Fermi Surfaces in General Codimension and a New Controlled Nontrivial Fixed Point

The energy of a $d$-dimensional Fermi system typically varies only along $d_c=1$ (“radial”) dimensions. We consider $d_c=1+\epsilon$ and study a transition to superconductivity in an $\epsilon$ expansion. The nontrivial fixed point describes a scale invariant theory with an effective space-time dimension $D=d_c+1$. Remarkably, the results can be reproduced by the Hertz-Millis action for the superconducting order parameter in higher effective space-time dimensions. We consider possible realizations of the transition at $\epsilon =1$, which corresponds to a linear Fermi surface in $d=3$.

Figure 1

Renormalization of the field that couples to the BCS order parameter at one loop. Open (closed) circle is the bare (renormalized) vertex.