Quantum critical collapse of a pinned vortex glass

Using plasmonic microwave spectroscopy on amorphous indium oxide films, this study reveals that a pinned vortex glass exhibits an anomalously slow, logarithmic decay of superfluid density driven by collective pinning enhanced by interactions, ultimately vanishing linearly at a continuous quantum critical point that governs the field-driven superconductor-insulator transition.

The Gist

Imagine a superconductor as a perfectly organized dance floor where pairs of electrons (the dancers) move in perfect unison without ever bumping into each other or losing energy. This is the state of "superconductivity."

Now, imagine two things trying to ruin this dance:

1. Disorder: The floor is covered in random obstacles (like spilled drinks or uneven tiles).
2. Magnetic Field: A strong wind is blowing across the floor, trying to push the dancers apart.

In a normal dance floor, the wind would create little whirlpools (called vortices) that spin around the dancers, causing chaos and stopping the dance. Usually, scientists thought that as you increased the wind (magnetic field), these whirlpools would multiply rapidly, the dancers would get stuck on the obstacles, and the superconductivity would crash down quickly.

The Big Surprise
This paper reports a discovery that completely changes that story. The researchers looked at a very "messy" superconductor (amorphous indium oxide) and found something unexpected:

Instead of the dance floor collapsing quickly as the wind got stronger, the dancers held on incredibly well. Even as the magnetic field increased by a factor of 1,000, the "superfluid" (the ability of the dancers to move together) only decreased very slowly, like a logarithmic slide rather than a steep cliff.

The "Cage" Analogy
Why did they hold on so well? The paper suggests a counter-intuitive reason.

Usually, we think of the obstacles (disorder) as the only thing stopping the whirlpools (vortices) from moving. But in this messy material, the whirlpools themselves started helping each other.

• The Old Idea: Whirlpools repel each other, which usually makes them harder to pin down.
• The New Discovery: In this specific "glassy" state, the whirlpools repel each other so strongly that they form a protective cage around one another.

Think of it like a crowd of people in a mosh pit. If everyone is pushing against each other, they actually get stuck in place because they can't move without pushing their neighbor. The "cage" formed by the whirlpools makes it much harder for them to move, effectively "pinning" them in place and protecting the superconductivity for much longer than expected.

The Final Collapse
Eventually, the wind (magnetic field) gets too strong. The researchers found that when the superconductivity finally breaks, it doesn't happen all at once. Instead, it fades away linearly, like a dimmer switch being slowly turned off, until it hits a critical point where the dance floor becomes an insulator (a place where no dancing happens at all).

The "Super-Stiff" Response
The paper also discovered a weird side effect. When they shook the system with microwaves (like shaking the dance floor), the whirlpools didn't just get loose; they actually got stiffer.

• Analogy: Imagine shaking a jar of jelly. Usually, shaking it makes it wobble more. Here, shaking the vortex glass made it act like a stiffer, more solid object. This is called a "positive Kerr effect," and it's a unique signature of this specific type of vortex glass.

Why It Matters (According to the Paper)
The authors conclude that this "pinned vortex glass" is the key intermediate state that controls how superconductors fail in a magnetic field. It solves a long-standing mystery about why some superconductors behave so differently when disorder is high.

They also note that because these materials can handle huge magnetic fields and have this unique "stiffening" response, they could be useful for quantum sensing (detecting very faint signals) and for building circuits that interact strongly with quantum systems, but the paper focuses primarily on explaining the physics of this collapse rather than detailing specific future devices.

In Summary:
The paper shows that in a very messy superconductor, the magnetic whirlpools don't destroy the superconductivity quickly. Instead, they trap each other in a "cage," allowing the superconductivity to survive much longer than anyone predicted, before finally fading away in a smooth, continuous transition.

Technical Summary

Technical Summary: Quantum Critical Collapse of a Pinned Vortex Glass

Problem Statement
The interplay between disorder and vortex-vortex interactions in strongly disordered superconductors under a magnetic field is known to stabilize a vortex-glass state, characterized by strong pinning and the absence of positional order. However, the specific role of this state in the destruction of superconductivity at the field-driven superconductor-insulator transition (SIT) has remained unresolved. A primary obstacle has been the lack of experimental probes capable of accessing the equilibrium superfluid stiffness deep in the mixed state and up to the critical field ($B_c$). Conventional DC transport measurements are dominated by vortex flow dissipation, providing limited insight into superconducting rigidity. Consequently, fundamental questions remain: How does the zero-temperature superfluid stiffness evolve with the magnetic field in a strongly disordered superconductor? Does it vanish linearly at the quantum critical point, as suggested by critical current measurements, or does it follow a different scenario? Furthermore, it is unclear whether the field-driven SIT follows the same mechanism as the disorder-driven SIT observed at zero field (which was recently identified as a first-order transition) or represents a fundamentally different route to quantum breakdown.

