Fig. 4: Band structure for realizing lasing edge modes with nonzero group velocities.

a Combining the Qi-Wu-Zhang (QWZ) model with large hoppings and the time-reversal QWZ model by a non-Hermitian coupling, we obtain the model with the edge band structure shown in the main panels. The edge dispersions between the pairs of EPs (red points) exhibit the nonzero imaginary part of the energy and the nonzero slope of the real part of the energy. Since the slope of the real part of the energy corresponds to the group velocity of the edge mode, these lasing edge modes have nonzero group velocity and propagate along the edge of the sample. The inset presents the enlarged view of the low momentum region indicated by the green dashed box and the red points represent the EPs in the edge dispersions. The parameters used are u = −1, β = 0.2, and \(\beta ^{\prime} =0.1\). b To obtain the lasing edge modes with nonzero group velocity, we utilize two edge modes which have opposite signs and different absolute values of the slope of the dispersion relation. By combining these edge modes by a non-Hermitian coupling term, we obtain exceptional edge modes, which can be applied to construct a topological insulator laser while Hermitian couplings open the gap owing to the avoided crossing.