Interplay of disorder and interactions in quantum Hall systems: from fractional quantum Hall liquids to Wigner crystals and amorphous solids

This paper investigates the interplay of disorder and interactions in two-dimensional quantum Hall systems, revealing a disorder-driven phase transition sequence from incompressible fractional quantum Hall liquids to locally ordered solids and finally to amorphous states, a progression that qualitatively aligns with recent scanning tunneling microscopy experimental observations.

The Gist

Imagine a crowded dance floor where everyone is trying to move in perfect sync, but the music is so loud (a strong magnetic field) that they can't really run around; they are stuck spinning in place. This is the world of Quantum Hall systems, a playground for electrons in a magnetic field.

In this paper, the authors act like detectives trying to figure out what happens to these electrons when two things mess with their dance:

1. Interactions: The electrons don't like each other; they push away (repel) to avoid getting too close.
2. Disorder: The dance floor isn't perfect. There are sticky spots, bumps, or random obstacles (impurities) scattered around.

The big question is: When the floor gets messy, do the electrons form a perfect crystal, a liquid, or a chaotic mess?

Here is the story of their findings, broken down into simple concepts:

1. The Two Extreme Dances

Before the mess starts, the electrons have two main ways to behave:

• The Fractional Quantum Hall (FQH) Liquid: Think of this as a super-coordinated, invisible fluid. The electrons move together in a complex, "topological" dance that is very stable and hard to break. It's like a school of fish moving as one giant organism. This happens at very specific, perfect numbers of electrons.
• The Wigner Crystal: If the electrons are pushed far enough apart, they stop flowing and lock into a rigid grid, like soldiers standing in perfect rows. This is a solid crystal.

2. The "Mess" Factor: What happens when we add disorder?

The authors wanted to see what happens when you start throwing random obstacles (impurities) onto the dance floor. They looked at three different scenarios:

Scenario A: The Classical Crystal (The "Snowball" Effect)

First, they looked at a simple, non-quantum version (like marbles on a table).

• No Mess: The marbles form a perfect hexagonal honeycomb.
• A Little Mess: If you put a few sticky spots on the table, the perfect honeycomb breaks into small, local patches. The marbles still form little crystals, but they are all facing different directions.
• Too Much Mess: If you scatter enough sticky spots, the marbles can't form any crystals at all. They get stuck in a random, jumbled pile. This is an amorphous solid (like glass or a pile of sand).

Scenario B: The Quantum Crystal (The "Ghost" Ring)

Next, they looked at electrons that aren't interacting with each other but are forced into a grid by the magnetic field.

• They found something weird. Even though the electrons are in a grid, their "fingerprint" (called a structure factor) shows a ring of light, not just sharp points.
• Analogy: Imagine a perfect crystal is like a lighthouse beam hitting a mirror (sharp points). A quantum crystal is like that beam hitting a prism; it spreads out into a ring. This ring is a purely quantum effect that doesn't exist in the classical world.

Scenario C: The Liquid vs. The Crystal (The Main Event)

This is the most important part. They studied the "Liquid" (FQH) state and added disorder to see if it would turn into a crystal or a mess.

• Low Disorder: The liquid is tough. It stays a liquid even with a few bumps.
• Medium Disorder: The liquid breaks. The electrons get pinned to the obstacles. They stop flowing and start forming local crystals. They are ordered in small neighborhoods, but the whole system isn't one big crystal.
• High Disorder: The obstacles are so strong that the electrons can't even agree on a local pattern. They get stuck in a random, amorphous state.
  • The "Arc" Discovery: The authors noticed that in this high-disorder state, the electrons form arc-like shapes. This is huge because a recent real-world experiment (using a super-microscope called STM) saw exactly these arc-like shapes in graphene. The authors say, "Aha! Those arcs aren't a mystery; they are just electrons getting stuck in a chaotic, disordered pile!"

3. The Temperature Twist (The "Hot Coffee" Effect)

Finally, they asked: "What if we heat it up?"

• At Absolute Zero: Electrons get stuck to the impurities, forming a pinned crystal.
• At Warm Temperatures: Heat gives the electrons energy. It's like shaking the table. The electrons can "break free" from the sticky spots.
• The Surprise: Sometimes, heating the system actually restores the liquid state! The electrons melt off the impurities and start flowing freely again. It's a "re-entrant" phase: Liquid $\to$ Solid (due to cold/disorder) $\to$ Liquid (due to heat).

The Big Takeaway

The paper unifies three different worlds (classical crystals, quantum crystals, and quantum liquids) under one rule:

As disorder increases, nature follows a predictable path:

1. Perfect Order (Crystal or Liquid)
2. Local Order (Small patches of crystals)
3. Total Chaos (Amorphous, disordered solid)

The authors conclude that the strange "arc" structures seen in recent experiments are likely just electrons in this third stage: a disordered, amorphous solid caused by too much mess on the dance floor. They also predict that if you heat these systems up just right, you might see the liquid dance return, even if the floor is still messy.

Technical Summary

1. Problem Statement

The paper addresses the complex interplay between electron-electron interactions and disorder in two-dimensional electron systems (2DES) under strong magnetic fields. While Wigner Crystals (WCs) (interaction-driven electron solids) and Fractional Quantum Hall (FQH) liquids (topologically ordered incompressible fluids) are well-studied individually, their behavior under simultaneous strong disorder remains unclear.

• Key Gap: Previous studies often treated these phases separately or used perturbative approaches. There was no unified, non-perturbative framework explaining how disorder drives transitions between these phases, particularly in light of recent Scanning Tunneling Microscopy (STM) experiments on bilayer graphene (Nature 628, 287, 2024) that observed both pristine crystalline order and "arc-like" amorphous structures.
• Open Questions: How does disorder fragment a coherent crystal into local domains and eventually an amorphous state? How do different disorder types (random short-range potentials vs. long-range charged impurities) affect correlation lengths? How does the system behave when slightly away from exact fractional filling (quasihole regimes)? Can thermal fluctuations restore an FQH state from a disorder-pinned WC?

