Title: Dynamical structure factor and a new method to measure the pairing gap in two-dimensional attractive Fermi-Hubbard model

Abstract: By calculating the dynamical structure factor along the high symmetry directions in the Brillouin zone, the dynamical excitations of attractive Fermi-Hubbard model in a two-dimensional square optical lattice are studied with random phase approximation. {Two kinds of collective modes are investigated, including a Goldstone phonon mode at transferred momentum ${\bf q}=\left[0,0\right]$ and a roton mode at ${\bf q}=\left[\pi,\pi\right]$. The phonon origins from the spontaneously U(1) symmetry breaking of pairing gap, and its speed is suppressed by the interaction strength. The collective roton mode origins from the breaking of a global pseudospin SU(2) symmetry.} Dynamical excitations at ${\bf q}=\left[\pi,\pi\right]$ consist of a sharp roton molecular peak in the low-energy region and a broad atomic excitation band in the higher energy region. Furthermore, the weight of the roton molecular peak decreases monotonically with increasing the hopping strength, while the weight of the atomic excitations increases quickly. Interestingly we check that the area covered by the roton molecular peak scales with the square of the pairing gap, which is also true in the system with spin-orbit coupling. This conclusion paves a potential way to measure the pairing gap of lattice system experimentally by measuring the dynamical structure factor at ${\bf q}=\left[\pi,\pi\right]$.