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Phase diagrams of bosonic ABn chains


The A B N − 1 chain is a system that consists of repeating a unit cell with N sites where between the A and B sites there is an energy difference of λ. We considered bosons in these special lattices and took into account the kinetic energy, the local two-body interaction, and the inhomogenous local energy in the Hamiltonian. We found the charge density wave (CDW) and superfluid and Mott insulator phases, and constructed the phase diagram for N = 2 and 3 at the thermodynamic limit. The system exhibited insulator phases for densities ρ = α/N, with α being an integer. We obtained that superfluid regions separate the insulator phases for densities larger than one. For any N value, we found that for integer densities ρ, the system exhibits ρ + 1 insulator phases, a Mott insulator phase, and ρ CDW phases. For non-integer densities larger than one, several CDW phases appear.

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  1. 1.

    I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)

    ADS  Article  Google Scholar 

  2. 2.

    I. Bloch, J. Dalibard, S. Nascimbène, Nat. Phys. 8, 267 (2012)

    Article  Google Scholar 

  3. 3.

    D. Jaksch, C. Bruder, J. Cirac, C. Gardiner, P. Zoller, Phys. Rev. Lett. 81, 3108 (1998)

    ADS  Article  Google Scholar 

  4. 4.

    M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, I. Bloch, Nature 415, 39 (2002)

    ADS  Article  Google Scholar 

  5. 5.

    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, M. Inguscio, Nature 453, 895 (2008)

    ADS  Article  Google Scholar 

  6. 6.

    M. Atala, M. Aidelsburger, J.T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, I. Bloch, Nat. Phys. 9, 795 (2013)

    Article  Google Scholar 

  7. 7.

    M. Di Liberto, T. Comparin, T. Kock, M. Ölschläger, A. Hemmerich, C. Morais Smith, Nat. Commun. 5, 5735 (2014)

    ADS  Article  Google Scholar 

  8. 8.

    M. Messer, R. Desbuquois, T. Uehlinger, G. Jotzu, S. Huber, D. Greif, T. Esslinger, Phys. Rev. Lett. 115, 115303 (2015)

    ADS  Article  Google Scholar 

  9. 9.

    P. Buonsante, A. Vezzani, Phys. Rev. A 70, 033608 (2004)

    ADS  Article  Google Scholar 

  10. 10.

    V.G. Rousseau, D.P. Arovas, M. Rigol, F. Hébert, G.G. Batrouni, R.T. Scalettar, Phys. Rev. B 73, 174516 (2006)

    ADS  Article  Google Scholar 

  11. 11.

    A. Dhar, T. Mishra, R.V. Pai, B.P. Das, Phys. Rev. A 83, 053621 (2011)

    ADS  Article  Google Scholar 

  12. 12.

    A. Dhar, M. Singh, R.V. Pai, B.P. Das, Phys. Rev. A 84, 033631 (2011)

    ADS  Article  Google Scholar 

  13. 13.

    M. Singh, A. Dhar, T. Mishra, R.V. Pai, B.P. Das, Phys. Rev. A 85, 051604(R) (2012)

    ADS  Article  Google Scholar 

  14. 14.

    S.R. White, Phys. Rev. Lett. 69, 2863 (1992)

    ADS  Article  Google Scholar 

  15. 15.

    V.V. França, K. Capelle, Phys. Rev. Lett. 100, 070403 (2008)

    ADS  Article  Google Scholar 

  16. 16.

    T.D. Kühner, H. Monien, Phys. Rev. B 58, R14741 (1998)

    ADS  Article  Google Scholar 

  17. 17.

    T.D. Kühner, S.R. White, H. Monien, Phys. Rev. B 61, 12474 (2000)

    ADS  Article  Google Scholar 

  18. 18.

    S. Ejima, H. Fehske, F. Gebhard, Europhys. Lett. 93, 30002 (2011)

    ADS  Article  Google Scholar 

  19. 19.

    S. Yamamoto, Phys. Rev. B 64, 140102(R) (2001)

    ADS  Article  Google Scholar 

  20. 20.

    F.X. Redl, K.S. Cho, C.B. Murray, S. O’Brien, Nature 423, 01702 (2003)

    Article  Google Scholar 

  21. 21.

    M.E. Torio, A.A. Aligia, G.I. Japaridze, B. Normand, Phys. Rev. B 73, 115109 (2006)

    ADS  Article  Google Scholar 

  22. 22.

    J. Silva-Valencia, A.M.C. Souza, Phys. Rev. A 85, 033612 (2012)

    ADS  Article  Google Scholar 

  23. 23.

    T. Li, H.M. Gou, S. Chen, S.Q. Shen, Phys. Rev. B 91, 134101 (2015)

    ADS  Article  Google Scholar 

  24. 24.

    G.J. Cruz, R. Franco, J. Silva-Valencia, in preparation

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Cruz, G., Franco, R. & Silva-Valencia, J. Phase diagrams of bosonic ABn chains. Eur. Phys. J. B 89, 106 (2016).

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