Quantum-geometric origin of superfluid weight in quasicrystals with critical states

This paper demonstrates that in quasiperiodic systems featuring critical states, the superfluid weight at zero temperature is primarily driven by quantum geometric contributions rather than conventional mechanisms, highlighting a fundamental interplay between superconductivity and criticality in quasicrystals.

The Gist

Imagine a world where the rules of a city's layout are different. In a normal city (a standard crystal), streets are laid out in a perfect, repeating grid. In a quasicrystal, the streets follow a complex, non-repeating pattern that still feels ordered, like a beautiful, intricate mosaic that never quite repeats itself.

In this paper, the researchers are exploring what happens when electrons (the "citizens" of this city) try to form a superfluid—a special state where they flow without any friction, like a superhighway with no traffic jams. This is the microscopic basis of superconductivity.

Here is the simple breakdown of their discovery:

1. The Three Types of "Citizens"

In these unique cities, electrons can behave in three ways:

• The Commuters (Extended States): They roam freely across the entire city.
• The Hermit Crabs (Localized States): They get stuck in a tiny corner and never leave.
• The Critical States (The Mystery Guests): These are the stars of this paper. They are neither fully roaming nor fully stuck. They are "in-between," wandering in a way that is neither free nor trapped. Think of them as people who are stuck in a crowd but can still shuffle around in a specific, fractal-like pattern.

2. The Old Map vs. The New Map

For a long time, scientists thought the ability of electrons to flow without friction (superfluid weight) depended only on how heavy the electrons felt (their "effective mass"). This is like saying a car's speed depends only on its engine size.

However, recent discoveries showed that geometry matters. Imagine the "shape" of the electron's path. If the path has a weird, twisted geometry, it can help the flow even if the engine is weak. This is called the quantum geometric contribution.

3. The Big Discovery

The researchers asked: What happens to this flow in a quasicrystal where those "Critical State" citizens exist?

They used two different methods to look at the problem:

• Method A (Real Space): Looking at the city with open borders, where the edges matter.
• Method B (Momentum Space): Looking at the city as if it were a perfect, repeating loop (a theoretical trick to measure the "shape" of the paths).

The Result:
They found that in quasicrystals, the geometric shape of the electron paths is the main reason the superfluid flows. The "old map" (conventional mass-based flow) barely matters. The "new map" (geometry) does almost all the work.

4. The Analogy: The Flat Band and the Critical State

To understand why, imagine a flat parking lot (a "flat band"). Usually, cars can't move on a flat surface because there's no slope to roll down. But in a topological flat band, the parking spots are arranged in a way that allows cars to "hop" over each other easily because their parking spots overlap.

The researchers found that Critical States in quasicrystals act like these special overlapping parking spots. Even though the electrons aren't in a perfect repeating grid, their "in-between" nature allows them to overlap and move freely. This overlap is purely a result of the geometry of the system.

5. The "Magic" Transition

They tested this on a specific model (the Aubry-André-Harper model) where they could tune the "chaos" of the city.

• When the city was too orderly or too chaotic, the flow was weak.
• But right at the tipping point where the electrons became "Critical" (the in-between state), the geometric contribution took over completely. The conventional flow vanished, and the geometric flow became the only thing keeping the superfluid moving.

Summary

The paper claims that in quasicrystals, the ability to conduct electricity without resistance isn't driven by how heavy the electrons are, but by the weird, fractal geometry of their "Critical" states. It's as if the electrons are dancing to a rhythm dictated by the shape of the city itself, rather than their own weight. This suggests that the "geometry" of the quantum world is a fundamental driver of superconductivity in these unique materials.

Technical Summary

Technical Summary: Quantum-geometric origin of superfluid weight in quasicrystals with critical states

Problem Statement
Quasiperiodic systems, such as quasicrystals, lack translational periodicity but possess long-range order, leading to a unique electronic structure where states are neither extended nor exponentially localized; these are known as critical states. While superconductivity has been theoretically and experimentally observed in quasicrystals, the role of these critical states in superconducting properties remains unclear. Specifically, it is unknown whether the quantum geometric effects, which have been shown to enhance superfluid weight in topological flat-band systems, play a significant role in quasiperiodic systems dominated by critical states. Conventional theory posits that superfluid weight is proportional to the inverse effective mass (intraband contribution), but in systems with flat or critical bands, interband geometric contributions may dominate.

Methodology
The authors investigate the superfluid weight at zero temperature in quasiperiodic systems using two complementary approaches to separate conventional and quantum geometric contributions:

1. Real-space formulation: Applied under open boundary conditions (OBC) using the Kubo formula. The superfluid weight $D_{\mu\nu}$ is derived from the response of the supercurrent to an external vector potential, calculated via the eigenstates of the Bogoliubov-de Gennes (BdG) Hamiltonian.
2. Momentum-space formulation: Applied under periodic boundary conditions (PBC). This approach allows for the explicit decomposition of the superfluid weight into intraband (conventional) and interband (geometric) contributions.

The study employs the mean-field theory for $s$-wave superconductivity with time-reversal symmetry, starting from a Hubbard model with onsite attractive interaction. Two specific models are analyzed:

• Ammann-Beenker Quasicrystal: A two-dimensional model generated via an inflation-deflation method. The system size is 2827 sites, and calculations are performed for various effective chemical potentials ($\tilde{\mu}$).
• Aubry-André-Harper (AAH) Model: A one-dimensional tight-binding model with a quasiperiodic potential. The study focuses on the localization-delocalization transition point ($\lambda = 2t'$), where all eigenstates are critical.

Key Contributions and Results

• Decomposition of Superfluid Weight: The authors successfully separate the superfluid weight into conventional ($D_{\mu\nu, \text{conv}}$) and geometric ($D_{\mu\nu, \text{geom}}$) components in aperiodic systems. They find that the geometric contribution exhibits a strong dependence on the interaction strength $U$, unlike the conventional contribution.
• Ammann-Beenker Results:
  • In the presence of confined states (zero-energy peaks in the density of states), the superfluid weight shows a linear dependence on weak interactions, a behavior characteristic of geometric superfluid weight in flat-band systems.
  • Crucially, even when the chemical potential is shifted away from the confined state peak (where confined states do not contribute), the superfluid weight remains strongly dependent on interaction.
  • Calculations under periodic boundary conditions reveal that the geometric contribution is dominant for all chemical potentials studied. In true quasicrystals (OBC), the wavefunctions acquire a more critical character, further enhancing the dominance of the geometric term.
• AAH Model Results:
  • At the localization-delocalization transition point ($\lambda = 2t'$), where all eigenstates are critical, the conventional contribution to the superfluid weight vanishes.
  • Consequently, the total superfluid weight is entirely accounted for by the geometric contribution ($D_{xx} \simeq D_{xx, \text{geom}}$).
  • For $\lambda > 2t'$ (localized regime), the contribution remains entirely geometric, though the total weight decreases due to strong inhomogeneity.

Significance
The paper claims that the geometric contribution to the superfluid weight is the primary mechanism in quasiperiodic systems with critical states. This finding reveals a fundamental interplay between superconductivity and critical states in quasicrystals, suggesting that the superfluid weight in these systems cannot be understood through conventional effective mass theory alone. The authors highlight that quantum geometry provides a primary contribution to superfluid weight in quasicrystals, offering a new perspective on the origin of superconductivity in these materials. Additionally, the paper notes that since optical lattices can realize quasicrystalline potentials with tunable interactions, they could serve as a platform for observing these geometric superfluid weight effects experimentally.