The truncated U(1) Abelian Higgs model and implication for its quantum simulation

A concrete experimental setup for quantum simulating the (1+1)d Abelian Higgs model has been proposed in J. Zhang et al. (2018), where the Hamiltonian in the plaquette quantum number (quantized electric field) representation can be implemented on a multi-leg ladder with a single atom each rung. The finite size scaling of the energy gap can be measured and its universal behavior extracted at large enough spin truncation. However the O(2) limit of this representation has no real BKT transition for any finite spin truncation, and properties of matter fields cannot be measured directly. We study the Hamiltonian in the link quantum number (charge) representation in detail with density matrix renormalization group. We find that the BKT transition in the O(2) limit is present in the smallest spin truncation (spin-1). The gap scaling to locate the BKT transition point and parameters b, C are studied for different spin truncations. We find universal functions relating the mass gap, the gauge coupling, and the spatial size in the smallest spin truncation, which does not exist in the field representation. We study the effect of g ∼ 1/N_s perturbing the BKT phase. The results for SU(2) spin operators are also compared. Possible experimental realization is also discussed.