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

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 and as the Berezinskii-Kosterlitz-Thouless (BKT) transition, while there are true BKT transitions for S2. The essential singularity in the correlation length for S=1 is different from that for S2. The exponential convergence of the phase-transition point is studied in both Lagrangian and Hamiltonian formulations. We discuss the effects of replacing the truncated { \^{}U}^=exp (\^{}) operators by the spin ladder operators { \^{}S}^ in the Hamiltonian. The marginal operators vanish at the Gaussian transition point for S=1, which allows us to extract the exponent with high accuracy.