Truncation effects in dual representations of the O(2) model

The classical O(2) model on a Euclidean space time can be obtained as the zero gauge coupling limit of scalar QED. This is a dual representation where the plaquette (field) quantum numbers determine the charge quantum numbers on the links according to Gauss's law. Alternatively, we can use the original charge representation of the O(2) model. Taking the time continuum limit, we study the spectra of the Hamiltonians in the two representations with a truncation to "spin S", where the quantum numbers have an absolute value less or equal to S. In the infinite S limit the spectra are identical however for quantum simulations, truncations are needed. The field representation is always gapped with finite spin truncations, while the charge representation preserves the BKT transition even for the smallest spin truncation S = 1. In the charge representation, the Hamiltonian with truncated exp( ±iθ) operators has the BKT critical point that converges exponentially with S, while that with SU(2) S^± has only algebraic convergence like 1/S² for the BKT point. The field representation can recover the gap scaling in the charge representation by gapping out the states that have a charge bigger than S.