Bound states in the one-dimensional two-particle Hubbard model with an impurity (original) (raw)

Abstract

We investigate bound states in the one-dimensional two-particle Bose-Hubbard model with an attractive (V>0) impurity potential. This is a one-dimensional, discrete analogy of the hydrogen negative ion (H−) problem. There are several different types of bound states in this system, each of which appears in a specific region. For given V, there exists a (positive) critical value Uc1 of U (the on-site atom-atom interaction), below which the ground state is a bound state. Interestingly, close to the critical value (U≲Uc1), the ground state can be described by the Chandrasekhar-type variational wave function, which was initially proposed for H−. For U>Uc1, the ground state is no longer a bound state. However, there exists a second (larger) critical value Uc2 of U, above which a molecule-type bound state is established and stabilized by the repulsion. We have also tried to solve for the eigenstates of the model using the Bethe ansatz. The model possesses a global Z2 symmetry (parity) which allows classification of all eigenstates into even and odd states. It is found that all states with odd parity have the Bethe form, but none of the states in the even-parity sector. This allows us to identify analytically two odd-parity bound states, which appear in the parameter regions −2V<U<−V and −V<U<0, respectively. Remarkably, the latter one can be embedded in the continuum spectrum with appropriate parameters. Moreover, in part of these regions, there exists an even-parity bound state accompanying the corresponding odd-parity bound state with almost the same energy.

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (43)

  1. E. Schrödinger, Ann. Phys. (Berlin, Ger.) 79, 361 (1926);
  2. W. Pauli, Z. Phys. 36, 336 (1926).
  3. H. Bethe, Z. Phys. 57, 815 (1929).
  4. E. A. Hylleraas, Z. Phys. 60, 624 (1930); 63, 291 (1930).
  5. S. Chandrasekhar, Astrophys. J. 100, 176 (1944).
  6. R. N. Hill, Phys. Rev. Lett. 38, 643 (1977);
  7. J. Math. Phys. 18, 2316 (1977).
  8. E. H. Lieb, Phys. Rev. Lett. 52, 315 (1984); Phys. Rev. A 29, 3018 (1984).
  9. A. R. P. Rau, Am. J. Phys. 80, 406 (2012);
  10. J. Astrophys. Astron. 17, 113 (1996).
  11. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Springer-Verlag, Berlin, 1977), p. 154.
  12. H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators (Springer-Verlag, Berlin, 1987).
  13. R. P. Feynman, R. B. Leighton, and M. Sands, The Feyn- man Lectures on Physics (Addison Wesley, Longman, 1970), Vol. 3.
  14. J. M. Zhang, D. Braak, and M. Kollar, Phys. Rev. Lett. 109, 116405 (2012).
  15. L. F. Santos and M. I. Dykman, New J. Phys. 14, 095019 (2012).
  16. A. Galindo and P. Pascual, Quantum Mechanics I, translated by J. D. García and L. Alvarez-Gaumé (Springer-Verlag, Berlin, 1990), p. 236.
  17. J. Daboul and M. M. Nieto, Phys. Lett. A 190, 9 (1994).
  18. M. M. Nieto, Phys. Lett. B 486, 414 (2000).
  19. D. C. Mattis, Rev. Mod. Phys. 58, 361 (1986);
  20. D. C. Mattis and S. Rudin, Phys. Rev. Lett. 52, 755 (1984).
  21. M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and B. Simon, J. Phys. A 16, 1125 (1983).
  22. J. von Neumann and E. Wigner, Z. Phys. 30, 465 (1929).
  23. H. Bethe, Z. Phys. 71, 205 (1931).
  24. J. B. McGuire, J. Math. Phys. 5, 622 (1964).
  25. K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. Hecker Denschlag, A. J. Daley, A. Kantian, H. P. Büchler, and P. Zoller, Nature (London) 441, 853 (2006).
  26. It should be stressed that, in this paper, by bound states or localized states we mean in fact normalizable states. They are not to be confused with extended molecule states in which the atoms are bound with respect to each other but the center of mass moves extendedly.
  27. The arguments here and below are based on the assumption that the lattice size is much larger than the characteristic lengths of the wave functions. Although this condition is satisfied for most eigenstates, for some eigenstates, it fails. For example, in Figs. 1 and 2, there is a data point with E n U and D. /L. 1 + 4/π 2 1.4, which deviates significantly from unity. This data point corresponds to two near-degenerate (one odd, one even) states in the third band. Like other states in the third band, these two states are molecule states. However, their wavelength is maximal. Specifically, the envelop of the wave functions is of a sinusoidal form with a node at the origin and maxima at the lattice edges. See Fig. 3 in Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pugatch, and Y. Silberberg, Phys. Rev. A 86, 011603 (2012) for a picture of these states.
  28. Of course, there exist many other quantities, such as the inverse participation ratio, which can serve the same purpose.
  29. J. M. Zhang and R. X. Dong, Eur. J. Phys. 31, 591 (2010).
  30. J. T. Edwards and D. J. Thouless, J. Phys. C: Solid State Phys. 5, 807 (1972).
  31. M. E. Fisher, M. N. Barber, and D. Jasnow, Phys. Rev. A 8, 1111 (1973).
  32. Coupling to the other state 1 √ 2 (|0,1 -|-1,0 ) is forbidden by symmetry.
  33. N. F. Mott, Adv. Phys. 16, 49 (1967).
  34. A. Albo, D. Fekete, and G. Bahir, Phys. Rev. B 85, 115307 (2012).
  35. D. Braak, Phys. Rev. Lett. 107, 100401 (2011).
  36. M. V. Berry and M. Tabor, Proc. R. Soc. London, Ser. A 356, 375 (1977).
  37. H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, London, 1999).
  38. As a far-fetched analogy, we would like to mention the reentrant behavior of the one-dimensional Bose-Hubbard model. There, the phase diagram contains regions in which increasing the hopping amplitude can drive the system from the insulator phase to the superfluid phase and then back to the insulator phase again. See T. D. Kühner and H. Monien, Phys. Rev. B 58, R14741 (1998);
  39. M. Pino, J. Prior, and S. R. Clark, Phys. Status Solidi B 250, 51 (2013).
  40. C. Weiss, Laser Phys. 20, 665 (2010).
  41. A. R. Kolovsky, J. Link, and S. Wimberger, New J. Phys. 14, 075002 (2012).
  42. V. L. Bulatov and P. E. Kornilovitch, Europhys. Lett. 71, 352 (2005).
  43. J. Callaway, Phys. Rev. A 26, 199 (1982).