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arXiv:2305.09685 · Chinese Physics B

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

Huaisong Zhao, Feng Yuan, Tianxing 天星 Ma 马, Peng Zou

Published Aug 15, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract The measurement of the pairing gap is crucial for investigating the physical properties of superconductors or superfluids. We propose a strategy to measure the pairing gap through the dynamical excitations. With the random phase approximation (RPA), we study the dynamical excitations of a two-dimensional attractive Fermi–Hubbard model by calculating its dynamical structure factor. Two distinct collective modes emerge: a Goldstone phonon mode at transferred momentum q = [0,0] and a roton mode at q = [ π , π ]. The roton mode exhibits a sharp molecular peak in the low-energy regime. Notably, the area under the roton molecular peak scales with the square of the pairing gap, which holds even in three-dimensional and spin–orbit coupled (SOC) optical lattices. This finding suggests an experimental approach to measure the pairing gap in lattice systems by analyzing the dynamical structure factor at q = [ π , π ].

DOI
10.1088/1674-1056/adfbd9

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Abstract The measurement of the pairing gap is crucial for investigating the physical properties of superconductors or superfluids. We propose a strategy to measure the pairing gap through the dynamical excitations. With the random phase approximation (RPA), we study the dynamical excitations of a two-dimensional attractive Fermi–Hubbard model by calculating its dynamical structure factor. Two distinct collective modes emerge: a Goldstone phonon mode at transferred momentum q = [0,0] and a roton mode at q = [ π , π ]. The roton mode exhibits a sharp molecular peak in the low-energy regime. Notably, the area under the roton molecular peak scales with the square of the pairing gap, which holds even in three-dimensional and spin–orbit coupled (SOC) optical lattices. This finding suggests an experimental approach to measure the pairing gap in lattice systems by analyzing the dynamical structure factor at q = [ π , π ].
By calculating the dynamical structure factor of the two-dimensional doped attractive Fermi-Hubbard model with Rashba spin-orbit coupling (SOC), we not only investigate collective modes and single-particle excitations of the system during the phase transition between the Bardeen-Cooper-Schrieffer superfluid and topological superfluid, but also propose a universal method to measure the pairing gap in an optical lattice system. Our numerical results show that the area of the molecular excitation peak at the transferred momentum $\mathbf{q}=[\ensuremath{\pi},\ensuremath{\pi}]$ 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. We verify 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.
We theoretically calculate the dynamic structure factor of a two-dimensional Rashba-type spin-orbit-coupled (SOC) Fermi superfluid with the random phase approximation and analyze the main characters of dynamical excitation shown by both density and spin dynamic structure factors during a continuous phase transition between a Bardeen-Cooper-Schrieffer superfluid and a topological superfluid. Generally we find three different excitations, including collective phonon excitation, two-atom molecular and atomic excitations, and pair-breaking excitations due to the two-branch structure of the quasiparticle spectrum. It should be emphasized that collective phonon excitation is overlapped with a gapless DDDDDD-type pair-breaking excitation at the critical Zeeman field hc{h}_{c}hc​ and is imparted a finite width to phonon peak when the transferred momentum q\mathit{q}q is around the Fermi vector kF{k}_{F}kF​. At a much larger transferred momentum (q=4kF)(\mathit{q}=4{k}_{F})(q=4kF​), the pair-breaking excitation happens earlier than two-atom molecular excitation, which is different from the conventional Fermi superfluid without the SOC effect.
We theoretically investigate the dynamic structure factor of a strongly interacting Fermi gas at the crossover from Bardeen–Cooper–Schrieffer superfluids to Bose–Einstein condensates, by developing an improved random phase approximation within the framework of a density functional theory (DFT)—the so-called superfluid local density approximation. Compared with the previous random-phase-approximation studies based on the standard Bogoliubov–de Gennes equations, the use of the DFT greatly improves the accuracy of the equation of state at the crossover, and leads to a better description of both collective Bogoliubov-Anderson-Goldstone phonon mode and single-particle fermionic excitations at small transferred momentum. Near unitarity, where the s -wave scattering length diverges, we show that the single-particle excitations start to significantly contribute to the spectrum of dynamic structure factor once the frequency is above a threshold of the energy gap at . The sharp rise in the spectrum at this threshold can be utilized to measure the pairing gap Δ. Together with the sound velocity determined from the phonon branch, the dynamic structure factor provides us some key information of the crossover Fermi superfluid. Our predictions could be examined in experiments with 6 Li or 40 K atoms using Bragg spectroscopy.
Interfacing unbiased quantum Monte Carlo simulations with state-of-the-art analytic continuation techniques, we obtain exact numerical results for dynamical density and spin correlations in the attractive Hubbard model, describing a spin-balanced two-dimensional cold Fermi gas on an optical lattice. We focus on half-filling, where on average one fermion occupies each lattice site, and the system displays an intriguing supersolid phase: a superfluid with a checkerboard density modulation. The coexistence of U(1)\text{U}(1)U(1) broken symmetry and the density modulations makes this regime very challenging and interesting for the calculation of dynamical properties. We compare our unbiased results with state-of-the-art generalized random phase approximation calculations: both approaches agree on a well-defined low-energy Nambu-Goldstone collective mode in the density correlations, while the higher energy structures appear to differ significantly. We also observe an interesting high-energy spin mode. We argue that our results provide a robust benchmark for generalized random phase approximation techniques, which are widely considered to be the method of choice for dynamical correlations in Fermi gases. Also, our calculations yield new physical insight into the high-energy behavior of the dynamical structure factor of the attractive Hubbard model, which is a well known prototype lattice model for superconductors and is a fertile field to target the observation of collective modes in strongly correlated systems.
Determining the center-of-mass (COM) momentum of Cooper pairs in unconventional superconductors or superfluids is a topic of great interest in condensed matter physics and ultracold atomic gases. Theoretically, we investigate the dynamical excitations of a two-dimensional spin-polarized attractive Hubbard model on a square optical lattice under an effective Zeeman field by computing the density and spin dynamical structure factors, focusing on phase transition from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid. In the FFLO superfluid, besides the phonon mode in the density channel, a low-energy bogolon mode emerges in the spin channel, which is associated with Bogoliubov quasiparticles on a Bogoliubov Fermi surface. Moreover, the dynamical excitations exhibit pronounced anisotropy in momentum space due to the finite COM momentum. At half filling, the roton mode around [ π , π ] evolves from a point-like minimum into a ring structure shifted by the COM momentum across the BCS-FFLO transition, providing a roton-based protocol to extract the COM momentum. These predictions provide key insights for confirming the existence of FFLO superfluids and understanding their dynamical excitation spectra.
Abstract Based on random phase approximation, we numerically calculate dynamical structure factors of a balanced two-dimensional (2D) Fermi superfluid, and discuss their energy, momentum and interaction strength dependence in the 2D BEC–BCS crossover. At a small transferred momentum, a stable Higgs mode is observed in the unitary 2D Fermi superfluid gas where the particle–hole symmetry is not satisfied. Stronger interaction strength will make the visibility of the dispersion of Higgs mode harder to be observed. We also discuss the dimension effect and find that the signal of the Higgs mode in two dimension is more obvious than that in 3D case. At a large transferred momentum regime, stronger interaction strength will induce the weight of the molecules excitation increasing, while in verse the atomic one decreasing, which shows the pairing information of Fermi superfluid. The theoretical results qualitatively agree with the corresponding quantum Monte Carlo data.