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

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