Demystifying quantum escapism on the honeycomb lattice

TL;DR

This work introduces minimally-augmented spin-wave theory (MAGSWT), a low-cost, physically motivated extension of spin-wave theory that stabilizes classically unstable magnetic states via a minimal shift of the magnon chemical potential. Applying MAGSWT to the $J_1$--$J_3$ FM-AF and $J_1$--$J_2$ AF-AF honeycomb models, the authors show that quantum fluctuations strongly favor collinear orders (FM, ZZ, dZZ, Iz) and suppress noncollinear spiral phases, producing quantum phase diagrams that closely match recent DMRG results. The Iz phase emerges as a robust escapist state driven by fluctuations, particularly in the partially anisotropic regime, while the Sp phase is largely eliminated except near the Heisenberg limit. The study demonstrates that MAGSWT can quantitatively capture phase boundaries and energetics, offering physical insight into quantum stabilization mechanisms and guiding future explorations of complex magnetic orders.

Abstract

We demonstrate the versatility, simplicity, and power of the minimally-augmented spin-wave theory in studying phase diagrams of the quantum spin models in which unexpected magnetically ordered phases occur or the existing ones expand beyond their classical stability regions. We use this method to obtain approximate phase diagrams of the two paradigmatic spin-$\frac{1}{2}$ models on the honeycomb lattice: the $J_1$-$J_3$ ferro-antiferromagnetic and $J_1$-$J_2$ antiferromagnetic $XXZ$ models. For the $J_1$-$J_3$ case, various combinations of the $XXZ$ anisotropies are analyzed. In a dramatic deviation from their classical phase diagrams, which host significant regions of the noncollinear spiral phases, quantum fluctuations stabilize several unconventional collinear phases and significantly extend conventional ones to completely supersede spiral states. These results are in close agreement with the available density-matrix renormalization group calculations. The applicability of this approach to the other models and its potential extension to different types of orders are discussed.

Figures (13)

• Figure 1: (a) Schematic illustration of the problem of the phase boundary within the standard SWT. The dashed and solid lines are classical and order $O(S)$ energies, respectively, dotted line marks classical phase boundary $J_{2,b}^{cl}$. (b) Schematics of $\varepsilon_{\nu,{\bf q}}^2$ calculated beyond the classical stability region.
• Figure 2: (a) The minimal $\mu$ for each of the three different states, FM, ZZ, and Iz, in the $J_1$--$J_3$ model for $\Delta\!=\!0$ and $\Delta_3\!=\!1$ as a function of $J_3$, see Sec. \ref{['Sec:J1J3']} for details. (b) The energy of the FM state for the same model at $J_3\!=\!0.3$ as a function of $\mu$. The minimal $\mu_{min}$ from (a) is indicated. Dashed line is the classical energy, solid lines are the quantum contribution $\delta E$, Eq. (\ref{['eq_dE']}), and the total energy ${\cal E}$, Eq. (\ref{['eq_E']}), respectively.
• Figure 3: (a) Classical phase diagram of the $J_1$--$J_3$ model (\ref{['eq_H']}) for any $0\!\leq\!\Delta_{1(3)}\!\leq\!1$. Sketches of the FM, Sp, and ZZ illustrate the spin order in each phase (red and blue arrows belong to two sublattices). The transition points $J_{3,c1}\!=\!0.25$ and $J_{3,c2}\!\approx\! 0.3904$ are indicated. (b) The sketch of the honeycomb lattice lattice with A and B sublattices, crystallographic axes, and the nearest- and third-neighbor vectors, ${\bm \delta}_\alpha$ and ${\bm \delta}^{(3)}_\alpha$. (c) Brillouin zone (BZ) of the honeycomb lattice with the high-symmetry $\Gamma$ and M points and the representative ${\bf Q}$ vector of the spiral. (d) Sketches of the Iz and dZZ states. Axes in the upper panel show out-of-plane spin direction in the Iz phase and in-plane for the other phases.
• Figure 4: Classical energies of the FM, ZZ, Sp, Iz, and dZZ states as a function of $J_3$ from Eqs. (\ref{['eq_Ecl']}), (\ref{['eq_Ecl_sp']}), and (\ref{['eq_Ecl_add']}) for $S\!=\!\frac{1}{2}$. Vertical dashed lines are the FM-Sp and Sp-ZZ transitions, see Fig. \ref{['Fig_1D']}(a). For the Iz state, two limiting cases are shown, $\Delta_1\!=\!\Delta_3\!=\!0$ and $\Delta_1\!=\!0$, $\Delta_3\!=\!1$.
• Figure 5: Energies of the FM, ZZ, Sp, Iz, and dZZ states as a function of $J_3$ for $\Delta_{3}\!=\!1$ and $S\!=\!1/2$. Dashed lines are the classical energies, Eqs. (\ref{['eq_Ecl']}), (\ref{['eq_Ecl_sp']}), and (\ref{['eq_Ecl_add']}), and solid lines are ${\cal E}\!=\!E_{cl}+\delta E$ from (\ref{['eq_E']}). Vertical dashed lines are classical transition boundaries from Fig. \ref{['Fig_1D']}. (a) $\Delta_{1}\!=\!1$ (all Heisenberg limit), (b) $\Delta_{1}\!=\!0.5$, and (c) $\Delta_{1}\!=\!0$ (partial $XY$ limit).