The lifetimes of spin excitations in complex magnets, including quantum-disordered spin chains and ladders and noncollinear antiferromagnets, have recently been the subject of considerable attention. The relative contributions of magnon-magnon scattering and various forms of disorder to the magnon lifetime have been of particular interest, as the same mechanisms are also responsible for the decay of thermal transport currents in magnets. Remarkably, however, our understanding of magnon lifetimes in simple, collinear, large-spin antiferromagnets has remained incomplete for decades. To provide a firm foundation for current research on more complex magnetic systems, we have reexamined this issue both experimentally and theoretically.

Dynamics in Heisenberg antiferromagnets have been studied very extensively by means of theoretical approaches such as hydrodynamics, dynamical scaling, and interacting spin-wave methods. In the dilute magnon gas in two and three dimensions, the dominant damping processes are identified by diagrammatic perturbation theory to be elastic pair-wise collisions of magnons; the transition rate is given by Fermi’s Golden Rule. Analytic formulae for the relaxation as a function of wavevector *q* and temperature *T* were derived for this mechanism. Following the discovery of high-temperature superconductivity in doped two-dimensional (2D) antiferromagnets, recent theoretical work has focused specifically on the differences between the behavior of magnons in two and three dimensions. Surprisingly, in recent experimental studies of both 2D and 3D antiferromagnets, the measured linewidths disagreed considerably with predicted values, and anomalous additional line broadening at low temperatures that eluded explanation was observed. These findings provide additional motivation for a detailed investigation of the lifetimes of magnons in 2D and 3D antiferromagnets.

**Fig. 1:**Linewidth data over the full Brillouin zone in (a) Rb

_{2}MnF

_{4}and (b) MnF

_{2}. The curves show self-consistent calculations, with no free parameters, of 4-magnon scattering, added to a small, nearly

*T*-independent contribution, peaked at low

*q*, representing 2-magnon scattering from crystal blocks (with one free parameter, fitted to the 3K data).

Rb_{2}MnF_{4} and MnF_{2} are near-ideal realizations of 2D and 3D antiferromagnets, respectively. Due to their structural and chemical similarity, they are a well-suited pair of compounds for comparison of linewidths in 2D and 3D. The magnetic interactions are dominated by the antiferromagnetic exchange coupling *J* between *S*=5/2 Mn^{2+} spins; the dispersion relations in the two systems are very similar. We used the high-resolution NRSE-TAS (neutron resonance spin-echo triple-axis spectroscopy) technique to measure magnon linewidths in the *ab*-plane of these antiferromagnets (see **Figs. 1– 3**). Temperatures ranged from 3K up to 0.8*T _{N}* (Rb

_{2}MnF

_{4}) and 0.6

*T*(MnF

_{N}_{2}); the wavevectors spanned the full magnetic Brillouin zone.

We initially compared the results with analytical approximations for the magnon linewidth originating from 4-magnon (two-in/two-out) collisions. The corresponding expressions derived by Harris *et al*. [1] and Tyc and Halperin [2] apply to restricted regions of small wavevectors at low temperatures. Their general low-temperature expressions, which contain no adjustable parameters and involve summations over the Brillouin zone, are intended for use at long wavelengths. Umklapp processes, which become increasingly important as *q* and *T* increase, are not treated properly in the scattering matrix elements.

We rederived the scattering matrix elements for general *q*, treating Umklapp processes correctly , and then computed the corresponding linewidths Γ_{mag}(*q*,*T*) numerically. It became clear that "on-shell" computations do not give correct line broadenings, particularly near the zone boundaries, because of the curvature of the dispersion relation and the large density of states present there. For this reason, we evaluated Γ_{mag}(*q*,*T*) self-consistently. We incorporated experimentally-measured values of the renormalized magnon energies into the calculation, and hence our results can be applied at higher temperatures than those considered by Harris *et al*. and Tyc and Halperin. Through such calculations, we determined that incorrect treatment of Umklapp processes yields inaccurate linewidth results, even at small wavevectors and low temperatures. Independent of how these are treated, our numerical results differ considerably in magnitude from the analytical approximations, and thus numerical evaluations are indispensable even for the limiting cases of small *q* and low *T*.

