Nematic Bond Theory of Heisenberg Helimagnets

We study classical two-dimensional frustrated Heisenberg models with generically incommensurate ground states. A new theory for the lattice-nematic “order by disorder” transition is developed based on the self-consistent determination of the effective exchange coupling bonds. In our approach, fluctuations of the constraint field imposing conservation of the local magnetic moment drive nematicity at low temperatures. The critical temperature is found to be highly sensitive to the peak helimagnetic wave vector, and vanishes continuously when approaching rotation symmetric Lifshitz points. Transitions between symmetry distinct nematic orders may occur by tuning the exchange parameters, leading to lines of bicritical points.

Figure 1

Schematic of the spin susceptibility ${\chi }_{\mathbf{q}}$ in the ordered phase with broken $C_{4v}$ symmetry. Here nematic order preserves mirror reflections about the diagonal ($q_x=q_y$) and antidiagonal ($q_x=-q_y$).

Figure 2

Left: Leading-order self-consistent self-energy diagram corresponding to Eq. (7). Right: The inverse constraint propagator $D^{-1}$ is itself a pure self-energy that contains, but is distinct from, the spin self-energy $\mathrm{\Sigma }$ implicitly; see Eq. (8).

Figure 3

Schematic phase diagram of the square-lattice ferromagnetic $J_1\text{−}{J}_2\text{−}{J}_3$ model, Eq. (13). Inset: Ground state phase diagram with the helimagnetic wave vector $\mathbf{Q}=(Q_x,Q_y)$ indicated. As one approaches the ferromagnetic (FM) phase the lattice-nematic transition temperature vanishes linearly. Open (solid) circles denote the onset of nematic order ${\sigma }_a$ (${\sigma }_d$) directed along the $x$ or $y$ axes (diagonal or antidiagonal). The intersection of critical surfaces, $T_c^{(a)}=T_c^{(d)}$, leads to a line of bicritical points.

Figure 4

$T_c$ versus $J_3$ for $J_2=0$, comparing Eq. (10) and Monte Carlo simulations from Ref. [12]. The linear vanishing of $T_c$ as $J_3\searrow 1/4$ is indicated by the red dashed lines, and is consistent with the general result Eq. (12).