## 3 Attosecond Transient Absorption of Monolayer _h_-BN
Therefore, the negative-mass band is expected to have a dominant contribution to the transient absorption spectrum in Figure 2.  
To elucidate the contribution from the negative-mass band at the \(M\) point, we compute the transient absorption spectrum with a parabolic two-band model. The parabolic two-band model is described in detail elsewhere [42]. In the parabolic two-band model, the electron dynamics at each \(k\) point in the Brillouin zone is described by the following Schrodinger equation for two-dimensional state vectors:  
\[i\hbar\frac{d}{dt}\left( \begin{array}{c}c_{\vec{x}}(t)\\ c_{\vec{x}}(t)\end{array} \right)=\left( \begin{array}{cc}0&h_{vc,\vec{x}}(t) \\ h_{vc,\vec{x}}^{*}(t)&0\end{array} \right)\left( \begin{array}{c}c_{\vec{v} \vec{x}}(t)\\ c_{\vec{x}}(t)\end{array} \right),\]  
where the off-diagonal matrix element of the Hamiltonian is given by  
\[h_{vc,\vec{x}}(t)=\frac{i\vec{p}_{vc,\vec{k}(t)}\cdot\vec{E}_{probe}(t)}{ \Delta\epsilon_{cv,\vec{k}(t)}}\frac{e\hbar}{m}e^{\frac{1}{\hbar}\int^{t}dt^{ \prime}\Delta\epsilon_{cv,\vec{k}(t^{\prime})}}, \tag{11}\]  
Figure 2: (**a**) real part of the transient conductivity of monolayer \(h\)-BN under the IR pump pulse for the \(x\)-direction in Figure 1a; (**b**) the time-profile of the applied pump electric field. The result is computed by TDDFT with ALDA. where \(\Delta\epsilon_{c,\vec{k}}\) is the transition energy between the valence and the conduction states at \(\vec{k}\), and \(\vec{p}_{vc,\vec{k}}\) is the transition momentum. In the Hamiltonian matrix, the Bloch wavevector is shifted based on the acceleration theorem, \(\vec{K}(t)=\vec{k}+e\vec{A}_{pump}(t)/\hbar c\), by the pump pulse [43]. Here, in order to elucidate the role of the pump-induced intraband transitions, only the probe electric field \(\vec{E}_{probe}(t)\) is coupled to the transition momentum \(\vec{p}_{vc}\), while only the pump vector potential \(\vec{A}_{pump}(t)\) is taken into account in the acceleration theorem. Thus, the probe pulse induces only the interband transitions, while the pump pulse is responsible for intraband transitions. For this model, we approximate the transition energy by the parabolic band as  
\[\Delta\epsilon_{c,\vec{k}}=\epsilon_{g}+\frac{\hbar^{2}}{2\mu_{eh}}k^{2}, \tag{12}\]  
where \(\epsilon_{g}\) is the band gap, and \(\mu_{eh}\) is the reduced mass of an electron-hole pair. To describe the negative-mass band at the \(M\) point, the reduced mass \(\mu_{eh}\) is set to \(\mu_{M}=-3.3m_{e}\), and the band gap \(\epsilon_{g}\) is set to \(\epsilon_{g,M}=14.1\) eV. These parameters are extracted from the parabolic fit shown as the blue-dashed line in in Figure 3.  
The negative-mass band causes an artificial crossing of the conduction and the valence bands. To eliminate the effect from the artificial band crossing of the model, we truncate the transition momentum matrix as follows:  
\[p_{vc,\vec{k}}= \begin{cases}p_{vc},&(\epsilon_{g,M}\leq\Delta\epsilon_{cv, \vec{k}}),\\ p_{vc}\sin^{2}\left(\frac{\pi\Delta\epsilon_{cv,\vec{k}}}{2\epsilon_{g}} \right),&(0<\Delta\epsilon_{cv,\vec{k}}<\epsilon_{g}),\\ 0,&\text{otherwise}.\end{cases}  \tag{13}\]  
Here, the transition momentum at the gap \(p_{vc}\) is set to 0.1 a.u. Note that the value of the transition momentum does not affect the structure of absorption spectra as long as the field strength of the probe pulse is weak enough to satisfy the linear response condition. We apply a pump-probe simulation to the parabolic two-band model and compute the transient absorption spectrum of the negative-mass band. Since the parabolic-band approximation is an appropriate approximation of the electronic structure only around the band edge, we truncate the  
Figure 3: Red lines show vertical transition energies from the highest valence band to conduction bands in monolayer \(h\)-BN around (**a**) the \(M\) point and (**b**) the \(\Gamma\) point.