## 3 Attosecond Transient Absorption of Monolayer _h_-BN  
In this section, we investigate transient optical properties of monolayer _h_-BN under intense IR pulses. For this purpose, we employ the TDDFT pump-probe approach explained in Section 2. For the pump-probe simulation, we first set the polarization direction of both the pump and probe pulses to the \(x\)-direction in Figure 1a. Thus, the polarization direction of the pump and probe pulses is perpendicular to a B-N bond. Note that _h_-BN has inversion symmetry along this direction. For the pump pulse, the peak field strength \(|\vec{E}_{pump}|\) is set to \(8.7\times 10^{8}\) V/m, the mean frequency \(\omega_{pump}\) is set to \(1.55\) eV/\(\hbar\), and the full pulse duration \(T_{pump}\) is set to \(20\) fs. The corresponding full width at half maximum (FWHM) of the pump pulse is about \(7.3\) fs. For the probe pulse, \(|\vec{E}_{probe}|\) is set to \(8.7\times 10^{7}\) V/m, \(\omega_{probe}\) is set to \(15\) eV/\(\hbar\), and \(T_{probe}\) is set to \(1\) fs. The corresponding FWHM of the probe pulse is about \(260\) as. Under these conditions, we repeat the TDDFT pump-probe simulations by changing the pump-probe time-delay, \(T_{delay}\). Figure 2a shows the real-part of the transient conductivity \(\sigma_{T}(\omega,T_{delay})\) computed by the TDDFT pump-probe simulations. The time-profile of the applied pump electric field is also shown in Figure 2b. In Figure 2a, one sees an oscillatory feature with a V-shaped energy dispersion around \(15\) eV, which we shall call _fishbone_ structure. As seen from the figure, the frequency of the oscillation in time domain is twice that of the pump frequency \(\omega_{pump}\). The \(2\omega_{pump}\)-oscillation is a direct consequence of the inversion symmetry of the material in the pump-probe direction, which forbids even-order nonlinear responses, because materials with inversion symmetry must show the same response regardless of the sign of the electric field along a symmetry direction. Similar features have been observed in attosecond transient absorption spectroscopy for semiconductors [10; 13], and transient terahertz absorption spectroscopy for GaAs quantum wells [39]. These features of bulk materials have been discussed on the basis of the pump-induced intraband transitions and the dynamical Franz-Keldysh effect [40; 41]. The dynamical Franz-Keldysh effect is a modification of optical properties of solids under the influence of oscillatory electric fields. It originates from the field-induced intraband transitions via the modification of the dynamical phase factor. In order to understand the microscopic origin of the fishbone structure in Figure 2, we investigate energies of photo-excited electron-hole pairs based on the independent particle approximation. For this purpose, we compute the vertical excitation energies from the highest valence band to the conduction bands. Note that in the independent particle approximation the transition energy from one state to another is identical to the difference of the single-particle energies of those states, \(\epsilon_{c\bar{k}}-\epsilon_{v\bar{k}}\). Figure 3 shows the computed vertical transition energies from the highest valence band to the conduction bands at each \(k\) point along high symmetry paths through Brillouin zone. Transitions around the \(M\) point are described in Figure 3a, while those around the \(\Gamma\) point are described in Figure 3b. In Figure 3a, one can see a negative electron-hole mass band around 14 eV at the \(M\) point. The negative-mass band is fitted by a parabola (blue-dashed line) to extract band parameters. The extracted reduced electron-hole mass at the \(M\)-point is \(-3.3m_{e}\). Furthermore, we numerically confirmed that all the other vertical transitions in Figure 3a at the \(M\) point are dipole forbidden.