spontaneous symmetry breaking within critical phases

There are many examples of the spontaneous symmetry breaking in discrete symmetries which result in the gapped phases, such as dimerization phase of the quantum antiferromagnetic spin-1/2 chain which breaks the single-lattice translation symmetry, or in continuous symmetries, such as ferromagnetism of ferromagnetic spin chains whose ground state breaks $SU(2)$ spin-rotation symmetry. However, neither of these two cases is critical.

My question is, assuming we have imposed a general global symmetry $G$ (discrete or continuous) for the Hamiltonian, whether there exist spontaneous $G$-symmetry breaking quantum critical phases? Does the answer depend on the symmetry or space-time dimensions?

• What is your definition of critical? E.g. is the ground state of the AFM Heisenberg model in 2D critical according to you? (Note that its ground state spontaneously breaks $SU(2)$ and has linearly dispersing gapless modes.) – Ruben Verresen Apr 19 '18 at 16:07
• @RubenVerresen I use the conventional definition of criticality that the correlation length diverges. If I remember correctly, SU(2) is restored when the system is fine-tuned to criticality from the Neél phase(?). – Smart Yao Apr 19 '18 at 23:39
• But the SU(2) antiferromagnet is gapless and has algebraic correlations, i.e. an infinite correlation length... – Ruben Verresen Apr 20 '18 at 0:02
• @RubenVerresen Thanks for the reminder! Then the 2d Néel phase can be a typical example although the nontrivial existence of 1d case seems not obvious. – Smart Yao Apr 20 '18 at 0:04