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Variational ansatz for the superfluid Mott-insulator transition in optical lattices

J. J. García-Ripoll, J. I. Cirac, P. Zoller, C. Kollath, U. Schollwöck, and J. von Delft

Open Access Open Access

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, and 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, and 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, and 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, and 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, 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]

2003 (1)

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]

2002 (3)

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]

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and 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]

2001 (3)

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]

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]

2000 (1)

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

1999 (2)

S. Rapsch, U. Schollwöck, and W. Zwerger, “Density matrix renormalization group for disordered bosons in one dimension,” Europhys. Lett. 46, 559–564 (1999).
[Crossref]

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

1998 (2)

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

1996 (1)

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]

1995 (3)

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]

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]

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]

1994 (2)

J. K. Freericks and H. Monien, “Phase diagram of the Bose Hubbard model,” Europhys. Lett. 26, 545–550 (1994).
[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]

1993 (1)

S. R. White, “Density-matrix algorithms for quantum renormalization groups,” Phys. Rev. B. 48, 10345–10356 (1993).
[Crossref]

1992 (2)

S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett. 69, 2863–2866 (1992).
[Crossref] [PubMed]

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]

1991 (2)

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

D. S. Rokhsar and B. G. Kotliar, “Gutzwiller projection for bosons,” Phys. Rev. B 44, 10328–10332 (1991).
[Crossref]

1989 (1)

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

1988 (1)

M. P. A. Fisher and G. Grinstein, “Quantum Critical Phenomena in Charged Superconductors,” Phys. Rev. Lett. 60, 208–211 (1988).
[Crossref] [PubMed]

Anglin, J. R.

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

Baltin, R.

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]

Batrouni, G. G.

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]

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]

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]

Bloch, I.

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

Bloch, Immanuel

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]

Bruder, C.

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]

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]

Burnett, K.

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]

Cha, Min-Chul

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

Chen, B. Y.

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]

Cirac, J. I.

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]

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]

Clarck, C. W.

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]

Drummond, P.

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

Edwards, M.

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]

Elstner, N.

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

Esslinger, T.

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

Fazio, R.

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]

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]

Fischer, M. P. A.

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

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

M. P. A. Fischer, G. Grinstein, and 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]

M. P. A. Fischer, G. Grinstein, and 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]

Fisher, D. S.

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

Fisher, M. P. A.

M. P. A. Fisher and G. Grinstein, “Quantum Critical Phenomena in Charged Superconductors,” Phys. Rev. Lett. 60, 208–211 (1988).
[Crossref] [PubMed]

Fong, C. Y.

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]

Freericks, J. K.

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]

J. K. Freericks and H. Monien, “Phase diagram of the Bose Hubbard model,” Europhys. Lett. 26, 545–550 (1994).
[Crossref]

Gardiner, C. W.

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]

Girvin, S. M.

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

M. P. A. Fischer, G. Grinstein, and 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]

Greiner, M.

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and 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]

Grinstein, G.

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

M. P. A. Fisher and G. Grinstein, “Quantum Critical Phenomena in Charged Superconductors,” Phys. Rev. Lett. 60, 208–211 (1988).
[Crossref] [PubMed]

M. P. A. Fischer, G. Grinstein, and 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]

Hallberg, K.

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

Hänsch, T. W.

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]

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

Jaksch, D.

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]

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]

Kaulke, M.

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

Kotliar, B. G.

D. S. Rokhsar and B. G. Kotliar, “Gutzwiller projection for bosons,” Phys. Rev. B 44, 10328–10332 (1991).
[Crossref]

Kühner, T. D.

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

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

Mahanti, S. D.

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]

Mandel, O.

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]

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

Monien, H.

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

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

J. K. Freericks and H. Monien, “Phase diagram of the Bose Hubbard model,” Europhys. Lett. 26, 545–550 (1994).
[Crossref]

Niyaz, P.

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]

Peschel, I.

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

Prokof’ev, N. V.

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

Rapsch, S.

S. Rapsch, U. Schollwöck, and W. Zwerger, “Density matrix renormalization group for disordered bosons in one dimension,” Europhys. Lett. 46, 559–564 (1999).
[Crossref]

Rey, A. M.

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]

Rokhsar, D. S.

D. S. Rokhsar and B. G. Kotliar, “Gutzwiller projection for bosons,” Phys. Rev. B 44, 10328–10332 (1991).
[Crossref]

Roth, R.

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]

Sachdev, S.

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

Scalettar, R. T.

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]

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]

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]

Schollwöck, U.

S. Rapsch, U. Schollwöck, and W. Zwerger, “Density matrix renormalization group for disordered bosons in one dimension,” Europhys. Lett. 46, 559–564 (1999).
[Crossref]

Schön, G.

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]

Smerzi, A.

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

Stoof, H. T. C.

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]

Svistunov, B. V.

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

Tupitsyn, I. S.

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

van der Straten, P.

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]

van der Zant, H.

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]

van Oosten, D.

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]

van Otterlo, A.

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]

Venturi, V.

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]

Wagenblast, K.-H.

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]

Wallin, M.

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

Wang, X.

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

Weichman, P. B.

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

White, S. R.

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

S. R. White, “Density-matrix algorithms for quantum renormalization groups,” Phys. Rev. B. 48, 10345–10356 (1993).
[Crossref]

S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett. 69, 2863–2866 (1992).
[Crossref] [PubMed]

Williams, C. J.

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]

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]

Young, A. P.

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

Yussouff, M.

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]

Zimanyi, G. T.

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]

Zoller, P.

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]

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]

Zwerger, W.

S. Rapsch, U. Schollwöck, and W. Zwerger, “Density matrix renormalization group for disordered bosons in one dimension,” Europhys. Lett. 46, 559–564 (1999).
[Crossref]

Europhys. Lett. (2)

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 (1)

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 (2)

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and 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 (1)

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

Phys. Rep. (1)

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 (2)

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 (9)

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, 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, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546–570 (1989).
[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, and A. P. Young, “Universal conductivity of two-dimensional films at the superconductor-insulator transition,” Phys. Rev. B 44, 6883–6902 (1991).
[Crossref]

Phys. Rev. B. (1)

S. R. White, “Density-matrix algorithms for quantum renormalization groups,” Phys. Rev. B. 48, 10345–10356 (1993).
[Crossref]

Phys. Rev. Lett. (6)

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]

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]

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 (4)

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

M. P. A. Fischer, G. Grinstein, and 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]

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

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

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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+n̄, and which behave like a set of chemical potentials acting on the barriers that connect neigboring sites.

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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 U≫J (dots).

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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 -n̄)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 (n̄=1), diamonds (n̄=2) and boxes (n̄=3).

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Fig. 4.
Fig. 4. (a) The ground state energy per site, ε, (b) nearest neighbor correlation, c 1=〈aj+1†aj 〉, and (c) variance of the number of atoms per site, σ 2=〈(nj -n̄)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].

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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 n̄=1. In Fig. (b) we show that the correction Δεest does not change much for large lattices.

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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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