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Green’s Function Approach to the Bose-Hubbard Model

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DOI: 10.4236/wjcmp.2013.32020    4,228 Downloads   6,980 Views   Citations

ABSTRACT

We use a diagrammatic hopping expansion to calculate finite-temperature Green functions of the Bose-Hubbard model which describes bosons in an optical lattice. This technique allows for a summation of subsets of diagrams, so the divergence of the Green function leads to non-perturbative results for the boundary between the superfluid and the Mott phase for finite temperatures. Whereas the first-order calculation reproduces the seminal mean-field result, the second order goes beyond and shifts the phase boundary in the immediate vicinity of the critical parameters determined by high-precision Monte-Carlo simulations of the Bose-Hubbard model. In addition, our Greens function approach allows for calculating the excitation spectrum both for zero and finite temperature and for determining the effective masses of particles and holes.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

M. Ohliger and A. Pelster, "Green’s Function Approach to the Bose-Hubbard Model," World Journal of Condensed Matter Physics, Vol. 3 No. 2, 2013, pp. 125-130. doi: 10.4236/wjcmp.2013.32020.

References

[1] M. P. A. Fisher, P. B. Weichman, G. Grinstein and D. S. Fisher, “Boson Localization and the Superfluid-Insulator Transition,” Physical Review B, Vol. 40, No. 1, 1989, pp. 546-570. doi:10.1103/PhysRevB.40.546
[2] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Physical Review Letters, Vol. 81, No. 15, 1998, pp. 3108-3111. doi:10.1103/PhysRevLett.81.3108
[3] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch and I. Bloch, “Quantum Phase Transition from a Superfluid to a Mott Insulator in a Gas of Ultracold Atoms,” Nature, Vol. 415, 2002, pp. 39-44. doi:10.1038/415039a
[4] S. Sachdev, “Quantum Phase Transitions,” 2nd Edition, Cambridge University Press, Cambridge, 2011.
[5] J. K. Freericks and H. Monien, “Strong-Coupling Expansions for the Pure and Disordered Bose-Hubbard Model,” Physical Review B, Vol. 53, No. 5, 1996, pp. 2691-2700. doi:10.1103/PhysRevB.53.2691
[6] N. Elstner and H. Monien, “Dynamics and Thermodynamics of the Bose-Hubbard Model,” Physical Review B, Vol. 59, No. 19, 1999, pp. 12184-12187. doi:10.1103/PhysRevB.59.12184
[7] D. van Oosten, P. van der Straten and H. T. Stoof, “Quantum Phases in an Optical Lattice,” Physical Review A, Vol. 63, No. 5, 2001, Article ID: 053601. doi:10.1103/PhysRevA.63.053601
[8] D. B. M. Dickerscheid, D. van Oosten, P. J. H. Denteneer and H. T. C. Stoof, “Ultracold Atoms in Optical Lattices,” Physical Review A, Vol. 68, No. 4, 2003, Article ID: 043623. doi:10.1103/PhysRevA.68.043623
[9] F. E. A. dos Santos and A. Pelsterr, “Quantum Phase Diagram of Bosons in Optical Lattices,” Physical Review A, Vol. 79, No. 1, 2009, Article ID: 013614. doi:10.1103/PhysRevA.79.013614
[10] N. Teichmann, D. Hinrichs, M. Holthaus, and A. Eckardt, “Bose-Hubbard Phase Diagram with Arbitrary Integer Filling,” Physical Review B, Vol. 79, No. 10, 2009, Article ID: 100503. doi:10.1103/PhysRevB.79.100503
[11] N. Teichmann, D. Hinrichs, M. Holthaus and A. Eckardt, “Process-Chain Approach to the Bose-Hubbard Model: Ground-State Properties and Phase Diagram,” Physical Review B, Vol. 79, No. 22, 2009, Article ID: 224515. doi:10.1103/PhysRevB.79.224515
[12] T. D. Kühner and H. Monien, “Phases of the One-Dimensional Bose-Hubbard Model,” Physical Review B, Vol. 58, No. 22, 1998, pp. R14741-R14744. doi:10.1103/PhysRevB.58.R14741
[13] B. Capogrosso-Sansone, N. V. Prokof'ev, and B. V. Svistunov, “Phase Diagram and Thermodynamics of the ThreeDimensional Bose-Hubbard Model,” Physical Review B, Vol. 75, No. 13, 2007, Article ID: 134302. doi:10.1103/PhysRevB.75.134302
