Solid State Communications, Vol. 18, PP. 1373—1375, 1976. Pergamon Press. Printed in Great Britain DIELECTRIC RESPONSE OF THE TWO-DIMENSIONAL COULOMB GAS J. Zittartz Department of Applied Physics, Stanford University,* Stanford, CA 94305, U.S.A. and B.A. Huberman Xerox Palo Alto Research Center, Palo Alto, CA 94304, U.S.A. (Received 28 November 1975 by B. Muhlschlegel) We calculate the temperature and wavevector dependent dielectric func- tion for the two-dimensional Coulomb gas in the low density limit. It is shown that this quantity describes a smooth change of the system from metallic to insulating behavior in a nice and simple way as the tempera- ture decreases. RECENTLY there has been some interest in the thermo- 2N /a2 + dynamic and electric properties of the two-component H2N = — e 1e, in (1) classical Coulomb gas in two dimensions. In particular a Hauge and Hemmer’ and Kosterlitz and Thouless 2 with e~= ±el and the charge neutrality condition emphasize that this system behaves like a neutral gas of ~ ~ e, = 0. The potential in equation (1) shows the typi- bound pairs of charges of opposite sign at low tempera- i = 1 tures and shows the usual properties of a free particle cal in-behavior for large separations r~= Ir, — rjI. Since a plasma at high temperatures. Both papers claim that the system with a pure in r potential (point charges) would change occurs via a phase transition, although they ob- not represent a thermodynamic system at low tempera- tam different values of the transition temperature in tures1 we use a potential with soft core and a character- their approximate treatments. To the contrary, it will be istic small length a.4 The thermodynamic properties3 are shown in a separate publication3 that no phase transition derived from the grand partition function occurs in the two-dimensional Coulomb gas in the sense ~ 2N /d2r ‘~ ~ ( IJ~_~3H~j~ (2) that the free energy is an analytic function for all finite Z = 1 + real values of the thermodynamic variables, temperature Nl(~)2S ~ \a~J T and fugacity z, or chemical potential respectively: where z = e~is the fugacity and .z the chemical poten- z = e~.This means that the system exhibits a smooth tial. The one component particle density follows from transition from an “insulating” state at low temperatures to a “metallic” state at high temperatures. <N) 1 -~- ln Z(z, 7). (3) In this note we calculate the wave vector and tem- P = = 2VZ az perature dependent dielectric function e(q, 1) in the low density limit. As we shall show, this quantity displays Within the linear response formulation the dielectric the change from metallic to insulating behavior in a nice function can be written as 4ir$3 and simple way which to our knowledge has not been e(q, 7) = 1 + —i- F(q, 7) (4) discussed so far in the literature of metal—non-metal q transitions. where the charge density correlation function is the We consider a gas of N positive and N negative grand canonical average charges interacting via the solution of Poisson’s equation in two dimensions, i.e., a Hamiltonian of the form 1 2N F(q, I) = ~ exp ~— iq(r1 — rj)}’>. (5) To calculate F in the low density regime (z -~ 0) we * Permanent address; Institut für Theoretische Physik, have to distinguish two temperature regions: (1) A <2, Universitat KOln, 5 Köln, Germany; work at Stanford (2) A > 2 with A = j3e 2. In the former case the function supported in part by Army Research Office, Durham, e~in rU = (rU)~is integrable for small distances. NC., U.S.A.