Emergent symmetries and Interactions: An isolated fixed point Vs a manifold of strongly interacting fixed points

In this article, we study conditions of continuous emergent symmetries in gapless states, either as topological quantum critical points (TQCPs) or a stable phase with protecting symmetries and connections to smooth deformations of the gapped states around. We illustrate that for a wide class of gapless states that can be associated with fully-isolated scale invariant fixed points, there shall always be emergent continuous symmetries that are directly related to smooth deformations of gapped states with symmetries lower than the protecting ones $G_p$. For a 3D TQCP in DIII classes with $G_p=Z^T_2$, $U_{EM}=U(1)$ and $N_f=\frac{1}{2}$ fermions but without charge $U(1)$ symmetry, we explicitly construct a corresponding boundary representation based on a $4D$ topological state with lattice symmetry $H=Z^T_2 \ltimes U(1)$ and $N_f={1}$ fermions. Although emergent continuous symmetries appear to be robust at weakly interacting TQCPs, we further show the breakdown of such one-to-one correspondence between deformations of gapped states and emergent continuous symmetries when gapless states become strongly interacting. In a strongly interacting limit, gapless states can be represented by a smooth manifold of conformal-field-theory fixed points rather than a fully isolated one. A smooth manifold of strong coupling fixed points hinders emergence of a continuous emergent symmetry in the strongly interacting gapless limit, as deformations no longer leave a gapless state or a TQCP invariant, unlike in the more conventional weakly interacting case. This typically reduces continuous emergent symmetries to a discrete symmetry originating from duality transformations under the protection symmetry $G_p$.