  In this article we are interested in the microscopic modeling of a
two-dimensional two-well problem which arises from the square-to-rectangular
transformation in (two-dimensional) shape-memory materials. In this discrete
set-up, we focus on the surface energy scaling regime and further analyze the
Hamiltonian which was introduced in \cite{KLR14}. It turns out that this class
of Hamiltonians allows for a direct control of the discrete second order
gradients and for a one-sided comparison with a two-dimensonal spin system.
Using this and relying on the ideas of Conti and Schweizer \cite{CS06},
\cite{CS06a}, \cite{CS06c}, which were developed for a continuous analogue of
the model under consideration, we derive a (first order) continuum limit. This
shows the emergence of surface energy in the form of a sharp-interface limiting
model as well the explicit structure of the minimizers to the latter.
