Fermi surfaces in general co-dimension and a new controlled non-trivial fixed point

Traditionally Fermi surfaces for problems in $d$ spatial dimensions have dimensionality $d-1$, i.e., codimension $d_c=1$ along which energy varies. Situations with $d_c>1$ arise when the gapless fermionic excitations live at isolated nodal points or lines. For $d_c>1$ weak short range interactions are irrelevant at the non-interacting fixed point. Increasing interaction strength can lead to phase transitions out of this Fermi liquid. We illustrate this by studying the transition to superconductivity in a controlled $ϵ$ expansion near $d_c=1$. The resulting non-trivial fixed point is shown to describe a scale invariant theory that lives in effective space-time dimension $D=d_c+1$. Remarkably, the results can be reproduced by the more familiar Hertz-Millis action for the bosonic superconducting order parameter even though it lives in different space-time dimensions.