Methodology
To address these questions, the authors employed plasmonic microwave spectroscopy on high-kinetic-inductance resonators patterned from amorphous indium oxide (a:InO) thin films. This technique allows for the direct probing of equilibrium superfluid stiffness while remaining insensitive to vortex flow dissipation.

• Device Architecture: L-shaped a:InO resonators were capacitively coupled to a 50 $\Omega$ impedance-matched gold feedline. An additional a:InO mesa on the same chip facilitated in situ DC transport measurements (resistance and critical current).
• Measurement Protocol: The study utilized two-tone microwave spectroscopy to track the frequency shift of the fundamental resonance as a function of magnetic field (up to multi-tesla) and temperature (down to 10 mK). The superfluid stiffness $\Theta(B)$ was extracted directly from the resonance frequency shift, $\Theta(B)/\Theta(0) = [f(B)/f(0)]^2$.
• Comparative Analysis: Control experiments were performed on a much less disordered superconductor (MoGe) to contrast the behavior of the vortex glass with standard vortex lattice regimes.
• Theoretical Support: The experimental findings were supported by numerical simulations of logarithmically interacting particles in a random potential, modeling the vortex glass as a 2D assembly of point-like vortices pinned by disorder.

Key Contributions and Results

1. Anomalous Logarithmic Suppression of Superfluid Stiffness:
   The study reveals that in strongly disordered a:InO films, the superfluid stiffness $\Theta(B)$ exhibits an unexpectedly resilient, logarithmic suppression over nearly three orders of magnitude in magnetic field. This contrasts sharply with the rapid power-law suppression expected for vortex lattices (observed in the MoGe control samples) and the standard collective pinning predictions where stiffness decays as $1/B$ or $1/B^2$.

2. Interaction-Induced Collective Pinning (Pinned Vortex Glass):
   The authors attribute this anomalously slow decay to a collective pinning mechanism in a pinned vortex glass (PVG). Unlike weakly disordered systems where vortex-vortex interactions stiffen the lattice and reduce pinning effectiveness, the authors find that in the strongly disordered regime, vortex-vortex repulsion counterintuitively enhances pinning. The mutual repulsion generates an effective confining "cage" for vortex motion, suppressing the deformation of the glass under supercurrents. Numerical simulations confirm that this behavior arises from the interplay of strong disorder and strong vortex-vortex interactions, leading to an effective pinning parameter $\tilde{k}(B)$ that increases with the magnetic field.

3. Linear Collapse at the Quantum Critical Point:
   As the magnetic field approaches the critical value $B_c$, the logarithmic suppression terminates, and the superfluid density vanishes linearly ($\Theta(B) \propto B_c - B$). This linear collapse is consistent with earlier indirect critical current measurements ($j_c \propto (B_c - B)^{3/2}$) and confirms a continuous quantum phase transition. This stands in stark contrast to the abrupt, first-order transition observed at zero magnetic field in similar disordered systems.

4. Nonlinear Electromagnetic Response:
   The vortex glass exhibits an exceptionally large nonlinear response. Contrary to the naive expectation that driving vortices would weaken pinning, the authors observe a pronounced positive-Kerr effect: the superfluid stiffness increases with microwave power. This is attributed to the overheating of quasiparticles within vortex cores (Caroli-de Gennes-Matricon states), which enhances the effective spring constant of the pinning potential. Additionally, the internal quality factor ($Q_i$) shows non-monotonic behavior, initially increasing with power due to dynamical localization effects in the discrete vortex-core states before decreasing at higher powers.

Significance and Claims
The paper claims to identify the pinned vortex glass as the key intermediate state governing the magnetic-field-induced superconductor-insulator transition in strongly disordered systems. The results demonstrate that disorder controls the critical magnetic field and that the transition is continuous, unlike the disorder-driven transition at zero field.

The authors highlight several implications:

• Mechanism of Transition: The findings challenge the conventional understanding of field-driven SITs, suggesting a unified mechanism where strong interactions enhance pinning rather than competing with it.
• Quantum Sensing Potential: The robustness of the resonators in multi-tesla fields, combined with the divergence of kinetic inductance (leading to ultra-high characteristic impedance), suggests a route toward circuit quantum electrodynamics (QED) sensing. The enhanced zero-point voltage fluctuations could facilitate strong or ultra-strong coupling to various quantum systems, including spin qubits and quantum Hall devices.
• Theoretical Framework: The work motivates the development of analytic theories for pinned vortex glasses beyond conventional approximations, potentially illuminating the broader physics of interacting quantum glasses.

The authors remain modest regarding the universality of these findings, noting that the transition occurs in a mixed 2D-3D regime where conventional quantum critical point universality is not assured, and that the microscopic details of the insulating state above $B_c$ require further clarification.