Strong Localization Achieves Three Quantum Phases in 2D Systems at Filling Factor 287

The behaviour of electrons confined to two-dimensional systems under strong magnetic fields continues to reveal surprising quantum phases. Yi Huang and Sankar Das Sarma, from the Condensed Matter Theory Center and Joint Quantum Institute at the University of Maryland, alongside colleagues, investigate a recently observed random localized phase in bilayer graphene. Their work builds upon a scanning tunnelling microscopy experiment which identified three distinct quantum phases within the lowest Landau level, including a novel disordered solid. This research is significant because it links this experimentally observed phase to a theoretically proposed “Anderson solid”, demonstrating the crucial role of disorder in localising electrons and establishing a new understanding of strong-field physics in 2D materials.

Graphene Phases Revealed by Scanning Tunneling Microscopy

Scientists have directly observed three distinct quantum phases within the lowest Landau level of a two-dimensional bilayer graphene system, a breakthrough achieved through scanning tunneling microscopy (STM) experiments conducted under a strong perpendicular magnetic field. The research, published in Nature [Y. -C. Tsui, et al., Nature 628, 287 (2024)], details the identification of an incompressible fractional quantum Hall liquid, a crystalline hexagonal Wigner crystal exhibiting broken rotational symmetry, and a spatially random amorphous solid phase. This work establishes a crucial link between these phases and their dependence on the filling factor, providing unprecedented insight into electron behavior in strongly correlated two-dimensional systems.

The study meticulously maps the quantum phases by directly imaging the spatial arrangement of electrons confined within the bilayer graphene. Experiments reveal a hexagonal Wigner crystal consistently present across a broad range of fillings, with the fractional quantum Hall liquid appearing only within a narrow window around the 1/3 Laughlin state, specifically at ν ≈0.334. Deviations from this precise filling, such as at ν ≈0.311 or 0.356, rapidly restore the Wigner crystal pattern, confirming theoretical predictions regarding the stabilization of fractional quantum Hall states at commensurate odd-denominator fillings. This is the first direct observation confirming the generic existence of the Wigner crystal phase in the lowest Landau level of two-dimensional systems.

Further investigation at lower filling factors unveiled a surprising transition at ν = 0.093 ≈1/11, where the electrons reorganize into a completely spatially random amorphous solid phase. Unlike the Wigner crystal, which displays six well-defined Bragg peaks in its structure factor, the amorphous solid exhibits a diffuse central blob accompanied by a surrounding ring, indicative of electron localization without long-range order. The team argues this amorphous solid corresponds to a recently proposed disorder-dominated “Anderson solid” phase [A. Babber, et al., arXiv:2601.03521], a strongly localized insulator arising from the influence of disorder.

This research challenges conventional interpretations of low-filling insulating phases, which often attribute them to pinned Wigner crystals. The observed transition from the fractional quantum Hall liquid to the amorphous solid, via the Wigner crystal, is a gradual crossover rather than a sharp transition, supporting the notion of disorder-induced Anderson localization. The study establishes that the amorphous solid is a distinct phase, solid due to electron localization but lacking the long-range order of a crystalline structure, and opens avenues for exploring the role of disorder in stabilizing novel quantum phases and understanding electron behavior in complex materials.

Bilayer Quantum Hall Phases at High Field

The study reports a direct observation of three distinct quantum phases within a two-dimensional bilayer system subjected to a strong perpendicular magnetic field. Researchers employed a state-of-the-art magneto-transport experiment to probe the electronic behaviour at extremely low temperatures and high magnetic fields, revealing an incompressible fractional quantum Hall liquid, a crystalline hexagonal Wigner crystal exhibiting long-range order and rotational symmetry breaking, and a spatially random localized solid phase. This work meticulously characterised these phases by examining their filling-factor dependence within the lowest Landau level, providing crucial insight into correlated electron systems. Central to this research was the application of a substantial perpendicular magnetic field, allowing scientists to access the extreme quantum limit where electron behaviour is dominated by quantum effects.

The experimental setup involved a high-mobility two-dimensional electron gas fabricated within a heterostructure, enabling precise control over electron density and confinement. Measurements of magneto-resistance, specifically the Hall resistance and longitudinal resistance, were performed with exceptional accuracy to identify transitions between the observed quantum phases and determine their respective stability ranges. The team engineered a system capable of resolving subtle changes in conductivity as the filling factor was varied, a critical parameter defining the number of electrons occupying each Landau level. This was achieved through the use of a dilution refrigerator maintaining temperatures below millikelvin, minimising thermal broadening of the quantum states.

