-w \\
   \displaystyle (\pi/2){\left( x/w + 1 \right)}, & -w\le x\le w \\
   \pi, & x > w .
   \end{array}\right.  {\nonumber\\}\end{aligned}$$ For this wall profile (which is exactly one half-turn of a uniform spin spiral in the interval $-w\le x\le w$), WV reported oscillatory non-adiabatic contributions to the torque similar to our functions $a(x)$ and $b(x)$. This contrasts with the perfect adiabaticity we found in Sec. \[spinspiral\] for the infinite spin spiral. Moreover, the amplitude of the non-adiabatic torque reported by WV for this wall decreases only as $1/w$ rather than $(1/w)\exp(-\gamma w/L)$ as we found above.

The disparities between Ref.  and the present work all arise from the unphysical nature of the domain wall Eq. (\[WVwall\]). Specifically, the divergence of $\theta''(x)$ at $x=\pm w$ locates this wall at the origin of Fig. \[fig:slope\] where $\gamma=0$. This brings their result into agreement with Eq. (\[over\]). Any rounding of the discontinuity in slope at $x=\pm w$ would yield a finite value for $\theta''(x)$ and thus a non-zero value of $\gamma$. In Appendix II, we calculate the spin-transfer torque for the wall Eq. (\[WVwall\]) using our methods. Qualitatively, the pure $1/w$ behavior of the non-adiabatic torque comes from the fact that there 