Novel dynamical excitations and roton-based measurement of Cooper-pair momentum in a two-dimensional Fulde-Ferrell-Larkin-Ovchinnikov superfluid on optical lattices

This paper theoretically investigates the dynamical excitations of a two-dimensional spin-polarized attractive Hubbard model on optical lattices to identify a roton-based protocol for measuring the center-of-mass momentum of Cooper pairs during the transition from a BCS to an FFLO superfluid.

The Gist

Imagine a ballroom dance floor filled with pairs of dancers. In a normal, calm dance (what physicists call a BCS superfluid), every couple holds hands and moves in perfect unison. They all stand still relative to the room, meaning their combined "center-of-mass" momentum is zero. They are perfectly paired, and no one is left alone.

Now, imagine a strong wind starts blowing across the dance floor (this is the Zeeman field). Suddenly, the dance changes. The couples don't just stand still anymore; they start drifting together in a specific direction. This new, drifting state is called an FFLO superfluid.

This paper is like a high-tech camera that films this dance floor to see exactly how the couples move when the wind blows. Here is what the researchers found, explained simply:

1. The Two Types of "Dancers"

In the normal dance (BCS), the couples are so tightly locked that it takes a lot of energy to break them apart. If you try to shake the floor, you only see the couples moving together as a group (a phonon).

But in the windy, drifting dance (FFLO), things get messy:

• The Drifting Couples: The pairs are still there, but they are moving with a specific speed and direction.
• The Solo Dancers: Because of the wind, some dancers get pushed out of their pairs. These "solo" dancers can move freely without needing a partner.
• The New Wave: Because of these solo dancers, a new type of ripple appears in the crowd, but only if you look at the "spin" (which way the dancers are facing). The researchers call this a bogolon. It's like a wave that only exists because some dancers are spinning in a different direction than the rest.

2. The "Ring" of Energy (The Roton)

In the normal dance, if you look at the energy of the moves, there is a specific spot on the dance floor where the energy is lowest, like a single dip in a bowl.

However, in the windy FFLO dance, that single dip doesn't stay in one spot. It stretches out and turns into a ring.

• The Analogy: Imagine a hula hoop lying on the floor. The dancers are most comfortable moving along the edge of that hoop.
• The Discovery: The size of this hula hoop (the ring) is exactly the same as the speed at which the couples are drifting.

3. The "Speedometer" Trick

This is the most exciting part of the paper. The researchers realized they could use that hula hoop to measure the wind speed without needing a wind gauge.

• The Problem: It's hard to measure how fast the Cooper pairs (the dancing couples) are drifting in a quantum system.
• The Solution: By looking at the "ring" of energy (the roton mode) in their data, they can measure how far the ring has shifted from the center.
• The Result: The distance the ring moves away from the center tells you exactly how much momentum the pairs have. It's like looking at a tire track on a road; the width of the track tells you how fast the car was going.

4. The "One-Way" Street

The paper also notes that this windy dance floor isn't the same in every direction.

• If you push the dancers in the direction they are drifting, they move easily.
• If you push them sideways, it's harder.
  This anisotropy (direction-dependence) is a clear sign that the system is in this special FFLO state, rather than the normal state.

5. What Happens When You Add More Dancers?

The researchers also tested what happens if you change the number of dancers on the floor (changing the "doping" or density).

• They found that the "ring" (the hula hoop) is very sensitive to how crowded the floor is.
• If you add or remove too many dancers, the ring changes shape or disappears. This means the "speedometer" trick only works best when the dance floor is perfectly full (at "half-filling").

Summary

In short, this paper uses computer simulations to predict how a special type of quantum fluid behaves when it is pushed by a magnetic field. They discovered that:

1. New types of waves appear because some particles are left without partners.
2. The energy patterns form a ring instead of a single point.
3. Most importantly: You can measure the speed of the drifting pairs simply by measuring how far that ring has shifted. This provides a new, direct way for scientists to prove that this exotic "FFLO" state actually exists in the lab.

Technical Summary

Technical Summary: Novel Dynamical Excitations and Roton-Based Measurement of Cooper-Pair Momentum in a 2D FFLO Superfluid

Problem Statement
Determining the center-of-mass (COM) momentum ($Q$) of Cooper pairs is a central challenge in identifying exotic superconducting states, specifically the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. Unlike conventional Bardeen-Cooper-Schrieffer (BCS) superfluids where $Q=0$, FFLO states possess a finite $Q$ stabilized by an external Zeeman field. While collective modes have been used to distinguish other exotic phases (e.g., Pair-Density Waves), a systematic investigation of the full dynamical excitation spectrum in two-dimensional (2D) lattice FFLO superfluids remains lacking. Specifically, the interplay between collective modes (phonons, rotons) and single-particle excitations, as well as a concrete experimental protocol to extract $Q$ from dynamical data, has not been fully established.

