Abstract
A dilute three-dimensional Fermi liquid is considered with an instantaneous attractive short-range interaction. Two sets of self-consistent equations for the temperature dependent fermion Greens functiong and the four-point vertex function Γ are derived by field theoretic means. The interaction is taken into account using the results of low energys-wave scattering theory. The crossover region between BCS superconductivity and Bose-Einstein condensation of tightly bound pairs is located near the threshold where in the two-particle scattering problem a virtual or quasi-stationary state turns into a bound state. We show how from our self-consistent equations the BCS theory and the theory of a superfluid Bose gas can be recovered in the weak and strong coupling limit, respectively. In the strong coupling limit we find that there is a repulsive interaction between the composite bosons due to the Pauli exclusion principle. It is described by a positive scattering lengtha B which is twice the scattering length of the fermions,a B =2a F . Furthermore we find that this interaction reduces the Bose-Einstein transition temperature to leading order by ΔT c /T c, BE =−(k F a F )3/3π. We show that most of the previous theories of the crossover scenario can be obtained from our theory in mean-field approximation neglecting self consistency.
Similar content being viewed by others
References
Leggett, A.J.: In: Pekalski, A., Przystawa, J. (eds.) Modern trends in the theory of condensed matter, p. 13 Berlin, Heidelberg, New York: Springer 1980
Randeria, M., Duan, J.-M., Shieh, L.-Y.: Phys. Rev. Lett.62, 981 (1989); Phys. Rev.41, 327 (1990)
Nozières, P., Schmitt-Rink, S.: J. Low Temp. Phys.59, 195 (1984)
Thouless, D.J.: Ann. Phys. (N.Y.)10, 553 (1960)
Drechsler, M., Zwerger, W.: Ann. Phys.1, 15 (1992)
Drechsler, M.: Diploma thesis, Universität Göttingen 1991 (unpublished)
Tokumitu, A., Miyake, K., Yamada, K.: J. Phys. Soc. Jpn.60, 380 (1991); Prog. Theor. Phys. Suppl.106 63 (1991)
Moriya, T.: Spin fluctuations in itinerant electron magnetism, Chap. 4. Berlin, Heidelberg, New York: Springer 1985
Scalettar, R.T., Loh, E.Y., Gubernatis, J.E., Moreo, A., White, S.R., Scalapino, D.J., Sugar, R.L., Dagotto, E.: Phys. Rev. Lett.62, 1407, (1989)
Sofo, J.O., Balseiro, C.A., Castillo, H.E.: Phys. Rev. B45, 9860 (1992)
Denteneer, P.J.H., Guozong, A., Leeuwen, J.M.J., van: Europhys. Lett.16, 5 (1991)
Frésard, R., Glaser, B., Wölfle, P.: (Preprint 1992)
Moulopoulos, K., Ashcroft, N.W.: Phys. Rev. Lett.66, 2915 (1991)
Galitskii, V.M.: Sov. Phys. JETP34, 104 (1958)
Matsubara, T.: Prog. Theor. Phys.14, 351 (1955)
Abrikosov, A.A., Gorkov, L.P., Dzyaloshinskii, I.E.: Methods of quantum field theory in statistical physics. New York: Dover 1963
Fetter, A.L., Walecka, J.D.: Quantum theory of many-particle systems. New York: McGraw Hill 1971
Nambu, Y.: Phys. Rev.117, 648 (1960)
Landau, L.D., Lifshitz, E.M.: Quantum mechanics, chapter XIV, London: Pergamon Press 1958
Gorkov, L.P., Melik-Barkhudarov, T.K.: Sov. Phys. JETP10, 1018 (1961)
Adawi, I.: J. Math. Phys.12, 358 (1971)
Handbook of mathematical functions. Abramowitz, M., Stegun, I.A. (eds.) New York: Dover 1965
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Haussmann, R. Crossover from BCS superconductivity to Bose-Einstein condensation: A self-consistent theory. Z. Physik B - Condensed Matter 91, 291–308 (1993). https://doi.org/10.1007/BF01344058
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01344058