Mean-field theory for extended Bose-Hubbard model with hard-core bosons. Mathias May, Nicolas Gheeraert, Shai Chester, Sebastian Eggert, Axel Pelster

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1 Mean-field theory for extended Bose-Hubbard model with hard-core bosons Mathias May, Nicolas Gheeraert, Shai Chester, Sebastian Eggert, Axel Pelster

2 Outline Introduction Experiment Hamiltonian Periodic patterns Main part Mean-field (MF) approximation method MF-Hamiltonian for unit cell in a periodic pattern (dependent on mean-field parameters) General calculation method for any pattern energy eigenvalues mean-field parameters Results (quadratic lattice, triangular lattice) Outlook Completing mean-field for triangular lattice Negative hopping J Other lattices Quantum corrections

3 Experiment T 0

4 Hamiltonian Extended Bose-Hubbard model Ĥ = J <i,j> â i âj + U i ˆn i (ˆn i 1) µ i ˆn i + <i,j> ˆn i ˆn j Hopping parameter J J = J (i, j) = [ ] d 3 x w (x x i ) + m ext (x) w (x x j ) On-site interaction U U = U (i) = 4πa m Next neighbour interaction d 3 x w (x x i ) 4 = (i, j) = d 3 x d 3 x dipol (x, x ) w (x x i ) w (x x j )

5 Feshbach resonance ( ) a (B) = a bg 1 B B 0 Hamiltonian for hard-core bosons: Ĥ = J <i,j> â i âj + Hard-core limit: a U n {0, 1} <i,j> ˆn i ˆn j µ i ˆn i

6 Patterns not frustrated frustrated Connectivity matrices: ( 0 4 N = 4 0 ) N =

7 Outline Introduction Experiment Hamiltonian Periodic patterns Main part Mean-field (MF) approximation method MF-Hamiltonian for unit cell in a periodic pattern (dependent on mean-field parameters) General calculation method for any pattern energy eigenvalues mean-field parameters Results (quadratic lattice, triangular lattice) Outlook Completing mean-field for triangular lattice Negative hopping J Other lattices Quantum corrections

8 Mean-field approximation Problem: bilocal terms in Hamiltonian Mean-field approximation: â i = â i + δâ i δâ i δâ j 0 i, j â i = â i + δâ i ˆn i = ˆn i + δˆn i δˆn i δˆn j 0 i, j Operators appearing in the Hamiltonian: â i âj â i â j + â j â i â i â j = ψ i â j + ψ j â i ψ i ψ j ˆn i ˆn j ˆn i ˆn j + ˆn j ˆn i ˆn i ˆn j ψ i := â i ϱ i := ˆn i = ϱ i ˆn j + ϱ j ˆn i ϱ i ϱ j

9 Mean-field Hamiltonian Hamiltonian for the whole lattice Ĥ MF i ĥ MF,i Ψ i := j NN i ψ j ) ĥ MF,i := J (â i Ψ i + â i Ψ i ψi Ψ i R i := j NN i ϱ j + (ˆn i R i ϱ i R i ) µˆn i

10 Mean-field Hamiltonian Hamiltonian for periodic patterns For periodic patterns: Hamiltonian reduces to a sum over the finite number of sites in the unit cell (UC). ĥ MF,UC = ( JΨ R) J X ( ) â X Ψ X + â X Ψ X + X ˆn ( X R ) X µ X {A, B,...}, Ψ := X ψ X Ψ X, R := X ϱ X R X

11 Usual calculation method Finding Hamiltonian for a special case (pattern) Defining a basis for the pattern Finding matrix representations for operators on this basis For n sites in the unit cell the size of the matrix is n! Combining them to the matrix representation of the Hamiltonian ) Solving the eigenvalue equation det (Ĥ E = 0 roots of n -order polynomial! no general solution for the energies E to expect!

12 Homogeneous pattern hard-core basis: B = { 0i, 1i} h MF,UC = B E±A = JΨ JΨ R JΨA R + RA JΨ µ ±A q JΨA R + RA RA µ µ + (J ΨA )

13 sites in the unit cell hard-core basis: B = { 0, 0i, 1, 0i, 0, 1i, 1, 1i} E±,± A B = JΨ R s µ + RA RA ±A s µ + RB ±B RB µ µ + (J ΨA ) + (J ΨB )

14 Energies General formula for the energies E ±A = E ±A,± B = (JΨ ) ( R + (JΨ R ) R A µ ) ± A ( R A µ ) + (J Ψ A ) ( + R A µ ) ( ± A R A µ ) + (J Ψ A ) ( + R B µ ) ( ± B R B µ ) + (J Ψ B ) E ±A,± B,... = (JΨ R ) ( ) + X + X R X µ ( (± X ) R X µ ) + (J Ψ X )

15 Finding the mean-field parameters leading to: ϱx E = 0 X, ψx E = 0 X ( ϱx 1 ) = 1 (± X ) ( ψ X = 1 (± X ) ( ( R X µ ) R X µ ), +(J Ψ X ) JΨ X R X µ ) +(J Ψ X )

