Degeneracy and ordering of the noncoplanar phase of the classical bilinear-biquadratic Heisenberg model on the triangular lattice

We investigate the zero-temperature behavior of the classical Heisenberg model on the triangular lattice in which the competition between exchange interactions of different orders favors a relative angle between neighboring spins Phi is an element of (0,2 pi/3). In this situation, the ground states are noncoplanar and have an infinite discrete degeneracy. In the generic case, i.e., when Phi not equal pi/2, arccos(-1/3), the ground-statemanifold is in one-to-one correspondence (up to a global rotation) with the set of noncrossing loop coverings of the three equivalent honeycomb sublattices into which the bonds of the triangular lattice can be partitioned. This allows one to identify the order parameter space as an infinite Cayley tree with coordination number 3. Building on the duality between a similar loop model and the ferromagnetic O(3) model on the honeycomb lattice, we argue that a typical ground state should have long-range order in terms of spin orientation. This conclusion is further supported by the comparison with the four-state antiferromagnetic Potts model [describing the Phi = arccos(-1/3) case], which at zero temperature is critical and in terms of the solid-on-solid representation is located exactly at the point of roughening transition. At Phi not equal arccos(-1/3), an additional constraint appears, whose presence drives the system into an ordered phase (unless Phi = pi/2, when another constraint is removed and the model becomes trivially exactly solvable).