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We propose a new approach to extract the important degrees of freedom inquantum dynamics induced by an external stimulus. We calculate the coefficientmatrix numerically, where the $i-l$ element of the matrix is the coefficient ofthe lth basis state at the ith discretized time in the solution of thetime-dependent Schr\"odinger equation induced by the external stimulus. Byperforming a randomized singular value decomposition of the coefficient matrix,a practically exact solution is obtained using a linear combination of theimportant modes, where the number of modes is much smaller than the dimensionsof the Hilbert space in many cases. We apply this method to analysis of thelight absorption spectrum in two-dimensional (2D) Mott insulators using aneffective model of the 2D Hubbard model in the strong interaction case. Fromthe dynamics induced by an ultrashort weak light pulse, we find that thepractically exact light absorption spectrum can be reproduced by as few as 1000energy eigenstates in the $1.7 \times 10^{7}$-dimension Hilbert space of a26-site cluster. These one-photon active energy eigenstates are classified intofree holon and doublon (H-D) and localized H-D states. In the free H-D states,the main effect of the spin degrees of freedom on the transfer of a holon (H)and a doublon (D) is the phase shift, and the H and the D move freely. In thelocalized H-D states, an H and a D are localized with relative distances of$\sqrt{5}$ or $\sqrt{13}$. The antiferromagnetic (AF) spin orders in thelocalized H-D states are much stronger than those in the free H-D states, andthe charge localization is of magnetic origin. There are sharp peaks caused byexcitations to the localized H-D states below the broad band caused byexcitations to the free H-D states in the light absorption spectrum.

New approach to extract important degrees of freedom in quantum dynamics  using singular value decomposition: Application to linear optical spectrum in  two-dimensional Mott insulators