Skip to main content
SpringerLink
Log in

From Hardcore Bosons to Free Fermions with Painlevé V

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We calculate zero temperature Green’s function, the density–density correlations and expectation values of a one-dimensional quantum particle which interacts with a Fermi-sea via a δ-potential. The eigenfunctions of the Bethe-Ansatz solvable model can be expressed as a determinant. This allows us to obtain a compact expression for the Green’s function of the extra particle. In the hardcore limit the resulting expression can be analyzed further using Painlevé V transcendents. It is found that depending on the extra particles momentum its Green’s function undergoes a transition of that for hardcore Bosons to that of free Fermions.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. There is a typo in Eq. (1.29c) of Ref. [41]. The square bracket has to read \([\theta+\frac{A^{2}}{4}-n^{2}-1]\).

References

  1. Gianmarchi, T.: Quantum Physics in One Dimension. Oxford University Press, Oxford (2004)

    Google Scholar 

  2. Imambekov, A., Schmidt, T.L., Glazman, L.I.: One-dimensional quantum liquids: beyond the Luttinger liquid paradigm. arXiv:1110.1374 (2011)

  3. Haldane, F.D.M.: Luttinger liquid theory of one-dimensional quantum fluids. J. Phys. C 14, 2585 (1981)

    Article  ADS  Google Scholar 

  4. Barak, G., Steinberg, H., Pfeiffer, L.N., West, K.W., Glazman, L., von Oppen, F., Yacoby, A.: Interacting electrons in one dimension beyond the Luttinger-liquid limit. Nat. Phys. 6, 489 (2010)

    Article  Google Scholar 

  5. Kinoshita, T., Wenger, T., Weiss, D.S.: A quantum newton’s cradle. Nature Lett. 900, 1005 (2006)

    Google Scholar 

  6. Lieb, E., Liniger, W.: Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Takahashi, M.: Thermodynamics of One-Dimensional Solvable Models. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  8. Cazalilla, M.A.: Bosonizing one-dimensional cold atomic gases. J. Phys. B 37, 1 (2004)

    Article  ADS  Google Scholar 

  9. Yang, C.N.: Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 19, 1312 (1967)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Gaudin, M.: Un systeme a une dimension de fermions en interaction. Phys. Lett. A 24, 55 (1966)

    Article  ADS  Google Scholar 

  11. Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions, 1st edn. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  12. Cheianov, V.V., Zvonarev, M.B.: Zero temperature correlation functions for the impenetrable fermion gas. Phys. Rev. A 37, 2261 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Imabekov, A., Demler, E.: Exactly solvable case of a one-dimensional Bose–Fermi mixture. Phys. Rev. A 73, 021602 (2006)

    Article  ADS  Google Scholar 

  14. McGuire, J.B.: Interacting fermions in one dimension. I. Repulsive potential. J. Math. Phys. 6, 432 (1965)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. McGuire, J.B.: Interacting fermions in one dimension. II. Attractive potential. J. Math. Phys. 7, 123 (1966)

    Article  MathSciNet  ADS  Google Scholar 

  16. Lai, C.K., Yang, C.N.: Ground-state energy of a mixture of fermions and bosons in one dimension with a repulsive δ-function interaction. Phys. Rev. A 3, 393 (1971)

    Article  ADS  Google Scholar 

  17. Imabekov, A., Demler, E.: Ann. Phys. 321, 2390 (2006)

    Article  ADS  Google Scholar 

  18. Gubbels, K.B., Stoof, H.T.C.: Renormalization group theory for the imbalanced Fermi gas. Phys. Rev. Lett. 100, 140407 (2008)

    Article  ADS  Google Scholar 

  19. Ying, Z.-J., Cuoco, M., Noce, C., Zhou, H.-Q.: Exact solution for a trapped Fermi gas with population imbalance and BCS pairing. Phys. Rev. Lett. 100, 140406 (2008)

    Article  ADS  Google Scholar 

  20. Klaus, M.: Bose condensates and Fermi gases at zero temperature. Phys. Rev. Lett. 80, 1804 (1998)

    Article  Google Scholar 

  21. Truscott, A.G., Strecker, K.E., McAlexander, W.I., Partridge, G.B., Hulet, R.G.: Observation of Fermi pressure in a gas of trapped atoms. Science 291, 2570 (2001)

    Article  ADS  Google Scholar 

  22. Pilch, K., et al.: Observation of interspecies Feshbach resonances in an ultracold Rb-Cs mixture. Phys. Rev. A 79, 042718 (2009)

    Article  ADS  Google Scholar 

  23. Olsen, M.L., Perreault, J.D., Cumby, T.D., Jin, D.S.: Coherent atom-molecule oscillations in a Bose-Fermi mixture. Phys. Rev. A 80, 030701 (2009)

    Article  ADS  Google Scholar 

  24. Lenard, A.: Some remarks on large Toeplitz determinants. Pac. J. Math. 42, 137 (1972)

    MathSciNet  MATH  Google Scholar 

  25. Widom, H.: Toeplitz determinants with singular generating functions. Am. J. Math. 95, 333 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jimbo, M., Miwa, T., Mori, Y., Sato, M.: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica D 1, 80 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Forrester, P.J., Frankel, N.E., Garoni, T.M., Witte, N.S.: Finite one-dimensional impenetrable Bose systems: occupation numbers. Phys. Rev. A 67, 043607 (2003)

