Truncation effects in the charge representation of the O(2) model

Abstract

The O(2) model in Euclidean space-time is the zero-gauge-coupling limit of the compact scalar quantum electrodynamics. We obtain a dual representation of it called the charge representation. We study the quantum phase transition in the charge representation with a truncation to ``spin $S$,'' where the quantum numbers have an absolute value less than or equal to $S$. The charge representation preserves the gapless-to-gapped phase transition even for the smallest spin truncation $S=1$. The phase transition for $S=1$ is an infinite-order Gaussian transition with the same critical exponents $\ensuremath{\delta}$ and $\ensuremath{\eta}$ as the Berezinskii-Kosterlitz-Thouless (BKT) transition, while there are true BKT transitions for $S\ensuremath{\ge}2$. The essential singularity in the correlation length for $S=1$ is different from that for $S\ensuremath{\ge}2$. The exponential convergence of the phase-transition point is studied in both Lagrangian and Hamiltonian formulations. We discuss the effects of replacing the truncated ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}}^{\ifmmode\pm\else\textpm\fi{}}=exp(\ifmmode\pm\else\textpm\fi{}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\theta}})$ operators by the spin ladder operators ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{\ifmmode\pm\else\textpm\fi{}}$ in the Hamiltonian. The marginal operators vanish at the Gaussian transition point for $S=1$, which allows us to extract the $\ensuremath{\eta}$ exponent with high accuracy.