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 $\Phi \in (0,2\pi /3)$. In this situation, the ground states are noncoplanar and have an infinite discrete degeneracy. In the generic case, i.e., when $\Phi \ne \pi /2,\arccos (-1/3)$, the ground-state manifold 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 $O\left(3\right)$ 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 $\Phi \ne \arccos (-1/3)$, 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).

Figure 1

Phase diagram of the classical bilinear-biquadratic Heisenberg model on the triangular lattice.

Figure 2

The loop representation: (a)–(c) allowed configurations of loops, (d) additional vertex allowed only at $\Phi =\arccos (-1/3)$, and (e) the vertex corresponding to an intersection of loops (allowed only at $\Phi =\pi /2$). The domain walls on different sublattices are shown with lines of different types.

Figure 3

From left to right, configurations obtained after successive flips of spins A, B, A, B. In the final configuration, the triad of spins is rotated around C by an angle $4\Psi \left(\Phi \right)$ with respect to the original one.

Figure 4

(Top) Example of loop covering. The numbers refer to the positions of different domains on the Cayley tree shown in the lower panel. The numbering is arbitrary and has been included to keep track of the relative positions on the tree.