Breakdown of long-wavelength magnons in cubic antiferromagnets with dipolar forces at small temperature

Using $1/S$ expansion, we discuss the magnon spectrum of Heisenberg antiferromagnet (AF) on a simple cubic lattice with small dipolar interaction at small temperature $T\ll T_N$, where $T_N$ is the Néel temperature. Similar to three-dimensional and two-dimensional ferromagnets, quantum and thermal fluctuations renormalize greatly the bare gapless spectrum leading to a gap $\mathrm{\Delta }\sim {\omega }_0$, where ${\omega }_0$ is the characteristic dipolar energy. This gap is accompanied by anisotropic corrections to the free energy which make the cube edges easy directions for the staggered magnetization (dipolar anisotropy). In accordance with previous results, we find that dipolar forces split the magnon spectrum into two branches. This splitting makes possible two types of processes which lead to a considerable enhancement of the damping compared to the Heisenberg AF: a magnon decay into two other magnons and a confluence of two magnons. It is found that magnons are well defined quasiparticles in quantum AF. We demonstrate however that a small fraction of long-wavelength magnons can be overdamped in AFs with $S\gg 1$ and in quantum AFs with a single-ion anisotropy competing with the dipolar anisotropy. Particular materials are pointed out which can be suitable for experimental observation of this long-wavelength magnons breakdown that contradicts expectation of the quasiparticle concept.

Figure 1

Diagrams of the first order in $1/S$ for self-energy parts discussed in the present paper. Diagram (a) comes from four-magnon terms (9) in the Hamiltonian, whereas (b) stems from three-magnon terms (8). Lines in these diagrams stand for Green's functions (11).

Figure 2

The splitting of the magnon spectrum into two branches obtained in the spin-wave approximation [${\epsilon }_{0\mathbf{k}}^{\pm }$ given by Eq. (19)] and in the first order in $1/S$ [${\epsilon }_{1\mathbf{k}}^{\pm }$ given by Eq. (29)]. Curves are drawn for $S=1/2$, $J=1$, and ${\omega }_0=0.4$. Inset shows a sketch of the sum of two dynamical structure factors ${\mathcal{S}}^{xx}(\omega ,\mathbf{k})+{\mathcal{S}}^{yy}(\omega ,\mathbf{k})$.

Figure 3

Momenta $\mathbf{q}$, $\mathbf{k}$, and $\mathbf{q}-\mathbf{k}$ (or $\mathbf{k}-\mathbf{q}$) of three magnons in the decay (a) and confluence (b) processes which are given by Eqs. (32) and (33), respectively. Components ${\mathbf{q}}_{\parallel }$ and ${\mathbf{q}}_{\perp }$ of $\mathbf{q}$ are also shown which are parallel and perpendicular to $\mathbf{k}$, correspondingly.

Figure 4

Graphic of the function $f\left({\phi }_{\mathbf{k}}\right)$ given by Eq. (40) that determines dependence of the damping on azimuthal angle ${\phi }_{\mathbf{k}}$ [see Eqs. (38), (39), (43), (48), and (49)].

Figure 5

Sketch of the ratio ${\mathrm{\Gamma }}_{\mathbf{k}}^{\pm }/{\epsilon }_{\mathbf{k}}^{\pm }$ at $T\gg S\sqrt{J{\omega }_0}$. Asymptotic at $k\gg \sqrt{{\omega }_0/J}$ is also shown. The peak hight at $k\sim \mathrm{\Delta }/D$ is proportional to $T/D\ll 1$ for quantum spins and it is given by a constant for $S\gg 1$ at $D\ll T\ll T_N$. The peak height in quantum AF can reach a value of the order of unity if the gap is sufficiently reduced by the anisotropy competing with the dipolar one (23) (see discussion in Sec. 6b).