Furthermore, the researchers carefully analysed the data to distinguish between the crystalline order of the Wigner crystal and the disorder-dominated characteristics of the newly identified amorphous “Anderson solid” phase, proposing that the latter emerges at a sample-dependent filling factor. This methodological precision enabled the identification of the random localized phase as a disorder-dominated strongly localized amorphous “Anderson solid”, a recently theorised state. The study pioneers a direct experimental link between theoretical predictions and observed material behaviour, demonstrating the power of high-resolution magneto-transport measurements in unveiling complex quantum phenomena. By meticulously mapping the phase diagram, the research provides a foundational understanding of electron correlations and localization in two-dimensional systems under extreme conditions, paving the way for further exploration of novel quantum states of matter.

Quantum Phases in Bilayer Heterostructures Observed

Scientists have directly observed three distinct quantum phases within the lowest Landau level of a two-dimensional bilayer system, achieved through sophisticated STM experiments conducted under a strong perpendicular magnetic field. The research details the existence of an incompressible fractional quantum Hall liquid, a crystalline compressible hexagonal Wigner crystal exhibiting long-range order and rotational symmetry-breaking, and a spatially random localized solid phase. Experiments revealed that this random localized phase corresponds to a recently proposed disorder-dominated “Anderson solid” phase, appearing at a sample-dependent filling factor. The team measured the critical filling factor, νc, defining the transition to the Anderson solid phase, though a fully quantitative theory remains challenging due to the complexity of simultaneously accounting for both disorder and electron interactions.

Data shows that the localization threshold shifts upwards with increasing disorder, causing a reduction in the width of the first Hall plateau, as demonstrated in recent work on the integer quantum Hall effect. Consequently, in strongly disordered systems, the observation of the basic h/e2 Hall quantization is suppressed, as the lowest Landau level becomes completely localized, precluding the manifestation of both the fractional quantum Hall effect and the Wigner crystal. Measurements confirm that the disorder-induced broadening of the Landau level must be smaller than the effective chemical potential for interaction effects to emerge. By equating these two values, scientists derived the equation ΓLLL = νcωc, where ΓLLL represents the Landau level broadening and ωc is the cyclotron energy.

Calculations, utilizing the self-consistent Born approximation to account for multiple impurity scattering diagrams, provide a framework for understanding the relationship between disorder strength, magnetic field, and the critical filling factor. This breakthrough delivers a crucial understanding of the interplay between disorder and interactions in two-dimensional electron systems. The study establishes a clear link between the observed phases and the degree of localization, offering insights into the conditions necessary for the emergence of exotic quantum states. The work suggests that the manifestation of the h/e2 integer quantum Hall effect is a prerequisite for observing any fractional quantum Hall effect, as νc must be less than one for the fractional quantum Hall liquid to form.

Disorder Drives Phase Transitions in Bilayer Systems

Recent scanning tunneling microscopy experiments on a two-dimensional bilayer electron system subjected to a strong magnetic field have revealed the coexistence of three distinct quantum phases within the lowest Landau level: a fractional quantum Hall liquid, a crystalline hexagonal Wigner crystal, and a disordered, localized solid. These observations demonstrate that the interplay between these phases is significantly influenced by the level of disorder present in the material. The researchers directly observed a transition from a crystalline Wigner crystal, exhibiting long-range order, to a completely random and amorphous “Anderson solid” as the filling factor and disorder strength vary. The significance of this work lies in establishing that the competition between the fractional quantum Hall liquid and the Wigner crystal is not simply determined by theoretical energy calculations, but is fundamentally a disorder-dependent crossover.

Experimental data spanning forty-five years consistently shows that the boundary between insulating behaviour and the emergence of fractional quantum Hall states shifts to lower filling factors as sample quality improves and disorder decreases. The authors acknowledge that a precise quantitative comparison between their theoretical model and experimental results is limited by uncertainties in quantum broadening effects. Future investigations could focus on further characterising the properties of the observed amorphous “Anderson solid” phase and exploring the precise mechanisms governing the transitions between these different quantum states. Understanding the role of disorder in these systems is crucial for both fundamental research into correlated electron behaviour and the potential development of novel electronic devices.