Canonical strong coupling spin wave expansion of Kondo lattice magnets. II. Itinerant ferromagnets and topological magnon bands

TL;DR

This paper develops and applies a canonical spin wave framework for itinerant Kondo lattice magnets in the strong coupling regime to ferromagnets and SOC-enabled systems. It demonstrates that first-order $t/J_K$ corrections extend the effective spin model from a nearest-neighbor Heisenberg form to $J_1$-$J_2$-$J_3$ type exchanges, with the sign and strength controlled by electron density, thereby lifting artificial degeneracies near magnetic phase boundaries. It also shows that spin-orbit coupling induces easy-axis Ising anisotropy and DM interactions, mapping the itinerant problem to a Heisenberg model with these anisotropies; on the Kane-Mele honeycomb lattice this yields topological magnon bands with a finite Chern number. Together, these results connect itinerant magnetism, extended exchange interactions, and magnon topology, with potential relevance to kagome and other lattice ferromagnets where SOC and electron density play crucial roles.

Abstract

In this paper we apply the canonical spin wave theory developed for itinerant Kondo lattice magnets in the strong coupling regime to Kondo ferromagnets, and address two general questions pertaining to their magnetic excitations. First, we compute corrections to the strong coupling (i.e., double-exchange) spin wave dispersion of itinerant ferromagnets. We show that the spin wave dispersion beyond the strong coupling limit can be mapped to the spin wave dispersion of a Heisenberg ferromagnet with farther neighbor exchange couplings, and discuss how this affects instabilities towards antiferromagnetism. Second, we examine the effect of including electronic spin-orbit coupling in the spin wave theory of Kondo ferromagnets. Including spin-orbit coupling is natural and straightforward in the formulation of the canonical spin wave expansion. Our key result is to demonstrate that the linear spin wave Hamiltonian of the itinerant Kondo ferromagnet can be mapped to the spin wave Hamiltonian of a Heisenberg ferromagnet with easy-axis Ising anisotropy and antisymmetric Dzyaloshinskii-Moriya exchange interaction. We show that in the case of the Kane-Mele honeycomb lattice Kondo ferromagnet this leads to topological magnon bands, and discuss the implications of this result for itinerant ferromagnets more broadly.

Figures (8)

• Figure 1: Effective exchange couplings $\tilde{J}_1$ and $\tilde{J}_2$ of the 1D Kondo ferromagnet, as given by Eq. \ref{['eq:J1-J2-1D']}. Solid blue and red curves correspond to $\tilde{J}_1$ and $\tilde{J}_2$ for $t/J_K=0.08$; the dashed curves correspond to $t/J_K=0.16$. For comparison, the grey solid curve corresponds to $\tilde{J}_1$ in the strong coupling limit ($t/J_K=0$), as given by Eq. \ref{['eq:tildeJ1']}. Here $x$ is the electron filling, i.e., the number of electrons per site.
• Figure 2: Effective exchange couplings $\tilde{J}_{1,2,3}$ as a function of filling fraction $x$. Shown are the (a) square lattice and (b) triangular lattice couplings, as given by Eqs. \ref{['eq:J1-square']}, \ref{['eq:J2-J3-square']}, and \ref{['eq:J1-tri']}. Here the blue, red, and magenta curves correspond to $\tilde{J}_{1}$, $\tilde{J}_{2}$, and $\tilde{J}_{3}$, respectively, and we have taken $t/J_K = 0.1$. For comparison, the gray curve corresponds to the strong coupling limit of $\tilde{J}_1$, i.e., $c_1 t/2$.
• Figure 3: Model of the itinerant zigzag chain ferromagnet with spin-orbit coupling. The zigzag structure gives rise to two sublattices (labeled $A$ and $B$ in the text). The model includes a nearest and next-nearest neighbor hopping $t_1$ and $t_2$, respectively. The gray dashed lines indicate a next-nearest neighbor spin-orbit coupling $\boldsymbol{\lambda} _{ij} = \lambda \nu_{ij} \hat{{\bf z}}$, with $\boldsymbol{\lambda} _{ij}$ defined in Eq. \ref{['eq:SOC-model']}. Here $\nu_{ij} = -\nu_{ji} = \pm 1$ denotes the sign structure of the spin-dependent hopping, which is opposite on the two sublattices (see text) and is indicated in the figure by arrows. Spin-orbit coupling introduces an easy-axis Ising anisotropy, such that the spins (shown as red arrows) point along the $\hat{{\bf z}}$ axis.
• Figure 4: Plot of the magnon spectrum $\omega^\pm_q$ of the 1D zigzag chain, as given by Eq. \ref{['eq:1d-dispersion']}, for different choices of exchange couplings $J$, $K^{xx}$, $K^{zz}$, and $D$ [as defined in Eq. \ref{['eq:H-HB-SOC']}]. Gray lines correspond to $D = K^{xx} = K^{zz} = 0$, while blue lines correspond to $D = 0.2J$, $K^{zz} = 0.15J$, and $K^{xx} = 0.05J$.
• Figure 5: Numerical evaluation of the dimensionless integrals $I_{1,2,3}$ defined in Eqs. \ref{['eq:I_1']} and \ref{['eq:I_23']} as a function of filling fraction $x$ (solid lines). Here we have used $\lambda /t_1= 0.2$. Approximations to the integrals, as given by Eq. \ref{['eq:Iapprox1D']}, are shown as dashed lines. Note that in this plot $x = 1$ corresponds to a fully filled lower electron band, or one electron per unit cell (i.e., one electron per two sites).