Spin Ice Thin Film: Surface Ordering, Partial Magnetic Wetting and Emergent Square Ice

Motivated by recent realizations of Dy$_{2}$Ti$_{2}$O$_{7}$ and Ho$_{2}$Ti$_{2}$O$_{7}$ spin ice thin films, and more generally by the physics of confined gauge fields, we study a model of spin ice thin film with surfaces perpendicular to the $[001]$ cubic axis. The resulting open boundaries make half of the bonds on the interfaces inequivalent. By tuning the strength of these inequivalent "orphan" bonds, dipolar interactions induce a surface ordering equivalent to a two-dimensional crystallization of magnetic surface charges. This surface ordering can also be expected on the surfaces of bulk crystals. In analogy with partial wetting in soft matter, spins just below the surface are more correlated than in the bulk, but \emph{not} ordered. For ultrathin films made of one cubic unit cell, once the surfaces are ordered, a square ice phase is stabilized over a finite temperature window, as confirmed by its entropy and the presence of pinch points in the structure factor. Ultimately, the square ice degeneracy is lifted at lower temperature and the system orders in analogy with the well-known $F$-transition of the $6$-vertex model.