Skip to main content
SpringerLink
Log in

One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries. II. N Particles

  • Published:
Few-Body Systems Aims and scope Submit manuscript

Abstract

This is the second in a pair of articles that classify the configuration space and kinematic symmetry groups for N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly–unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article analyzes the symmetries of N particles in asymmetric, symmetric, and harmonic traps; the prequel article treats the one, two and three particle cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hamermesh M.: Group Theory and Its Application to Physical Problems. Dover, New York (1989)

    MATH  Google Scholar 

  2. Sagan B.E.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd edn. Graduate Texts in Mathematics, vol. 203. Springer, Berlin (2000)

    Google Scholar 

  3. Chen J.-Q., Ping J., Wang F.: Group Representation Theory for Physicists, 2nd edn. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  4. Ma Z.-Q.: Group Theory for Physicists. World Scientific, Singapore (2007)

    Book  MATH  Google Scholar 

  5. Guan L., Chen S., Wang Y., Ma Z.-Q.: Exact solutions for infinitely strongly interacting Fermi gases in tight waveguides. Phys. Rev. Lett. 102, 160402 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Ma Z.-Q., Chen S., Guan L., Wang Y.: Mathematical calculation for exact solutions of infinitely strongly interacting Fermi gases in tight waveguides. J. Phys. A 42, 385210 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Fang B., Vignolo P., Gattobigio M., Miniatura C., Minguzzi A.: Exact solution for the degenerate ground-state of a strongly interacting one-dimensional Bose–Fermi mixture. Phys. Rev. A 84, 023626 (2011)

    Article  ADS  Google Scholar 

  8. Jakubczyk P., Kravets Y., Jakubczyk D.: Entanglement of one-magnon Schur–Weyl states. Eur. Phys. J. D 61, 507–512 (2011)

    Article  ADS  Google Scholar 

  9. Cui X., Ho T.-L.: Ground-state ferromagnetic transition in strongly repulsive one-dimensional Fermi gases. Phys. Rev. A 89, 023611 (2014)

    Article  ADS  Google Scholar 

  10. Yurovsky V.A.: Permutation symmetry in spinor quantum gases: selection rules, conservation laws, and correlations. Phys. Rev. Lett. 113, 200406 (2014)

    Article  ADS  Google Scholar 

  11. Yurovsky V.A.: Sum rules for spin-1/2 quantum gases in well-defined-spin states: spin-independent interactions and spin-dependent external fields. Phys. Rev. A 91, 053601 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  12. Fernández, F.M.: On the Symmetry of the Quantum-Mechanical Particle in a Box. arXiv:1310.5136 (2013)

  13. Coxeter H.S.M.: Regular Polytopes. Dover, New York (1973)

    MATH  Google Scholar 

  14. Gaudin M.: Boundary energy of a Bose gas in one dimension. Phys. Rev. A 4, 386–394 (1971)

    Article  ADS  Google Scholar 

  15. Gaudin, M.: The Bethe Wavefunction. Cambridge University Press, Cambridge (2014)

  16. Frame J.S.: Orthogonal group matrices of hyperoctahedral groups. Nagoya Math. J. 27, 585–590 (1966)

    MathSciNet  MATH  Google Scholar 

  17. Goodman R.: Alice through looking glass after looking glass: the mathematics of mirrors and kaleidoscopes. Math. Soc. Am. Mon. 111, 281–298 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Avery J.: Hyperspherical Harmonics: Application in Quantum Theory, pp. 35. Kluwer, Dordrecht (1989)

    Book  MATH  Google Scholar 

  19. Harshman N.L.: Spectroscopy for a few atoms harmonically trapped in one dimension. Phys. Rev. A 89, 033633 (2014)

    Article  ADS  Google Scholar 

  20. Crescimanno M., Landsberg A.S.: Spectral equivalence of bosons and fermions in one-dimensional harmonic potentials. Phys. Rev. A 63, 035601 (2001)

    Article  ADS  Google Scholar 

  21. Girardeau M.: Relationship between systems of impenetrable bosons and fermions in one dimension. J. Math. Phys. 1, 516–523 (1960)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Leyvraz F., Frank A., Lemus R.: Accidental degeneracy in a simple quantum system: a new symmetry group for a particle in an impenetrable square-well potential. Am. J. Phys. 65, 1087–1094 (1997)

    Article  ADS  Google Scholar 

  23. Baker G.A.: Degeneracy of the n-dimensional, isotropic, harmonic oscillator. Phys. Rev. 103, 1119 (1956)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Louck J.D.: Group theory of harmonic oscillators in n-dimensional space. J. Math. Phys. 6, 1786 (1965)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Brouzos I., Schmelcher P.: Construction of analytical many-body wave functions for correlated bosons in a harmonic trap. Phys. Rev. Lett. 108, 045301 (2012)

