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Spin crossover (SCO) materials are reversible molecular switches found in awide range of transition metal complexes and metal organic frameworks (MOFs).They exhibit diverse spin state orderings and transitions between them. Wepresent an exact mapping from an elastic lattice mismatch model to a long-rangeIsing model, with an inverse square decay of the interaction strengths at largedistances (on the square lattice). This provides a microscopic justificationfor an Ising model description, which has previously only been justified onphenomenological grounds. Elastic frustration is required for non-zero Isinginteractions, but whether or not the short-range interactions in the Isingmodel are geometrically frustrated depends on the ratio of the bulk and shearmoduli or equivalently Poisson's ratio. We show that, for a simple squarelattice model with realistic parameters, sweeping the enthalpy differencebetween the two spin-states at zero temperature leads to a large (probablyinfinite) number of spin-state orderings and corresponding steps in thefraction of high-spin ions, consistent with a Devil's staircase. The staircasecan also be climbed by varying the temperature, but then some of the steps arehidden and only a finite number remain, consistent with experiments on relevantframework materials, such as {(Fe[Hg(SCN)$_3$]$_2$(4,4'-bipy)$_2$)}$_n$. Ourresults are also relevant to other binary systems with lattice mismatch, e.g.,heterogeneous solids.

Hidden Devil's staircase in a two-dimensional elastic model of spin crossover materials