Topological Quantum Critical Points in Strong Coupling limits: Global Symmetries and Strongly Interacting Majorana Fermions

In this article, we discuss strong coupling limits of topological quantum critical points (TQCPs) where quantum phase transitions between two topological distinct superconducting states take place. We illustrate that while superconducting phases on both sides of TQCPs spontaneously break same symmetries, universality classes of critical states can be identified only when global symmetries in topological states are further specified. In dimensions $d=2,3$, we find that continuous $(d+1)$th order transitions at weakly interacting TQCPs that were pointed out previously in the presence of emergent Lorentz symmetry can be terminated by strongly interacting fixed points of majorana fields. For $2d$ time reversal symmetry breaking TQCPs, termination points are supersymmetric with ${\mathcal N}=4N_f={1}$ (where $N_f$ is the number of four-component Dirac fermions and ${\mathcal N}$ is the number of two-component real fermions) beyond which transitions are discontinuous first order ones. For $2d$ time reversal symmetric TQCPs without other global symmetries, termination points of $(d+1)$th order continuous transition lines are generically conformal invariant without supersymmetry. Beyond these strong coupling fixed points, there are first-order discontinuous transitions as far as the protecting symmetry is not spontaneously broken but no direct transitions if the protecting symmetry is spontaneously broken in the presence of strong interactions. In $3d$, strong coupling termination points can be further effectively represented by new emergent gapless real bosons weakly coupled with free gapless majorana fermions. However, in $1d$, time reversal symmetric $(d+1)$th continuous transition lines of TQCPs are terminated by simple free majorana fermion fixed points.