The solution of the dirty bosons problem

Transcription

1 The solution of the dirty bosons problem Nikolay Prokof ev (UMass, Amherst) Boris Svistunov (UMass, Amherst) Lode Pollet (Harvard, ETH) Matthias Troyer (ETH) Victor Gurarie (U. Colorado, Boulder) Frontiers of Ultracold Atoms and Molecules, October 13, 2010

2 Theorem of inclusions: Phase transitions in disordered systems; system, dimension Superfluid -- Bose/Mott Glass -- Mott insulator transitions: Absence of direct SF- transition Harris criterion & vortex phase mechanism in d=1,2 T=0 phase diagram in d=3 Weakly interacting gas in a random speckle potential: Critical temperature. Do we have any analytic theory?

3 Theorem of Inclusions Mathematical background required Lemma: Product of finite number of non-zero numbers is non-zero

4 Theorem of Inclusions A Def: Generic disorder = non-zero prob. density i (, ) P() For an arbitrary transition in a system with generic disorder with transition point C depending on disorder properties 0 there exist P rare, () but arbitrarily large, inclusions of A (B) inside B (A) across the transition line. 0 P() 0 P( ) Models of disorder (inf.- dim. space) Phase A 0 δ phase boundary Phase B There exist rare, but arbitrarily large, inclusions of A (B) inside B (A) across the transition line.

5 Theorem of Inclusions B P() Transition point independent of disorder properties! 0 P() 0 P( ) Models of disorder Phase A phase boundary Phase B 0 Transition is driven by statistically rare fluctuations when locally disorder emulates a regular external field with amplitude, e.g. == E /2 = Griffiths type transition i GAP

6 Phase A Phase B

7 Consequences: - For generic lines: if A is gapless then B is gapless too and vise versa. - All transitions between gapfull and gapless phases are of the Griffiths type. - All phases next to the gapfull one are insulating SF-to-Mott insulator transition is forbidden (any D). an intermediate phase (BG) must separate the two - For generic lines: if A is superfluid then B is compressible (in the absence of particle-hole symmetry) gapless (possibly incompressible in particle-hole symmetric case)

8 Bose Hubbard model with (, ) at n = 1 (or other integer) i U H = t a a + n ( μ ) n 2 i j i i i < ij> 2 i i i Superfluid (SF) zt > U Mott insulator () gapped insulator Bose glass (BG) compressible insulator U zt 0 T. Giamarchi, H. Schulz, 87 M.P.A. Fisher, P.B. Weichman, G. Grinstein, D.S. Fisher, 89 / t BG?? BG W. Krauth, N. Trivedi, D. Ceperley 91 SF? Griffiths type? = E /2 = GAP G. Falco, T. Nattermann, V. Pokrovsky 08 C U 1/4 SF? U / t

9 Bose Hubbard model with (, ) at n = 1 (or other integer) i U H = t a a + n ( μ ) n 2 i j i i i < ij> 2 i i i / t BG? SF BG = U C 1/4? U/ t

10 Standard Harris criterion: Random fluctuations in the critical region δμ~ / d ξ smaller then the distance to the critical point δμ ( μ μ ) ξ C 1/ ν ν D = d + 1 > 2/ d δμ ~ ξ d /2 At the tip critical point fluctuation δu ( δμ) ξ 1/ ν 1/ ν d/2ν smaller then the distance to the critical point δu ( U U C ) 1/ ν ξ violated in d=1,2 d > 2

11 Euclidean SF hydrodynamic action with disorder: r= ( r, cτ ) S ϕ K = dr in( x) + ϕ y 2π 2 cτ γ space pair x 2 = 2 π n( x) dx x 1 average density a multiple at space point x of γ 2π = = i i 0 ImS 2 π q n( x) dx x i 2D XY-type mod( γ,2 π ) = 0 x1 x2 - ideal system with n(x)=integer standard KT (isotropic pairs) x - disordered system R R γ pair R = 2 π δn( x) dx = 2 πκ( R) δμ( x) dx R U 0 0 1/2 KT with vertical pairs

12 One-dimensional diagram tip /t Svistunov 96 = # e t UU C SF BG C UC / t U/ t

13 2 γ = ImS = 2 π q i n( ρ) d ρ i A i Two-dimensional diagram tip τ projected vortex loop area x A y 2 2 γ = 2 π n( ρ) d ρ 2 πκ( R) ( ρ) d ρ κ( R) R loop R 2 2 A R A R κ ( R) 1/ U(1)-criticality in d+1=3 has different scales contribute equally R γ R R ξ 1 ln ln ln U a U a U U UC

14 Two-dimensional diagram tip /t C ln 1 U BG = ( U U ) ν C SF UC / t U/ t

15 3D phase diagram tip (Worm Algorithm Monte Carlo)

16 Robust SF region, but fragile superfluidity! D/ t BG SF U/ t

17 Reality of finite traps D/ t 7 with N ~10 particles at temperature T ~ t N BG N SF N U/ t

18 Critical temperature dependence on speckle disorder S. Pilati, S. Giorgini, NP 09 n l i.e. a = l S C 3 (4 p / 3) 1 T ( D ) C Theoretical understanding of (Bose or Fermi)?

19 Conclusions: 1. Theorem of inclusions & its consequences apply to system in dim. 2. SF- transition is forbidden but will be observed for weak disorder at the SF-BG boundary 3. Proper understanding of the SF-BG boundary in d=3 and critical temperature dependence on disorder in the WIBG is still lacking (though known).