Abstract

We develop a variational wave function for the ground state of a one-dimensional bosonic lattice gas. The variational theory is initally developed for the quantum rotor model and later on extended to the Bose-Hubbard model. This theory is compared with quasi-exact numerical results obtained by Density Matrix Renormalization Group (DMRG) studies and with results from other analytical approximations. Our approach accurately gives local properties for strong and weak interactions, and it also describes the crossover from the superfluid phase to the Mott-insulator phase.

© 2004 Optical Society of America

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References

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  1. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, I. Bloch, �??Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,�?? Nature 415, 39 (2002).
    [CrossRef] [PubMed]
  2. M. Greiner, O. Mandel, T. W. Hänsch and Immanuel Bloch, �??Collapse and revival of the matter wave field of a Bose�??Einstein condensate,�?? Nature 419, 51 (2002).
    [CrossRef] [PubMed]
  3. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner and P. Zoller, �??Cold Bosonic Atoms in Optical Lattices,�?? Phys. Rev. Lett. 81, 3108-311 (1998).
    [CrossRef]
  4. D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams and P. Zoller, �??Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator,�?? Phys. Rev. Lett. 89, 040402 (2002).
    [CrossRef] [PubMed]
  5. B. Y. Chen, S. D. Mahanti and M. Yussouff, �??Helium atoms in zeolite cages: Novel Mott-Hubbard and Bose-Hubbard systems,�?? Phys. Rev. Lett. 75, 473-476 (1995).
    [CrossRef] [PubMed]
  6. Min-Chul Cha, M. P. A. Fischer, S. M. Girvin, M. Wallin, A. P. Young, �??Universal conductivity of two-dimensional films at the superconductor-insulator transition,�?? Phys. Rev. B 44, 6883-6902 (1991).
    [CrossRef]
  7. M. P. A. Fischer, G. Grinstein, S. M. Girvin, �??Presence of quantum diffusion in two dimensions: Universal resistance at the superconductor-insulator transition,�?? Phys. Rev. Lett. 64, 587-590 (1990); M. P. A. Fischer, �??Quantum phase transitions in disordered two-dimensional superconductors,�?? Phys. Rev. Lett. 65, 923-926 (1990).
    [CrossRef]
  8. S. Sachdev, Quantum Phase Transitions, (Cambridge Univ. Press, Cambridge, 2001).
  9. M. P. A. Fischer, P. B. Weichman, G. Grinstein, D. S. Fisher, �??Boson localization and the superfluid-insulator transition,�?? Phys. Rev. B 40, 546-570 (1989).
    [CrossRef]
  10. J. K. Freericks and H. Monien, �??Phase diagram of the Bose Hubbard model,�?? Europhys. Lett. 26, 545-550 (1994).
    [CrossRef]
  11. J. K. Freericks and H. Monien, �??Strong-coupling expansions for the pure and disordered Bose-Hubbard model,�?? Phys. Rev. B 53, 2691-2700 (1996).
    [CrossRef]
  12. D. van Oosten, P. van der Straten and H. T. C. Stoof, �??Quantum phases in an optical lattice,�?? Phys. Rev. A 63, 053601 (2001).
    [CrossRef]
  13. A. M. Rey, K. Burnett, R. Roth, M. Edwards, C. J. Williams and C. W. Clarck, �??Bogoliubov Approach to Super-fluidity of Atoms in an Optical Lattices,�?? J. Phys. B, 36, 825-841 (2003).
    [CrossRef]
  14. R. Fazio and H. van der Zant, �??Quantum phase transitions and vortex dynamics in superconducting networks,�?? Phys. Rep. 355, 235 (2001) and ref. therein.
    [CrossRef]
  15. A. van Otterlo, K.-H. Wagenblast, R. Baltin, C. Bruder, R. Fazio and G. Schön, �??Quantum phase transitions of interacting bosons and the supersolid phase,�?? Phys. Rev. B 52, 16176-16186 (1995).
    [CrossRef]
  16. M. P. A. Fisher and G. Grinstein, �??Quantum Critical Phenomena in Charged Superconductors,�?? Phys. Rev. Lett. 60, 208-211 (1988).
    [CrossRef] [PubMed]
  17. S. R. White, �??Density matrix formulation for quantum renormalization groups,�?? Phys. Rev. Lett. 69, 2863-2866 (1992).
    [CrossRef] [PubMed]
  18. S. R. White, �??Density-matrix algorithms for quantum renormalization groups,�?? Phys. Rev. B. 48, 10345-10356 (1993).
    [CrossRef]
  19. I. Peschel, X. Wang, M. Kaulke and K. Hallberg, Density-matrix Renormalization, (Springer-Verlag, Berlin, 1998).
  20. T. D. Kühner and S. R. White and H. Monien, �??One-dimensional Bose-Hubbard model with nearest-neighbor interaction,�?? Phys. Rev. B 61, 12474-12489 (2000).
    [CrossRef]
  21. T. D. Kühner, Diploma work (1997), University of Bonn.
  22. S. Rapsch, U. Schollwöck and W. Zwerger, �??Density matrix renormalization group for disordered bosons in one dimension,�?? Europhys. Lett. 46, 559-564 (1999).
    [CrossRef]
  23. G. G. Batrouni, R. T. Scalettar and G. T. Zimanyi, �??Supersolids in the Bose-Hubbard Hamiltonian,�?? Phys. Rev. Lett. 74, 2527-2530 (1995).
    [CrossRef] [PubMed]
  24. G. G. Batrouni and R. T. Scalettar, �??World-line quantum Monte Carlo algorithm for a one-dimensional Bose model,�?? Phys. Rev. B 46, 9051 (1992).
    [CrossRef]
  25. P. Niyaz, R. T. Scalettar, C. Y. Fong and G. G. Batrouni, �??Phase transitions in an interacting boson model with near-neighbor repulsion,�?? Phys. Rev. B 50, 362-373 (1994).
    [CrossRef]
  26. N. V. Prokof�??ev, B. V. Svistunov and I. S.Tupitsyn, Phys. Lett. A 238, 253 (1998).
    [CrossRef]
  27. N. Elstner and H. Monien, �??Dynamics and thermodynamics of the Bose-Hubbard model,�?? Phys. Rev. B 59, 12184-12187 (1999) and ref. therein.
    [CrossRef]
  28. D. S. Rokhsar and B. G. Kotliar, �??Gutzwiller projection for bosons,�?? Phys. Rev. B 44, 10328-10332 (1991).
    [CrossRef]
  29. J. R. Anglin, P. Drummond and A. Smerzi, �??Exact quantum phase model for mesoscopic Josephson junctions,�?? Phys. Rev. A 64, 063605 (2001).
    [CrossRef]