Methodology
The authors theoretically investigate a 2D spin-polarized attractive Fermi-Hubbard model on a square optical lattice under an effective Zeeman field. The study proceeds through the following steps:

1. Mean-Field Theory: The system is first treated within a mean-field approximation to derive Green's functions and determine the ground state parameters (chemical potential $\mu$, pairing gap $\Delta$, and COM momentum $Q$) by minimizing the thermodynamic potential. This identifies the phase transition from BCS ($Q=0$) to FFLO ($Q>0$).
2. Random Phase Approximation (RPA): To incorporate quantum fluctuations beyond the mean-field level, the authors compute the density and spin dynamical structure factors ($S_D(q, \omega)$ and $S_S(q, \omega)$) using the RPA framework. This involves constructing a $4\times4$ response function matrix coupling normal spin densities and anomalous pairing operators.
3. Numerical Simulation: Calculations are performed at half-filling ($n=1$) and various doping levels, focusing on the evolution of excitation spectra across the BCS-FFLO transition and along high-symmetry paths in the Brillouin zone.

Key Contributions and Results

• Emergence of Bogolon Modes: In the FFLO phase, the Fermi surface persists, forming a Bogoliubov Fermi surface (BG-FS). Consequently, single-particle excitations are gapless. The study reveals a distinct low-energy collective mode in the spin channel, termed the "bogolon," associated with Bogoliubov quasiparticles on the BG-FS. Unlike the phonon mode in the density channel, the bogolon exhibits a stronger signal in spin excitations due to the strong spin polarization of unpaired atoms near the Fermi surface.
• Competition and Damping: The gapless single-particle excitations in the FFLO state strongly compete with collective modes. This competition leads to spectral broadening and damping of the phonon and bogolon modes, contrasting with the well-separated modes in gapped BCS superfluids.
• Anisotropy: Due to the finite COM momentum $Q$, the dynamical excitations exhibit pronounced anisotropy in momentum space. Both the sound speed ($c_s$) and bogolon speed ($c_b$) vary with the angle relative to $Q$.
• Roton Mode Evolution and $Q$-Measurement Protocol: A central finding is the behavior of the roton mode near the $[\pi, \pi]$ point.
  • In the BCS phase, the roton minimum is a point-like feature at $[\pi, \pi]$.
  • In the FFLO phase, the roton minimum evolves into a ring structure centered at $[\pi, \pi]$ with a radius equal to the magnitude of the COM momentum $|Q|$.
  • The authors propose a roton-based measurement protocol: by detecting the displacement of the roton minimum from $[\pi, \pi]$ in the density dynamical structure factor, one can directly extract the magnitude of the Cooper-pair momentum $Q$. This correlation holds across different hopping strengths ($t$) and Zeeman fields ($h$).
• Doping Dependence: The study analyzes the doping dependence of these excitations. The excitation gap at $[\pi, \pi]$ exhibits an "open-close-reopen" behavior as doping increases (density $n$ decreases). At low densities ($n=0.6$), the collective modes and single-particle continua become well-separated again, whereas at intermediate doping, they overlap significantly.

Significance and Claims
The paper claims that these theoretical predictions provide key insights for confirming the existence of FFLO superfluids in ultracold atomic gases. Specifically:

1. Identification: The distinct anisotropic behavior and the presence of the bogolon mode serve as fingerprints to discriminate FFLO superfluids from conventional BCS superfluids.
2. Experimental Feasibility: The proposed protocol to measure $Q$ via the roton ring displacement offers a direct method to determine the Cooper-pair momentum, a quantity that is otherwise difficult to measure.
3. Theoretical Validation: The authors assert that their RPA-based strategy provides qualitatively accurate predictions for 2D lattice systems in intermediate-coupling regimes, consistent with previous findings in 3D systems and quantum Monte Carlo calculations.

The work concludes that while the competition between gapless excitations and collective modes complicates observation, the specific signatures of the roton ring and the bogolon mode offer a viable path for experimental verification using techniques such as two-photon Bragg spectroscopy.