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arXiv:2401.17488 · Physical Review A

Universal pairing-gap measurement proposal by dynamical excitations in a two-dimensional doped attractive Fermi-Hubbard model with spin-orbit coupling

Huaisong Zhao, Rui Han, Ling Qin, Feng Yuan, Peng Zou

Published Dec 30, 2024• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.110.063326

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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 q=[\ensuremathπ,\ensuremathπ]\mathbf{q}=[\ensuremath{\pi},\ensuremath{\pi}]q=[\ensuremathπ,\ensuremathπ] 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.
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 ].
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.
Abstract 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 (DOSs) 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.
We investigate the effects of attractive Hubbard interaction on two-component fermionic atoms in narrow two-dimensional (2D) energy bands that exhibit Rashba spin-orbit coupling (SOC) in the presence of an applied Zeeman field. This narrow-band 2D spin-orbit-coupled attractive Fermi-Hubbard model can potentially be realized in cold-atom systems in optical lattices with artificially engineered Rashba SOC and Zeeman field. Employing a self-consistent mean-field approximation for the pairing potential, we uncover a complex phase diagram featuring various topological superfluid (TS) phases, dependent on the chemical potential and the Zeeman field. We focus on the pairing potential and the corresponding quasiparticle gap characterizing the TS phases, which are notably small for a wide-band model with quadratic dispersion near the $\mathrm{\ensuremath{\Gamma}}$ point, as found in earlier work. We identify the parameter regimes that maximize the gap. We find that, while generally the value of the pairing potential increases with the reduction of the fermionic bandwidth, as expected for narrow- or flat-band systems, the magnitude of the topological gap characterizing the TS phases reaches a significantly improved maximum of about 10−−12.510--12.5%10−−12.5 of the interaction strength at finite values of the hopping amplitude, Rashba coupling, and Zeeman field. Our work establishes a robust operating regime for experimenting with topological superfluids in cold-atom optical lattices.
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 the single-particle spectrum, and the cross single-particle excitation is much easier to see in the spin dynamic structure factor at a small transferred momentum. Also we find a new rotonlike collective mode emerges at a fixed transferred momentum $q\ensuremath{\simeq}2{k}_{F}$, and it only appears once the system enters the topological superfluid state. The occurrence of this rotonlike excitation is related to the switch of the global minimum in the single-particle spectrum from k=0k=0k=0 to $k\ensuremath{\simeq}2{k}_{F}$.
We investigate the single-particle and collective excitations of a Rashba spin-orbit-coupled atomic Fermi gas with attractive interaction, loaded in a two-dimensional (2D) square optical lattice, in the presence of an effective out-of-plane Zeeman field. Our numerical calculations show that the many-body physics of the Bardeen-Cooper-Schrieffer (BCS) side is strongly modified compared to the Fermi gases in the free space. The physics behind this statement lies in the fact that in the 2D lattice case the single-particle ground-state energy is four times degenerated, while without a lattice structure the ground state is infinitely degenerate, and because of that, the spin-orbit coupling (SOC) in optical lattices gives rise to unusual properties entirely different from the continuum. For example, in the continuum, the pairing gap as well as the condensate fraction is strongly enhanced by the SOC strength on the BCS side. In the presence of lattice geometry the gap and the condensate fraction increase as a function of the SOC strength only at small fillings and a weak attraction limit. Moreover, we found that the speed of sound also exhibits different behavior: in the 3D and 2D continua, the slope of the Goldstone mode decreases as a function of the SOC strength, while in the lattice case the speed of sound increases monotonically with the SOC strength.