Magnon pair emission from a relativistic domain wall in antiferromagnets

Magnon emission and excitation by a relativistic domain wall at a constant velocity in antiferromagnets is theoretically studied. A pair emission due to a quadratic magnon coupling is shown to be dominant. The emission corresponds in the comoving frame to a vacuum polarization induced by a zero-energy instability of the Lorentz-boosted anomalous response function. The emission rate is sensitive to the magnon dispersion and wall profile, and is significantly enhanced for a thin wall with velocity close to the effective light velocity. The Ohmic damping constant due to magnon excitation at low velocity is calculated.

Figure 1

Schematic figure showing magnon processes due to moving domain wall (black curved line) with a velocity $v_{\mathrm{w}}$; there is a scattering, with the amplitude $〈a^{†}a〉$, a pair emission ($〈a^{†}{a}^{†}〉$) and a pair annihilation ($〈aa〉$). Red and blue arrows denote created and annihilated magnons, respectively. A wave vector $k$ is that of one of the magnons, and $q$ is the transferred wave vector from the wall.

Figure 2

Effect of Lorentz boost on the anomalous response function $\mathrm{Im}\left[{\mathrm{\Gamma }}_{q,\mathrm{\Omega }}\right]$ at $v_{\mathrm{w}}/c=0.8$ for a hyperbolic dispersion ($\mu =5$), $\widetilde{{\lambda }_0}=2, \tilde{\mathrm{\Delta }}=0.1, \tilde{\eta }=0.01$, and $\tilde{T}=0.2$ in dimensionless arbitrary units. Blue is the amplitude at rest frame, which is localized at $\mathrm{\Omega }\gtrsim 2\mathrm{\Delta }$ with negligibly small weight at $\mathrm{\Omega }=0$. In the boosted frame, shown in red, the amplitude extends to the zero-energy regime at finite $q$, inducing spontaneous vacuum polarization, which corresponds to a pair emission in the laboratory frame. The right plot is a global plot.

Figure 3

Energy conservation conditions in (a) scattering and (b) emission and absorption of two magnons. The emission is regarded as a scattering from a hole state with energy $-{\omega }_k$ to a particle state with energy ${\omega }_{-k+q}$. The slope of dotted straight lines is $v_{\mathrm{w}}$.

Figure 4

Dispersions considered: relativistic (rel; black line) and hyperbolic dispersions with $\mu =5,10$. The maximum slope (maximum group velocity) is $c(=1)$ and the gap is $\tilde{\mathrm{\Delta }}=0.1$ for both cases.

Figure 5

Wall profiles, ${sin}^2\theta \left(x\right)$, for tanh and linear walls, and their Fourier transforms (form factors) $W_q$. Linear wall has slow power-law decay at large $q$, resulting in the enhancement of the anomalous response function (see Figs. 6 and 9).

Figure 6

(a) $\mathrm{Im}{\mathrm{\Gamma }}_q$ for relativistic (dotted lines) and hyperbolic (solid lines) dispersion for $v_{\mathrm{w}}/c=0.6,0.7,0.8,0.96$ with $\mu =5$ and ${\lambda }_0/a=4$. At high velocity the hyperbolic dispersion shows a sharp peak at finite $q(\equiv q^{*})$, while the response function for the relativistic dispersion, with low and broad peak near $q=0$, is not strongly affected by the velocity. (b) The peak amplitude $I^{*}$ and position $q^{*}$ (black dotted line) of $\mathrm{Im}{\mathrm{\Gamma }}_q$ for different ${\lambda }_0/a$. Wall profiles are linear (solid lines) and tanh (dashed lines). The linear wall has larger amplitudes due to a slower decay of $W_q$ at large wave vector $q$. The breakdown velocities $v_c$ of the continuum approximation are shown as vertical dotted lines.

Figure 7

Plot of the real and the imaginary parts of response functions ${\mathrm{\Pi }}_q$ and ${\mathrm{\Gamma }}_q$ for relativistic (rela) dispersion and a hyperbolic (hyp) dispersion with $\mu =5$. Calculated for $v_{\mathrm{w}}/c=0.2,0.8,0.9$ at $\tilde{T}=0.8$ with $\tilde{\lambda }(\equiv {\lambda }_0/a)=2$ and $\tilde{\mathrm{\Delta }}=0.1, \tilde{\eta }={10}^{-2}$. The effect of temperature is shown for relativistic dispersion at $\tilde{v}=0.8$. The amplitude is reduced by about a factor of 2 at lower temperature of $\tilde{T}=0.4$ compared to $\tilde{T}=0.8$.

Figure 8

Comparison of ${\mathrm{\Pi }}_q$ and ${\mathrm{\Gamma }}_q$ for ${\tilde{\lambda }}_0=2$ and 4 for hyperbolic dispersion with $\mu =5$.

Figure 9

Comparison of $\mathrm{Im}{\mathrm{\Gamma }}_q$ for the wall profiles tanh (tanh w) and linear (linear w) walls for ${\tilde{\lambda }}_0=4,6,8$ and for hyperbolic dispersion with $\mu =5$. Peak positions are shown by small circles.

Figure 10

(a) Plot of the force $F$ as functions of wall velocity for tanh and linear wall with ${\lambda }_0/a=4$. The normal ($\mathrm{\Pi }$) and anomalous ($\mathrm{\Gamma }$) contributions to the force are shown by dashed and solid lines with axis at the left and right, respectively. Shaded region ($\tilde{v}>0.97$) shows a breakdown of continuum description, i.e., $\lambda \lesssim a$, for ${\lambda }_0/a=4$. (b) Friction constant $\alpha$ (left axis) and contribution to the Gilbert damping constant (right axis) determined at small wall velocity ($\tilde{v}\sim 0$) as a function of wall thickness.

Figure 11

Plot of the force $F$ for tanh wall with ${\lambda }_0/a=4$ for different temperatures. The normal ($\mathrm{\Pi }$) and anomalous ($\mathrm{\Gamma }$) contributions to the force are shown by dashed and solid lines, respectively.

Figure 12

Normal response function at $v_{\mathrm{w}}/c=0.8$ plotted in the $k-q$ plane for ${\tilde{\lambda }}_0=2, \tilde{\mathrm{\Delta }}=0.1, \tilde{\eta }=0.01$, and $\tilde{T}=0.8$. Blue straight lines, $k=0.13$ and $k+q=0.13$, represent the wave vectors $k$ and $k+q$, respectively, where the group velocity $v_{\mathrm{g}}\left(k\right)=\frac{\partial {\omega }_k}{\partial k}$ is equal to the wall speed, 0.8. Shaded regime below those lines is therefore magnon excitations behind the wall.

Figure 13

The Lorentz boost effects on the imaginary parts of the normal (upper row) and the anomalous (lower row) response functions, $\mathrm{Im}\left[{\mathrm{\Pi }}_{(q+v\mathrm{\Omega })/\gamma ,(\mathrm{\Omega }+vq)/\gamma }\right]$ and $\mathrm{Im}\left[{\mathrm{\Gamma }}_{(q+v\mathrm{\Omega })/\gamma ,(\mathrm{\Omega }+vq)/\gamma }\right]$, respectively, for the relativistic dispersion at $v/c=0,0.4,0.8$ from the left to the right.