[14] B. Capogrosso-Sansone, S. G. Soyler, N. Prokof'ev and B. Svistunov, “Monte Carlo Study of the Two-Dimensional Bose-Hubbard Model,” Physical Review A, Vol. 77, No. 1, 2008, Article ID: 015602. doi:10.1103/PhysRevA.77.015602
[15] H. Kleinert, S. Schmidt and A. Pelster, “Reentrant Phenomenon in the Quantum Phase Transitions of a Gas of Bosons Trapped in an Optical Lattice,” Physical Review Letters, Vol. 93, No. 16, 2004, Article ID: 160402. doi:10.1103/PhysRevLett.93.160402
[16] P. Buonsante and A. Vezzani, “Phase Diagram for Ultracold Bosons in Optical Lattices and Superlattices,” Physical Review A, Vol. 70, No. 3, 2004, Article ID: 033608. doi:10.1103/PhysRevA.70.033608
[17] K. V. Krutitsky, A. Pelster and R. Graham, “Mean-Field Phase Diagram of Disordered Bosons in a Lattice at Nonzero Temperature,” New Journal of Physics, Vol. 8, 2006, p. 187. doi:10.1088/1367-2630/8/9/187
[18] S. Folling, A. Widera, T. Müller, F. Gerbier and I. Bloch, “Formation of Spatial Shell Structure in the Superfluid to Mott Insulator Transition,” Physical Review Letters, Vol. 97, No. 6, 2006, Article ID: 060403. doi:10.1103/PhysRevLett.97.060403
[19] A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski, “Methods of Quantum Field Theory in Statistical Physics,” Dover Publications, New York, 1963.
[20] J. Zinn-Justin, “Quantum Field Theory and Critical Phenomena,” 4th Edition, Oxford University Press, Oxford, 2002. doi:10.1093/acprof:oso/9780198509233.001.0001
[21] W. Metzner, “Linked-Cluster Expansion around the Atomic Limit of the Hubbard Model,” Physical Review B, Vol. 43, No. 10, 1993, pp. 8549-8563. doi:10.1103/PhysRevB.43.8549
[22] H. Kleinert and V. Schulte-Frohlinde, “Critical Properties of Φ4-Theories,” World Scientific, Singapore, 2001. doi:10.1142/9789812799944
[23] F. Gerbier, et al., “Phase Coherence of an Atomic Mott Insulator,” Physical Review Letters, Vol. 95, 2005, Article ID: 050404. doi:10.1103/PhysRevLett.95.050404
[24] A. Hoffmann and A. Pelster, “Visibility of Cold Atomic Gases in Optical Lattices for Finite Temperatures,” Physical Review A, Vol. 79, No. 5, 2009, Article ID: 053623. doi:10.1103/PhysRevA.79.053623
[25] F. Gerbier, S. Trotzky, S. Foelling, U. Schnorrberger, J. D. Thompson, A. Widera, I. Bloch, L. Pollet, M. Troyer, B. Capogrosso-Sansone, N. V. Prokof’ev and B. V. Svistunov, “Expansion of a Quantum Gas Released from an Optical Lattice,” Physical Review Letters, Vol. 101, No. 15, 2008, Article ID: 155303. doi:10.1103/PhysRevLett.101.155303
[26] F. Gerbier, “Boson Mott Insulators at Finite Temperatures,” Physical Review Letters, Vol. 99, No. 12, 2007, Article ID: 120405. doi:10.1103/PhysRevLett.99.120405
[27] W. Metzner and D. Vollhardt, “Correlated Lattice Fermions in d = ∞ Dimensions,” Physical Review Letters, Vol. 62, No. 3, 1989, pp. 324-327. doi:10.1103/PhysRevLett.62.324
[28] L. Amico and V. Penna, “Dynamical Mean Field Theory of the Bose-Hubbard Model,” Physical Review Letters, Vol. 80, No. 10, 1998, pp. 2189-2192. doi:10.1103/PhysRevLett.80.2189
[29] K. Byczuk and D. Vollhardt, “Correlated Bosons on a Lattice: Dynamical Mean-Field Theory for Bose-Einstein Condensed and Normal Phases,” Physical Review B, Vol. 77, No. 23, 2008, Article ID: 235106. doi:10.1103/PhysRevB.77.235106
[30] B. Bradlyn, F. E. A. dos Santos and A. Pelster, “Effective Action Approach for Quantum Phase Transitions in Bosonic Lattices,” Physical Review A, Vol. 79, No. 1, 2009, Article ID: 013615. doi:10.1103/PhysRevA.79.013615
[31] T. D. Grass, F.E.A. dos Santos and A. Pelster, “Excitation Spectra of Bosons in Optical Lattices from the Schwinger-Keldysh Calculation,” Physical Review A, Vol. 84, No. 1, 2011, Article ID: 013613. doi:10.1103/PhysRevA.84.013613

  
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