Europhys. Lett.

J. K. Freericks and H. Monien, �??Phase diagram of the Bose Hubbard model,�?? Europhys. Lett. 26, 545-550 (1994).
[CrossRef]

S. Rapsch, U. Schollwöck and W. Zwerger, �??Density matrix renormalization group for disordered bosons in one dimension,�?? Europhys. Lett. 46, 559-564 (1999).
[CrossRef]

J. Phys. B

A. M. Rey, K. Burnett, R. Roth, M. Edwards, C. J. Williams and C. W. Clarck, �??Bogoliubov Approach to Super-fluidity of Atoms in an Optical Lattices,�?? J. Phys. B, 36, 825-841 (2003).
[CrossRef]

Nature

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, I. Bloch, �??Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,�?? Nature 415, 39 (2002).
[CrossRef] [PubMed]

M. Greiner, O. Mandel, T. W. Hänsch and Immanuel Bloch, �??Collapse and revival of the matter wave field of a Bose�??Einstein condensate,�?? Nature 419, 51 (2002).
[CrossRef] [PubMed]

Phys. Lett. A

N. V. Prokof�??ev, B. V. Svistunov and I. S.Tupitsyn, Phys. Lett. A 238, 253 (1998).
[CrossRef]

Phys. Rep.

R. Fazio and H. van der Zant, �??Quantum phase transitions and vortex dynamics in superconducting networks,�?? Phys. Rep. 355, 235 (2001) and ref. therein.
[CrossRef]

Phys. Rev. A

D. van Oosten, P. van der Straten and H. T. C. Stoof, �??Quantum phases in an optical lattice,�?? Phys. Rev. A 63, 053601 (2001).
[CrossRef]

J. R. Anglin, P. Drummond and A. Smerzi, �??Exact quantum phase model for mesoscopic Josephson junctions,�?? Phys. Rev. A 64, 063605 (2001).
[CrossRef]

Phys. Rev. B

N. Elstner and H. Monien, �??Dynamics and thermodynamics of the Bose-Hubbard model,�?? Phys. Rev. B 59, 12184-12187 (1999) and ref. therein.
[CrossRef]

D. S. Rokhsar and B. G. Kotliar, �??Gutzwiller projection for bosons,�?? Phys. Rev. B 44, 10328-10332 (1991).
[CrossRef]

G. G. Batrouni and R. T. Scalettar, �??World-line quantum Monte Carlo algorithm for a one-dimensional Bose model,�?? Phys. Rev. B 46, 9051 (1992).
[CrossRef]

P. Niyaz, R. T. Scalettar, C. Y. Fong and G. G. Batrouni, �??Phase transitions in an interacting boson model with near-neighbor repulsion,�?? Phys. Rev. B 50, 362-373 (1994).
[CrossRef]

