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

Optics Express, Vol. 12, Issue 1, pp. 42-54 (2004)

http://dx.doi.org/10.1364/OPEX.12.000042

Acrobat PDF (187 KB)

### 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

## 1. Introduction

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]

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]

## 2. Quantum phase model

### 2.1. Relation to the Bose-Hubbard model

*n̄*[6

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]

*M*is the number of lattice sites and

*N*=

*n̄M*the number of atoms. Both the Bose-Hubbard model and the quantum rotor model show a phase transition due to the interplay between the kinetic term proportional to

*J*and the interaction term proportional to

*U*. For convenience we have subtracted the ground state energy in the perfect insulator limit

*U/J*→∞.

*n*

_{k}>1, we may approximate the hopping terms as follows

*A*±|

*n*〉=|

*n*±1〉 are ladder operators and

*𝓟*projects on states with non-negative occupation numbers,

*n*

_{k}≥0. To lowest order the error |Δ

_{lj}〉 is

*n*

_{l}-

*n̄*)

^{2}〉 is the variance in the number of particles per lattice site. For the approximation (3) to be valid, the uncertainty in the number of atoms must be small compared to the mean value,

*n̄*≫σ, and the interaction energy must exceed the neglected terms,

*Ū*

*n*(

*n̄*-1)≫

*Jσ*.

*a†*

_{j}

*a*

_{j}

*-n̄*is essentially the number operator and we have used that ∑j

*n*

_{k}=

*n̄*, the usual step now is to drop the projector,

*𝓟*, and move to the basis of phase states, defined by

*i*

_{∂}/

*∂ϕ*

_{j}, which produces the usual representation of the quantum rotor model

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

### 2.2. Variational ansatz

*H*

_{QR}variationally. Due to the previous splitting (10), any wavefunction Ψ(

*ϕ⃗*) can only depend on the phase difference between neighboring wells,

*ξ*

_{j}=

*ϕ*

_{j+1}-

*ϕj*. These new quantum variables describe the connections between neigboring sites. In the limit of large lattices it seems reasonable to assume that these connections become independent from each other adopting the product state

*U/J*→∞, where

*h*

_{mott}(

*ξ*)=1, and in the superfluid limit,

*U/J*→0, where

*h*

_{sf}(

*ξ*)=∑

_{n∈ℤ}δ(ξ-2

*πn*), as can be verified by direct substitution in Eq. (10).

*ξj*=

*ϕ*

_{j+1}-

*ϕ*

_{j}. These are the new quantum numbers,

*w*

_{k}, given by

*w*

_{k}play the roles of chemical potentials which are established between different wells: the difference between the potentials on the extremes of a site gives the fluctuations over the mean and commesurate occupation

*n̄*. In this picture

*w*〉=|

*w*±1〉 are new ladder operators and ∑

^{z}|

*w*〉=

*w*|

*w*〉.

*H*

_{QR}over all states within a given ansatz we can both obtain an upper bound to the energy of the ground state and approximate its wave function. A simple computation with our product ansatz leads to the result [compare (7)]

^{z}〉=∑

*w*

^{w}|

*h*

_{w}|

^{2}, and the wavefunctions are assumed to be normalized, ∑

_{w}|

*h*

_{w}|

^{2}=1. Since the stationary states have a well defined parity,

*h̃*(-

*w*)=(-1)

^{P}

*h̃*

_{w}, the optimal variational state must satisfy the linear equation

*H*

_{QR}, we can also compute other properties of the ground state. For instance, the variance of the number of atoms per lattice site

*γ1*,

*ε*…) are properly estimated even if long–range ones are not.

*U*≫

*J*we compare with a first order perturbative calculation around the solution Ψ(

*ϕ⃗*)=1, which is possible thanks to an energy gap of order

*𝒪*(

*U*) in the excitation spectrum. In the limit

*J*≫

*U*we rather use a spin wave or harmonic approximation in which the cosineterms of the Hamiltonian

*H*

_{QR}are expanded up to second order in the phase difference between neighboring sites [see Sec. 2

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]

*J*→0. This differs from the expected behavior of the ground state of the original Bose-Hubard Hamiltonian, and it reminds us that

*H*

_{QR}can only model the atomic ensemble when the variance, σ, is small compared to the mean occupation number,

*n̄*.

### 2.3. Harmonic approximations to the quantum phase model

*J*≫

*U*it is possible to estimate the ground state of the rotor model (6) analytically. Since we are deep in the superfluid regime, the wavefunction will be concentrated around the line

*ϕ*1=

*ϕ*2=…=

*ϕM*, and we can approximate

*ϕ*

_{i}and change variables, the preceding Hamiltonian may be diagonalized,

*ω*

_{k}=(8

*ρJU*)

^{1/2}|sin(

*πk/M*)|, where

*k*is an integer in the range -

*M*+1<2

*k*<

*M*and labels the different values of the momentum in the lattice. The ground state energy [Fig. 4(a)] may be estimated as the zero-point energy of the harmonic oscillator. For large

*M*, the sum over

*k*may be replaced with an integral, giving

## 3. The Bose-Hubbard model

*ε*,

*γ*1, σ), which are to be validated with DMRG calculations.

