Melting and the vector Coulomb gas in two dimensions

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 $K_R^i\mathrm{}\left(T\right)-K_R^i\left(T_c^{-}\right)\sim t^{\overline{\nu }}$, 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, $\theta$ part and $t$ is the reduced temperature, $t=\frac{(T_c-T)}{T_c}$. The exponent $\overline{\nu }$ varies as the ratio $\frac{K_R^{\theta }\left(T_c^{-}\right)}{K_R^r\left(T_c^{-}\right)}$ is changed and is equal to 0.3696... for the physical value of $K^{\theta }=K^r$. 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 $K_R^i=0\mathrm{}$ for $T>\mathrm{}{T}_c$, corresponding to metallic behavior of the vector Coulomb gas.