Pairing transition of a one-dimensional classical plasma

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Abstract
A classical one-dimensional gas of charges ei=+or-1 with interaction potential - Sigma eiejIn(1+ mod xi-xj mod ) is shown to undergo a transition from a metallic (or plasma) state at high temperature to an insulating (or dielectric) state at low temperature due to the formation of pairs of oppositely charged particles. This transition is the 1D analogue of the pairing transition of a 2D Coulomb plasma. The early method (Baxter 1964) of mapping real gas statistics in 1D to ordinary quantum mechanics is applied, and is demonstrated to work fairly well even when the latter involves an infinite number of degrees of freedom. By variational treatments of this quantum mechanics a first-order phase transition is obtained. At small fugacity z the transition line starts perpendicular to the inverse temperature axis at beta =2 and turns to the right for large z. An exact relation is derived between the dielectric constant and the effective mass of a Bloch band problem. On the insulator side the dielectric constant is argued to be identical to one. The equivalence of the model gas to a sine-Gordon type theory is also established and soliton solutions of the latter are given. By function space analysis the gas is again found to be metallic at sufficiently large fugacity but arbitrary temperature.