Colloquium: Spontaneous magnon decays

A theoretical overview of the phenomenon of spontaneous magnon decays in quantum antiferromagnets is presented. The intrinsic zero-temperature damping of magnons in quantum spin systems is a fascinating many-body effect, which has recently attracted significant attention in view of its possible observation in neutron-scattering experiments. An introduction to the theory of magnon interactions and a discussion of necessary symmetry and kinematic conditions for spontaneous decays are provided. Various parallels with the decays of anharmonic phonons and excitations in superfluid ${}^4\mathrm{He}$ are extensively used. Three principal cases of spontaneous magnon decays are considered: field-induced decays in Heisenberg antiferromagnets, zero-field decays in spiral antiferromagnets, and triplon decays in quantum-disordered magnets. Analytical results are compared with available numerical data and prospective materials for experimental observation of the decay-related effects are briefly discussed.

Figure 1

Zero-temperature self-energy diagrams due to the (a) and (b) three-magnon, and (c) and (d) four-magnon interactions.

Figure 2

Magnon decay threshold boundaries for the cubic antiferromagnet in an external field shown in one-eighth of the Brillouin zone for several values of the field. Decays are possible within the part of the Brillouin zone enclosing the $M_{\pi }$ point.

Figure 3

Sketches of the quasiparticle spectra ${\epsilon }_{\mathbf{k}}$ with (a) negative and (b) positive curvature of the acoustic mode, respectively. (c) Spectrum with several acoustic branches.

Figure 4

Field-induced canted spin structure in the square-lattice Heisenberg antiferromagnet.

Figure 5

Decay regions in one-fourth of the Brillouin zone for representative values of $H/H_s$ indicated by numbers. Solid and dashed lines are threshold boundaries for two- and three-magnon decays, respectively.

Figure 6

Magnon spectral function for the spin-$\frac{1}{2}$ antiferromagnet at $H=0.9H_s$ for $\mathbf{k}=\pi (q,q)$. Dashed and solid lines correspond to Born approximation and restricted self-consistent born approximation, respectively. From 142.

Figure 7

QMC results for the $S^{zz}(\mathbf{q},\omega )$ component of the dynamical structure factor for the $S=1/2$ square-lattice antiferromagnet at $H=0.875H_s$ along the $\Gamma XM\Gamma$ path. Black lines are the magnon linewidths in the Born approximation (118). The box marks the $\mathbf{k}\mathrm{\text{−}}\omega$ region of Fig. 6 for comparison. From 118.

Figure 8

Decay region and associated singularities for $H=0.9H_s$. Solid, dashed, and dotted lines denote the decay channels into a pair of magnons with equal momenta and different momenta, and emission of an acoustic magnon, respectively. Thick portions of solid and dashed lines indicate the saddle-point singularity in the continuum, while the thin parts correspond to local minima.

Figure 9

Magnon energy and decay rate in units of $J$ for the spin-$\frac{1}{2}$ square-lattice antiferromagnet along the $M\Gamma X$ path at $H=0.9H_s$. The two upper curves show the harmonic ${\epsilon }_{\mathbf{k}}$, dashed line, and the renormalized magnon energy ${\overline{\epsilon }}_{\mathbf{k}}$, solid line. The lower curve gives the decay rate ${\Gamma }_{\mathbf{k}}$.

Figure 10

Intensity maps of the magnon decay rate ${\Gamma }_{\mathbf{k}}$ calculated within the iSCBA for several values of external magnetic field and for two values of the spin: $S=1$ (upper panels) and $S=5/2$ (lower panels).

Figure 11

Intensity plots of the dynamical structure factor $S(\mathbf{k},\omega )$, for fields $0.9H_s$ and $0.95H_s$, $S=1/2$, and for the $\mathbf{k}$ path in the $k_z=0$ plane, along the $\Gamma \to \mathrm{M}\to \mathrm{X}\to {\mathrm{X}}^{\prime }\to \Gamma$ line; see Fig. 2 for notations. From 44.

Figure 12

The ordered 120° spin structure of the triangular-lattice Heisenberg antiferromagnet.

Figure 13

Brillouin zone of the triangular lattice. The magnon decay region is shown as a shaded area. Solid and dashed lines indicate saddle-point van Hove singularities of the two-magnon continuum.

Figure 14

Magnon energy and decay rate in units of $J$ for the spin-$\frac{1}{2}$ triangular-lattice antiferromagnet. Dotted and dashed lines are the harmonic and the $1/S$ renormalized results, respectively. Solid lines are solutions of the Dyson equation (44) in the complex plane.

Figure 15

Geometry of exchange bonds for (a) symmetric spin ladder and (b) asymmetric ladder with extra diagonal coupling.

Figure 16

Structure of the energy spectrum of a quantum spin-gap magnet. The thin solid line is the bare triplon energy ${\epsilon }_{\mathbf{k}}$, the full solid line is the renormalized dispersion ${\overline{\epsilon }}_{\mathbf{k}}$, and the shaded region is the two-particle continuum. The decay threshold is marked by $k_c$ and ${\epsilon }_c$.

Figure 17

Time-of-flight inelastic neutron-scattering data measured in $\mathrm{IPA}\mathrm{\text{−}}{\mathrm{CuCL}}_3$ at $T=1.5\mathrm{K}$ (83). The arrow indicates the termination point of the magnon branch. From Andrey Zheludev.