On the Dirty Boson Problem

Transcription

1 On the Dirty Boson Problem Axel Pelster 1. Experimental Realizations of Dirty Bosons 2. Theoretical Description of Dirty Bosons 3. Huang-Meng Theory (T=0) 4. Shift of Condensation Temperature 5. Hartree-Fock Mean-Field Theory 6. Summary and Outlook SFB/TR 12: Symmetries and Universality in Mesoscopic Systems 1

2 1.1 Overview of Set-Ups Superfluid Helium in Porous Media: (persistence of superfluidity) Reppy et al., PRL 51, 666 (1983) Laser Speckles: (controlled randomness) Billy et al., Nature 453, 891 (2008) Wire Traps: (undesired randomness) Krüger et al., PRA 76, (2007) Fortàgh and Zimmermann, RMP 79, 235 (2007) Localized Atomic Species: (theoretical suggestion) Gavish and Castin, PRL 95, (2005) Incommensurate Lattices: (quasi-randomness) Roati et al., Nature 453, 895 (2008) 2

3 1.2 Laser Speckles Lye et al., PRL 95, (2005) global condensate vanishes 3

4 1.3 Wire Trap +5 B/B (10 5 ) 0 arbitrary units B/B (10 6 ) Longitudinal Position (µm) Distance: d = 10 µm Wire Width: 100 µm Magnetic Field: 10 G, 20 G, 30 G Deviation: B/B 10 4 Krüger et al., PRA 76, (2007) Fortàgh and Zimmermann, RMP 79, 235 (2007) 4

5 2.1 Model System Action of a Bose Gas: β [ A = dτ d 3 x {ψ τ 2 2M 0 + U(x) + V (x) µ ]ψ + g2 ψ 2 ψ 2 } Properties: harmonic trap potential: U(x) = M 2 ω2 x 2 disorder potential: V (x) ; bounded from below, i.e. V (x) V 0 V (x 1 ) = 0, V (x 1 )V (x 2 ) = R(x 1 x 2 ),... chemical potential: µ repulsive interaction: periodic Bose fields: g = 4π 2 a M ψ(x,τ + β) = ψ(x,τ) 5

6 2.2 Random Potential Disorder Ensemble Average: = DV P[V ], DV P[V ] = 1, P [V < V 0 ] = 0 Assumption: V (x 1 ) = 0, V (x 1 )V (x 2 ) = R (2) (x 1 x 2 ) = (x R 1 x 2 ) 2 (2πξ 2 e 2ξ 2 ) 3/2 Characteristic Functional: exp { i } d D xj(x)v (x) = exp { n=2 i n n! d D x 1 d D x n R (n) (x 1,...,x n ) j(x 1 ) j(x n ) } 6

7 2.3 Grand-Canonical Potential Aim: Ω = 1 β ln Z Z = DψDψ e A[ψ,ψ]/ Problem: ln Z ln Z Solution: Replica Trick Ω = 1 β lim N 0 Z N 1 N 7

8 2.4 Replica Trick Disorder Averaged Partition Function: { } N Z N = D 2 ψ α e P N α=1 A([ψα,ψα])/ = α =1 { } N D 2 ψ α α=1 e A(N) / Replicated Action: A (N) = Z β 0 dτ Z + g 2 ψ α(x, τ) 4 ) + NX α 1 =1 NX αn=1 d D x X n=2 NX α=1 1 n! " (ψ α (x, τ) # τ 2 2M + U(x) µ «1 n 1 Z β dτ 1 0 Z β 0 dτ n Z R (n) (x 1,..., x n ) ψ α1 (x 1, τ 1 ) 2 ψ α n(x n, τ n ) 2 = Disorder amounts to attractive interaction for n = 2 Z d D x 1 ψ α (x, τ) = Higher-order disorder cumulants negligible in replica limit N 0 d D x n 8

9 3.1 Condensate Density Assumptions: homogeneous Bose gas: U(x) = 0 δ-correlated disorder: R(x) = R δ(x) Bogoliubov Theory: background method: ψ α (x,τ) = Ψ α + δψ α (x,τ) replica symmetry: Ψ α = n 0 Result: n 0 = n 8 3 π Huang and Meng, PRL 69, 644 (1992) Falco, Pelster, and Graham, PRA 75, (2007) a n0 3 M2 R 8π 3/2 4 n0 a 9

10 3.2 Superfluid Density Galilei Boost: A = β 0 dτ d 3 xψ (x,τ)u i ψ(x,τ) dω = S dt p dv N dµ p du p = Ω(T, V,µ,u) u = MV n n u +... T,V,µ Result: n s = n n n = n 4 3 Huang and Meng, PRL 69, 644 (1992) Falco, Pelster, and Graham, PRA 75, (2007) M 2 R 8π 3/2 4 n0 a 10

11 3.3 Collective Excitations Hydrodynamic Equation in Trap With Disorder: [ ] m 2 t δn(x,t) gn 2 TF (x) δn(x,t) = 2[ ] 3gn R (x)δn(x,t) n R (x) : Huang-Meng depletion in trap n TF (x) = [µ V (x)]/g : Thomas-Fermi density Violation of Kohn Theorem: [ ] 4g 3 n R(x) δn(x,t) Surface dipole mode (n = 0,l = 1): δω dip (ξ = 0) ω dip = 5π 16 M 2 R 8π 3/2 4 n TF (0)a Falco, Pelster, and Graham, PRA 76, (2007) 11

