Hyperuniform charge distributions and phase transitions in a generalized Aubry-André model

This paper demonstrates that a generalized Aubry-André model exhibits a third-order phase transition between distinct inhomogeneous charge distributions, where the hyperuniformity class of the many-body system is determined by the localization properties of single-particle states near the Fermi energy.

The Gist

Imagine a crowded dance floor where everyone is trying to move around. In a normal, chaotic crowd, people bump into each other randomly, creating messy, unpredictable clusters. But in this paper, the researchers are studying a very special kind of dance floor: one that is aperiodic. This means the floor tiles aren't arranged in a simple repeating pattern (like a checkerboard) or a completely random mess. Instead, they follow a complex, non-repeating rule, similar to the patterns found in a quasicrystal or a spiral shell.

The scientists wanted to understand how electrons (the dancers) arrange themselves on this strange floor. Specifically, they looked at two things:

1. How "clumpy" the crowd is: Do the electrons spread out evenly, or do they bunch up in weird spots?
2. How "stuck" the dancers are: Can the electrons move freely across the floor (extended), or are they trapped in one small corner (localized)?

The "Hyperuniform" Rule

The researchers used a special measuring tool called Hyperuniformity. Think of this as a "perfect order" detector.

• Normal disorder: If you look at a random crowd, the number of people in a specific area fluctuates wildly as you change the size of your viewing window.
• Hyperuniform: In these special systems, the crowd is so well-organized that even though it looks messy up close, the fluctuations in density are anomalously suppressed over long distances. It's like a crowd that looks chaotic but somehow manages to keep a perfect balance when you step back and look at the whole room.

The paper claims that no matter what, the electrons in this specific model always form a "hyperuniform" crowd. They never become truly random.

The Three Types of "Perfect Order"

However, just because the crowd is "perfectly ordered" doesn't mean they all look the same. The researchers found three distinct "classes" of this order, which act like different uniforms for the crowd:

1. Class I (The Smooth Crowd): The density is very smooth. There are no sudden jumps or gaps. This happens when the electrons are free to move (extended) or when there is a "gap" in the energy levels that forces them into a specific, stable arrangement.
2. Class II (The Bumpy Crowd): The density has a logarithmic "bumpiness." It's still ordered, but there are more fluctuations. This happens when the electrons are "stuck" or localized in specific spots.
3. Class III (The Fractal Crowd): A more complex, self-similar pattern (though the paper focuses mostly on the switch between Class I and II).

The Big Discovery: The "Fingerprints" of the Fermi Energy

The most exciting finding is about what determines which "uniform" the crowd wears.

In this model, there is a special boundary called a mobility edge. Imagine a line drawn across the dance floor.

• On one side of the line, dancers can run freely (extended states).
• On the other side, dancers are stuck in place (localized states).

The researchers found that the entire crowd's "uniform" (Hyperuniformity Class) is dictated solely by the dancers standing right at the "Fermi Energy" (the energy level of the most energetic dancers currently on the floor).

• If the dancers at the edge of the energy crowd are free runners, the whole system looks like Class I (smooth).
• If the dancers at the edge are stuck, the whole system looks like Class II (bumpy).

It doesn't matter what the dancers deep in the "sea" of energy are doing; only the ones at the very top edge decide the pattern for everyone else.

The "Third-Order" Phase Transition

Finally, the paper describes what happens when you tweak the system (like changing the strength of the floor's potential) so that the "mobility edge" moves across the Fermi energy.

• As the edge crosses, the dancers at the top switch from "free" to "stuck."
• Consequently, the whole crowd's uniform instantly switches from Class I to Class II.

The researchers calculated that this switch is a third-order phase transition.

• Analogy: Imagine a car accelerating.
  • A first-order transition is like hitting a brick wall (sudden stop, huge crash).
  • A second-order transition is like slamming on the brakes (you feel the deceleration, but it's smooth).
  • A third-order transition is like gently easing off the gas pedal. The car doesn't jerk or even noticeably slow down at first; the rate at which it slows down changes smoothly, but if you look at the change in that rate, you see a sudden jump. It's a very subtle, smooth, yet distinct change in the system's behavior.

Summary

In simple terms, the paper shows that in these complex, non-repeating systems, the electrons always arrange themselves in a highly ordered way (hyperuniform). However, the type of order they choose depends entirely on whether the electrons at the very top of the energy pile are free to move or stuck in place. When they switch from free to stuck, the entire system undergoes a very smooth, subtle, yet distinct "third-order" transformation in its pattern.

