Dobrushin states for classical spin systems with complex interactions

We consider a classical spin system on the hypercubic lattice with a general interaction of the form $$ H = \frac{\beta } {4}\sum\limits_{\begin{array}{*{20}c} {x,y:} \\ {|x - y| = 1} \\ \end{array} } {|s_x - s_y | - h} \sum\limits_x {x{}_x + } \sum\limits_A {\lambda _A \prod\limits_{y \in A} {S_y } } $$ are the spin variables, Β is the inverse temperature,h is the magnetic field, andλ _A are translation-invariant coupling constants satisfyingλ _A = 0 if diamA > l. No symmetry relating the configurationss ={sinx} and-s=-s _x is assumed. In dimension d-3, we construct low-temperature States which break the translation invariance of the system by introducing so-called Dobrushin boundary conditions which force a horizontal interface into the system. In contrast to previous constructions, our methods work equally well for complex interactions, and should therefore be generalizable to quantum spin systems.