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Metal oxides such as VO$_2$ undergo structural transitions to low-symmetryphases characterized by intricate crystalline order, accompanied by richelectronic behavior. We derive a minimal ionic Hamiltonian based on symmetryand local energetics which describes structural transitions involving all fourobserved phases, in the correct order. An exact analysis shows that complexityresults from the symmetry-induced constraints of the parent phase which forcesionic displacements to form multiple interpenetrating groups usinglow-dimensional pathways and distant neighbors. Displacements within each groupexhibit independent, quasi two-dimensional order, which is frustrated andfragile. This selective ordering mechanism is not restricted to VO$_2$: itapplies to other oxides which show similar complex order.

Complex Quasi-Two-Dimensional Crystalline Order Embedded in VO$_2$ and Other Crystals