The electromagnetic response of topological insulators and superconductors is governed by a modified set of Maxwell equations that derive from a topological Chern-Simons (CS) term in the effective Lagrangian with coupling constant $kappa$. Here we consider a topological superconductor or, equivalently, an Abelian Higgs model in $2+1$ dimensions with a global $O(2N)$ symmetry in the presence of a CS term, but without a Maxwell term. At large $kappa$, the gauge field decouples from the complex scalar field, leading to a quantum critical behavior in the $O(2N)$ universality class. When the Higgs field is massive, the universality class is still governed by the $O(2N)$ fixed point. However, we show that the massless theory belongs to a completely different universality class, exhibiting an exotic critical behavior beyond the Landau-Ginzburg-Wilson paradigm. For finite $kappa$ above a certain critical value $kappa_c$, a quantum critical behavior with continuously varying critical exponents arises. However, as a function $kappa$ a transition takes place for $|kappa| < kappa_c$ where conformality is lost. Strongly modified scaling relations ensue. For instance, in the case where $kappa^2>kappa_c^2$, leading to the existence of a conformal fixed point, critical exponents are a function of $kappa$.
Download