Many-body localization-delocalization transition for 1D bosons in the quasiperiodic potential

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1 Many-body localization-delocalization transition for 1D bosons in the quasiperiodic potential Gora Shlyapnikov LPTMS, Orsay, France University of Amsterdam, The Netherlands Introduction. Many-body localization-delocalization transition Classical and degenerate bosons Low temperature regime Phase diagram Conclusions Collaborations B.L. Altshuler (Columbia Univ.), V. Michal (LPTMS, Orsay) Seattle, May 28, p.1/16

2 Quantum gases in disorder One-dimensional disordered bosons at finite temperature DOGMA No finite temperature phase transitions in 1D as all spatial correlations decay exponentially There is a non-conventional phase transition between two distinct states Fluid and Insulator I.l. Aleiner, B.L. Altshuler, GS, (2010). p.2/16

3 Old question How does the interparticle interaction influences Anderson localization? Crucial for charge transport in electronic systems Appears in a new light for disordered ultracold bosons Palaiseau, LENS, Rice, Urbana experiments. More underway. p.3/16

4 Experiments BEC V BEC in a harmonic + weak random potential V(z) ng small density modulations of the static BEC. Switch off the harmonic trap, but keep the disorder What happens? First experiments (Orsay, LENS, Rice) The expansion stops and BEC gets stacked in between 2 large peaks z a) ω τ = 10 L TF ψ 2 / N p.4/16

5 Orsay experiment 0.8 s 1.0 s 2.0 s 0.8 a) b) z (mm) t (s) Atom density (at/µm) Localization length L loc (mm) p.5/16

6 LENS experiment 1D quasiperiodic potential Single-particle state J(ψ n+1 +ψ n 1 )+V cos(2πβn)ψ n = εψ n V > 2J all single-particle states are localized. p.6/16

7 AAH model for interacting bosons H int = U j n j (n j 1)/2 Localization length ζ min = a; ζ max = av/(v 2J). p.7/16

8 ρ(ε) = 1 L AAH model. Density of states π β 1 quasiclassical DOS π dk 2π ( ρ(ε) = 2 π 2 Va ln L/2 L/2 dxδ(ε 2J coska V cos2βx) 2 2V ε V +2J ) ; V ε > (V 2J). p.8/16

9 Many-body localization-delocalization transition (Aleiner, Altshuler, Basko ) Analyze how different states of two particles α, β hybridize due to the interaction The probability P(ε α ) that for a given state α there exist β, α, β such that α,β and α,β are in resonance: α,β H int α,β exceeds α β αβ ε α +ε β ε α ε β MBLDT criterion P(ε α ) 1 α β α β. p.9/16

10 MBLDT Matrix element for large occupation numbers α,β H int α,β = U N β N α +N β N β N α N β ψ α (j) ψ β (j) ψ α (j)ψ β (j) j ε α ε α ; ε β ε β N β N α +N β N β N α N β N β ψ α (j) ζ 1/2 α ; j j α < ζ α /2 ψ α (j) 0 j j α > ζ α /2 α,β H int α,β = UN β a ζ max Energy mismatch δ α ε α ε α [ζ α ρ(ε α ] 1 ; δ β ζ β ρ(ε β ] 1 α β αβ = δ α δ β 1 ζ α ρ(ε α ) 1 ζ β ρ(ε β ) 1 (ζρ) min. p.10/16

11 MBLDT criterion The probability that α,β H int α,β exceeds α β P(ε α ) = P α β a(ζρ) min αβ UN β ζ max αβ =U P α β β,α,β Critical coupling strength U c αβ dε β ρ(ε β )ζ β N β a(ζρ) min ζ max [ dε β ρ(ε β )ζ β N β a(ζρ) min ζ max T 2J disregard deeply localized low-energy states ] 1 U c min α ρ(ε) and ζ(ε) decrease with increasing energy [ εα ] 1 dε β N β ρ(ε β )ρ(ε α )aζ α + dε β N β ρ 2 (ε β ) aζ2 β V+2J ε α ζ α. p.11/16

12 Classical regime, Weak interactions U J n = ( T T > T d µ = T ln T d ) 0 ρ(ε)dε exp[(ε µ)/t] 1 ; T d = n ρ(t d ) ρ 2 (ε β ζ 2 β ε β peaks at low energies ε V 2J T U c ν T π2 2 ln(v/t d ) ln 2 [V/(V 2J)] Increasing T favors localization!. p.12/16

13 Quantum degenerate regime T < T < T d decoherent system ( ) 2 δn Tερ(ε) T n ε 2 +2εU ν dε 1 0 ln(v/t d ) T T d ln(v/u ν) [ ] 1 µ = T ln 1 exp( T d /T) Multiple occupation of the states U c ν (V 2J)+ µ ) T d T π 2 2 Normal behavior ln(v/t d ) ln 2 [V/(V 2J)]. p.13/16

14 Low-temperature regime T < T T = 0 the boson density is fragmented into lakes of size ζ if ε > U ν ε [ζ ρ(ε )] 1 π2 V 2J 2 ln(v/ε ) Energy cost of bringing a boson to lake i is E i = ǫ i +gn i /ζ = µ. Occupation ζ (µ ǫ i )/Ua n = µ 2 /2gE µ 2ε U ν U ν < ε small fraction of low-energy states is occupied Interlake distance l > ζ insulator l(κ) ζ * U ν > ε the interlake coupling drives the system to superfluid state x ε U ν π2 2 V 2J ln[v/(v 2J)] U ν. p.14/16

15 Phase diagram Low-T criterion coincides with that in the quantum decoherent regime at T T U fluid U c insulator insulator insulator T * T d T. p.15/16

16 Conclusions Atoms in quasiperiodic potentials interesting system to study Increasing temperature may favor localization What I did not show high-temperature regime (T 2J) What we did not yet do strongly interacting regime Thank you for attention!. p.16/16