Melting and the vector Coulomb gas in two dimensions

TLDR

The dislocation theory of two‑dimensional melting on a triangular lattice, as developed by Kosterlitz and Thouless, is examined with particular focus on equal‑magnitude angular and radial forces between dislocation pairs. By extending the dislocation Hamiltonian to a vector Coulomb gas with distinct radial and angular interactions, the authors derive a renormalized coupling that scales as a power of the reduced temperature, with separate radial and angular components. The study finds that the critical exponent depends on the ratio of angular to radial couplings, taking the value 0.3696 when they are equal, and that both shear and bulk elastic constants follow the same temperature dependence, with the radial coupling remaining finite at the transition and vanishing above it, signaling metallic behavior of the vector Coulomb gas.

Abstract

The dislocation theory of two-dimensional melting due to Kosterlitz and Thouless is investigated for the triangular lattice, paying special attention to angular forces between dislocation pairs, which are equal in magnitude to the radial forces. Generalizing the dislocation Hamiltonian to an arbitrary vector Coulomb gas with different radial and angular interactions we find ${K}_{R}^{i}(T)\ensuremath{-}{K}_{R}^{i}({T}_{c}^{\ensuremath{-}})\ensuremath{\sim}{t}^{\overline{\ensuremath{\nu}}}$, where ${K}_{R}^{i}$ is a renormalized coupling which includes the screening effect of bound dislocation pairs, the superscript $i$ signifies either a radial, $r$, or angular, $\ensuremath{\theta}$ part and $t$ is the reduced temperature, $t=\frac{({T}_{c}\ensuremath{-}T)}{{T}_{c}}$. The exponent $\overline{\ensuremath{\nu}}$ varies as the ratio $\frac{{K}_{R}^{\ensuremath{\theta}}({T}_{c}^{\ensuremath{-}})}{{K}_{R}^{r}({T}_{c}^{\ensuremath{-}})}$ is changed and is equal to 0.3696... for the physical value of ${K}^{\ensuremath{\theta}}={K}^{r}$. In this case the shear and bulk elastic constants have the same temperature dependence as the ${K}_{R}^{i}$. We find that ${K}_{R}^{r}$ has finite universal value at ${T}_{c}$ and ${K}_{R}^{i}=0$ for $T>{T}_{c}$, corresponding to metallic behavior of the vector Coulomb gas.