Topologically-Protected Remnant Vortices in Confined Superfluid $^3$He

This paper reports that in confined superfluid $^3$He, remnant vortices form with a density determined by channel size rather than quench time, challenging standard Kibble-Zurek predictions by proposing that confinement prevents vortex reconnection, thereby fixing the defect separation to the wall spacing.

The Gist

The Big Picture: When Things Freeze, They Crack

Imagine you are making a giant batch of ice cream. As the liquid cools down and turns into solid ice, it doesn't freeze all at once perfectly. Instead, little pockets of ice form in different spots, growing outward like bubbles.

When these growing pockets of ice meet, they don't always line up perfectly. Sometimes, the "grain" of the ice in one pocket points north, while the neighbor points east. Where they crash into each other, you get a crack or a kink. In physics, we call these cracks defects.

Usually, scientists have a rulebook (called the Kibble-Zurek theory) that predicts how many of these cracks will form. The rule says: The faster you freeze the ice cream, the more cracks you get. If you cool it slowly, the ice has time to line up nicely, and you get fewer cracks. If you shock-freeze it, you get a mess of cracks.

The Experiment: A Tiny, Narrow Hallway

The scientists in this paper decided to test this rulebook using a very special kind of "ice cream": Superfluid Helium-3. This is a liquid so cold it acts like a quantum fluid with zero friction.

But here's the twist: They didn't let the liquid freeze in a big open bowl. They squeezed it into nano-channels—tiny hallways so narrow they are thinner than a human hair (about 1,000 times thinner).

They expected the rulebook to work. They thought: "If we cool the helium slowly, we should get a few cracks. If we cool it fast, we should get a lot."

The Surprise: The Rulebook Broke!

When they ran the experiment, something weird happened.

1. The Speed Didn't Matter: They tried cooling the helium at different speeds (slowly, quickly, very quickly). According to the old rulebook, the number of cracks should have changed. But it didn't. The number of cracks stayed exactly the same, no matter how fast they cooled it.
2. The Size of the Hallway Mattered: Instead of speed, the width of the hallway determined the number of cracks.
   • In the narrowest hallway (636 nanometers wide), they found a huge number of cracks.
   • In the widest hallway (1,067 nanometers wide), they found fewer cracks.

The New Theory: The "Hallway Constraint"

Why did this happen? The scientists came up with a new idea using a simple analogy: The Dance Floor.

Imagine a dance floor where couples (the "domains" of ordered helium) are trying to form.

• In a big room (Bulk System): Couples can form anywhere. If the music stops suddenly (the phase transition), they freeze in random positions. Where they bump into each other, they create a "defect" (a vortex). The number of bumps depends on how fast the music stopped.
• In a narrow hallway (Confined System): The hallway is so narrow that the couples can't really spread out sideways. They are forced to line up in a single file or a very thin line.
  • Because the hallway is so tight, the "couples" hit the walls almost instantly.
  • The walls act like a guide. They force the defects to form in a specific pattern based on the width of the hallway, not how fast the music stopped.
  • It's like trying to fold a long piece of paper. If the paper is very wide, you can fold it many ways. If the paper is a thin strip, you can only fold it one way. The width of the strip dictates the fold, not how fast you folded it.

The Result: "Remanent" Vortices

The defects they found are called remnant vortices. Think of them as tiny, permanent tornadoes trapped in the liquid.

Because the hallway is so narrow, these tornadoes get "stuck" or "pinned" to the walls. They can't shrink away or disappear because the walls hold them in place. This is why they remain even after the experiment is over.

Why This Matters

This discovery is a big deal because:

1. It breaks the old rule: It shows that the Kibble-Zurek theory (which works for big, open systems) doesn't work when you squeeze things into tiny spaces.
2. It helps us understand the Universe: The same math used to describe these tiny helium vortices is used to describe cosmic strings (giant defects in the fabric of space-time) and the early universe. If the rules change in tiny boxes, it might mean our understanding of how the universe formed needs a tweak, too.
3. It's about control: If we want to build future quantum computers (which use these superfluids), we need to know how to control these defects. This paper tells us that to control the defects, we need to control the size of the container, not just the cooling speed.

Summary in One Sentence

The scientists found that when you squeeze a super-cold liquid into a tiny hallway, the number of "cracks" (vortices) that form depends entirely on how narrow the hallway is, completely ignoring how fast you cooled it down, proving that geometry can override the usual laws of freezing.

Technical Summary

1. Problem Statement and Context

The paper addresses the formation of topological defects (specifically quantized vortices) during second-order phase transitions. According to the Kibble-Zurek Mechanism (KZM), the density of defects formed during a rapid quench is determined by the system's freeze-out time ($\hat{t}$). At this moment, the correlation length ($\hat{\xi}$) and relaxation time diverge, causing the system to fall out of equilibrium. The KZM predicts that the defect density ($L$) scales with the quench rate ($dT/dt$) as $L \propto \sqrt{dT/dt}$, based on the size of independent domains formed at the freeze-out time.

While KZM has been tested in various systems (liquid crystals, Bose-Einstein condensates), experiments in superfluid $^4$He have historically failed to match predictions, and results in superfluid $^3$He have been mixed. A critical gap exists in understanding how geometric confinement affects defect formation when the confinement dimension is significantly smaller than the KZM correlation length ($\hat{\xi}$).

