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We study the zero-temperature phase diagrams of Majorana-Hubbard models withSO($N$) symmetry on two-dimensional honeycomb and $\pi$-flux square lattices,using mean-field and renormalization group approaches. The models can beunderstood as real counterparts of the SU($N$) Hubbard-Heisenberg models, andmay be realized in Abrikosov vortex phases of topological superconductors, orin fractionalized phases of strongly-frustrated spin-orbital magnets. In theweakly-interacting limit, the models feature stable and fully symmetricMajorana semimetal phases. Increasing the interaction strength beyond a finitethreshold for large $N$, we find a direct transition towards dimerized phases,which can be understood as staggered valence bond solid orders, in which partof the lattice symmetry is spontaneously broken and the Majorana fermionsacquire a mass gap. For small to intermediate $N$, on the other hand, phaseswith spontaneously broken SO($N$) symmetry, which can be understood asgeneralized N\'eel antiferromagnets, may be stabilized. These antiferromagneticphases feature fully gapped fermion spectra for even $N$, but gapless Majoranamodes for odd $N$. While the transitions between Majorana semimetal anddimerized phases are strongly first order, the transitions between Majoranasemimetal and antiferromagnetic phases are continuous for small $N \leq 3$ andweakly first order for intermediate $N \geq 4$. The weakly-first-order natureof the latter transitions arises from fixed-point annihilation in thecorresponding effective field theory, which contains a real symmetric tensorialorder parameter coupled to the gapless Majorana degrees of freedom, realizinginteresting examples of fluctuation-induced first-order transitions.

Phase diagrams of SO($N$) Majorana-Hubbard models: Dimerization, internal symmetry breaking, and fluctuation-induced first-order transitions