Technical Summary

Technical Summary: Hyperuniform Charge Distributions and Phase Transitions in a Generalized Aubry–André Model

Problem Statement
While symmetry-breaking and topological classifications effectively describe electronic phases in periodic and certain aperiodic systems, they fail to fully characterize inhomogeneous electron distributions in quasiperiodic and random systems. In these systems, spatial patterns of charge density can change without altering symmetry or topological invariants. The authors address the need for a framework to quantify and classify these global inhomogeneous features, specifically investigating whether changes in spatial patterns constitute phase transitions and how to characterize the resulting phases. The study focuses on the limitations of existing hyperuniformity (HU) classifications in the standard Aubry–André (AA) model, where all single-particle states share the same localization character, and seeks to extend this framework to systems exhibiting mobility edges, where extended and localized states coexist.

Methodology
The authors employ a generalized Aubry–André (GAA) model, a one-dimensional tight-binding system with a quasiperiodic potential $V_i$ that breaks self-duality and introduces a mobility edge. The Hamiltonian includes a parameter $\alpha$ controlling the potential's shape, allowing for the coexistence of extended and localized single-particle states at the same potential strength $\lambda$.

The study utilizes the following methods:

1. Single-Particle Analysis: The localization characteristics of eigenstates are quantified using the inverse participation ratio (IPR) and fractal dimension ($D_f$) to distinguish between extended, localized, and critical (multifractal) states.
2. Many-Body Charge Density: At zero temperature, the many-body charge density $n_i$ is calculated by summing contributions from all occupied single-particle states below the Fermi energy $E_F$.
3. Hyperuniformity (HU) Classification: The global uniformity of the charge distribution is analyzed via the variance of the number of particles in a window of radius $R$, $\sigma^2(R)$. The system is classified as hyperuniform if the variance grows slower than the volume ($R^d$). The authors distinguish three HU classes based on the large-$R$ behavior of the integrated variance $\bar{B}(R)$ and the small-$k$ scaling of the structure factor $S(k)$ or integrated intensity $Z(k)$:
   • Class I: $\bar{B}(R) \to \text{const}$ (suppressed fluctuations).
   • Class II: $\bar{B}(R) \sim \ln R$.
   • Class III: $\bar{B}(R) \sim R^{1-\gamma}$.
4. Phase Transition Analysis: The order of transitions between HU classes is determined by examining the continuity of the total energy per site ($E_{tot}$) and its partial derivatives with respect to model parameters ($\lambda, \alpha$).

Key Contributions and Results

• Universal Hyperuniformity: The study demonstrates that the many-body charge distribution in the GAA model is always hyperuniform, regardless of whether the system is gapped or gapless, or the specific localization properties of the occupied states. This contrasts with systems under random potentials, which are non-hyperuniform.
• Dependence on Fermi-Level States: In gapped phases, the charge distribution invariably belongs to Class I HU, irrespective of the localization properties of the states at the band edges. In gapless phases, the HU class is determined solely by the localization properties of the single-particle states near the Fermi energy:
  • If states at $E_F$ are extended, the distribution is Class I.
  • If states at $E_F$ are localized, the distribution is Class II.
  • If states at $E_F$ are critical (at the mobility edge), the distribution is Class II.
    Crucially, the HU class is insensitive to the localization properties of states far from the Fermi energy.
• Mobility Edge Signatures: The presence of a mobility edge allows for mixed scenarios where extended and localized states coexist below $E_F$. The results show that the HU class is dictated by the nature of the states at the Fermi level. For instance, even if localized states exist deep in the band, if the states at $E_F$ are extended, the system remains Class I.
• Third-Order Phase Transitions: The transition between different HU classes (specifically from Class I to Class II) as the mobility edge crosses the Fermi energy is identified as a third-order phase transition. The total energy $E_{tot}$ and its first and second derivatives remain continuous across the transition, while a discontinuity (jump) appears in the third derivative. This extends previous findings in the AA model to systems with mobility edges.

Significance and Claims
The paper claims that the hyperuniformity framework provides a robust method to distinguish distinct inhomogeneous electron phases in aperiodic systems where symmetry and topology are insufficient. Specifically:

1. Fingerprint of Localization: In gapless phases, the HU class serves as a direct "fingerprint" of the localization character of low-energy single-particle states, allowing one to infer localization properties without calculating localization lengths or conductance.
2. Mechanism of Insulation: The HU classification distinguishes between insulating phases based on their microscopic origin. Gapped phases (spectral gap) are universally Class I, while gapless localized phases (mobility gap) are Class II. This suggests that HU scaling encodes information about whether insulation arises from a lack of states at $E_F$ or from wave localization.
3. Extension of AA Model Results: The findings generalize the HU classification and the nature of the phase transition observed in the standard Aubry–André model to the more complex GAA model, which features mobility edges.
4. Limitations: The authors note that the current framework, which neglects quantum fluctuations, cannot distinguish between gapped and gapless phases within the extended regime. They suggest that extending the framework to include quantum fluctuations is necessary for a finer classification of delocalized quantum phases.

The study concludes that the HU framework offers a unified perspective on charge fluctuations, linking spatial patterns directly to the underlying single-particle localization properties and the nature of the phase transition.