Self-Consistent Renormalized Spin-Wave Theory of Magnetic and Topological Transitions in Two-Dimensional Honeycomb Ferromagnets

TL;DR

The study develops an extended self-consistent renormalized spin-wave theory for 2D honeycomb ferromagnets, incorporating higher-order Holstein-Primakoff corrections to analyze how anisotropy, Zeeman field, NNN exchange, and Dzyaloshinskii-Moriya interaction shape magnetic and topological transitions. By mapping to a renormalized magnon problem with $\mathcal{H}\approx \mathcal{H}_0+\mathcal{H}_4+\mathcal{H}_6$ and solving for the renormalized spectrum $\varepsilon_\pm(\mathbf{k})$ and magnetization $m$ self-consistently, the authors reveal that SRSWT tends to predict first-order magnetic transitions due to enhanced self-energy corrections, though appropriate tuning (e.g., Zeeman field and antiferromagnetic $J'$) can suppress the metastable region. Crucially, topological transitions signaled by gap closings at Dirac points can occur below $T_{\rm Curie}$ on the thermodynamically stable branch, with a representative CrI$_3$-like parameter set giving $T_c\approx 2.4J$ while $T_{\rm Curie}\approx 2.5J$. The work thus provides actionable guidance for realizing thermally driven topological magnon transitions in 2D honeycomb magnets and clarifies the predictive limits of SRSWT, suggesting complementary methods for benchmarking.

Abstract

We investigate finite-temperature magnetic and topological phase transitions in two-dimensional honeycomb ferromagnets using an extended self-consistent renormalized spin-wave theory (SRSWT) that incorporates higher-order corrections from the Holstein--Primakoff expansion. Focusing on the combined effects of single-ion anisotropy, Zeeman field, next-nearest-neighbor (NNN) exchange, and Dzyaloshinskii--Moriya interaction, we analyze how these parameters influence the magnetization curves and magnon spectra. This work serves two main goals. First, we critically examine the limitations of SRSWT, showing that in the absence of external or interaction tuning, the theory tends to overestimate magnon self-energy corrections, often predicting first-order magnetic transitions with multivalued magnetization and metastable solution branches (i.e., self-consistent but thermodynamically unstable states). Second, we demonstrate that topological transitions -- signaled by magnon gap closings at the Dirac points -- can be tuned to occur below the magnetic transition temperature and within the thermodynamically stable regime. In particular, we identify two practical tuning strategies: applying an external Zeeman field of appropriate sign depending on the anisotropy strength, and introducing a small antiferromagnetic NNN exchange coupling. These findings not only clarify the predictive scope and limitations of SRSWT but also provide experimentally relevant guidance for realizing thermally driven topological transitions in two-dimensional honeycomb magnetic insulators.

Figures (5)

• Figure 1: (a) The illustration of a honeycomb lattice, showing the three nearest-neighbor and next-nearest-neighbor vectors, labeled as $\boldsymbol{\delta}_i$ and $\boldsymbol{\zeta}_i$ ($i=1,2,3$), respectively. We set the out-of-plane direction to be along $\hat{\bm{z}}$. The blue and red sites are denoted as $A$ and $B$ sites, respectively. (b) Bubble diagrams corresponding to Hartree-type self-energy contributions in the self-consistent renormalized spin-wave theory. (c) Feynman diagram representing six-operators interaction terms.
• Figure 2: (a) Magnetization $m$ as a function of $T/J$ for three values of the external Zeeman field: $h/J = 0$, 0.1, and $-0.07$. All curves exhibit a first-order phase transition, indicated by the characteristic bend-back feature. (b) Temperature dependence of the gap $\Delta_K/J$ at the Dirac point for the same fields. For $h = 0$ and 0.1, the gap closes on the thermodynamically unstable branch, whereas for $h = -0.07$, the gap closes on the stable branch, marking a physically accessible topological transition. (c) Magnon band structures at three representative temperatures for $h = -0.07$, showing gap closing and reopening near the Dirac point. Solid lines in (a) and (b) are guides to the eye.
• Figure 3: (a) Magnon spectral gap at the $\Gamma$ point, $\Delta_\Gamma/J$, as a function of temperature for the three external fields shown in Fig. 2(a), with fixed anisotropy $A/J = 0.25$. The gap remains finite across the temperature range, increasing with Zeeman field. (b) Magnetization $m$ as a function of $T/J$ for three combinations of anisotropy and external field: $(A/J, h/J) = (0.2, -0.18)$, (0.25, $-0.07$), and (0.3, 0.1). In all cases, the transition remains first-order, with stronger anisotropy and more positive field enhancing the stability of the ordered phase. (c) Same quantity as in panel (a), but for the parameter sets shown in panel (b), illustrating how combined effects of anisotropy and Zeeman field influence the spectral gap. The gap does not close in any case, consistent with finite-temperature ferromagnetism in the presence of easy-axis anisotropy. Solid lines in all panels are guides to the eye.
• Figure 4: (a) Magnetization $m$ as a function of $T/J$ for two next-nearest-neighbor (NNN) exchange couplings, $J'/J = 0$ and $0.1$, with fixed anisotropy $A/J = 0.25$ and zero external field $h = 0$. (b) Same quantity as in (a), but with external field fixed at $h/J = 0.1$, comparing $J'/J = 0$ and $-0.07$. The case with $J'/J = -0.07$ exhibits a first-order phase transition with an extremely narrow coexistence region; the bend-back feature in the magnetization curve is nearly imperceptible, making the transition appear almost continuous. (c) Effect of the Dzyaloshinskii–Moriya interaction on the magnetization, comparing $D/J = 0.1$ and $0.2$, with $A/J = 0.25$, $h/J = 0.1$, and $J'/J = -0.07$ held fixed. Increasing $D$ enhances the first-order character of the transition, making the coexistence region more visible. Solid lines in all panels are guides to the eye.
• Figure 5: Free energy for the case $h=0$ in Fig. \ref{['fig:2']}(a). The red and blue curves correspond to the stable (lower free energy) and unstable branches, respectively.