In QFT, are there any restrictions on spontaneous breaking $G\to H$, due to "spontaneity"?

Question

For simplicity, let us restrict to the spontaneous breaking of global symmetries. Given any pair of groups $G\supset H$, is it always possible to find a $G$-invariant Lagrangian that gives a QFT such that the vacuum state is only $H$-invariant?

For example, at least perturbatively at tree level, $$\mathcal L=\frac12(\partial_\mu\phi^a)(\partial^\mu\phi^a)+\frac12m^2\phi^a\phi^a-\frac14\lambda\phi^a\phi^a\phi^b\phi^b,\quad a,b=1,\cdots,N$$ has a spontaneous $O(N)\to O(N-1)$ breaking. It has the familiar Mexican hat effective potential $V(\phi)$ (i.e., effective action for field configurations constant throughout spacetime). However, is it possible to modify the Lagrangian so that the vacuum only has an $O(N-2)$ symmetry? My guess is that we should introduce another $O(N)$-vector field and carefully setup the interactions, but after thinking for a couple of minutes I have got nothing... [If I later find a way to do this, I will leave a comment below.]

More generally, I wonder if there is a general strategy to construct a QFT with spontaneous $G\supset H$ breaking.

Any comment is welcome!

Answer 1

For half a century, students have been going (running!) to LF Li 1974 which also treats your problem as a warmup, a homework problem infiltrating lots of standard QFT texts like yours, apparently.

Two scalar fields, which are O(N) vectors, will give you the generic O(N)-invariant potential (trash the disorienting superfluous pseudo-mass terms, for crying out loud!), $$ \lambda (\vec\phi^2-v^2)^2 + \lambda' (\vec\chi^2-w^2)^2 +\lambda'' (\vec\phi \cdot \vec \chi-cvw)^2, $$ where c is the cosine of the v.e.v. vectors, of lengths v and w, respectively.

You can see the minimum of the potential at its vanishing vanishing for these lengths and vanishing cosine (so the kinetic term need not be rediagonalized...). Of course, $c=1$ aligns the two vectors and gives you nothing new; but any other c is in the domain of vanishing c, w.l.o.g. So, choose $\langle \phi^i\rangle =v\delta_{1i}$, $\langle \chi^i\rangle=w\delta_{2i}$, hence their cosine vanishes, w.l.o.g.

The vanishing remaining components 3,...,N span an N-2-dimensional space, invariant under the unbroken O(N-2).

The strategy you are asking for is evident in the reference. You contrive potentials invariant under G with a minimum invariant under H, which the reference illustrates systematically for the standard classical Lie algebras/groups. I know of no inaccessible unbroken subgroup H.