2. Methodology

The authors designed an experiment using superfluid $^3$He confined in nanofluidic channels to probe defect formation under extreme geometric constraints.

• Experimental Setup: They utilized Helmholtz resonators fabricated with parallel-plate capacitors. These devices confined superfluid $^3$He in channels with thicknesses ($H$) of 636 nm, 805 nm, and 1067 nm.
• Confinement Scale: The channel thickness ($H \approx 10^{-3} \mu m$) is orders of magnitude smaller than the predicted KZM coherence length ($\hat{\xi} \approx 60\text{--}160 \mu m$) for the experimental cooling rates.
• Measurement Technique:
  • The devices were driven into a Helmholtz resonance mode using electrostatic forces.
  • The resonance frequency ($f_0$) was used to determine the superfluid density ($\rho_s \propto f_0^2$).
  • The linewidth ($\Delta f$) was measured to quantify energy dissipation.
  • The authors identified vortex mutual friction as the dominant dissipation mechanism in the linear regime, ruling out surface-bound states, normal fluid slip, and second viscosity.
• Calibration: By fitting the ratio $\Delta f / f_0^2$ against the theoretical mutual friction parameter $\alpha(T)$, they extracted the vortex line density ($L$).
• Variables: Experiments were conducted across a range of pressures (0–30 bar) and temperature ramp rates (0.03 to 0.36 mK/hr).

3. Key Contributions and Findings

A. Deviation from KZM Scaling

The experimental results showed a stark contradiction to standard KZM predictions:

• Magnitude: The measured vortex densities were $10^{10} \text{--} 10^{11} \text{ m}^{-2}$, which is 3 to 4 orders of magnitude higher than the KZM prediction ($L \sim 10^7 \text{--} 10^8 \text{ m}^{-2}$) for the given ramp rates.
• Ramp Rate Independence: Contrary to the KZM prediction that $L \propto \sqrt{dT/dt}$, the measured vortex density was independent of the temperature ramp rate across the tested range.

B. Geometric Truncation Model

The authors propose a new mechanism where the defect density is determined by the physical dimensions of the confinement rather than the quench rate.

• Truncation Hypothesis: Because the channel height $H$ is much smaller than the KZM length scale $\hat{\xi}$, the growth of correlation domains is "truncated" by the walls.
• Effective Length Scale: The authors suggest replacing the KZM length scale $\hat{\xi}$ with an effective truncation length of $2H$ (the distance between walls).
• New Scaling Law: The vortex density is estimated as $L \approx (2H)^{-2}$.
  • This model predicts densities within an order of magnitude of the experimental data.
  • It correctly predicts the observed trend: the 636 nm channel (most confined) had the highest defect density, while the 1067 nm channel had the lowest.

C. Nature of the Defects

• Topological Protection: The vortices are "remanent," meaning they persist after the phase transition. In the confined geometry, vortices are wall-terminated (connecting one wall to the other). Unlike vortex loops, which can shrink and annihilate, wall-terminated lines are topologically stable and cannot shrink to zero length without breaking the boundary conditions.
• Suppression of Complex Textures: Theoretical analysis in the supplementary material suggests that defects requiring gradients in the orbital angular momentum vector ($\hat{\ell}$) across the thin slab are energetically suppressed. The observed dissipation is consistent with simple, static, or collectively pinned vortex lines.

D. Pressure Dependence

While the density is independent of ramp rate, it shows a dependence on pressure. The authors attribute this to vortex recombination. As pressure decreases, the coherence length (vortex core size $\xi_0$) increases. When the average separation between vortices ($\sim 2H$) becomes comparable to the core size, vortices are more likely to recombine and annihilate shortly after formation. The data fits a model $L \propto (2H)^{-2} e^{-a(\xi_0/H)}$.

4. Significance

• Challenge to Universal KZM: This work demonstrates that the Kibble-Zurek mechanism is not universal in all regimes. When the system size is smaller than the KZM correlation length, geometric constraints dominate the defect formation process, overriding the quench-rate dependence.
• New Paradigm for Confinement: It establishes that in quasi-2D or highly confined systems, the "freeze-out" is spatially determined by the container walls rather than temporally determined by the cooling rate.
• Implications for Cosmology and Quantum Simulation: The findings suggest that in the early universe (if confined by topological structures) or in engineered quantum systems, defect densities may be controlled by geometry rather than dynamics. This offers a pathway to engineer specific defect densities in quantum fluids.
• Experimental Validation: The study provides a robust experimental framework using nanofluidic Helmholtz resonators to measure vortex densities with high precision, resolving discrepancies seen in previous bulk $^3$He and $^4$He experiments.

Conclusion

The paper concludes that in confined superfluid $^3$He, the density of topological defects is not determined by the Kibble-Zurek freeze-out time but is instead set by the channel thickness. The defects are topologically protected remanent vortices that terminate at the channel walls, resulting in a defect density that scales with $H^{-2}$ and is independent of the cooling rate. This represents a fundamental modification of phase transition dynamics in non-infinite systems.