Robust semiclassical magnetization plateau in the kagome lattice

TL;DR

The paper addresses the robustness of the $1/3$ magnetization plateau in the kagome $J_1$-$J_2$ Heisenberg model under a magnetic field. It combines classical analysis, thermal Monte Carlo, and both linear and nonlinear spin-wave theories to show that thermal and quantum fluctuations select the collinear uud state via order-by-disorder, producing a plateau whose width is only weakly sensitive to $J_2$. A notable finding is the magnetization jump at saturation that occurs only at $J_2=0$ due to a flat magnon band, a feature that disappears with finite $J_2$ or at finite temperature. Overall, the semiclassical framework reliably captures the plateau physics and provides a useful lens for interpreting experiments and numerical studies of kagome magnets.

Abstract

Inspired by recent observations of the $1/3$ magnetization plateau in kagome-based magnets, we investigate the $J_1-J_2$ Heisenberg model on the kagome lattice under the influence of an external magnetic field. Although the classical ground state at zero field depends on the sign of $J_2$, we find a robust $1/3$ semiclassical magnetization plateau in both cases. The mechanism that stabilizes this plateau is analogous to that observed in the triangular lattice, where quantum fluctuations select a collinear state from the degenerate classical manifold. We calculate the plateau width, which shows a weak dependence on $J_2$, using nonlinear spin-wave theory. Additionally, we find that a straightforward approach based on linear spin-wave yields quantitatively accurate results even for $S=1/2$. Furthermore, we identify a magnetization jump at the saturation field when $J_2=0$; this jump is related to the presence of a flat band and disappears for $J_2 \neq 0$. Our study demonstrates that a semiclassical approach effectively captures the $1/3$ plateau in the kagome lattice and serves as a valuable tool for interpreting experimental and numerical results alike.

Figures (8)

• Figure 1: (a)Kagome lattice showing its three sublattices, $A$, $B$, and $C$. Here, $J_1$ is the nearest-neighbor exchange coupling and $J_2$ the next-nearest neighbor; (b) Network connecting next-nearest neighbours forming a kagome supperlattice; Semiclassical ground states of the $J_1-J_2$ model: (c) $\vec{Q}=0$ for $J_2>0$, and (d) $\sqrt{3} \times \sqrt{3}$ for $J_2 < 0$. The shaded green area indicates the magnetic unit cell; (e) Possible semiclassical states of the $J_1-J_2$ Heisenberg model in a kagome lattice for a finite field. The three colors represent the three different spins on an elementary triangle. From left to right: Y state, up-up-down (uud) state, V state, and polarized state.
• Figure 2: Magnon dispersions $\omega_{\textbf{k}}$, obtained with the linear spin-wave theory, for the $J_1-J_2$ Heisenberg model in the kagome lattice inside the polarized phase at $h=h_c$. The dispersions are plotted along the indicated path in the Brillouin zone for different values of $J_2$: (a) $J_2<0$, (b) $J_2=0$, and (c) $J_2>0$.
• Figure 3: Classical MC results for the magnetization $m/S$ along the field direction, and its derivative $\chi=dm/dh$, as a function of the field $h$ for $k_{B}T=0.07J_1$ and $L=24$. We have $h_c/S=6J_1$ for $J_2\le0$ and $h_c/S=6\left(J_1+J_2\right)$ for $J_2>0$. For $J_2 > 0$ we renormalized the $\chi$ curve by $\left(J_1 + J_2\right)/J_1$ to account for the change in $h_c$ due to $J_2$.
• Figure 4: (a) Ground-state energy $E_{\text{gs}}$ as function of the magnetic field $h$ obtained with LSWT, Eq. \ref{['eq:Enegs']}, for $S=1/2$ and $J_2/J_1=-0.2$; The uud energy was calculated with Eq. \ref{['eq:linextrap']}. (b) Same as (a), for $S=1/2$ and $J_2/J_1=0.2$; (c) Magnetization along the field direction $m$ as a function of $h$, obtained with Eq. \ref{['eq:mh-dEgsdh']}, for $S=1/2$ and different values of $J2$: $J_2/J_1=-0.2$, $J_2/J_1=0$, and $J_2/J_1=0.2$; (d) Same as (c) for $S=1$. The $1/3$ plateau was inserted phenomenologically as discussed in Sec. \ref{['sec:MagnetizationPlateau']}.
• Figure 5: Critical fields $h_1$ and $h_2$ for the $\vec{Q}=0$ state as function of $J_2$ for: a) spin $S=1/2$; and $b)$ spin $S=1$. The open symbols indicate the result of the angle renormalization given by Eqs. \ref{['eq:chubukov1_def']}, and \ref{['eq:chubukov2_def']}. The solid symbols with dashed lines represent the result of the gap closing calculated with Eq. \ref{['eq:defhcgap']}. The yellow star on the $J_2$ axis indicates the breakdown of LSWT stability for $h=0$, see Sec. \ref{['sec:unstable']}; Critical fields of the plateau phase as a function of $1/S$ for: c) $J_2\geq 0$; and d) $J_2\leq 0$. The dashed red (black) lines correspond to results with the phenomenological method discussed in Sec. \ref{['sec:MagnetizationPlateau']} for $J_2/J_1=\pm0.1 (J_2/J_1=\pm 0.2)$. The solid symbols indicate the result of the angle renormalization given by Eqs. \ref{['eq:chubukov1_def']}, and \ref{['eq:chubukov2_def']}. The gray lines following the solid symbols are a guide for the eye.