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Physical review. B, Condensed matter

Results for the phase diagram of the two-dimensional Coulomb gas

Petter Minnhagen, Mats Wallin

Published Sep 1, 1989• 46 citations
Publisher

Abstract

A derivation of a self-consistent set of renormalization equations for the two-dimensional Coulomb gas is given within a systematic approximation scheme. Results for the phase-transition properties of the Coulomb gas are inferred from high-precision numerical integration of these equations. Connections to and extensions of earlier-known results are given. Critical exponents for a new critical point in the phase diagram are calculated from the equations and are found to correspond to those of a Ginzburg-Landau model in a Gaussian approximation. It is suggested that the true character of the new critical point is of Ising type.

DOI
10.1103/physrevb.40.5109

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A derivation of a self-consistent set of renormalization equations for the two-dimensional Coulomb gas is given within a systematic approximation scheme. Results for the phase-transition properties of the Coulomb gas are inferred from high-precision numerical integration of these equations. Connections to and extensions of earlier-known results are given. Critical exponents for a new critical point in the phase diagram are calculated from the equations and are found to correspond to those of a Ginzburg-Landau model in a Gaussian approximation. It is suggested that the true character of the new critical point is of Ising type.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.
A new set of renormalization equations for the two-dimensional Coulomb gas is derived and solved numerically. These equations give a new picture of the Kosterlitz-Thouless transition. A new temperature T∗{T}^{\mathrm{*}}T∗ is found. Above T∗{T}^{\mathrm{*}}T∗ the value of the dielectric constant at the transition is ${\ensuremath{\epsilon}}_{c}$=1/(4Tc{T}_{c}Tc​) as before whereas below T∗{T}^{\mathrm{*}}T∗ the new result ${\ensuremath{\epsilon}}_{c}1/(41/(41/(4{T}_{c}$) is obtained. The singular behavior at T∗{T}^{\mathrm{*}}T∗ is explored. The results are translated into a renormalization-group flow diagram. An interesting consequence is that the universal jump prediction for the Kosterlitz-Thouless transition turns into a nonuniversal jump prediction for critical temperatures below T∗{T}^{\mathrm{*}}T∗.
A new critical point is established through Monte Carlo simulations for a class of two dimensional XY models with a modified nearest-neighbor spin interaction. The critical exponents found are \ensuremath{\nu}\ensuremath{\approxeq}1.4, \ensuremath{\beta}\ensuremath{\approxeq}0.6, and an \ensuremath{\alpha}0, consistent with hyperscaling. The phase transition is interpreted as a vortex-unbinding transition. Comparisons are made with the phase diagram of the two dimensional Coulomb gas and with the spin-1/2 XY model.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.
The second-order phase transition in the two-dimensional (2D) classical Coulomb gas of half-integer charges on a square lattice is investigated by using Monte Carlo simulations. Based on the finite-size scaling analysis, we estimate the second-order phase transition temperature Tc and the static critical exponents β and ν with a new numerical analysis method. More precise critical temperature Tc = 0.1311(2) and critical exponents β/ν = 0.1152(12) and ν = 0.857(15) are obtained. The estimated value of ν indicates that the charge lattice melting transition is different from the pure 2D Ising transition.