T. D. Kühner and S. R. White and H. Monien, �??One-dimensional Bose-Hubbard model with nearest-neighbor interaction,�?? Phys. Rev. B 61, 12474-12489 (2000).
[CrossRef]

M. P. A. Fischer, P. B. Weichman, G. Grinstein, D. S. Fisher, �??Boson localization and the superfluid-insulator transition,�?? Phys. Rev. B 40, 546-570 (1989).
[CrossRef]

S. R. White, �??Density-matrix algorithms for quantum renormalization groups,�?? Phys. Rev. B. 48, 10345-10356 (1993).
[CrossRef]

A. van Otterlo, K.-H. Wagenblast, R. Baltin, C. Bruder, R. Fazio and G. Schön, �??Quantum phase transitions of interacting bosons and the supersolid phase,�?? Phys. Rev. B 52, 16176-16186 (1995).
[CrossRef]

J. K. Freericks and H. Monien, �??Strong-coupling expansions for the pure and disordered Bose-Hubbard model,�?? Phys. Rev. B 53, 2691-2700 (1996).
[CrossRef]

Min-Chul Cha, M. P. A. Fischer, S. M. Girvin, M. Wallin, A. P. Young, �??Universal conductivity of two-dimensional films at the superconductor-insulator transition,�?? Phys. Rev. B 44, 6883-6902 (1991).
[CrossRef]

Phys. Rev. Lett.

M. P. A. Fischer, G. Grinstein, S. M. Girvin, �??Presence of quantum diffusion in two dimensions: Universal resistance at the superconductor-insulator transition,�?? Phys. Rev. Lett. 64, 587-590 (1990); M. P. A. Fischer, �??Quantum phase transitions in disordered two-dimensional superconductors,�?? Phys. Rev. Lett. 65, 923-926 (1990).
[CrossRef]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner and P. Zoller, �??Cold Bosonic Atoms in Optical Lattices,�?? Phys. Rev. Lett. 81, 3108-311 (1998).
[CrossRef]

D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams and P. Zoller, �??Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator,�?? Phys. Rev. Lett. 89, 040402 (2002).
[CrossRef] [PubMed]

B. Y. Chen, S. D. Mahanti and M. Yussouff, �??Helium atoms in zeolite cages: Novel Mott-Hubbard and Bose-Hubbard systems,�?? Phys. Rev. Lett. 75, 473-476 (1995).
[CrossRef] [PubMed]

M. P. A. Fisher and G. Grinstein, �??Quantum Critical Phenomena in Charged Superconductors,�?? Phys. Rev. Lett. 60, 208-211 (1988).
[CrossRef] [PubMed]

S. R. White, �??Density matrix formulation for quantum renormalization groups,�?? Phys. Rev. Lett. 69, 2863-2866 (1992).
[CrossRef] [PubMed]

G. G. Batrouni, R. T. Scalettar and G. T. Zimanyi, �??Supersolids in the Bose-Hubbard Hamiltonian,�?? Phys. Rev. Lett. 74, 2527-2530 (1995).
[CrossRef] [PubMed]

Other

T. D. Kühner, Diploma work (1997), University of Bonn.

I. Peschel, X. Wang, M. Kaulke and K. Hallberg, Density-matrix Renormalization, (Springer-Verlag, Berlin, 1998).

S. Sachdev, Quantum Phase Transitions, (Cambridge Univ. Press, Cambridge, 2001).

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Figures (5)

Fig. 1.
Fig. 1.

Instead of working directly with the population of each well, nk , we can use other quantum numbers, wk , defined by the relation nk =wk -wk -1+, and which behave like a set of chemical potentials acting on the barriers that connect neigboring sites.

Fig. 2.
Fig. 2.

Estimates for (a) energy energy per lattice site and (b) density fluctuations of the quantum rotor Hamiltonian (6) obtained with the variational method (solid), and perturbative calculations for U≪J (dashed) and UJ (dots).

Fig. 3.
Fig. 3.

(a) The ground state energy per site, ε, (b) nearest neighbor correlation, c 1=〈aj+1aj 〉, and (c) variance of the number of atoms per site, σ 2=〈(nj -)2〉. Plots (b) and (c) use a log-log scale. The results of the DMRG (solid line) are obtained on a system with 128 sites, a maximum occupation number of 9 bosons per site and a reduced space of states of about 200 states. The estimates from the variational theory are plotted using dashed lines. The vertical lines mark the location of the phase transition according to [11]. The mean occupation numbers are denoted with circles (=1), diamonds (=2) and boxes (=3).

Fig. 4.
Fig. 4.

