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

TL;DR

This paper explores Fermi surfaces with higher codimension, revealing a new non-trivial fixed point through an $ ext{epsilon}$ expansion that describes a scale-invariant theory in an effective space-time dimension.

Contribution

It introduces a controlled $ ext{epsilon}$ expansion to study phase transitions in systems with higher codimension Fermi surfaces, uncovering a novel fixed point and connecting it to known bosonic theories.

Findings

Identification of a new non-trivial fixed point for higher codimension Fermi surfaces.

Demonstration that the fixed point describes a scale-invariant theory in effective space-time.

Equivalence of results with the Hertz-Millis action for superconductivity.

Abstract

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 $\epsilon$ 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.