For 2D SU(2) gauge theory with one massless Majorana fermion, only one of the two overlap zero-mode classes survives the continuum limit depending on boundary conditions, and a zero-mode-excluding 'topological condensate' is nonzero without spontaneous symmetry breaking.
Creutz ,\ 10.1103/PhysRevD.36.515 journal journal Phys
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Lattice study of two-dimensional SU(2) gauge theories with a single massless Majorana fermion
For 2D SU(2) gauge theory with one massless Majorana fermion, only one of the two overlap zero-mode classes survives the continuum limit depending on boundary conditions, and a zero-mode-excluding 'topological condensate' is nonzero without spontaneous symmetry breaking.