### 3.1. Coherent states

*ϕ*〉 are defined by

*ϕ*|θ〉=exp[

*e*

^{i(θ-ϕ)}], but they form a complete basis and an expansion like (10) is still possible. A nice property of the coherent states is that we can rewrite the operators

*a*,

*a*

^{†},

*a*

^{†}

*a*, etc, in terms of the phases very easily. For instance,

*ϕ⃗*), i.e.

*H*

_{coh}. This operator was already used in Ref. [29

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

### 3.2. Variational procedure for non-Hermitian operators

*Oa*

_{j}

*O*

^{-1}=

*A-*

_{j}and

*Oa*

^{†}

*jO-1*=

*A+*

_{j}, we find

*H*

_{1}, and terms which involve pairs of connections,

*H*

_{2}.

*H*

_{1}. The difference now is that, since the operator

*H*

_{coh}is not Hermitian, we cannot establish a variational principle and that argument is no longer valid. Nevertheless, we will again propose a variational ansatz which is an eigenstate of the local operator

*H*

_{1}|

*h̃*〉⊗

^{M}=

*Mε*

_{est}|

*h̃*〉⊗

^{N}. Using the following equality

*χ*〉=|

*h̃*〉⊗

^{M}, we arrive to an upper bound for the lowest eigenvalue of the Bose-Hubbard Hamiltonian, expressed in terms of the non-Hermitian one

*J*and

*U*we compute the lowest eigenstate of

*H*

_{1}and this way obtain

*h̃*. After the equivalence (14), finding the ground state of the local Hamiltonian

*H*

_{1}becomes equivalent to solving a modified Mathieu equation

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

*ε*

_{est}we must still compute the correction

*Δε*

_{est}using a rather straightforward expansion which is shown in Sec. 3.4. Surprisingly, Δ

*ε*

_{est}happens to be negative, so that it is actually an improvement over the simple estimate given by

*ε*

_{est}[See Fig. 5].

*ψ*〉=

*O*-1|

*h̃*〉⊗

^{M}, and the estimate for the energy,

*ε*

_{var}=

*ε*

_{est}+

*Δε*

_{est}, we may compute other observables. For the density fluctuations and nearest neighbor correlations we use the virial theorem

*h̃*〉⊗M correlations decay exponentially, opposite to what is expected in the superfluid phase, whose correlations should decay algebraically. Nevertheless, as we will see next, this family of states does estimate accurately the local properties of the optical lattice.

### 3.3. Comparison to DMRG results

*M*containing the most relevant states.

*M*is chosen to be small enough to be handled numerically, but large enough to obtain the desired accuracy; numerical results can be extrapolated in

*M*to the exact limit of infinite

*M*in the thermodynamic limit. However, results presented here have converged for the largest

*M*considered and no further extrapolation was necessary.

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]

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]

*n̄*=1,2, and 3. In Figs. 3(a–c) we show the mean energy per site

*ε*, the nearest neighbor correlations,

*c*

_{1}=〈

*a*

_{j}〉, and the variance σ of the density, calculated both with the DMRG and with the variational estimates developed above. As expected, there are no indications of the phase transitions in these quantities, neither in the variational results nor in the numerical solutions. Rather, an inflexion of the nearest neighbor correlation points out the location of the superfluid-insulator transition which lies roughly between 3

*n̄*and 4

*n̄*(see [20

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]

*n̄*=1. In this figure we plot together results from the DMRG, the variational ansatz derived above, the quantum rotor model and the well-known Gutzwiller ansatz. The Gutzwiller ansatz [28

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

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]

_{G}〉=Π

^{M}

_{j=1}|Φ

*j*〉, where |Φ

*j*〈=∑∞

_{m=0}

*f*

^{(j)}|

*m*

_{j}〉 and

*𝒪*(

*J*/

*U*) are lost and all correlations become zero. However this ansatz gives good results in the superfluid regime, where the long-range order is well described by |Ψ

_{G}〉, and we can use these results and those of the DMRG to assert the accuracy of our variational estimates. As Fig. 4 shows, in the Mott insulator regime, the DMRG results agree perfectly with our variational theory for the Bose-Hubbard model and for the quantum rotor model. Close to the phase transition is the point at which the quantum rotor model no longer describes well the atoms in the optical lattice due to the growth of density fluctuations. At this point we also observe a small disagreement between the DMRG and the coherent states, which is due to the growth of long range correlations and vanishes as we get deeper into the superfluid regime.

