Lattice symmetries and regular magnetic orders in classical frustrated antiferromagnets

We consider some classical and frustrated lattice spin models with global $\mathrm{O}(3)$ spin symmetry. No general analytical method to find a ground state exists when the spin dependence of the Hamiltonian is more than quadratic (i.e., beyond the Heisenberg model) and/or when the lattice has more than one site per unit cell. To deal with these cases, we introduce a family of variational spin configurations, dubbed “regular magnetic orders” (RMO’s), which respect all the lattice symmetries modulo global $O(3)$ spin transformations (rotations and/or spin flips). The construction of these states is made explicit through a group-theoretical approach, and all the RMO’s on the square, triangular, honeycomb, and kagome lattices are listed. Their equal-time structure factors and powder averages are shown for comparison with experiments. Well known Néel states with one, two, or three sublattices on various lattices are RMO’s, but the RMO’s also encompass exotic nonplanar states with cubic, tetrahedral, or cuboctahedral geometry of the $T=0$ order parameter. Regardless of the details of the Hamiltonian (with the same symmetry group), a large fraction of these RMO’s are energetically stationary with respect to small deviations of the spins. In fact, these RMO’s appear as exact ground states in large domains of parameter space of simple models that we have considered. As examples, we display the variational phase diagrams of the $J_1$-$J_2$-$J_3$ Heisenberg model on all the previous lattices as well as that of the $J_1$-$J_2$-$K$ ring-exchange model on square and triangular lattices.

Figure 1

A lattice symmetry $X\in S_L$ acts on a spin configuration $c$ to give a new configuration $c^{\prime }=Xc$. If $c$ is regular, there is a spin symmetry $G_X\in S_S$ such that one gets back the initial state: $G_X{c}^{\prime }=c$.

Figure 2

Generators of the lattice symmetries group $S_L$ for the triangular, kagome, honeycomb, and square lattices. For the first three lattices: the two translations $T_1$ and $T_2$ (along the two basis vectors ${\mathbf{T}}_1$ and ${\mathbf{T}}_2$), the reflection $\sigma$ and the rotation $R_6$ of angle $\pi /3$. For the square lattice, generators of $S_L$ are $T_1$, $T_2$, $\sigma$, and the rotation $R_4$ of angle $\pi /2$.

Figure 3

Regular magnetic orders on the triangular lattice. The sublattice arrangements (labeled by colors) and the spin directions on each sublattice are displayed in the left and center columns. A spin unit cell is surrounded with green lines. The positions and weights of the Bragg peaks in the hexagonal Brillouin zone of the lattice are in the right column. The energy per site of each structure is given as a function of the parameters of the models described in Sec. 6.

Figure 4

Regular magnetic orders on the kagome lattice and their equal-time structure factors in the EBZ (see text). The energies (per site) of these states are given for the $J_1$-$J_2$-$J_3$-$J_3^{\prime }$ model described in Sec 6.

Figure 5

Regular magnetic orders on the honeycomb lattice and their equal-time structure factors in the EBZ (see text). The energies (per site) are given for a $J_1$-$J_2$-$J_3$ Heisenberg model.

Figure 6

Regular magnetic orders on the square lattice and their equal-time structure factors in the square BZ. The energy per site of each structure is given as a function of the parameters of the models described in Sec. 6.

Figure 7

Phase diagram of the $J_1$-$J_2$-$J_3$ Heisenberg model on the honeycomb lattice. Labels refer to the RMO’s described in Fig. 5. In each colored region (black excluded), the RMO is an exact GS. In the black region, a generalized spiral state (SS) has an energy strictly lower than the RMO’s, but the actual GS energy might still be lower.

Figure 8

Phase diagram of the $J_1$-$J_2$-$J_3$-$J_3^{\prime }$ model on the kagome lattice. In each colored region (white included, gray and black excluded), the RMO is an exact GS. Labels refer to the RMO’s described in Fig. 4. In the gray regions, the nearby RMO does not reach the lower bound of Sec. 6b, but no SS is energetically lower. In the black regions, an SS has a lower energy than the RMO’s, but the actual GS might yet be lower.

Figure 9

Phase diagrams on the square lattice with $J_1$-$J_2$-$J_3$ Heisenberg interactions (top line) and $J_1$-$J_2$-$K$ model (bottom line). Labels refer to RMO’s defined in Fig. 6. In each colored region (black excepted), the RMO has the lowest energy of the set of all regular and generalized spiral states. In the black regions, an SS has a lower energy than the RMO’s. For pure Heisenberg interactions, we know that we obtain the GS energy, but for non-Heisenberg interactions the actual GS might be lower. In the $J_1$-$J_2$-$J_3$ model, the coplanar (orthogonal four-sublattice) phase is degenerate with nonregular collinear states, which will win upon introductions of fluctuations. On the contrary, the coplanar (orthogonal four-sublattice) phase is stable in a large range of parameters in the $J_1$-$J_2$-$K$ model.

Figure 10

Phase diagrams on the triangular lattice with $J_1$-$J_2$-$J_3$ Heisenberg interactions (top line) and $J_1$-$J_2$-$K$ model (bottom line). Labels refer to RMO’s defined in Fig. 3. In each colored region (except black), the RMO has the lowest energy of the set of all regular and generalized spiral states. In the black regions, an SS has a lower energy than the RMO’s. For pure Heisenberg interactions, we know that we obtain the GS energy, but for non-Heisenberg interactions the actual GS might be lower.

Figure 11

Powder-averaged equal-time structure factors $S(|\mathbf{Q}|)$ of the RMO’s on the triangular lattice [$\left|\mathbf{Q}\right|$ is in units of $2\pi$, $S(|\mathbf{Q}|)$ in arbitrary units].

Figure 12

Powder-averaged equal-time structure factors $S(|\mathbf{Q}|)$ of the RMO’s on the kagome lattice [$\left|\mathbf{Q}\right|$ is in units of $2\pi$, $S(|\mathbf{Q}|)$ in arbitrary units].