Hidden supersymmetry of a $P,T-$invariant $3D$ fermion system

Abstract

We show that a (2+1)-dimensional $P,T-$invariant free fermion system, relevant to $P,T-$conserving models of high-$T_c$ superconductivity, has a U(1,1) dynamical symmetry as well as an $N=3$ supersymmetry with the even generator being a quadratic function of the spin operator and of the generator of chiral ${\rm U_c(1)}$ transformations. We demonstrate that the hidden supersymmetry leads to a non-standard superextension of the (2+1)-dimensional Poincaré group. As a result, the one particle states of the $P,T-$invariant fermion system realize an irreducible representation of the Poincaré supergroup labelled by the zero eigenvalue of the superspin operator.