Abstract
The particle and charge correlations of the two-dimensional Coulomb gas are studied in the dielectric phase. A term-by-term analysis of the low-fugacity expansions suggests that the large-distance behaviors of the particle correlations are governed by multipolar interactions, similar to what happens in a system of permanent dipoles. These behaviors are compatible with the asymptotic structure of the BGY hierarchy equations; on the other hand, a new identity for the dielectric constant ɛ is used to show that the four-particle correlations decay as the dipole-dipole potential 1/r2 when two neutral pairs are separated by a large distancer. Near the zero-density critical point of the Kosterlitz-Thouless transition, we resum the low-fugacity expansions of both 1/ɛ and the charge correlation C(r). We thus retrieve the coupling constant flow equations of the renormalization group as well as the effective interaction energy of the iterated mean-field theory by Kosterlitz and Thouless. The coupling constant at the RG fixed point is then identified with 1/ɛ. The nonanalyticity of 1/ɛ at the transition turns out to coincide with the divergence of the low-fugacity series for this quantity. The leading term in the large-distance behavior of C(r) is found to be the same as for external charges. Moreover, we exhibit the subleading terms which also contribute to 1/ɛ.
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References
E. M. Hauge and P. C. Hemmer,Phys. Norv. 5:209 (1971).
W. Yang,J. Stat. Phys. 49:1 (1987).
D. Brydges and P. Federbush,Commun. Math. Phys. 73:197 (1980).
J. Imbrie,Commun. Math. Phys. 87:515 (1983).
M. Gaudin,J. Phys. (Paris) 46:1027 (1985).
F. Cornu and B. Jancovici,J. Stat. Phys. 49:33 (1987).
J. M. Kosterlitz and D. J. Thouless,J. Phys. C 6:1181 (1973).
J. Frohlich and T. Spencer,Commun. Math. Phys. 81:527 (1981); D. H. U. Marchetti, A. Klein, and J. F. Perez,J. Stat. Phys. 60:137 (1990).
N. D. Mermin and H. Wagner,Phys. Rev. Lett. 17:1133 (1966); S. Coleman,Commun. Math. Phys. 31:259 (1973).
B. Nienhuis, inPhase Transitions, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1987).
J. L. Cardy, inFields, Strings and Critical Phenomena (1988 Les Houches Lectures), E. Brézin and J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1990).
D. J. Amit, Y. Y. Goldschmidt, and G. Grinstein,J. Phys. A 13:585 (1980).
D. Chandler,J. Chem. Phys. 67:1113 (1977).
J. M. Kosterlitz,J. Phys. C 7:1046 (1974).
P. Minnhagen,Rev. Mod. Phys. 59:1001 (1987).
J. Fröhlich and Y. M. Park,Commun. Math. Phys. 59:235 (1978).
F. H. Stillinger and R. Lovett,J. Chem. Phys. 48:3858 (1968);49:1991 (1968).
Ph. A. Martin and Ch. Gruber,J. Stat. Phys. 31:691 (1983).
E. R. Speer,J. Stat. Phys. 42:895 (1986).
J. V. Jose, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson,Phys. Rev. B 16:1217 (1977).
J. S. Høye and G. Stell,J. Chem. Phys. 61:562 (1974).
K. Gawedski and A. Kupiainen,Ann. Phys. 147:198 (1983).
G. Benfatto, G. Gallavotti, and F. Nicolo,J. Stat. Phys. 55:739 (1988).
J. Dimock and T. R. Hurd,Commun. Math. Phys. 137:263 (1991).
G. Stell, inModem Theoretical Chemistry, Vol. 5, B. J. Berne, ed. (Plenum, New York, 1977).
J. M. Caillol and D. Levesque,Phys. Rev. B 33:499 (1986).
J. R. Fontaine and Ph. A. Martin,J. Stat. Phys. 36:163 (1984).
Ph. A. Martin,Rev. Mod. Phys. 60:1075 (1988).
Ch. Gruber, Ch. Lugrin, and Ph. A. Martin,J. Stat. Phys. 22:193 (1980); L. Blum, Ch. Gruber, J. L. Lebowitz, and Ph. A. Martin,Phys. Rev. Lett. 48:1769 (1982).
A. Alastuey and Ph. A. Martin,J. Stat. Phys. 39:405 (1985).
A. Alastuey and Ph. A. Martin,Phys. Rev. A 40:6485 (1989).
E. Meeron,J. Chem. Phys. 28:630 (1958); E. E. Salpeter,Ann. Phys. 5:183 (1958); R. Abe,Progr. Theor. Phys. 22:213 (1959).
A. P. Young and T. Bohr,J. Phys. C 14:2713 (1981).
Ph. Choquard, B. Piller, and R. Rentsch,J. Stat. Phys. 43:197 (1985);46:599 (1987); Ph. Choquard, B. Piller, R. Rentsch, and P. Vieillefosse,J. Stat. Phys. 55:1185 (1989).
J. Clerouin, J. P. Hansen, and B. Piller,Phys. Rev. A 36:2793 (1987).
A. Alastuey, F. Cornu, and B. Jancovici,Phys. Rev. A 38:4916 (1988).
Ph. Choquard, B. Piller, R. Rentsch, J. Clerouin, and J. P. Hansen,Phys. Rev. A 40:931 (1989).
A. Alastuey,J. Stat. Phys. 48:839 (1987); A. Alastuey, J. Clerouin, and J. P. Hansen,J. Stat. Phys. 52:395 (1988).
P. J. Forrester,J. Stat. Phys. 42:871 (1986);51:457 (1988).
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Alastuey, A., Cornu, F. Correlations in the Kosterlitz-Thouless phase of the two-dimensional Coulomb gas. J Stat Phys 66, 165–231 (1992). https://doi.org/10.1007/BF01060065
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DOI: https://doi.org/10.1007/BF01060065