Multiple correlation lengths and type-1.5 superconductivity in $U(1)$ superconductors due to hidden competition between irreducible representations of nonlocal pairing

This paper demonstrates that nominally single-component $U(1)$ superconductors on a square lattice can exhibit multiple correlation lengths and type-1.5 superconductivity with vortex clustering due to hidden competition between $s$-wave and $d$-wave irreducible representations of nonlocal pairing, even when the subdominant order parameter is suppressed in the ground state.

The Gist

Imagine a superconductor as a perfectly organized dance floor where electrons pair up and move in perfect unison. Usually, physicists think of this dance floor as having just one "rhythm" or style of movement. They measure how far a disturbance (like a bump in the floor) travels before the dancers recover their rhythm; this distance is called the coherence length.

For decades, the rule was simple:

• If the dancers recover quickly (short distance), they repel each other like magnets with the same pole (Type-II superconductor).
• If they recover slowly (long distance), they attract and clump together (Type-I superconductor).

The Big Discovery
This paper reveals a surprising twist: Even if a superconductor looks like it only has one rhythm, it might actually be hiding a second, suppressed rhythm underneath.

Think of it like a band playing a song. The lead singer (the main superconducting state) is loud and clear. But there's a backup singer (a "subdominant" state) who is so quiet you can't hear them in the main song. However, the paper shows that this quiet backup singer still exists in the background. Their mere presence changes how the band reacts to outside noise.

The "Hidden Competition"
The researchers studied a specific type of superconductor where electrons can pair up in two different ways (called "s-wave" and "d-wave").

• In their model, one style wins completely in the ground state (the quietest, most stable state). The other style is completely suppressed—it doesn't show up in the main song.
• The Surprise: Even though the losing style is "silenced," it leaves a ghostly imprint. It creates a second, hidden distance scale for how the system recovers from disturbances.

The "Type-1.5" Dance
This leads to a new, weird behavior the authors call Type-1.5 superconductivity.

Imagine the magnetic field trying to enter the dance floor creates "vortices" (little whirlpools in the dance).

1. Short Range: Because of the main rhythm, the whirlpools push each other away (repulsion).
2. Long Range: Because of the hidden, suppressed rhythm, the whirlpools pull each other together (attraction).

The result? The whirlpools don't just spread out evenly, and they don't just crash into a single pile. Instead, they form clusters—like a group of friends who want to stay close to each other but still need a little personal space. They form tight little groups that float around the dance floor.

The "Skyrmion" Twist
In some cases, these clusters aren't just simple whirlpools. The paper describes them as having a "fractionalized" core, like a single whirlpool made of two smaller, intertwined ones. The authors call this a skyrmion (a fancy name for a specific type of magnetic knot). It's as if the dance floor has a secret texture that only reveals itself when you look closely at these clusters.

Why It Matters (According to the Paper)
The paper claims this isn't just a theoretical curiosity. It suggests that:

1. It's more common than we thought: You don't need a complex system with multiple broken symmetries to get this behavior. Even a "simple" superconductor with a hidden competitor can do it.
2. It changes how magnets interact: The magnetic field penetrates the material in a unique way, sitting right in the middle of the two different recovery distances.
3. It helps build better detectors: The paper mentions that this "clustering" behavior could be useful for single-photon detectors (devices that catch single particles of light). The specific way these vortex clusters form and interact could help reduce "dark counts" (false alarms) in these sensitive instruments.

In Summary
The paper argues that a superconductor doesn't need to be complex on the surface to have complex behavior underneath. A "hidden" competitor, even if it's completely suppressed in the main state, can create a second distance scale. This forces the magnetic whirlpools (vortices) to form clusters, creating a new "Type-1.5" state of matter that sits between the traditional types.

Technical Summary

Technical Summary: Multiple Correlation Lengths and Type-1.5 Superconductivity in U(1) Superconductors

Problem Statement
Conventionally, the coherence length ($\xi$) is treated as a single characteristic scale in superconductors, determining the spatial recovery of the Cooper pairing field. While multicomponent superconductors (those breaking multiple symmetries or possessing multiple bands) are known to exhibit multiple coherence lengths, leading to the "type-1.5" regime where the magnetic penetration depth ($\lambda$) lies between these lengths, this behavior is generally not expected in nominally single-component superconductors with a single broken $U(1)$ symmetry. The central question addressed is whether a system with a single ground-state order parameter can nonetheless exhibit multiple correlation lengths and type-1.5 behavior due to hidden competition with a suppressed pairing channel.

