A conjugate-representation SU(3) model on the honeycomb lattice: exact single-site entropy and the operator structure of the chiral ring exchange

Abstract

We establish the exact algebraic structure of an SU(3) spin model on the bipartite honeycomb lattice whose two sublattices carry conjugate representations. Requiring that every nearest-neighbor bond be able to form a singlet forces the fundamental representation on one sublattice and the antifundamental on the other, and because the resulting bond representation is multiplicity-free, the two-site interaction is then unique up to an additive constant and a scale. In any finite-size singlet ground state the single-site reduced entropy is exactly ln 3, independently of the lattice and of the form of the interaction. The one-step cyclic advance around a hexagonal plaquette fails to commute with the global SU(3) generators, commuting only with the real subgroup SO(3), so the two-step rotation R = P^2 is the smallest advance the sublattice structure permits. Its time-reversal-odd part decomposes exactly into the SU(3) scalar chiralities of the two sublattice triangles, each dressed by the time-reversal-even ring exchange of its partner, with a Schur-Weyl argument fixing the underlying three-site identity. Every identity is verified by direct computation on the full hexagon Hilbert space. Symmetry alone does not single out the chiral operator we adopt; a transport picture of the elementary move and a choice of orientation do, and given those the two-step form is forced. A fermionic parton mean field, with the six-site ring exchange decoupled by Wick contraction and the constraint enforced self-consistently, then indicates that the constraint-induced mass is overcome beyond a finite chirality, placing each color band at unit Chern number and pointing to an SU(3)_1 chiral spin liquid with total quantum dimension sqrt(3), Z_3 fusion, chiral central charge two, and topological entanglement entropy ln sqrt(3). We present that identification as a mean-field conjecture, state the limitations of the treatment producing it, and cast it as a well-posed numerical test.