Phase diagram of spin 1/2 bosons on optical lattices

Transcription

1 Phase diagram of spin 1/ bosons on optical lattices Frédéric Hébert Institut Non Linéaire de Nice Université de Nice Sophia Antipolis, CNRS Workshop Correlations, Fluctuations and Disorder Grenoble, December 1 ni S i stitut non linéaire de nice O P H I A A N T I P O L I S

2 Laurent De Forges De Parny and George Batrouni Institut Non Linéaire de Nice, UNS, CNRS People Valéry Rousseau Department of Physics and Astronomy Louisiana State University, Baton Rouge Richard Scalettar Department of Physics University of California, Davis

3 Motivation Effect of an internal degree of freedom on the physics of interacting bosons with hyperfine F = 1 spin Impossible to study with magnetic traps (lift degeneracy) Possible with purely optical traps ferromagnetic 87 Rb and antiferromagnetic 3 Na (Stamper-Kurn group) Possibility to study interplay between superfluid/solid and magnetic behaviors

4 ? Optical lattice to go in the strongly interacting regime Standing EM wave not resonant with atomic transition classical periodic potential for the atoms The laser also couples the internal F z = ±1 states Two low energy states left : F z = and Λ states E internal states and Λ bosons! Krutitsky et. al. PRA 7, 6361 (4) and PRA 71, 3363 (5) Ashhab, J. Low. Temp. Phys. 14, 51 (5)

5 A simple ( < 3) model for bosons with internal degrees of freedom. H = t L =1 σ=,λ (a +1σ a σ + a σa +1 σ ) n (n 1) +U + U n n Λ µ U ) (a Λ a Λ a a + a a a Λa Λ (1) n () (3)

6 A simple ( < 3) model for bosons with internal degrees of freedom. H = t L =1 σ=,λ (a +1σ a σ + a σa +1 σ ) n (n 1) +U + U n n Λ µ U ) (a Λ a Λ a a + a a a Λa Λ (1) n () (3) -1 +1

7 A simple ( < 3) model for bosons with internal degrees of freedom. H = t L =1 σ=,λ (a +1σ a σ + a σa +1 σ ) n (n 1) +U + U n n Λ µ U ) (a Λ a Λ a a + a a a Λa Λ (1) n () (3) +

8 A simple ( < 3) model for bosons with internal degrees of freedom. H = t L =1 σ=,λ (a +1σ a σ + a σa +1 σ ) n (n 1) +U + U n n Λ µ U ) (a Λ a Λ a a + a a a Λa Λ (1) n () (3)

9 A simple ( < 3) model for bosons with internal degrees of freedom. H = t L =1 σ=,λ (a +1σ a σ + a σa +1 σ ) n (n 1) +U + U n n Λ µ U ) (a Λ a Λ a a + a a a Λa Λ (1) n () (3)

10 A simple ( < 3) model for bosons with internal degrees of freedom. H = t L =1 σ=,λ (a +1σ a σ + a σa +1 σ ) n (n 1) +U + U n n Λ µ U ) (a Λ a Λ a a + a a a Λa Λ (1) n () (3)

11 One site approach When the density ρ is integer and U t Mott phase

12 One site approach When the density ρ is integer and U t Mott phase

13 One site approach When the density ρ is integer and U t Mott phase

14 When the density ρ is integer and U t Mott phase. One site approach One site approach for the Mott phases for integer densities. G(n = ) = + E G (n = ) = U U µ

15 approach One site coupled to a reservoir of condensed bosons with order parameter ψ = a Hamiltonian ) H MF = t (ψ a + ψ a + ψ Λ a Λ + ψ Λa Λ n(n 1) +U + U n n Λ µn U ) (a Λ a Λ a a + a a a Λa Λ

16 approach One site coupled to a reservoir of condensed bosons with order parameter ψ = a Hamiltonian ) H MF = t (ψ a + ψ a + ψ Λ a Λ + ψ Λa Λ n(n 1) +U + U n n Λ µn U ) (a Λ a Λ a a + a a a Λa Λ

17 approach One site coupled to a reservoir of condensed bosons with order parameter ψ = a Hamiltonian ) H MF = t (ψ a + ψ a + ψ Λ a Λ + ψ Λa Λ n(n 1) +U + U n n Λ µn U ) (a Λ a Λ a a + a a a Λa Λ

18 results t/u =.3, U /U =.5, µ/u = E (a) ψ ψ Λ.5 1 Solid phase for strong interaction

19 results t/u =.8, U /U =.5, µ/u = 1.35 E (b) ψ ψ Λ Solid phase + metastable polar superfluid

20 results t/u =.8, U /U =.5, µ/u = 1.35 E (c) ψ ψ Λ.5 1 Polar superfluid + metastable solid

21 results t/u =.9, U /U =.5, µ/u = 1.35 E (d) ψ ψ Λ.5 1 Polar superfluid

22 results () ρ ρ Λ ρ ρ s =ρ s U /U =.1, t/u =.5, d= µ/u Mott Phases and polar superfluid

23 results () µ/u MI MI MI MI ρ=4 ρ=3 ρ= ρ=1 U /U =.1, d=1 metastable SF polar,,4,6,8,1,1 t/u Mott Phases and polar superfluid

24 Green function algorithm 1 Measure densities, superfluid and condensed densities at finite (inverse) temperature β. β large enough to reach the ground state limit 1D and D simulations 1 V.G. Rousseau, Phys. Rev. E77, 5675 (8), Phys. Rev. E78, 5677 (8)

25 1D results ρ, L= ρ, L=4 ρ, L=6 ρ s, L= ρ s, L=6 t/u =.1, U /U =.1, β=l µ/u Mott phase and superfluid, no 1st order transition

26 1D results ρ = 1 P(ρΛ) t/u ρ Λ (a) P(ρ) t/u ρ.75 (b) Evolution from non polarised (NP) to polarised (P)

27 µ/u 1 MI - NP MI ρ= MI P NP ρ=1 1D results QMC L=, β=4 QMC L=4, β=8 QMC L=6, β=1 MF U /U =.1, d=1 SF polar t/u Polarisation in the SF and also in the ρ = 1 Mott

28 D results QMC, S=8*8, β=16 QMC, S=1*1, β= QMC, S=1*1, β=4 MF µ/u MI NP MI NP ρ= ρ=1 MI P SF polar U /U = t/u Polarisation in the SF and also in the ρ = 1 Mot

29 D results ρ s QMC MF.5 β=4 L*L=1*1.1. t/u U /U =. U /U =.5 U /U =.1 U /U =.5 U /U =.5 U /U =1..1. t/u 1st order phase transition found in QMC and MF at ρ =

30 D results..1 L*L=8*8, β=16, ρ= U /U =.5, t/u =.4 ρ µ/u 1st order phase transition found in QMC and MF at ρ =

31 Complete phase diagram t/u U /U µ/u 1.5 Phase diagram in D for different U /U

32 Summary Simple model of bosons with internal degrees of freedom MF predicts polar SF and 1st order MI-SF transition for ρ = QMC confirms polar SF QMC confirms ρ = 1st order transition in D (but not 1D) QMC shows a polar/non polar transition in the ρ = 1 MI

33 Other investigations Equivalent study of the U < case Finite temperature study, QPT points... k B T/U U /U =.1, L*L=1*1, ρ= MI Spin 1 as a future prospect 1 Liquid Superfluid polar BEC t/u Thank you! 1 Batrouni et. al. PRL (9)