16 Resulting formulas Relation between ϱ and ψ independent of other sites ( ϱx 1 ) + ψx = 1 4 ψ X = 0 ϱ X {0, 1} no hopping bosons localized in valleys expectation value for density: 0 (absent) or 1 (present)

17 Resulting formulas Relation between ϱ s and ψ s of different sites N X1,X 1 N X1,X... N X,X 1 N X,X } {{ } reduced connectivity matrix ψ X1 ψ X. Ψ X η X := ψ X = η X η X ψ X1 ψ X. ϱ X1 ϱ X. = J η X η X..... reduced connectivity matrix 1 {}}{ N X1,X. + 1 N X1,X... N X,X 1 N X,X Jη X1 + µ j N Y j X 1 (01) Yj 1 Jη X + µ j N Y j X (01) Yj. X i M ψ 0 i Y j M ψ=0 j ϱ Yj = (01) Yj {0, 1}

18 Resulting formulas Energies ( ( E ϱ J A, µ ) (, ϱ J B, µ ) ),..., η A, η B,... = 1 µ N XY ϱ Y ϱ X ϱ X X M ψ=0 Y M ψ=0 X M ψ=0 }{{} =: E ψ=0 + 1 N XY ϱ Y ϱ X X M ψ=0 Y M ψ 0 }{{} =: E mix 1 ( J ϱ X η X + µ ) X M ψ 0 }{{} =: E ψ 0

19 Possible phases % Mott Density wave Superfluid Supersolid ψ X, Y %X = %Y X X, Y %X 6= %Y X, Y %X = %Y X, Y ψx = ψy 6= 0 X, Y %X 6= %Y X, Y ψx 6= ψy 6= 0 X ψx = 0 ψx = 0

20 Quadratic lattice ( = 1) Μ 5 MOTT INSULATOR SUPERFLUID: ΡA=ΡB>0, Ψ>0 4 3 DENSITY WAE: ΡA=1, ΡB=0, Ψ=0 1 0 MOTT INSULATOR -1 J

21 Triangular lattice ( = 1) Μ Mott Insulator:ΡA ΡB ΡC 1,Ψ 0 6 Superfluid:ΡA ΡB ΡC 0,Ψ 0 4 Density Wave:ΡA ΡB 1,ΡC 0,Ψ 0 Supersolid: 0 ΡA ΡB ΡC 0, 0 ΨA Ψb Ψc 0 Density Wave:ΡA ΡB 0,ΡC 1,Ψ J Mott Insulator:ΡA ΡB ΡC 0,Ψ 0 X.-F. Zhang, R. Dillenschneider, Y. Yu and S. Eggert, Phys. Rev. B 84, (011) G. Murthy, D. Arovas, A. Auerbach, Phys. Rev. B 55, 3104 (1996)

22 Outline Introduction Experiment Hamiltonian Periodic patterns Main part Mean-field (MF) approximation method MF-Hamiltonian for unit cell in a periodic pattern (dependent on mean-field parameters) General calculation method for any pattern energy eigenvalues mean-field parameters Results (quadratic lattice, triangular lattice) Outlook Completing mean-field for triangular lattice Negative hopping J Other lattices Quantum corrections

23 Negative hopping J Quadratic lattice ( = 1) Μ 5 MOTT INSULATOR DENSITYWAE:ΡA 1,ΡB 0,Ψ 0 SUPERFLUID:ΡA ΡB 0,Ψ 0 SF1: ψ A = ψ B signψ A = signψ B SF: ψ A = ψ B signψ A = signψ B 1 MOTT INSULATOR J A. Eckardt, C. Weiss, and M. Holthaus, Phys. Rev. Lett., 95, (005) A. Zenesini, H.Lignier, C. Sias, O. Morsch, D. Ciampini, E. Arimondo, Phys. Rev. Lett., 10, (009)

24 Other lattices Kagome, 1D Stripe, decorated triangular,... G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. ishwanath and D. M. Stamper-Kurn, PRL 108, (01)

25 Quantum corrections inspired by variational perturbation theory (H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific Publishing Company, 5th edition (009)) ) Ĥ (ξ) = Ĥ MF + ξ (ĤBH Ĥ MF Free energy: F (ξ) = 1 β ln Z (ξ), Z (ξ) = Tr ( e βĥ(ξ) ) Expansion with respect to ξ: F (ξ) = F MF + ξ Ĥ ĤMF MF +..., Ô MF ) := 1 βĥmf Z MF Tr (Ôe Truncation after n-th order depends artificially on ϱ X, ψ X for ξ = 1: F (n) (ξ = 1) = F (n) (ϱ X, ψ X ) Principle of minimal sensitivity: ϱx F (n) = 0, ψx F (n) = 0 Quantum corrections: n = 0: mean-field n = 1: first quantum corrections n = :... F. E. A. dos Santos and A. Pelster, Phys. Rev. A 79, (009)