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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?

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  • $\begingroup$ 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.) $\endgroup$ – Ruben Verresen Apr 19 '18 at 16:07
  • $\begingroup$ @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(?). $\endgroup$ – Smart Yao Apr 19 '18 at 23:39
  • $\begingroup$ But the SU(2) antiferromagnet is gapless and has algebraic correlations, i.e. an infinite correlation length... $\endgroup$ – Ruben Verresen Apr 20 '18 at 0:02
  • $\begingroup$ @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. $\endgroup$ – Smart Yao Apr 20 '18 at 0:04

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