ψ σ ( ) I 0 Ψ UΨ = Ψ + iα Ψ I γ 0 Chen & Kaplan, PRL (2003).

7 Pairing source ( Jψ T σ 2 ψ + J ψσ 2 ψt ) J = ( Jσ2 0 0 J σ 2 ) A = K + J {γ 0, K} = 0, {γ 0, J } = 0, {γ k γ 5, A} = 0 conserves U(1) breaks U(1) conserves SU(2)

8 Reduced Gor kov basis L = (ψ 1, ψ 2 ) ( ij K K ij )( ψ2 ψ 1 ) φ2 η Ã η φ2 Ã = K + J {σ 3, K} = 0, {σ 3, J } = 0 conserves U(1) Z = breaks U(1) Dφ detã e S φ

9 Pseudofermions Z = Dφ Dζ exp [ ( ζ Ã 1 ζ φ2 )] HMC: Molecular dynamics + accept/reject step Requires inversion of sparse matrix But matrix becomes singular in physical limit

10 Banks-Casher for superfluidity Order parameter (negl. signs, limits...) log Z Σ J = 1 J=0 2 ψt σ 2 ψ + ψσ ψt 2 = 1 2 ΨT σ 31 Ψ = ηη Kη n = λ n η n det à = e.v. real, paired if nonzero (λ n ij) Σ = lim J 0 lim V 1 V n 1 λ n ij n

11 Banks-Casher for superfluidity Σ = lim J 0 lim V 1 V n 1 λ n ij = lim J 0 0 2i J ρ(λ) λ 2 + J 2 dλ = 2iπρ(0) Non-vanishing condensate implies accumulation of zeromodes in the thermodynamic limit Interesting, but makes inversions for HMC increasingly difficult

12 Tc

13 Extrapolation to zero external source 0.2 µ = 0.4, m 2 = Fermion pairing condensate Σ ξ = 1.1 ξ = 1.2 ξ = 1.25 ξ = 1.3 ξ = J External pairing source strength

14 Phase transition Order parameter Σ(J 0) , nb m 2 = 0.14 m 2 = m 2 = m 2 = C ξ Anisotropy ( Temperature) MW, cond-mat/

15 Diagrammatic Determinant Monte Carlo Burovski, Prokof ev, Svistunov, Troyer, New J. Phys 8, 153 (2003) Z = =

16 Critical Temperature Burovski, Prokof ev, Svistunov, Troyer, New J. Phys 8, 153 (2003) rescaled order parameter R(L,T) fermions on a 6 3 lattice 80 fermions on a 8 3 lattice 240 fermions on a 12 3 lattice Figure 4. A typical crossing of the R(L, T ) curves. The errorbars are 2σ, and solid lines are the linear fits to the MC points. Inset shows the finite-size scaling of the filling factor (ν vs 1/L), which yields ν =0.148(1). From this plot and Eq. (4.3) one obtains 1/T c (ν) = 4.41(5)/t! 1/T

17 ! Figure 4. A typical crossing of the R(L, T ) curves. The errorbars are 2 solid lines are the linear fits to the MC points. Inset shows the finite-size s of the filling factor (ν vs 1/L), which yields ν =0.148(1). From this plot a (4.3) one obtains 1/T c (ν) = 4.41(5)/t Continuum limit Burovski, Prokof ev, Svistunov, Troyer, New J. Phys 8, 153 (2003). T c /# F A. Sewer et al., 2002 HMC stuck here T.A. Maier et al., " 1/3 Figure 5. The scaling of the lattice critical temperature with filling (circles). ν = 1 corresponds to the half filling. The errorbars are one sta deviation. The results of Ref. [42, 43] at quarter filling and ν = 0.25 ar

18 Our investigations with DDMC with Olga Goulko Implemented own version of code Found long autocorrelations in cases Update of result for Tc in balanced case Investigation of imbalanced gas, Tc vs. Δn/n, in continuum limit Final results later this spring Lattice 2009 proceedings, arxiv:

19 Trade-offs Can work directly with J=0 Fine for finite volume determination of Tc Scales like V 2

20 Superfluid EFT

21 Phonon ψ T σ 2 ψ = ψ T σ 2 ψ e 2iθ X χ ( τ 2 χ(τ, x) e iθ(τ, x) 2m ) χ = " ( i τ θ + 1 ) 2m θ 2 Initial power counting: ( θ) n O(1), m ( θ) n O(p m ) L eff = c 0 m 3/2 X 5/2 c 0 ¼ 25=2 15p 2 n 3=2 ¼ n 3 5 Fn. NLO derived in D T Son & M W, Annals Phys (2006)

22 Expand about ground state ϕ = θ iµτ X = µ + i τ ϕ + ϕ 2 2m ϕ(τ, x) π(τ, x) F m. L eff = c 0 m 3/2 [ µ 5/2 5µ3/2 2F m + 15µ1/2 4mF 2 ( i τ π + ( i τ π + ) 1 2m 3/2 F π 2 1 2m 3/2 F π 2 ) ]

23 Include pairing source J e 2iα J work done with J-W Chen Y = J(χ ) 2 + J χ 2 Y = 2J cos 2ϕ = 2J(1 2ϕ ) L eff = c 0 m 3/2 X 5/2 + d 0 m 3/2 X 3/2 Y [( L eff = m 3/2 c 0 + 2d 0J µ + µ1/2 4mF 2 ( 15c d 0J µ ) µ 5/2 µ3/2 F m )( i τ π + ( 5c d 0J µ 1 2m 3/2 F π 2 )( i τ π + ) 2 4d 0Jµ 3/2 mf 2 π 2 ) 1 2m 3/2 F π 2 ]

24 Phonon mass B = 2d 0 m 1/2 µ 3/2 Σ LO phonon mass M 2 0 = 4B F 2 c 4 s J Correlation length ξ = 1 2mMc 2 s Must use power counting: ( n θ) O(p n ), M k O(p k )

25 Pressure Constant part of L P 0 (µ) = (c 0 µ 5/2 +2d 0 Jµ 3/2 )m 3/2 n = P 0(µ) = ( 5 2 c 0µ 3/2 +3d 0 Jµ 1/2 )m 3/2 Canonical norm n for kinetic term L eff = P 0 (µ) + in F m τ π ( τ π) 2 + c2 s 2 π M 2 0 π 2 + L int Speed of sound c 2 s = 1 m n(µ) n (µ) = 2 3 µ m ( ( d 0 J J 2 c 0 µ + O µ 2 ))

26 Finite volume effects p regime: small effects, e -ML 1 F 2 ξ L ε regime: phonon zero mode dominates, must be resummed. 1 F 2 L ξ δ regime: symmetry restored L 1 F 2 ξ

27 Random Matrix Theory

28 Random matrix theory Low energy EFT is universal given symmetries and symmetry breaking pattern In ε regime uniform phonon zero mode dominates over fluctuating nonzero modes Replace Gor kov operator with random matrix with appropriate global symmetries (and no spacetime dependence) Spectral quantities on scale of average level spacing are thought to be universal Compare to Monte Carlo Guidance for algorithmic improvements

29 Random Matrix Model

30 Random matrix model Going beyond universal results... Use RMT as toy model (with same low energy EFT as Fermi gas) Explore larger population imbalances analytically Interesting phase diagram?

31 Summary Numerical calculation of critical temperature Need to improve Monte Carlo calculations (HMC) EFT study of SSB in finite volume Importance of low-lying eigenmodes Random matrix theory for unitary Fermi gas

32 Photo: Stefan Meinel