## 3 Attosecond Transient Absorption of Monolayer _h_-BN
The transition energies are computed by the difference of the Kohn–Sham eigenenergies of the highest valence band and conduction bands, \(\epsilon_{c\vec{k}}-\epsilon_{c\vec{k}^{\prime}}\) corresponding to excitations in the independent particle approximation. Blue lines show the parabolic fits of specific bands. signal to reduce the contribution from inappropriate regions by applying the following attenuation function to the transient conductivity  
\[w_{M}(\omega)= \begin{cases}1,&(\omega>\epsilon_{g,M}/\hbar),\\ \exp\left[-\frac{1}{2\sigma^{2}}(\hbar\omega-\epsilon_{g,M})^{2}\right],&\text{ otherwise.}\end{cases}  \tag{14}\]  
Here, \(\epsilon_{g,M}\) is the corresponding band gap at the \(M\)-point, which is 14.1 eV. We set the width of the attenuation \(\sigma\) to 1 eV. Figure 4a shows the truncated transient conductivity, \(\sigma^{T}(\omega,T_{probe})w_{M}(\omega)\), of the negative-mass band at the \(M\) point. One may see that the transient absorption spectrum of the negative-mass parabolic band model in Figure 4a nicely reproduces the upper-half part of the transient absorption spectrum of the full ab initio simulation in Figure 2. This fact indicates that the upper-half part of the fishbone structure originates from this specific band around the \(M\) point, which has the anomalous band dispersion with the negative reduced electron-hole mass. To reveal the other half contribution in the transient absorption spectrum of \(h\)-BN, we investigate vertical transitions around the \(\Gamma\) point. Figure 3b shows the vertical transition energies from the highest valence band to conduction bands around the \(\Gamma\) point. Among those of transitions around the energy of 13 eV at the \(\Gamma\)-point, we pick up a single band, which has the largest transition momentum for the polarization direction of the pump and probe pulses. The chosen band is fitted by a parabola that is shown as the blue dashed line in Figure 3b. The extracted band parameters are as follows: the band gap \(\epsilon_{g}\) is 13.1 eV, and the reduced electron-hole mass \(\mu_{ch}\) is \(1.1m_{e}\). Employing the parabolic two-band model with these extracted parameters, we computed the transient optical conductivity based on the pump-probe simulation. As explained above, the parabolic fit is only accurate around the band edge. Therefore, we apply the following attenuation function to the transient optical conductivity to reduce the influence from irrelevant energy regions;  
\[w_{\Gamma}(\omega)= \begin{cases}1,&(\omega<\epsilon_{g,\Gamma}/\hbar),\\ \exp\left[-\frac{1}{2\sigma^{2}}(\hbar\omega-\epsilon_{g,\Gamma})^{2}\right],& \text{otherwise.}\end{cases}  \tag{15}\]  
Here, \(\epsilon_{g,\Gamma}\) is the corresponding band gap, which is 13.1 eV. Figure 4b shows the truncated transient conductivity, \(\sigma^{T}(\omega,T_{probe})w_{\Gamma}(\omega)\), with the parabolic two-band model of the \(\Gamma\) point. One can clearly see that the result of the parabolic two-band model in Figure 4b nicely reproduces the half bottom part of the full ab initio simulation result in Figure 2.  
Figure 4: Transient surface conductivities of parabolic two-band models under the femtosecond IR pulse. The panels (**a**) and (**b**) show the result of the parabolic band shown as the blue-dashed line in Figure 3a, b, respectively. To provide the complete description, we combine the results of the two-band models for the \(M\) and \(\Gamma\) points by simply summing the transient conductivities in Figure 4a,b. The combined transient conductivity is shown in Figure 5. One sees that the combined result nicely reproduces the full ab initio result in the whole investigated photon-energy range. Therefore, we can conclude that the fishbone structure in the transient absorption spectrum of monolayer \(h\)-BN consists of two major contributions: One is the contribution from the negative-mass electron-hole band at the \(M\) point, accounting for the upper half of the fishbone structure.