In all the examples of SSB in our course/the books (even in our SUSY course) we have just looked at minima of the scalar potential. Why do we restrict ourselves to the scalars, why not also minimise the fermions and vector bosons?
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Qmechanic
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Toby Peterken
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Only scalars can have vevs due to Lorentz invariance. In the vacuum, all fermions and gauge bosons must vanish. – Prahar Mar 18 '20 at 12:47
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Related: Why a field theory containing only fermions does not show spontaneous symmetry breaking? , Why is a nonzero VEV for a spinor field said to break Lorentz invariance? – SRS Apr 28 '20 at 15:12