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

The measurement of the pairing gap plays an essential role in studying the physical properties of superconductors or superfluids. We develop a strategy for measure the pairing gap through the dynamical excitations. With the random phase approximation (RPA), the dynamical excitations of a two-dimensional attractive Fermi-Hubbard model are studied by calculating the dynamical structure factor. Two distinct collective modes are investigated: a Goldstone phonon mode at the transferred momentum ${\bf q}=\left[0,0\right]$ and a roton mode at ${\bf q}=\left[\pi,\pi\right]$. The roton mode demonstrates a sharp molecular peak in the low-energy regime. Remarkably, the area under the roton molecular peak scales with the square of the pairing gap, which persists even in three-dimensional and spin-orbit coupled (SOC) optical lattices. This result provides a potential strategy to measure the pairing gap of lattice systems experimentally by measuring the dynamical structure factor at ${\bf q}=\left[\pi,\pi\right]$.