2. Methodology

The authors employ a unified theoretical framework combining classical energy minimization and exact diagonalization (ED) of quantum Hamiltonians within the Lowest Landau Level (LLL).

• Classical Benchmark: They first model classical Wigner crystals with charged impurities at zero temperature to establish a baseline for how disorder fragments order.
• Quantum Non-Interacting Crystals: They study non-interacting electrons in periodic potentials to distinguish quantum mechanical effects (like exchange terms) from classical behavior.
• Interacting Quantum Systems: They analyze FQH liquids ($\nu = 1/3$) subject to:
  • Random short-range disorder (Gaussian white noise).
  • Long-range Coulomb impurities (charged point defects).
• Finite Temperature: They investigate thermal effects using the density matrix $\rho = e^{-H/T} / \text{Tr}(e^{-H/T})$ to study thermal ionization of impurity-bound electrons.
• Diagnostic Tools:
  • Real-space density profiles: To visualize localization and structural order.
  • Projected Structure Factor ($\bar{S}(q)$): To distinguish between long-range order, local order, and amorphous states.
  • Particle Entanglement Spectrum (PES): To identify topological order (FQH) vs. localized order (WC) via entanglement gaps and state counting (generalized Pauli principle).

3. Key Contributions and Results

A. Classical Wigner Crystals with Charged Impurities

• Transition Sequence: As impurity concentration increases, the system transitions from a single global hexagonal crystal $\to$ local crystalline domains with varying orientations $\to$ an amorphous state.
• Structure Factor: The structure factor $S(q)$ evolves from sharp Bragg peaks (long-range order) to broadened peaks (local order) and finally to a ring structure with faint vestiges of peaks (amorphous). The ring radius increases as repulsive impurities compress the electron lattice.

B. Quantum vs. Classical Crystals

• Quantum Signatures: Non-interacting electron crystals in the LLL exhibit a projected structure factor $\bar{S}(q)$ containing both Bragg-like peaks and a quantum ring.
• Origin of the Ring: Unlike classical WCs, the ring in quantum crystals arises from "exchange" terms in the Slater determinant (analytically derived as $\bar{S}(q) \propto e^{q^2 l_B^2/4} - e^{q^2 l_B^2/2}$ in the $\nu \to 0$ limit). This distinguishes quantum electron crystals from classical ones.

C. FQH Liquids with Random Disorder

• Phase Transitions:
  1. Incompressible Liquid: Robust against weak disorder; density remains uniform.
  2. Localized Ordered State: As disorder increases, the topological gap is destroyed (loss of degeneracy), and electrons localize into short-range ordered structures (resembling local WCs).
  3. Amorphous Solid: At strong disorder, the system becomes a localized, disordered amorphous state.
• Quasihole Sensitivity: Systems slightly away from exact filling (e.g., $\nu = 1/3$ with quasiholes) are significantly more susceptible to disorder, localizing at much lower disorder strengths than the commensurate case.
• Experimental Connection: The "arc-like" amorphous structures observed in the strong-disorder limit closely mimic the experimental STM images in bilayer graphene.

D. FQH Liquids with Charged Impurities (Long-Range Disorder)

• Pinning Mechanism: Attractive impurities trap electrons, effectively creating local Wigner crystals pinned to the impurity sites.
• Phase Diagram:
  • Low Impurity Density: The FQH state remains stable (or pinned FQH with quasiholes).
  • Intermediate Density: Formation of locally ordered structures (pinned WCs) with long correlation lengths.
  • High Density/Strength: Transition to a disordered amorphous solid.
• Thermodynamic Limit: While STM may show long-range ordering due to a large Larkin length, the authors argue (via Imry-Ma argument) that true long-range order does not survive in the thermodynamic limit; the system is a collection of local domains.

E. Finite Temperature Effects

• Thermal Re-entrant FQH: The authors predict a counter-intuitive crossover where increasing temperature can restore an FQH liquid from a pinned WC.
  • Mechanism: At $T=0$, impurities trap electrons, reducing itinerant carrier density and suppressing FQH order. As $T$ increases, thermal energy ionizes these trapped electrons, restoring the itinerant density required for the FQH state.
  • Condition: This occurs only if the ionization temperature $T_e$ is lower than the temperature scale where the FQH state itself melts ($T_{FQH}$). If disorder is too strong, the system remains a localized amorphous solid regardless of temperature.

4. Significance and Conclusion

• Unified Framework: The paper provides a comprehensive, non-perturbative explanation for the entire spectrum of phases in disordered quantum Hall systems, linking Wigner crystals, FQH liquids, and amorphous solids through a common progression driven by disorder strength.
• Interpretation of STM Data: The findings offer a definitive explanation for recent STM observations in bilayer graphene: the "arc-like" structures are not a new exotic phase but rather a disorder-induced amorphous solid resulting from the fragmentation of the Wigner crystal.
• Generic Behavior: The transition from coherent order $\to$ local order $\to$ amorphous disorder is identified as a robust, generic feature of 2DES in strong magnetic fields, applicable to both short-range and long-range disorder.
• Thermal Dynamics: The prediction of a thermally induced re-entrant FQH liquid challenges the conventional view that disorder and temperature always act to destroy quantum order, highlighting the competition between energy (pinning) and entropy (delocalization).

In summary, the work establishes that disorder is the primary driver converting topological liquids and crystalline solids into disordered amorphous phases, while also revealing that thermal fluctuations can, under specific conditions, reverse this localization to restore topological order.