EXACT SO(8) SYMMETRY IN THE WEAKLY-INTERACTING 2-LEG LADDER

We revisit the problem of interacting electrons hopping on a two-leg l adder. A perturbative renormalization-group analysis reveals that at h alf-filling the model scales onto an exactly soluble Gross-Neveu model for a;arbitrary finite-ran,oed interactions, provided they are suffic iently weak. The Gross-Neveu model has an enormous global SO(8) symmet ry, manifest in terms of eight real Fermion fields that, however, are highly nonlocal in terms of the electron operators. For generic repuls ive interactions, the two-leg ladder exhibits a Mott insulating phase at half-filling with d-wave pairing correlations. Integrability of the Gross-Neveu model is employed to extract the exact energies, degenera cies, and quantum numbers of all the low-energy excited states, which fall into degenerate SO(8) multiplets. One SO(8) vector includes two c harged Cooper pair excitations, a neutral s=1 triplet of magnons, and three other neutral s=0 particle-hole excitations. A triality symmetry relates these eight two-particle excitations to two other degenerate octets, which are comprised of single-electron-like excitations. In ad dition to these 24 degenerate ''particle'' states costing an energy (m ass) m to create, there is a 28-dimensional antisymmetric tensor multi plet of ''bound'' states with energy root 3m. Doping away from half-fi lling liberates the Cooper pairs, leading to quasi-long-range d-wave p air field correlations, but maintaining a gap to spin and single-elect ron excitations. For very low doping levels, integrability allows one to extract exact values for these energy gaps. Enlarging the space of interactions to include attractive interactions reveals that there are four robust phases possible for the weak coupling two-leg ladder. Whi le each of the four phases has a (different) SO(8) symmetry, they are shown to all share a common SO(5) symmetry-the one recently proposed b y Zhang as a unifying feature of magnetism and superconductivity in th e cuprates.