2D Superconductivity: Classification of Universality Classes by Infinite Symmetry

The quantum order of superconducting condensates is characterized by their incompressibility in the infinite gap limit. Classical 2D incompressible fluids possess the dynamical symmetry of area-preserving diffeomorphisms. I show that the corresponding infinite dynamical symmetry of 2D superconducting fluids is the coset ${{W_{1+\infty} \otimes \bar W_{1+\infty}} \over U(1)_{\rm diagonal}}$, with $W_{1+\infty}$ the chiral algebra of quantum area-preserving diffeomorphisms and I derive its minimal models. These define a discrete set of 2D superconductivity universality classes which fall into two main categories: conventional superconductors with their vortex excitations and unconventional superconductors. These are characterized by a broken $U(1)_{\rm vector} \otimes U(1)_{\rm axial}$ symmetry and are labeled by an integer level $m$. They possess neutral spinon excitations of fractional spin and statistics $S = {\theta \over 2\pi} = {{m-1} \over 2m}$ which carry also an $SU(m)$ isospin quantum number; this hidden $SU(m)$ symmetry implies that these anyon excitations are non-Abelian. The simplest unconventional superconductor is realized for $m=2$: in this case the spinon excitations are semions (half-fermions). My results show that spin-charge separation in 2D superconductivity is a universal consequence of the infinite symmetry of the ground state.