Methodology
The authors employ a mean-field Hubbard model on a square lattice with attractive nearest-neighbor interactions ($V > 0$). The superconducting order parameter on links is mapped to a linear superposition of two irreducible representations: extended $s$-wave ($A_{1g}$) and $d$-wave ($B_{1g}$).

1. Microscopic Calculation: The authors solve the self-consistency equations for the superconducting gap numerically to generate a phase diagram in chemical potential ($\mu$) and temperature ($T$) coordinates. Analytical approximations are also derived for specific limits (near critical temperatures and low temperatures) to characterize phase boundaries.
2. Ginzburg-Landau (GL) Derivation: A GL free energy functional is derived from the microscopic model in the weak-coupling limit near the $U(1)$ phase transition. This functional includes anisotropic gradient terms and mixed gradient couplings ($\gamma_{12}$) between the $s$ and $d$ components, as well as interband potential couplings ($\beta_3, \beta_4$).
3. Length Scale Analysis: The authors linearize the GL free energy to calculate the magnetic penetration depth ($\lambda$) and coherence lengths ($\xi_i$). They analyze the eigenvalues of the Hessian matrix of the potential energy coupled with the gradient matrix to determine the hierarchy of length scales.
4. Numerical Simulation: To verify the analytical estimates, the authors perform numerical simulations of the GL model to study vortex configurations and interaction energies in three distinct ground states: $s+id$, pure $s$-wave, and pure $d$-wave.

Key Contributions and Results

• Emergence of Multiple Length Scales in Single-Component States: The paper demonstrates that even when the ground state is dominated by a single order parameter (e.g., pure $s$-wave or pure $d$-wave) and the competing component is completely suppressed, the system possesses multiple correlation lengths. This arises from the structure of the gradient terms in the GL functional, specifically the mixed gradient coupling ($\gamma_{12}$). The suppressed component, though zero in the bulk, nucleates near perturbations (such as vortex cores), imprinting an additional length scale on the system.
• Type-1.5 Superconductivity in Single-Breaking-Symmetry Systems: The authors identify a regime where the magnetic penetration depth $\lambda$ falls between the multiple coherence lengths ($\xi_1 < \lambda < \xi_2$). This hierarchy leads to type-1.5 superconductivity, characterized by long-range attraction and short-range repulsion between vortices, resulting in vortex clustering.
• Phase Diagram and Transitions: The calculated phase diagram reveals regions of pure $s$-wave, pure $d$-wave, and an $s+id$ state with a relative phase difference of $\pi/2$. A distinct feature is a sharp, vertical transition between the $s+id$ and $s$-wave phases, which is analytically shown to be independent of temperature under certain approximations.
• Vortex Clustering and Skyrmions: Numerical simulations confirm that in the estimated type-1.5 regime, vortices form stable clusters.
  • In the $s+id$ state, vortex clusters exhibit nontrivial morphology (e.g., chains along diagonals) and the vortex core carries a nontrivial skyrmionic topological charge ($Q=2$), composed of two bound $Q=1$ skyrmions.
  • In pure $s$-wave and pure $d$-wave ground states (where the subdominant component is suppressed), vortex clustering still occurs, confirming that the type-1.5 behavior is a generic feature of systems with competing pairing channels, even if only one is realized in the ground state.
• Anisotropy: The coherence lengths are shown to be direction-dependent due to the anisotropic nature of the gradient terms, although the type-1.5 hierarchy persists along various directions.

Significance and Claims
The authors claim that this work fundamentally alters the understanding of superconducting typology. They demonstrate that the presence of a competing, suppressed order parameter belonging to a different irreducible representation of nonlocal pairing can principally alter the magnetic response of a system, introducing multiple correlation lengths without requiring multiple broken symmetries or multiple electronic bands in the ground state.

The paper asserts that type-1.5 behavior is a generic characteristic of "double-transition superconductors" (systems with a single $U(1)$ transition but proximity to a second symmetry-breaking transition). The authors suggest that these findings could be relevant for applications involving vortex instabilities, such as in single-photon detectors, where vortex clustering might influence dark counts. The work provides a theoretical basis for diagnosing pairing symmetries and mapping phase diagrams through the observation of vortex cluster morphologies.