Phenomenological Ginzburg-Landau theory for triple-Q magnetic orders on a hexagonal lattice

TL;DR

This work develops a symmetry-based Ginzburg-Landau framework for triple-$Q$ magnetic orders on hexagonal lattices, employing three $O(N)$ order parameters to capture single-, double-, and triple-$Q$ states. By enforcing hexagonal ($D_3$) lattice symmetry and internal $O(N)$ rotations, it derives complete phase diagrams for $N=2$ and $N=3$, and analyzes the associated real-space spin textures. The paper further resolves the collective excitation spectra, revealing Goldstone and amplitude modes tied to the residual symmetries across phases, with explicit mode-counting in each case. These results illuminate how microscopic interactions and lattice symmetry generate complex magnetic orders and provide concrete predictions for neutron and Raman probes in frustrated hexagonal magnets, offering directions for including DMI, extending to $U(N)$, and material-specific mappings.

Abstract

We develop a comprehensive Ginzburg-Landau theory describing triple-Q magnetic orders on hexagonal lattices, focusing on $O(N)$ models with $N=2$ and $N=3$. Through systematic analysis of symmetry-allowed terms in the free energy, we establish complete phase diagrams governed by competing interaction parameters. Our theory reveals distinct magnetic configurations including single-Q, double-Q, and triple-Q states, each characterized by unique symmetry breaking patterns and collective excitations. The framework provides fundamental insights into complex magnetic orders recently observed in materials such as Na$_2$Co$_2$TeO$_6$, where the interplay between geometric frustration and multiple ordering vectors leads to exotic magnetic states. Our results establish clear connections between microscopic interactions, broken symmetries, and experimentally observable properties, offering a powerful tool for understanding and predicting novel magnetic phases in frustrated magnets.

Figures (5)

• Figure 1: (a) The triple Q-vectors $\bm{Q}_1$, $\bm{Q}_2$, and $\bm{Q}_3$ are specified at the $M$ points of the first Brillouin Zone on a hexagonal lattice. Note that $-\bm{Q}_{l}$ is equivalent to $\bm{Q}_{l}$ at each $M$ point, where $l=1,2,3$. Multiple Q-vector magnetic configurations for $O(N)$ models include: single-Q (b) $\mathrm{I}_{O(N=2,3)}$, double-Q (c) $\mathrm{II}^{A}_{O(N=2,3)}$ and (d) $\mathrm{II}^{B}_{O(N=2,3)}$, and triple-Q (e) $\mathrm{III}^{A}_{O(N=2,3)}$ (collinear), (f) $\mathrm{III}^{B}_{O(3)}$ (orthogonal), (g) $\mathrm{III}^{C}_{O(2)}$ (120$^{\circ}$ coplanar), and (h) $\mathrm{III}^{D}_{O(2)}$ (orthogonal-collinear) states. The vectors that are collinear at distinct $\bm{Q}_l$ in (c), (e) and (h) may align in parallel or opposite directions. The configuration in (f) has three mutually orthogonal vectors, which is allowed only if $N\geq{}3$. The configurations in (g) and (h) are coplanar states.
• Figure 2: Phase diagrams for (a) $O(3)$ and (b) $O(2)$ models. Notations I, II, and III denote single-, double-, and triple-Q phases, respectively. Labels $A$, $B$, and $C$ indicate collinear, orthogonal, and 120$^\circ$ phases. Model (a) $O(3)$ is stable under conditions $\beta_1>0$, $\beta_1+\beta_2>0$, and $\beta_1+\beta_2+\beta_3>0$, while model (b) $O(2)$ is stable with $\beta_1>0$, $2\beta_1+\beta_2>0$, $\beta_1+\beta_2+\beta_3>0$, and $4(\beta_1+\beta_2)+\beta_3>0$. Each phase has a unique magnetic configuration, displayed in Fig. \ref{['fig:QVec']}(b)-(g).
• Figure 3: Real-space configurations of the order parameters for the phases (a) $\mathrm{I}_{O(N=2,3)}$, (b) $\mathrm{II}^B_{O(2)}$, (c) $\mathrm{III}^{A}_{O(N=2,3)}$, (d) $\mathrm{III}^{C}_{O(2)}$ and (e) $\mathrm{III}^B_{O(3)}$ in Fig. \ref{['fig:PhaseDiagram']}.
• Figure 4: Unit vectors that form the basis for the decomposition of $\bm{\Delta}_{\bm{Q}_l}$ and $\bm{\eta}_{\bm{Q}_l}$ for phase $\mathrm{III}^C_{O(2)}$.
• Figure 5: The phase diagram and the stable region for $\mathcal{F}_{U(1)}$.