EQUATION OF STATE OF THE UNITARY GAS

Transcription

1 EQUATION OF STATE OF THE UNITARY GAS DIAGRAMMATIC MONTE CARLO Kris Van Houcke (UMass Amherst & U of Ghent) Félix Werner (UMass Amherst) Evgeny Kozik Boris Svistunov & Nikolay Prokofev (UMass Amherst) (ETH Zurich) EXPERIMENT Andre Schirotzek Ariel Sommer Mark Ku Martin Zwierlein (MIT) Seattle, March 17, 2010

2 What is the unitary gas: Spin 1/2 fermions { zero range Interactions have infinite scattering length (3D) Universality hypothesis: V(r) a = b r Zero-range limit { n 1/3 b λ b λ (quasi) equilibrium homogeneous phase 2π 2 mk B T properties independent on V(r) N = N n(t, µ) n 0 (T, µ) = universal function of βµ [β 1/T ]

3 { V (r) = g 0 δ 3 ( r ) 1 k < Λ g 0 = m 4π 2 a k <Λ d k m (2π) 3 k 2 Zero-range limit: Λ g 0 (Λ) s.t. a fixed Σ = Γ 0 = Σ = + + +

4 Diagrammatic Monte-Carlo Random walk in the space of all possible diagram topologies and all values of internal and external variables. Each configuration is visited with a probability proportional to the absolute value of its contribution to Σ( p, τ)

5 Diagram order MC update MC update MC update { p, τ, ( p i ), (τ i )} Diagram topology

6 Diagram order MC update MC update MC update { p, τ, ( p i ), (τ i )} Diagram topology

7 Diagram order MC update MC update MC update { p, τ, ( p i ), (τ i )} Diagram topology

8 Diagram order MC update MC update MC update { p, τ, ( p i ), (τ i )} Diagram topology

9

10 after each MC update: τ ±1 p histogram for Σ(p, τ)

11 Previous applications of DiagMC: Solution of Fermi-polaron problem [Prokofev&Svistunov, PRB 2008] Doped Hubbard model [Van Houcke et al., 2008; Kozik et al. arxiv 2009] A new way to fight the sign problem: traditional QMC: error bars ~ exp{#β Volume} DiagMC: Volume = error bars ~ exp{# diagram order}

12 non-degenerate degenerate limit limit T T F T T F nλ 3 1 nλ 3 1 βµ 0 Normal Superfluid (βµ) c = 3.2(2) [Burovski et al.]

13 βµ = n! / (diagram order)

14 Resummation Σ bare (N) (p, τ) = N n=1 Σ resummed (N) (p, τ) = Σ (n) (p, τ) N n=1 F (N) n Σ (n) (p, τ) F (N) n 1 N 1 Cesaro: F (N) n = 1 n 1 N 1 Riesz: F (N) n = ( 1 n 1 ) δ N δ = 3 F (N) n F (N) n 0 1 n N N n N N+1

15 βµ = bare series Riesz-2 resummation Riesz-3 resummation advanced resummation n! / (diagram order) nλ 3 = 2.90(2) i.e. T/T F = 0.646(6)

16 βµ = n(k) k 4! k!

17 βµ = bare series Riesz-3 resummation Riesz-4 resummation advanced resummation n! / (diagram order) nλ i.e. T/T F 0.46

18 n(t,µ) / n 0 (T,µ) DiagMC (preliminary) Virial order 2 Virial order 3 [Liu et al.] T=0, " 0.4 [Juillet; Gezerlis et al.,...] Bulgac et al. critical point [Burovski et al.] ! µ

19 P(µ,T) / P 0 (µ,t) 3 2 Comparison with experiments: MIT Diag MC}(preliminary) Virial 3 [Liu et al.] ENS [Nascimbene et al.] AFQMC [Bulgac et al.] µ / βµ (k T) B T c [Burovski et al.]

20 How were these experimental EOS obtained: y z ρ = x 2 + y 2 file is trapping potential V (ρ, z) local density n 3D (ρ, z) local density approximation : n 3D (ρ, z) = n homogeneous (T, µ = µ g V (ρ, z)) p 3D (ρ, z) = p homogeneous (T, µ = µ g V (ρ, z)) n 2D (y, z) = + ENS protocol: p 3D (ρ = 0, z) = mω2 ρ 2π dx n 3D (ρ = x 2 + y 2, z). + dy n 2D (y, z)? p homogeneous (T, µ) for harmonic trap fitting the wings with virial (2nd order) (T, µ g ) p 3D our protocol: p(ρ 0, z) = 1 π [Ho&Zhou] ρ 0 dy n 2D (y, z) [ V ρ (y, z) y2 ρ 2 0 fitting the wings with virial (3rd order) (T, µ g ) + y ρ 0 dρ ρ V ρ V (y, z) y ρ (ρ, z) (y 2 ρ 2 ) 3/2 ] for any trap

21 What we are working on

22 Bold Diagrammatic Monte Carlo G = G0 + G0 Σ G Γ Γ 0 + Γ0 Π Γ = G Σ = + + Γ G G 0 Π = - + G G 0 +

23 Combine theory with experiment to obtain accurately EOS down to T=0 DiagMC fit

24 Next projects: Finite scattering length Finite imbalance Doped Hubbard model

25 2 0 Virial expansion (order 2 in fugacity) Virial expansion (order 3 in fugacity) Determinant MC (Burovski et al.) DiagMC -2 µ/e F T/T F