(a) The ground state energy per site, ε, (b) nearest neighbor correlation, c 1=〈aj+1aj 〉, and (c) variance of the number of atoms per site, σ 2=〈(nj -)2〉. Plot (b) and (c) are in log-log scale. Using filling factor n̄=1, we show results from the variational model for the Bose-Hubbard model using phase coherent states (solid), the quantum rotor model (dashed), the Gutzwiller ansatz for the Bose-Hubbard Hamiltonian (dots) and DMRG (circles). Vertical dash-dot lines mark the location of the phase transition according to [11].

Fig. 5.
Fig. 5.

The energy of the product ansatz contains a contribution from each connection, εest , plus the interaction between neighbouring connections, Δεest . In Fig. (a) we show that Δεest (dash) is actually negative, and improves the estimate εest moving it towards the exact value, εDMRG (circles). Everything has been computed for =1. In Fig. (b) we show that the correction Δεest does not change much for large lattices.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

H B H = j = 1 M [ J ( a j + 1 a j + a j a j + 1 ) + U 2 a j a j a j a j ] U 2 M n ¯ ( n ¯ 1 ) ,
ψ = n c n n = n c n n 1 n M ,
a l a j ψ = n ¯ ( n ¯ + 1 ) 𝓟 A l + A j n + Δ l j
Δ l j = n c n ( n ¯ + 1 ) ( n j n ¯ ) + n ( n l n ) 2 n ¯ ( n ¯ + 1 ) n ,
Δ l j n ¯ 2 + ( n ¯ + 1 ) 2 2 n ¯ ( n ¯ + 1 ) σ l ,
H Q R = 𝓟 j [ 2 ρ J ( A j + 1 + A j + A j + A j + 1 ) + U 2 ( A j z ) 2 ]
ε 1 M H Q R .
n ϕ = e i n · ϕ ( 2 π ) M 2 , ϕ [ π , π ] M .
H Q R = j [ 2 ρ J cos ( ϕ j ϕ j + 1 ) U 2 2 ϕ j 2 ] ,
ψ = ( 2 π ) M 2 e i n ¯ Σ ϕ k Ψ ( ϕ ) | ϕ d M ϕ .
Ψ ( ϕ ) = j = 1 M h ( ϕ j ϕ j + 1 ) .
n k = w k w k 1 + n ¯ .
ψ = h ˜ ( M 1 ) = w h ˜ w 1 h ˜ w M 1 w 1 w M 1 ,
h ˜ m = h ( ξ ) e i m ξ d ξ .
H Q R = k = 1 M 1 [ 2 ρ J ( k + + k ) + U 2 ( k z k 1 z ) 2 ] ,
ε [ h ˜ ] 4 ρ J Re Σ + + U ( Σ z ) 2 U Σ z 2 ,
2 ρ J ( h ˜ j + 1 + h ˜ j 1 ) + U j 2 h ˜ j = ε est h ˜ j ,
[ U 2 2 ξ 2 2 ρ J cos ( ξ ) ] h ( ξ ) = ε est h ( ξ ) .
σ j 2 = ( a j a j n ¯ ) 2 = 2 ( Σ z ) 2 ,
a j + 1 a j = ρ j ρ γ 1 ,
a j + l a j = ρ k = j j + l 1 k = ρ ( γ 1 ) l ,
H Q R j [ U 2 2 ϕ j 2 ρ J ( ϕ j ϕ j + 1 ) 2 ] .
E g 2 M π 2 ρ J U .
σ 1 π 8 J ρ U .
n | ϕ = e i n ϕ n ! .
H coh t = J i , j [ 2 ( n ¯ + 1 ) cos ( ϕ i ϕ j ) i e i ( ϕ i ϕ j ) ϕ j ] + U 2 j ( 2 ϕ j 2 ) .
O n = k = 1 M n k ! n .
H coh = O H B H O 1
= J i , j ( A i z + n ¯ ) A i + A j + U 2 j ( A j z ) 2 .
H coh = H 1 + H 2 ,
H 1 = j [ J n ¯ Σ x + i J j z j y + U ( j z ) 2 ] ,
H 2 = j [ J ( j 1 z j + j + 1 z j ) + U j z j + 1 z ] ,
ε 0 = min ψ 0 ψ H B H ψ ψ 2 = min χ 0 χ O 2 H coh χ χ O 2 χ ,
ε 0 ε est + 1 N h ˜ M O 2 H 2 h ˜ M h ˜ M O 2 h ˜ M ε est + Δ ε est .
[ U 2 ξ 2 2 J ( n ¯ + 1 ) cos ( ξ ) 2 J sin ( ξ ) ξ ] h = ε est h ,
a j + 1 a j = J ε var ,
σ 2 = U ε var n ¯ 2 ,
ϕ = h ˜ 1 h ˜ k 1 A h ˜ k B h ˜ k + 1 h ˜ k + 2 h ˜ M ,
a k + Δ a k ψ = u t O ( H O ) k 1 ( H O ) Δ ( H O ) M k + 1 u u t O ( H O ) M u .

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