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

Abstract: We investigate the interplay of disorder and interactions in two-dimensional electron systems in a strong magnetic field, focusing on the transition between Wigner crystals and fractional quantum Hall liquids. We first study classical Wigner crystals with charged impurities, revealing a transition from a single crystal to local crystals and eventually to an amorphous state as impurity concentration increases. We then analyze noninteracting electron crystals created by periodic potentials, showing that their structure factor exhibits both peaks and a ring, distinct from classical Wigner crystals. Finally, we explore fractional quantum Hall liquids with random disorders and charged impurities, demonstrating that the ground state can transition from an incompressible liquid to a localized ordered state and eventually to an amorphous state as disorder strength increases. Our findings highlight the rich interplay between disorder and interactions in quantum Hall systems and provide insights into experimental observations of these phenomena. Comparing qualitatively with a recent STM experiment [Nature \textbf{628}, 287 (2024)], we conclude that the 2D system makes a transition from an incompressible homogeneous fractional quantum Hall liquid to a generic locally ordered solid and eventually to a disordered amorphous solid at large disorder.

• The paper demonstrates how disorder modulates electron phases in two-dimensional quantum Hall systems using classical energy minimization and exact diagonalization.
• The paper shows a transition from homogeneous FQH liquids to locally pinned Wigner crystals, evolving into disorder-dominated amorphous states as impurity strength increases.
• The paper identifies finite-temperature reentrant FQH regimes via thermal activation and correlates numerical predictions with experimental STM observations.

Interplay of Disorder and Interactions in Quantum Hall Systems

Overview

This work provides a systematic study of the interplay between disorder and interactions in two-dimensional electron systems (2DES) under strong magnetic fields, focusing on the transition and competition among Wigner crystals (WCs), fractional quantum Hall (FQH) liquids, and amorphous solid phases. Using a combination of classical energy minimization and exact diagonalization in the lowest Landau level (LLL), the authors analyze the evolution of electronic phases under increasing disorder, both short-range (random potential) and long-range (charged impurities), at zero and finite temperatures.

Classical Wigner Crystals and Disorder Effects

The authors begin with a detailed analysis of classical Wigner crystals subjected to charged impurities. In the pristine case, electrons form a commensurate hexagonal array. Introduction of impurities leads to fragmentation of the global crystal structure into locally correlated domains, characterized by orientational correlation and long-range order persisting up to impurity concentrations where $N_{\mathrm{imp}}/N_e \approx 1/2$.

With further increase in impurity density, a transition to an amorphous state occurs, evidenced by the structure factor’s evolution from sharp Bragg peaks to broadened features and ultimately a diffuse ring indicative of short-range order only. The existence of this intermediate regime of local crystalline order prior to amorphization is robust and highlights the role of disorder in setting finite correlation lengths for electronic order.

Quantum Regime: Noninteracting Crystals and FQH Liquids

Extending to quantum WCs in the LLL, the structure factor $\bar S(\mathbf{q})$ is shown to feature not only Bragg peaks due to crystalline order but also a quantum-fluctuation-induced ring, distinguishing it from the purely classical case. In the limit $\nu\to0$, the analytic form of this ring is well characterized. The competition between direct (classical) and exchange (quantum) terms in the structure factor underscores the nontrivial collective behavior even in nominally uncorrelated quantum states.

The investigation of FQH liquids with random short-range disorder utilizes exact diagonalization for finite-size systems at and near $\nu=1/3$ filling. The incompressible FQH state, identified via uniform density and a robust entanglement spectrum, is shown to withstand weak disorder. For disorder strengths exceeding a critical threshold, localization occurs, leading to the breakdown of topological order (loss of ground state degeneracy and gap in the PES). The spatial signature evolves from homogeneity to localized, short-range-ordered electron packets, and ultimately to amorphous arrangements with disorder-dominated correlations.

Impact of Long-Range Disorder: Charged Impurities

Analysis with long-range Coulomb impurities introduces a further layer of complexity. Impurities act as traps for electrons; sufficient impurity strength leads to the creation of locally ordered, pinned WCs, as supported by real-space densities and structure factors. The fate of the electronic state is controlled by impurity number $N_{\text{imp}}$ and charge $Z$:

• Few, weak impurities: The incompressible FQH phase persists.
• Intermediate regime: Impurities induce local WCs, pinned to impurity locations; correlation lengths can remain large depending on $N_{\text{imp}}$.
• Large $N_{\text{imp}}$ and/or $|Z|$: The system transitions to an amorphous, disorder-dominated state, mirroring the findings with short-range disorder.

Theoretical considerations based on the Imry-Ma argument and the Larkin length suggest that, while local order may persist over finite distances, true long-range crystalline order is not generic in the thermodynamic limit in the presence of sufficient disorder.

Finite Temperature Effects and Entropic Stabilization

At finite temperature, the interplay between energy and entropy can induce crossovers between the pinned and itinerant regimes. For FQH states with impurities, thermal activation can release electrons from pinned states, restoring itinerant quasihole density and enabling the (re)establishment of FQH order above a characteristic temperature $T_e$. This effect is reflected in the closure of entanglement gaps and the modification of real-space densities with temperature.

A detailed analysis of the particle entanglement spectrum (PES) as a function of impurity strength and temperature maps out the finite-temperature phase diagram, revealing robust entropic stabilization of the FQH regime at intermediate temperatures, and providing a clear diagnostic for experimental probes.

Numerical and Experimental Corroboration

The main numerical results show the evolution of real-space density, structure factor, energy spectra, and PES across various regimes of disorder and interaction strength. The phase diagram is constructed based on entanglement gap closures, and density profiles at finite temperature demonstrate the melting of pinned electron structures and the emergence of liquid-like behavior.

A key assertion is that the observed progression—from a homogeneous incompressible FQH phase, through a locally ordered (Wigner crystal-like) phase, to a disorder-driven amorphous phase—is generic for two-dimensional quantum Hall systems with disorder. The signature "arc-like" amorphous structures observed experimentally in STM studies of bilayer graphene [Nature 628, 287 (2024)] are reproduced in the high-disorder limit, supporting the assertion that disorder is the principal driver of these states.

Conclusion

This work provides a unified, nonperturbative framework for understanding the interplay between disorder and interactions in quantum Hall systems. By systematically tracing the evolution from FQH liquids to (pinned) WCs and then to amorphous phases, the study offers clear predictions for experimental observations and delineates the phase boundaries in both zero and finite temperature regimes.

The principal implications are as follows:

• Theoretical models and experimental data converge on the view that both short- and long-range disorder generically drive quantum Hall systems through a sequence of incompressible liquid → locally ordered WC → amorphous insulator.
• Disorder stabilizes compressible Wigner crystal phases at partial filling, but sufficiently strong disorder ultimately destroys both FQH and WC order, regardless of microscopic disorder details.
• The thermally induced reentrant FQH regime, driven by entropy, is a nontrivial and verifiable prediction relevant for finite-temperature experiments and device engineering in quantum Hall platforms.

Future research directions include scaling analysis in larger system sizes, extension to multicomponent quantum Hall fluids, and investigation of disorder-interaction interplay in more complex topological settings. These results set the stage for more detailed experimental and theoretical exploration of disorder-driven phenomena in strongly correlated electron systems.