Emergent symmetries and interactions in high dimensions: An isolated fixed point versus a manifold of strongly interacting fixed points

In this article, we explore 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. For a wide class of gapless states that can be associated with fully isolated scale-invariant fixed points, we illustrate that there shall be emergent continuous symmetries that are directly related to smooth deformations of gapped states with symmetries lower than the protecting ones Gₚ. A short-distance invariance of gapped states under deformations can descend to be an emergent continuous symmetry when approaching the gapless limit. Around one TQCP with Gₚ=Z₂T symmetry, we construct these deformations explicitly and show emergence of symmetries via fully gapped quaternion superconducting states that break the protecting symmetry Gₚ=Z₂T. For a 3D TQCP in DIII classes with Gₚ=Z₂T, UEM=U(1), and Nf=1/2 fermions but without charge $U(1)$ symmetry, we further explicitly construct a corresponding boundary representation based on a 4D topological state with lattice symmetry H=Z₂TU(1) and Nf=1 fermions. The lattice model is shown to be dual to a conventional 4D topological insulator. 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 Gₚ.