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By calculating dynamical structure factor of two-dimensional doped attractive Fermi-Hubbard model with Rashba spin-orbit coupling, we not only investigate collective modes and single-particle excitations of the system during the phase transition between Bardeen-Cooper-Schrieffer superfluid and topological superfluid, but also propose a universal method to measure pairing gap measurement in an optical lattice system. Our numerical results show that the area of the molecular excitation peak at the transferred momentum ${\bf q}=\left[\pi,\pi\right]$ is proportional to the square of the pairing gap in the system with Rashba SOC. In particular, this method is very sensitive to the pairing gap. This goes on verifying that this method is universal to measure the pairing gap in a doped optical lattice with Rashba SOC. These theoretical results are important for experimentally measuring the pairing gap and studying the topological superfluid in an optical lattice., Comment: 12 pages, 8 figures

We theoretically investigate a one-dimensional Fulde-Ferrell Fermi superfluid at a finite effective Zeeman field $h$, and study entire dynamical excitations related to density perturbation. By calculating the density dynamic structure factor, we find anisotropic dynamical excitations in both collective modes and single-particle excitations. Along the direction of centre-of-mass momentum $p$, there are two obvious gapless collective modes with different speed. The lower collective modes is from the usual gauge symmetry breaking and has a larger speed than the one in the negative direction of $p$. The higher one is due to the direction spontaneous symmetry breaking of centre-of-mass momentum $p$, and separates two kinds of single-particle excitations in the positive $p$ direction. However, this higher mode disappears in the opposite direction of $p$, where two single-particle excitations overlap with each other. These signals of dynamical excitations can do help to distinguish Fulde-Ferrell superfluid from the conventional Bardeen-Cooper-Schrieffer superfluid in the future experiment., Comment: 7 pages, 5 figures

We theoretically calculate the dynamic structure factor of two-dimensional Rashba-type spinorbit coupled (SOC) Fermi superfluid with random phase approximation, and analyse the main characters of dynamical excitation sh own by both density and spin dynamic structure factor during a continuous phase transition between Bardeen-Cooper-Schrieffer superfluid and topological superfluid. Generally we find three different excitations, including collective phonon excitation, two-atom molecular and atomic excitations, and pair-breaking excitations due to two-branch structure of quasi-particle spectrum. It should be emphasized that collective phonon excitation is overlapped with a gapless DD type pair-breaking excitation at the critical Zeeman field hc, and is imparted a finite width to phonon peak when transferred momentum q is around Fermi vector kF. At a much larger transferred momentum (q = 4kF ), the pair-breaking excitation happens earlier than two-atom molecular excitation, which is different from the conventional Fermi superfluid without SOC effect., Comment: 10 pages, 8 figures. arXiv admin note: text overlap with arXiv:2210.10407

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 ${\bf q}=\left[\pi,\pi\right]$., Comment: 12 pages, 9 figures

Based on the two-dimensional (2D) attractive Fermi-Hubbard model with Rashba spin-orbit coupling (SOC), the SOC strength and Zeeman field dependences of the phase diagram are investigated by calculating the pairing gap self-consistently. The results reveal that the phase transition from the BCS superfluid to the topological superfluid happens under proper Zeeman field strength and SOC strength. In particular, in contrast to the BCS superfluid decreasing monotonically as the SOC strength increasing, the topological superfluid region shows a dome with the SOC strength increasing. An optimal region in the phase diagram to find the topological superfluid can be found, which is important to realize the topological superfluid in optical lattice experimentally. Then we obtain the change of both band structure and density of states (DOS) during the topological phase transition, and explain the four peaks of DOS in the topological superfluid by the topology change of the low-energy branch of quasiparticle energy spectra. Moreover, the topological superfluid can be suppressed by the doping concentration., Comment: 15 pages,9 figures

We theoretically calculate the density dynamic structure factor of one-dimensional Fermi superfluid with Raman-type spin-orbit coupling, and analyze its main dynamical character during phase transition between Bardeen-Cooper-Schrieffer superfluid and topological superfluid. Our theoretical results display four kinds of single-particle excitations induced by the two-branch structure of single-particle spectrum, and the cross single-particle excitation is much easier to be seen in the spin dynamic structure factor at a small transferred momentum. Also we find a new roton-like collective mode emerges at a fixed transferred momentum $q \simeq 2k_F$, and it only appears once the system enters the topological superfluid state. The occurrence of this roton-like excitation is related to switch of global minimum in single-particle spectrum from $k=0$ to $k \simeq 2k_F$.