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

Abstract

We study the zero-temperature phase diagrams of Majorana-Hubbard models with SO(𝑁) symmetry on twodimensional honeycomb and 𝜋-flux square lattices, using mean-field and renormalization group approaches. The models can be understood as real counterparts of the SU(𝑁) Hubbard-Heisenberg models, and may be realized in Abrikosov vortex phases of topological superconductors, or in fractionalized phases of strongly-frustrated spin-orbital magnets. In the weakly-interacting limit, the models feature stable and fully symmetric Majorana semimetal phases. Increasing the interaction strength beyond a finite threshold for large 𝑁, we find a direct transition towards dimerized phases, which can be understood as staggered valence bond solid orders, in which part of the lattice symmetry is spontaneously broken and the Majorana fermions acquire a mass gap. For small to intermediate 𝑁, on the other hand, phases with spontaneously broken SO(𝑁) symmetry, which can be understood as generalized Néel antiferromagnets, may be stabilized. These antiferromagnetic phases feature fully gapped fermion spectra for even 𝑁, but gapless Majorana modes for odd 𝑁. While the transitions between Majorana semimetal and dimerized phases are strongly first order, the transitions between Majorana semimetal and antiferromagnetic phases are continuous for small 𝑁 ≤ 3 and weakly first order for intermediate 𝑁 ≥ 4. The weakly-first-order nature of the latter transitions arises from fixed-point annihilation in the corresponding effective field theory, which contains a real symmetric tensorial order parameter coupled to the gapless Majorana degrees of freedom, realizing interesting examples of fluctuation-induced first-order transitions.