    Article  ADS  Google Scholar 

  28. Forrester, P.J., Frankel, N.E., Garoni, T.M., Witte, N.S.: Painlevé transcendent evaluations of finite systems density matrices for 1d impenetrable Bosons. Commun. Math. Phys. 238, 257–285 (2003)

    MathSciNet  ADS  MATH  Google Scholar 

  29. Fetter, A.L., Walecka, J.D.: Quantum Theory of Many Particle Systems. McGraw-Hill, New York (1971)

    Google Scholar 

  30. Lenard, A.: Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons. J. Math. Phys. 5, 930 (1964)

    Article  MathSciNet  ADS  Google Scholar 

  31. Vaidya, H.G., Tracy, C.A.: One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature. J. Math. Phys. 20, 2291 (1979)

    Article  ADS  Google Scholar 

  32. Vaidya, H.G., Tracy, C.A.: One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature. Phys. Rev. Lett. 42, 3 (1979)

    Article  ADS  Google Scholar 

  33. Forrester, P.J., Frankel, N.E., Garoni, T.M.: Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions. J. Math. Phys. 44, 4157 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Forrester, P.J., Frankel, N.E., Makin, I.M.: Analytic solutions of the one-dimensional finite-coupling delta-function Bose gas. Phys. Rev. A 74, 043614 (2006)

    Article  ADS  Google Scholar 

  35. Papenprock, T.: Ground-state properties of hard-core bosons in one-dimensional harmonic traps. Phys. Rev. A 67, 41601 (2003)

    Article  ADS  Google Scholar 

  36. Metha, M.L.: Random Matrices. Academic Press, New York (2004)

    Google Scholar 

  37. Forrester, P.J., Witte, N.S.: Application of the τ-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Commun. Math. Phys. 219, 357 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. Okamato, K.: Studies on the Painlevé equations I. Sixth Painlevé equation PVI. Ann. Mat. Pura Appl. 146, 337 (1986)

    Article  Google Scholar 

  39. Okamoto, K.: Studies on the Painlevé equations. II. Fifth Painlevé equation P V . Jpn. J. Math. (N.S.) 13(1), 7 (1987)

    Google Scholar 

  40. McCoy, B.M., Tang, S.: Connection formulae for Painlevé V functions. Physica D 19, 42–72 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. McCoy, B.M., Tang, S.: Connection formulae for Painlevé V functions II, the δ function Bose gas problem. Physica D 20, 187–216 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. McCoy, B.M., Tang, S.: Connection formulae for Painlevé functions. Physica D 18, 190–196 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Gangardt, D.M., Shlyapnikov, G.: Local correlations in a strongly interacting one-dimensional Bose gas. New J. Phys. 5, 79 (2003)

    Article  ADS  Google Scholar 

  44. Creamer, D.B., Thacker, H.B., Wilkinson, D.: Some exact results for the two point function of an integrable quantum field theory. Phys. Rev. D 23, 3081 (1981)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Acknowledgements

We acknowledge fruitful discussion with B. Gutkin, A.M. Lunde and V. Osipov. For helpful comments we thank V. Leiss. CH acknowledges financial support by Studienstifutung des deutschen Volkes. HK acknowledges financial support by the CSIC through the JAE program.

Author information

Authors and Affiliations

  1. Instituto de Ciencias Materiales de Madrid, CSIC, Sor Juana Inés de la Cruz 3, 28049, Madrid, Spain

    Christian Recher & Heiner Kohler

  2. Fakultät für Physik, Universität Duisburg-Essen, Lotharstrasse 1, 47048, Duisburg, Germany

    Christian Recher

Authors
  1. Christian Recher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Heiner Kohler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Recher.

Appendix: Representation as Töplitz Determinant

Appendix: Representation as Töplitz Determinant

For finite interaction strength the determinant in Eq. (40) representing the Green’s function is not a Töplitz determinant due to the non constant diagonal entries. However, in the limit c→∞ the Green’s function has a representation as Töplitz determinant. To reveal it we write the quantities g jl in Eq. (42) as

(91)

Here we use the notations

(92)

and furthermore assume that the quantum number n j are given by the set in Eq. (15). Now the diagonal entries are constant, i.e. independent of k j . For the linear combination of the quantities g nm as it appears in Eq. (41) this yields

(93)

where we have defined

(94)

The full Green’s function (see Eq. (41)) then reads

(95)

Employing the normalization condition G(0)=1/L this evaluates to

(96)

Correspondingly the interaction part G I(t) reads

(97)

For λ=0 Eq. (96) is equivalent to the representation of the Green’s function for hardcore Bosons as Töplitz-determinant [27, 30]. If on the other hand λ→∞ we obtain from Eq. (96)

(98)

Expanding the determinant in the equation above yields the following recursion

(99)

With help of the relation 2cos(x)sin(x)=sin(x−y)+sin(x+y) we see that its solution is given by

(100)

Consequently Eq. (98) acquires the form

(101)

Analogously one obtains for the interaction part

(102)

Equations (101) and (102) correspond to the free Fermion result.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Recher, C., Kohler, H. From Hardcore Bosons to Free Fermions with Painlevé V. J Stat Phys 147, 542–564 (2012). https://doi.org/10.1007/s10955-012-0482-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-012-0482-1

Keywords

Search

Navigation