Order-by-disorder from Schwinger bosons in a frustrated honeycomb ferromagnet

TL;DR

This work addresses how a simple $J_1$-$J_3$ Heisenberg model on the honeycomb lattice can realize a double-zigzag ground state via order-by-disorder. The authors develop a generalized Schwinger-boson mean-field theory ($g$-SBMFT) that accommodates both ferromagnetic and antiferromagnetic couplings by allowing triplet pairing on ferromagnetic bonds, and they validate their mean-field results with exact diagonalization. The phase diagram features a narrow $dZZ$ window between the ferromagnetic and zig-zag phases, stabilized by quantum fluctuations, consistent with DMRG results. They also predict distinctive dynamical spin structure factors and momentum-space spectral weight transfer, offering concrete INS signatures for BaCo2(AsO4)2 (BCAO) and related cobaltates. The approach generalizes SBMFT to frustrated FM systems and provides a framework to connect microscopic spin interactions to experimental observables in quasi-2D magnets.

Abstract

The cobalt-based honeycomb magnet BaCo$_2$(AsO$_4$)$_2$ (BCAO) has recently emerged as a promising platform for studying frustrated magnetism beyond conventional paradigms. Neutron-scattering experiments and first-principles calculations have revealed an unexpected double-zigzag (dZZ) magnetically ordered ground state, whose microscopic origin remains under active debate. Here, we investigate the emergence of such dZZ phase in a ferro-antiferromagnetic $J_1$-$J_3$ Heisenberg model on the honeycomb lattice, using a generalized Schwinger boson mean-field theory ($g$-SBMFT) that treats ferromagnetic and antiferromagnetic interactions on equal footing. Based on $g$-SBMFT and exact-diagonalization (ED) techniques, we find that the dZZ is selected by an order-by-disorder mechanism in a narrow $J_3/|J_1|$ range, in agreement with recent density-matrix renormalization-group calculations. The magnetic excitation spectra within the dZZ phase displays a distinctive smearing out in momentum space due to quantum fluctuations which may be probed through inelastic neutron-scattering experiments.

Figures (10)

• Figure 1: (a) Phase diagram of the ferro-antiferromagnetic $J_1$-$J_3$ model at $T=0$. (b) Honeycomb lattice with two sublattices $u$ and $v$, and the three nearest-neighbor displacement vectors $\{ \bf{d}_i \}$ for a 2-site unit-cell and its primitive vectors $(\bf{e}_1,\bf{e}_2)$, and for an 8-site unit-cell with $(\bf{e}_1',\bf{e}_2')$. (c) Corresponding first (solid lines) and second (dashed lines) Brillouin zones together with high-symmetry points. The dynamical structure factors are analyzed along the path $K\Gamma M \Gamma' M' K \Gamma$.
• Figure 2: Ansätze of the ZZ and the dZZ structures within the $s$- and $g$-SBMFT frameworks. (a) Mean-field structure of the ZZ phase obtained from the $s$-SBMFT. Despite ferromagnetic nearest-neighbor bonds, nonzero singlet pairings are present. (b) Corresponding ZZ Ansatz obtained from the $g$-SBMFT, where ferromagnetic bonds (solid lines) host nonzero triplet pairings $P_{2,3} \neq 0$, while antiferromagnetic $J_3$ bonds (dashed lines/arrows) carry both singlet $p_0$ and triplet hopping amplitudes $h_3$. This Ansatz achieves a lower energy than (a); see text for discussion. (c) Structure of the dZZ phase, accessible only within the $g$-SBMFT formalism and on the 8-site unit-cell.
• Figure 3: Lowest-energy spinon dispersions for (a) the ZZ phase obtained from $s$-SBMFT on a two-site unit cell with $S=0.35$ and $J_3 = 1.0$, (b) the same ZZ Ansatz on an eight-site unit cell, and (c,d) the ZZ and dZZ Ansätze from $g$-SBMFT on the same eight-site cell. For all cases, only one quarter of the first (solid lines) and second (dashed lines) Brillouin zones shown in Fig. \ref{['fig:lattice']} are shown.
• Figure 4: Dependence of the ground state energy of the ferro-antiferromagnetic $J_1$-$J_3$ Heisenberg model on $J_3$. Energies obtained from ED (thick solid gray line) for the model with $S=1/2$ are compared with our $g$-SBMFT approach with $S = 0.25$ (symbols). Results are shown for the ZZ (red) and dZZ (blue) Ansätze, and compared with the ZZ solution obtained from the conventional $s$-SBMFT (dashed blue line). Dashed gray lines indicate the classical energies of the FM, dZZ and ZZ orders. The red shaded region marks the finite parameter window in which the dZZ phase is selected through order-by-disorder. Energies per site normalized to $S^2$ for fixed $J_1 = -1$ are shown in this plot.
• Figure 5: Real space spin-spin correlations for (a) the ZZ and (b) the dZZ phases, calculated for $\kappa =0.5$ at $J_3 / |J_1| = 0.3$ on a $8 \times 6 \times 6$ site cluster. The size of the disk corresponds to the strength of the correlation (in arbitrary units) and the colors correspond to positive (blue), negative (red) correlations and the reference site is at the lower-left corner (gray). Black arrows show the translation vectors of the 8-site unit-cell compatible with all ordered phases of the present work.