  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 $\delta$
and $\eta$ as the Berezinskii-Kosterlitz-Thouless (BKT) transition, while there
are true BKT transitions for $S \ge 2$. The essential singularity in the
correlation length for $S = 1$ is different from that for $S \ge 2$. The
exponential convergence of the phase-transition point is studied in both
Lagrangian and Hamiltonian formulations. We discuss the effects of replacing
the truncated $\hat{U}^{\pm} = \exp(\pm i \hat{\theta})$ operators by the spin
ladder operators $\hat{S}^{\pm}$ in the Hamiltonian. The marginal operators
vanish at the Gaussian transition point for $S = 1$, which allows us to extract
the $\eta$ exponent with high accuracy.
