Vortices in Two-Dimensional Chiral Superfluids

This paper investigates the orbital angular momentum of two-dimensional chiral superfluids with multiply quantized vortices using Bogoliubov-de Gennes theory, revealing that while the angular momentum follows a universal formula in the BEC regime, it is significantly suppressed in the BCS regime due to spectral asymmetry and unpaired fermions, with the degree of reduction depending on the specific pairing symmetry and vortex vorticity.

The Gist

Imagine a superfluid as a giant, invisible dance floor where particles (fermions) pair up to move in perfect unison. In a "chiral" superfluid, these pairs don't just dance; they spin in a specific direction, like a synchronized line of dancers all turning clockwise. This paper investigates what happens when you introduce a "twist" or a "vortex" into this dance floor—a whirlpool where the dancers spin around a central point.

The authors, Yan He and Wenxing Nie, are asking a simple but tricky question: If we spin this dance floor, how much total "spin" (Orbital Angular Momentum, or OAM) does the whole system have?

Here is the breakdown of their findings using everyday analogies:

1. The Two Dance Styles: The "Easy" Way vs. The "Hard" Way

The paper looks at two different regimes (conditions) for the dancers:

• The BEC Regime (The "Tight-Knit" Dance): Imagine the dancers are holding hands so tightly they act like a single, solid unit. In this state, the math is simple. If you have a vortex spinning with strength $k$ and the dancers naturally spin with strength $\nu$, the total spin of the room is exactly what you'd expect: $(k + \nu)$ times the number of dancers. It's a perfect, predictable calculation.
• The BCS Regime (The "Loose" Dance): Now imagine the dancers are holding hands loosely, just barely connected. They are more independent. In this state, things get messy. The paper finds that the total spin is often less than the "perfect" number calculated above.

2. The Mystery of the Missing Spin

Why does the spin disappear in the "loose" dance? The authors use a concept called Spectral Asymmetry (or "Spectral Flow").

Think of the energy levels of the dancers as a set of stairs. In a perfect world, for every dancer going up a step, another goes down, keeping the balance. But in these superfluids with vortices, the stairs get messed up. Some dancers get "stuck" on the stairs or end up unpaired.

• The Unpaired Fermions: These are the dancers who lost their partners. Instead of spinning with the group, they spin in the opposite direction.
• The Cancellation: These "rogue" dancers spin backward, canceling out some of the forward spin of the paired dancers. This is why the total spin drops.

3. The Different Types of Twists (Vortices)

The paper tests two main variables: how strong the pairing is (p-wave, d-wave, etc.) and how strong the vortex is (single twist vs. multiple twists).

• The "Perfect" Twist (Single Twist, p-wave):
  If the dancers are doing a simple "p-wave" dance (spinning once) and the vortex is a single twist ($k=1$), the system behaves beautifully. Even in the "loose" dance regime, the total spin remains perfect. The "rogue" dancers don't appear to cancel anything out.

  • However, there is a twist within the twist: If the vortex spins the opposite way ($k=-1$), the total spin becomes zero. But the paper notes that even though the total is zero, the distribution of spin is complex. It's like a room where half the people are spinning left and half are spinning right, canceling out globally, but locally, the movement is very active and different from a calm room.

• The "Messy" Twists (Multiple Twists or Complex Dances):
  If you make the vortex spin twice or more ($|k| \ge 2$) OR if the dancers do a more complex dance (like d-wave, where they spin twice naturally), the "rogue" dancers appear.

  • Multiple Twists ($|k| \ge 2$): The "rogue" dancers gather in the very center of the vortex (the core). Their backward spin is moderate but depends on how big the core is.
  • Complex Dances ($\nu \ge 2$): The "rogue" dancers gather near the walls of the room (the edge). Their backward spin is sharp and significant.

4. The "Counter-Flow" Surprise

One of the most interesting findings is the existence of counter-flows.
Imagine the main dance floor is spinning clockwise. The paper found that in certain complex scenarios, there are small pockets of dancers spinning counter-clockwise.

• In the center of a strong vortex, some dancers spin backward.
• Near the walls of the room, other dancers spin backward.
  These backward-spinning pockets are the "unpaired fermions" mentioned earlier. They act like a brake, reducing the total spin of the system.

Summary

The paper essentially says:

1. Simple is predictable: If you have a simple dance and a simple twist, the total spin is exactly what you calculate.
2. Complexity creates chaos: If you add more twists or make the dance more complex, "rogue" dancers appear.
3. The rogues cancel the spin: These unpaired dancers spin the wrong way, reducing the total spin of the system.
4. Location matters: Depending on whether the twist is strong or the dance is complex, these "rogues" hide either in the center of the vortex or near the walls.

The authors didn't propose any new machines or medical uses for this; they simply mapped out exactly how these quantum dancers behave when you spin the room, proving that the "perfect" spin only happens under very specific, simple conditions.