### 3.4. Numerical evaluation of the upper bound

*Δε*

_{est}, from Eq. (31). We basically need a method to compute expectation values of the operator

*O*

^{-2}around product states which have the form

*ψ*〉=

*O*

^{-1}|

*h̃*〉⊗

^{M}, where

*A*and

*B*are just the identity, but also of

*H*

_{2}|

*ψ*〉, where

*A*and

*B*are ∑

^{+}, ∑- or ∑z. After some manipulations it is possible to write

*Δε*

_{est}for different lattice sizes and found small or no differences for more than 30 sites. Intuitively, this is because in the limit of large powers the matrices (

*HO*)

^{k}become projectors on the eigenvector with the largest eigenvalue. Using the same type of expansion we may compute other correlations

*γΔ*

_{1}, where

*γ*

_{1}is the largest eigenvalue of the matrix

*H*_

*O*.

## 4. Conclusions

## Acknowledgments

## References and links

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 |

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 |

3. | D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. |

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. |

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. |

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 |

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, |

9. | M. P. A. Fischer, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B |

10. | J. K. Freericks and H. Monien, “Phase diagram of the Bose Hubbard model,” Europhys. Lett. |

11. | J. K. Freericks and H. Monien, “Strong-coupling expansions for the pure and disordered Bose-Hubbard model,” Phys. Rev. B |

12. | D. van Oosten, P. van der Straten, and H. T. C. Stoof, “Quantum phases in an optical lattice,” Phys. Rev. A |

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 , |

14. | R. Fazio and H. van der Zant, ‘Quantum phase transitions and vortex dynamics in superconducting networks,” Phys. Rep. |

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 |

16. | M. P. A. Fisher and G. Grinstein, “Quantum Critical Phenomena in Charged Superconductors,” Phys. Rev. Lett. |

17. | S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett. |

18. | S. R. White, “Density-matrix algorithms for quantum renormalization groups,” Phys. Rev. B. |

19. | I. Peschel, X. Wang, M. Kaulke, and K. Hallberg, |

20. | T. D. Kühner, S. R. White, and H. Monien, “One-dimensional Bose-Hubbard model with nearest-neighbor interaction,” Phys. Rev. B |

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. |

23. | G. G. Batrouni, R. T. Scalettar, and G. T. Zimanyi, “Supersolids in the Bose-Hubbard Hamiltonian,” Phys. Rev. Lett. |

24. | G. G. Batrouni and R. T. Scalettar, “World-line quantum Monte Carlo algorithm for a one-dimensional Bose model,” Phys. Rev. B |

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 |

26. | N. V. Prokof’ev, B. V. Svistunov, and I. S. Tupitsyn, Phys. Lett. A |

27. | N. Elstner and H. Monien, “Dynamics and thermodynamics of the Bose-Hubbard model,” Phys. Rev. B |

28. | D. S. Rokhsar and B. G. Kotliar, “Gutzwiller projection for bosons,” Phys. Rev. B |

29. | J. R. Anglin, P. Drummond, and A. Smerzi, “Exact quantum phase model for mesoscopic Josephson junctions,” Phys. Rev. A |

**OCIS Codes**

(000.2690) General : General physics

(000.3860) General : Mathematical methods in physics

**ToC Category:**

Focus Issue: Cold atomic gases in optical lattices

**History**

Original Manuscript: November 7, 2003

Revised Manuscript: December 1, 2003

Published: January 12, 2004

**Citation**

Juan Jose García-Ripoll, J. Cirac, P. Zoller, C. Kollath, U. Schollwöck, and J. von Delft, "Variational ansatz for the superfluid Mott-insulator transition in optical lattices," Opt. Express **12**, 42-54 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-42

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### References

- 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]
- 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]
- 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]
- 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]
- S. Sachdev, Quantum Phase Transitions, (Cambridge Univ. Press, Cambridge, 2001).
- 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]
- J. K. Freericks and H. Monien, �??Phase diagram of the Bose Hubbard model,�?? Europhys. Lett. 26, 545-550 (1994). [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]
- 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]
- 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]
- 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]
- 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]
- S. R. White, �??Density-matrix algorithms for quantum renormalization groups,�?? Phys. Rev. B. 48, 10345-10356 (1993). [CrossRef]
- I. Peschel, X. Wang, M. Kaulke and K. Hallberg, Density-matrix Renormalization, (Springer-Verlag, Berlin, 1998).
- 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]
- T. D. Kühner, Diploma work (1997), University of Bonn.
- S. Rapsch, U. Schollwöck and W. Zwerger, �??Density matrix renormalization group for disordered bosons in one dimension,�?? Europhys. Lett. 46, 559-564 (1999). [CrossRef]
- 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]
- 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]
- N. V. Prokof�??ev, B. V. Svistunov and I. S.Tupitsyn, Phys. Lett. A 238, 253 (1998). [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]
- D. S. Rokhsar and B. G. Kotliar, �??Gutzwiller projection for bosons,�?? Phys. Rev. B 44, 10328-10332 (1991). [CrossRef]
- J. R. Anglin, P. Drummond and A. Smerzi, �??Exact quantum phase model for mesoscopic Josephson junctions,�?? Phys. Rev. A 64, 063605 (2001). [CrossRef]

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