Emergent U(1) symmetries in gapless fermionic superfluids and superconductors

Abstract

A superfluid spontaneously breaks the usual $U(1)$ symmetry because of condensation. In this article, we illustrate six linearly independent families of emergent $U(1)$ symmetries that naturally appear in infrared limits in a broad class of generic gapless topological superfluids (that either belong to a stable phase or are quantum critical). In gapless states we have considered, emergent $U(1)$ symmetry groups are embedded in a $Spin(4)=SU(2)\ensuremath{\bigotimes}SU(2)$ group that double covers (and algebraically is isomorphic to) an $SO(4)$ group. All $U(1)$ charges associated with symmetries are further invariant under an $SU(2)$ spin group or an equivalent of it but always break preexisting higher space-time Lorentz symmetry of the $SO(3,1)$ group. Emergent $U(1)$ symmetries can be further spontaneously broken only if interactions are strong enough and resultant strong-coupling states become fully gapped. However, if states remain gapless, emergent $U(1)$ symmetries are always present, despite that these states may exhibit much lower space-time symmetries compared to their weakly interacting gapless Lorentz symmetric counterparts. In the limit of our interests, we have identified all possible gapless real fermions with or without Lorentz symmetries and find that they all display emergent $U(1)$ symmetries in the infrared limit. We argue emergent $U(1)$ symmetries are intrinsic in a broad class of interacting gapless superfluid or superconducting states and are typically well defined in high dimensions where there are infrared stable fixed points dictating emergent properties.