Technical Summary

Technical Summary: Vortices in Two-Dimensional Chiral Superfluids

Problem Statement
This study investigates the orbital angular momentum (OAM), denoted as $L_z$, of two-dimensional chiral superfluids (SFs) characterized by $(p_x + ip_y)^\nu$-wave pairing symmetry. Specifically, the authors examine the system in the presence of an axisymmetric multiply quantized vortex (MQV) with vorticity $k$ on a disk at zero temperature. The central problem addresses the "orbital angular momentum paradox," where different theoretical approximations yield conflicting estimations for the total OAM. While a naive picture suggests $L_z = \nu N/2$ (where $N$ is the total number of fermions and $\nu$ is the pairing order), Fermi liquid theory and recent Bogoliubov-de Gennes (BdG) studies suggest significant suppression in the weak-pairing Bardeen-Cooper-Schrieffer (BCS) regime due to spectral asymmetry and unpaired fermions. The paper seeks to determine how the coexistence of chiral pairing and vortex circulation affects the total OAM and its spatial distribution, particularly distinguishing between singly quantized vortices ($|k|=1$) and MQVs ($|k|>1$) across different pairing symmetries ($\nu=1$ for $p+ip$, $\nu=2$ for $d+id$).

Methodology
The authors employ the framework of the Bogoliubov-de Gennes (BdG) theory for a two-dimensional chiral fermionic superfluid confined within an infinite circular well (a disk with a specular wall).

• Hamiltonian: The system is described by a BCS mean-field Hamiltonian with a symmetrized pairing operator $\hat{\Delta} \sim (\hat{p}_x + i\hat{p}_y)^\nu$.
• Vortex Configuration: A vortex with integer vorticity $k$ is placed at the center of the disk. The gap function is modeled as $\Delta(r) = \Delta_0 \tanh(r/\xi)e^{ik\theta}$, where $\xi$ is the coherence length.
• Numerical Solution: The BdG equations are solved by expanding the quasiparticle wavefunctions ($u_n, v_n$) in a basis of Bessel functions, which are eigenfunctions of the kinetic energy operator in a circular well. This reduces the problem to a matrix eigenvalue equation.
• Observables: The authors calculate the particle density $n(r)$, the azimuthal mass current, and the local OAM distribution $L_z(r)$. The total OAM is derived from the ground state expectation values.
• Spectral Asymmetry: A key analytical tool is the spectral flow (or spectral asymmetry) $\eta_l$, defined as the difference between the number of positive and negative energy eigenvalues for a given angular momentum channel $l$. The authors utilize a generalized Bogoliubov transformation to relate the deviation of the total OAM from its "full" value to this spectral asymmetry.

Key Contributions and Results
The study reveals distinct behaviors of OAM depending on the pairing order $\nu$, the vortex vorticity $k$, and the coupling regime (BEC vs. BCS):

1. BEC Regime: In the Bose-Einstein Condensation regime ($\mu/E_F < 0$), the energy spectrum is fully gapped with no in-gap states. Consequently, the spectral flow $\eta_l$ is zero for all $l$. The total OAM follows the "full" value:
   $$L_z = \frac{(k + \nu)N}{2}$$
   This holds for any integer $\nu$ and $k$, indicating that every Cooper pair contributes an angular momentum of $k+\nu$.

2. BCS Regime ($p+ip$ with $k=\pm 1$): For chiral $p+ip$ superfluids ($\nu=1$) with singly quantized vortices ($k=\pm 1$), the spectral flow remains zero. The total OAM retains the full value $L_z = (k+\nu)N/2$.

   • For $k=1$, $L_z = N$.
   • For $k=-1$ (antivortex), $L_z = 0$. Although the total OAM is zero, the spatial distribution $L_z(r)$ is non-trivial: the negative contribution from the antivortex core exactly cancels the positive edge current contribution from the $p+ip$ pairing.

3. BCS Regime ($\nu \ge 2$ or $|k| \ge 2$): Significant deviations occur when the pairing order is higher than $p$-wave ($\nu \ge 2$) or the vortex vorticity is greater than 1 ($|k| \ge 2$).

   • Spectral Flow: In these cases, in-gap states (edge modes) appear, leading to non-zero spectral flow $\eta_l$.
   • OAM Suppression: The total OAM is remarkably reduced from the "full" value. The suppression is attributed to unpaired fermions in the ground state, which carry OAM opposite to that of the Cooper pairs.
   • Mechanism of Suppression:
     • For $\nu \ge 2$ (e.g., $d+id$ with $k=1$), the suppression is associated with edge modes near the system boundary.
     • For $|k| \ge 2$ (e.g., $p+ip$ with $k=2$), the suppression is moderate and core-size dependent, arising from in-gap vortex modes.
   • Counter-flows: The presence of unpaired fermions manifests as "counter-flows" (negative OAM currents) either in the vortex core (for $|k| \ge 2$) or near the boundary (for $\nu \ge 2$). These counter-flows explain the reduction in total OAM.

Significance and Claims
The paper claims to resolve the interplay between vortex-induced circulation and chiral pairing in determining the macroscopic OAM of superfluids. The authors demonstrate that the "full" OAM value is robust only when the system lacks spectral asymmetry (i.e., no in-gap edge modes crossing the Fermi level).

• They establish that the reduction of OAM in the BCS regime is fundamentally linked to spectral flow and the existence of unpaired fermions in the ground state of the BdG Hamiltonian.
• The study highlights a qualitative difference between singly quantized vortices and MQVs in chiral superfluids, noting that MQVs induce core-dependent counter-flows that suppress OAM.
• The work provides a unified explanation for OAM suppression in both high-order pairing ($\nu \ge 2$) and high-vorticity ($|k| \ge 2$) scenarios, attributing the phenomenon to the same underlying mechanism: spectral asymmetry leading to unpaired fermions.

The authors do not propose new experimental setups or future applications but frame their results as a theoretical clarification of topological properties (spectral flow, edge modes) in chiral superfluids with vortices, supporting previous findings regarding the orbital angular momentum paradox.