12 Æ ¼ µ Æ ¼ ¼µ Typical Values: 3.4 Comparison With Experiment Inguscio et al., PRL 95, (2005) ξ = 10 µm R TF = 100 µm l HO = 10 µm } ξ = ξr TF 7 lho 2 2 ½ ¼ ¼ ¼ ¼ ¾ n = 0, l = 1 n = 0, l = 2 ¼ ½ ¾ ¼ = Disorder effect vanishes in laser speckle experiment Improvement: laser speckle setup with correlation length ξ = 1 µm Aspect et al., NJP 8, 165 (2006) = Disorder effect should be measurable Falco, Pelster, and Graham, PRA 76, (2007) 12

13 3.5 Rederivation of Huang-Meng Depletion Gross-Pitaevskii Equation: } { 2 + V (x) + g Ψ(x) 2 Ψ(x) = µψ(x) 2M Perturbative Expansion: µ Ψ(x) = g + Ψ 1(x) + Ψ 2 (x) +... = Condensate density: n 0 (µ) = Ψ(x) 2 = Particle density: n(µ) = Ψ(x) 2 Disorder-Induced Depletion: d 3 k R(k) n 0 = n n (2π) 3 ( ) k 2 2M + 2gn R(k) = 1 = Huang-Meng depletion Krumnow, von Hase, and Pelster (to be published) 13

14 4.1 Earlier Results trapped Bose gas T (0) c T c T (0) c = ω g k B [ ] 1/3 N T c (0) ζ(3) = a λ (0) c homogeneous Bose gas = 2π 2 k B M T c T (0) c [ n ζ(3/2) = 1.3an 1/3 ] 2/3 Giorgini et al., PRA 54, R4633 (1996) Kleinert, MPLB 17, 1011 (2003) Gerbier et al., PRL 92, (2004) Kastening, PRA 69, (2004) R(x) =? T c T (0) c =? T c T c (0) R(x) = R δ(x) M 2 R = 3π[ζ(3/2)] 2/3 2 n 1/3 Lopatin and Vinokur, PRL 88, (2002) Procedure: n = n(µ), µ ր µ c T c 14

15 4.2 Our Results ½ ÌÆ ¼µ Ì Ê solid: Gaussian ¼º¼ ¼º¼¾ dashed: Lorentzian ¼º¼ ¼º¼½ ½ ¾ Length Scale: l HO =, ω g = (ω 1 ω 2 ω 3 ) 1/3 Mω g Æ ½ Dimensionless Units: ξ R ξ =, R = ( ) l 2 HO l 3 2 Ml 2 HO HO = M3/2 R 7/2 ω 1/2 g Timmer, Pelster, and Graham, EPL 76, 760 (2006) 15

16 Definition: Note: 5.1 Order Parameters lim x x ψ(x,τ)ψ (x, τ) = n 0 lim x x ψ(x,τ)ψ (x,τ) 2 = (n 0 + q) 2 q is similar to Edwards-Anderson order parameter of spin-glass theory Hartree-Fock Mean-Field Theory: Self-consistent determination of n 0 and q for R(x x ) = R δ(x x ) Phase Classification: gas Bose glass superfluid q = n 0 = 0 q > 0,n 0 = 0 q > 0,n 0 > 0 16

17 5.2 Hartree-Fock Results Isotherm: T = const. Phase Diagram: µ = const. disorder strength R = const. n R < R c R > R c superfluid R Bose glass continuous n m n BEC gas µ BEC µ m Bose glass µ M µ R c superfluid T BEC gas first order T c T Graham and Pelster, IJBC 19, 2745 (2009) 17

18 5.3 Properties of Bose Glass Absence of ODLRO and phase coherence (n 0 = 0), presence of long-range glassy order (q > 0) Localization length of excitations: Life time (width) of excitations: Mξ k ξ = 2π 4 M 2 R Superfluid density: Localized part of condensate q contributes to normal but not to superfluid density Note: Localized condensation (q > 0) without global condensate, but not global condensation n 0 > 0 without localized part 18

19 6.1 Summary and Outlook Frozen Disorder Potential: arises both artificially (laser speckles) or naturally (wire trap) Bosons: local condensates in minima + global condensate + thermally excited Localization Versus Transport: disorder reduces superfluidity Phase Diagram: yet unknown for strong disorder Navez, Pelster, and Graham, APB 86, 395 (2007) Disordered Bosons in Lattice: Bose Glass versus Mott phase 19

20 6.2 Quantum Phase Transition Gross-Pitaevskii Equation: } { 2 2M + V (x) + gψ2 (x) Ψ(x) = µ Ψ(x) Nonperturbative Approach: Gaussian approximation for random fields V (x) and Ψ(x) V (x) = 0, V (x)v (x ) = R(x x ), Ψ(x) = n 0 Ψ(x)Ψ(x ) = G ΨΨ (x x ), V (x)ψ(x ) = G V Ψ (x x ) = Self-consistency equations Result: n 0 (r) = n (1 r2 2 r 2 r c = 1, r = 2 π 2gn R ) r2 + 4 ( M 2π 2 ) 3/2 Krumnow, von Hase, and Pelster (to be published) 20

21 6.3 Disordered Bosons in Lattice Bose-Hubbard Hamilton Operator: Ĥ BH = t â i âj + [ ] U 2 ˆn i(ˆn i 1) + (ǫ i µ) ˆn i <i,j> i = i + p(ǫ i) dǫ i, p(ǫ i ) = { 1/ ; ǫi [ /2, + /2] 0 ; otherwise Mean-Field Phase Diagram: 2dt/U MI SF T = 0 /U = 0.5 MI µ/u Krutitsky, Pelster, and Graham, NJP 8, 187 (2006) 2dt/U BG MI SF BG kt/u = 0.01 /U = 0.5 MI Stochastic Mean-Field Theory: Bissbort and Hofstetter, EPL 86, (2009) 21 µ/u