Discrete Abelian lattice gauge theories on a ladder and their dualities with quantum clock models

We study a duality transformation from the gauge-invariant subspace of a $Z_N$ lattice gauge theory on a two-leg ladder geometry to an $N$-clock model on a single chain. The main feature of this mapping is the emergence of a longitudinal field in the clock model, whose value depends on the superselection sector of the gauge model, implying that the different sectors of the gauge theory can show quite different phase diagrams. To investigate this and see if confined phases might emerge, we perform a numerical analysis for $N=2,3,4$, using both exact diagonalization and the density matrix renormalization group algorithm.