Conformal properties of 1D quantum systems with long-range interactions

Abstract

We investigate conformal properties of a one-dimensional quantum system with a long-range, Coulomb-like potential of the form $\frac{1}{|x|^σ}$, with $σ>0$. We compute the conformal anomaly $c$ as function of $σ$. We obtain $c=0$ for $σ<1$ (Wigner crystal) and $c=1$ for $σ>1$ (Luttinger liquid). By studying the scaling regime of the system when it is deformed by the inclusion of a term $t ρ(x)$, (with $ρ(x)$ a scaling operator), we show that, for $σ>1$, the correlation length $ξ$ gets an additional interaction dependent factor. This leads to a necessary redefinition of $ξ$ in order to avoid an unphysical dependence of the central charge on the coupling constant.