**Fig. 2:**Linewidths at 3K for Rb

_{2}MnF

_{4}(red) and MnF

_{2}(blue), with the calculated 4-magnon scattering subtracted off (this is less than 0.4µeV for Rb

_{2}MnF

_{4}and 0.05µeV for MnF

_{2}, for all

*q*). The dashed curves show fits to 2-magnon scattering (see text); the fits shown by the solid curves also allow for a constant offset.

An additional potential scattering process consists of a non-momentum-conserving collision of a magnon with a boundary (e.g. a grain boundary, antiferromagnetic domain wall, or microcrack in the crystal). A rough estimate of the associated magnon lifetime for such a process is τ_{b}=*L*/(2*v*(*q*)), where *L* is the size of the grain, domain, or microcrystallite and *v(q)* is the magnon velocity. The corresponding additive contribution Γ_{b}=*h*τ_{b}^{-1} to the overall linewidth exhibits a peak at low *q* that is similar to that in the 3K data (see Fig. 2). We fitted these data to the corresponding expression after first subtracting off the calculated Γ_{mag}. Using the measured magnon dispersion data, we obtained an average *L* of 0.58±0.04µm for Rb_{2}MnF_{4} and 0.50±0.04µm for MnF_{2}. The latter is consistent with the antiferromagnetic domain size of 0.63±0.05µm obtained from independent high-resolution neutron Larmor diffraction measurements performed on the same MnF_{2} crystal at 5K.

The total calculated linewidth Γ=Γ_{mag}+Γ_{b} agrees well with the data (Figs. 1–3). In Rb_{2}MnF_{4}, the agreement worsens with increasing *T* at intermediate *q* (0.025–0.15r.l.u.); in MnF_{2}, for all *q*, and the discrepancy is largest at intermediate *q* (0.2–0.4r.l.u.). Additional scattering channels, such as 6-magnon (three-in/three-out) scattering, may become important at higher temperatures.

**Fig. 3:**(a) Linewidth data vs.

*T*in Rb

_{2}MnF

_{4}(symbols), measured by NRSE-TAS at wavevectors along the (

_{2}MnF

_{4}(open symbols) and MnF

_{2}(filled symbols). The fitted contribution from 2-magnon scattering from crystal blocks at 3K (dashed curves from Fig. 2) has been subtracted off. The lines (solid for Rb

_{2}MnF

_{4}and dashed for MnF

_{2}) are power-law fits to the data. The data and curves are color-coded by

*q/q*. Inset: the exponent corresponding to such power-law fits in Rb

_{ZB}_{2}MnF

_{4}(red symbols) and MnF

_{2}(blue symbols).

Selected linewidth data from which the fitted curves to the 3K data (Fig. 2) have been subtracted are compared in the main panel of Fig. 3b. The exponents from (phenomenological) power-law fits for these and additional values of *q/q _{ZB}* , where

*q*is the zone-boundary wavevector, are shown in the inset. The

_{ZB}*q*-dependence of the exponents is non-monotonic; the values for the two compounds agree within error at low

*q*, and diverge near the respective zone boundaries. The exponent for Rb

_{2}MnF

_{4}there is 2.6±0.1, which is not too different from Kopietz’ analytical result of 3 for the isotropic case (derived from approximations made to simplify the general expression of Tyc and Halperin at the zone boundary) [3], though the region of validity of his approximations lies lower in temperature than our data.

In summary, comprehensive comparison of experimental data and numerical results suggests that the processes that determine the magnon lifetime in 2D and 3D antiferromagnets are two-in/two-out magnon collisions and boundary scattering. The difference between 2D and 3D systems is thus quantitative, not qualitative: it originates from the difference between near-cylindrical and ellipsoidal (near-spherical) surfaces of constant magnon energy with respect to the conservation of energy and momentum in 4-magnon collisions. The seminal theories of 4-magnon scattering considered here could not previously be compared realistically and quantitatively with experimental data. This limitation is relieved here through proper inclusion of the Umklapp scattering and the use of numerical evaluations, permitting the first complete experimental and theoretical description of magnon linewidths over broad ranges of wavevector and temperature in two and three dimensions. The quantitative understanding of prototypical antiferromagnets that we have obtained provides a solid basis for investigations of the influence of noncollinearity, disorder, and doping in more complex magnetic systems. Comprehensive, momentum-resolved linewidth data that are described accurately by theory also enable the first calculation of the magnon-mediated thermal conductivity without adjustable parameters.