Confined states of a spinon gas in a mixed-dimensional XXZ model*

Here we study a two-dimensional system where we consider the usual XXZ interactions in the x-direction and a simple Ising interaction in the y-direction, which we refer to as a mixed-dimensional XXZ model. This is in analog to the mixed-dimensional t-J models where hopping is restricted to one dimension. By using DMRG we study phase diagrams at different magnetizations of the system and uncover an extended region of an AFM ordered state at the zero magnetization, and a spin-stripe order at finite magnetization. By considering spin-spin correlations in the ground state and at finite temperature, we uncover melting of the stripes into a meson gas at low but finite temperature. Our research is motivated by the recent advancement of quantum simulation with cold atoms that carry magnetic dipoles, where such systems could be simulated. Dipole-dipole interactions in such systems could be used to engineer two-dimensional spin systems with tunable interactions in different spatial directions. In addition, we find a mapping of our spin system to a one-dimensional lattice gauge theory (LGT) with superconducting pairing terms, by considering the spin domain walls as hard-core bosons. We show that the confining term in the LGT basis emerges from the short-ranged spin-spin correlations on a mean-field level, and thus explain the confining signatures of the meson gas. Interpretation of the mixed-dimensional XXZ model in terms of LTGs thus provides simple understanding of the rich phase diagrams of these systems. In addition, we sample snapshots from our numerical calculations and study which experimental observables are best suited to probe confinement in a cold atom setup.