    Article  ADS  Google Scholar 

  26. Harshman N.L.: Symmetries of three harmonically trapped particles in one dimension. Phys. Rev. A 86, 052122 (2012)

    Article  ADS  Google Scholar 

  27. García-March M.A., Juliá-Díaz B., Astrakharchik G.E., Boronat J., Polls A.: Distinguishability, degeneracy, and correlations in three harmonically trapped bosons in one dimension. Phys. Rev. A 90, 063605 (2014)

    Article  ADS  Google Scholar 

  28. Loft N.J.S., Dehkharghani A.S., Mehta N.P., Volosniev A.G., Zinner N.T.: A variational approach to repulsively interacting three-fermion systems in a one-dimensional harmonic trap. Eur. Phys. J. D 69, 65 (2015)

    Article  ADS  Google Scholar 

  29. Avakian M.P., Pogosyan G.S., Sissakian A.N., Ter-Antonyan V.M.: Spectroscopy of a singular linear oscillator. Phys. Lett. A 124, 233–236 (1987)

    Article  ADS  Google Scholar 

  30. Busch T., Englert B.-G., Rzążewski K., Wilkens M.: Two cold atoms in a harmonic trap. Found. Phys. 28, 549–559 (1998)

    Article  Google Scholar 

  31. Jonsell S.: Interaction energy of two trapped bosons with long scattering length. Few-Body Sys. 31, 255–260 (2002)

    Article  ADS  Google Scholar 

  32. Oelkers N., Batchelor M.T., Bortz M., Guan X.-W.: Bethe ansatz study of one-dimensional Bose and Fermi gases with periodic and hard wall boundary conditions. J. Phys. A 39, 1073–1098 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Hao Y., Zhang Y., Guan X.-W., Chen S.: Ground-state properties of interacting two-component Bose gases in a hard-wall trap. Phys. Rev. A 79, 033607 (2009)

    Article  ADS  Google Scholar 

  34. Sutherland B.: Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems. World Scientific, Singapore (2004)

    Book  MATH  Google Scholar 

  35. Girardeau M.D., Minguzzi A.: Soluable models of strongly interacting ultracold gas mixtures in tight waveguides. Phys. Rev. Lett. 99, 230402 (2007)

    Article  ADS  Google Scholar 

  36. Deuretzbacher F., Fredenhagen K., Becker D., Bongs K., Sengstock K., Pfannkuche D.: Exact solution of strongly interacting quasi-one dimensional spinor Bose gases. Phys. Rev. Lett. 100, 160408 (2008)

    Article  Google Scholar 

  37. Yang C.N.: Ground state of fermions in a 1D trap with \({\delta}\) function interaction. Chin. Phys. Lett. 26, 120504 (2009)

    Article  ADS  Google Scholar 

  38. Girardeau M.D., Astrakharchik G.E.: Wave functions of the super-Tonks–Girardeau gas and the trapped one-dimensional hard-sphere Bose gas. Phys. Rev. A 81, 061601 (2010)

    Article  ADS  Google Scholar 

  39. Girardeau M.D.: Two super-Tonks–Girardeau states of a trapped one-dimensional spinor Fermi gas. Phys. Rev. A 82, 011607 (2010)

    Article  ADS  Google Scholar 

  40. Girardeau M.D.: Tonks–Girardeau and super-Tonks–Girardeau states of a trapped one-dimensional spinor Bose gas. Phys. Rev. A 83, 011601(R) (2011)

    Article  ADS  Google Scholar 

  41. Deuretzbacher F., Becker D., Bjerlin J., Reimann S.M., Santos L.: Quantum magnetism without lattices in strongly interacting one-dimensional spinor gases. Phys. Rev. A 90, 013611 (2014)

    Article  ADS  Google Scholar 

  42. Volosniev A.G., Fedorov D.V., Jensen A.S., Valiente M., Zinner N.T.: Strongly interacting confined quantum systems in one dimension. Nat. Commun. 5, 5300 (2014)

    Article  ADS  Google Scholar 

  43. Levinsen J., Massignan P., Bruun G.M., Parish M.M.: Strong-coupling ansatz for the one-dimensional Fermi gas in a harmonic potential. Sci. Adv. 1, e1500197 (2015)

    Article  ADS  Google Scholar 

  44. Gharashi S.E., Yin X.Y., Yan Y., Blume D.: One-dimensional Fermi gas with a single impurity in a harmonic trap: perturbative description of the upper branch. Phys. Rev. A 91, 013620 (2015)

    Article  ADS  Google Scholar 

  45. Yang L., Guan L., Pu H.: Strongly interacting quantum gases in one-dimensional traps. Phys. Rev. A 91, 043634 (2015)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, American University, 4400 Massachusetts Ave. NW, Washington, DC, 20016-8058, USA

    N. L. Harshman

Authors
  1. N. L. Harshman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. L. Harshman.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Harshman, N.L. One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries. II. N Particles. Few-Body Syst 57, 45–69 (2016). https://doi.org/10.1007/s00601-015-1025-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00601-015-1025-5

Keywords

Search

Navigation