Confinement in the Three-dimensional Anisotropic $t$-$J$ Model

The normal-state resistivity of the cuprate superconductors is metallic in the $ab$ plane but is characteristic of an insulator along the $c$ direction, leading to the possibility of the remarkable phenomenon of confinement. By continuity, such a behavior is not expected to occur in a Fermi liquid. We consider the anisotropic $t$-$J$ model with the c-axis parameters $t_c$ and $J_c$ different from their in-plane counterparts, $t$ and $J$. Within the slave-fermion mean-field approximation it is shown that the spiral state exhibits charge-confinement in the intermediate $\delta$ regime for a range of values of $t_c/t$. In the confined state the hopping amplitude $<c^{\dag}_{i\sigma}c_{j\sigma}> = 0$ along the c direction so that c-axis resitivity is infinite at $T = 0$.