Hokkaido University Team Models Many-Body Quantum Hall Phase Diagram for Fermion Systems

Giovanna Marcelli from Roma Tre University and colleagues at Hokkaido University, Sapienza University of Rome, University of Tübingen reveal that the familiar coloured Hofstadter butterfly holds a key interpretation for interacting quantum systems. The team identify locally unique gapped ground states within a parameter space defined by magnetic field, chemical potential, and interaction strength. Their findings rigorously show that the integer ‘colours’ of the non-interacting butterfly continue to represent quantized Hall conductivity in these interacting systems, offering a new understanding of many-body quantum Hall phases and extending this understanding to both commensurate and incommensurate magnetic fluxes.

Quantised Hall conductivity directly links to Hofstadter butterfly features

A quantifiable persistence of the Hofstadter butterfly’s features is now apparent. Hall conductivity, previously reliant on non-interacting models, now maintains quantized values, specifically satisfying the relationship 2πσH ∈Z, even with weak electron interactions. Establishing this quantized behaviour in interacting systems proved impossible previously due to the complexities of modelling electron-electron interactions and their impact on energy levels; a method to definitively link the butterfly’s ‘colours’ to measurable electrical properties beyond simplified scenarios was lacking. The significance of this lies in the fact that the quantum Hall effect, and its associated quantized Hall conductivity, is a topologically protected phenomenon with potential applications in metrology and quantum computing. However, understanding its behaviour in systems where electrons interact, a more realistic depiction of materials, has remained a substantial challenge. Previous theoretical approaches often relied on perturbative methods, which are limited to weak interactions and can obscure the underlying physics. This new work provides a non-perturbative framework for understanding the quantum Hall effect in weakly interacting systems.

Marcelli and colleagues at Hokkaido University rigorously proved that these distinct ‘colours’ within the diagram directly correspond to macroscopic Hall conductivity, extending the understanding of quantum Hall phases to both commensurate and incommensurate magnetic fluxes. Hokkaido University scientists have substantiated this connection, extending it to both commensurate and incommensurate magnetic fluxes, situations where traditional modelling techniques struggle due to the absence of a repeating magnetic unit cell. Commensurate flux corresponds to a rational fraction of magnetic flux per unit cell (e.g., 1/2, 2/3), leading to a well-defined periodic potential, while incommensurate flux (irrational fractions like π/e) results in an aperiodic potential. The ability to demonstrate quantized Hall conductivity in both regimes is crucial for broadening the applicability of this understanding to a wider range of materials. This demonstrates that gapped regions, indicative of stable quantum phases, persist even with weak interactions between electrons, a key step towards understanding real-world materials. These gapped regions represent insulating states, and their persistence suggests that the topological order responsible for the quantized Hall effect is robust against weak perturbations. Translating these findings into practical, room-temperature devices, or demonstrating this persistence at stronger interaction strengths, remains a significant challenge, however. The primary obstacle lies in finding materials that exhibit strong electron correlations while maintaining the necessary conditions for observing the quantum Hall effect, such as high magnetic fields and low temperatures.

Establishing the Hofstadter Butterfly via Quasi-Adiabatic Evolution of Interacting Lattice Fermions

Quasi-adiabatic continuation proved central to establishing the enduring relevance of the Hofstadter butterfly; the technique involves gradually changing a system’s parameters, like magnetic field or interaction strength, while carefully tracking its energy levels. The principle behind quasi-adiabatic continuation is to ensure that the changes to the system are slow enough that the system remains close to its ground state at all times. This allows researchers to track the evolution of the energy spectrum and identify any topological changes that might occur. Beginning with a simplified, non-interacting model, a lattice fermion system where electrons behave as isolated particles on a grid, the team slowly introduced interactions between them, carefully monitoring the energy landscape’s evolution. The lattice fermion system is a standard model in condensed matter physics, representing electrons confined to a periodic lattice structure. The introduction of interactions is typically modelled using a Hubbard-like Hamiltonian, which accounts for both kinetic energy and on-site Coulomb repulsion between electrons. This approach is akin to smoothly morphing one shape into another, allowing the team to confidently predict system behaviour at each stage, avoiding abrupt transitions that might obscure the underlying physics. They investigated a lattice fermion system, focusing on the interplay between magnetic field, chemical potential, and interaction strength, parameters defining a three-dimensional space vital to understanding the system’s behaviour. The magnetic field induces the Hofstadter butterfly spectrum, the chemical potential controls the electron density, and the interaction strength governs the strength of electron-electron repulsion. The team meticulously mapped out the parameter space (b, μ, λ), where ‘b’ represents the magnetic flux, ‘μ’ the chemical potential, and ‘λ’ the interaction strength, to identify regions where stable gapped states exist.

Mapping electronic states in magnetic fields reveals connections to measurable conductivity

The confirmation of the Hofstadter butterfly’s relevance to interacting systems offers a powerful new perspective through which to view complex materials. Understanding how electrons behave in magnetic fields is vital for designing future electronic devices. The researchers from the Institute of Quantum Optics and the University of Heidelberg have demonstrated that the intricate patterns within the Hofstadter butterfly, a famous visualisation of electron behaviour in magnetic fields, extend to systems where electrons interact. The Hofstadter butterfly arises from the interplay between the periodic lattice potential and the magnetic field, leading to a fractal energy spectrum. This work demonstrates that the distinct ‘colours’ within this diagram, traditionally understood for non-interacting systems, accurately represent quantized Hall conductivity even when electrons influence each other; Hall conductivity is a measure of electrical flow perpendicular to a magnetic field. Specifically, each ‘colour’ corresponds to a quantized Hall plateau, where the Hall conductivity remains constant over a range of magnetic field strengths. By employing quasi-adiabatic continuation, scientists tracked how energy levels evolved as interactions were gradually introduced, confirming the persistence of stable energy states. The current work focused on weakly interacting electrons, raising a critical question regarding the fate of these meticulously mapped ‘colours’ and gapped phases when interactions become sharply stronger. Future research will need to explore the behaviour of the system at higher interaction strengths, potentially revealing new quantum phases and phenomena. Investigating the impact of disorder and imperfections in the lattice structure is also crucial for assessing the robustness of these findings in real-world materials.

The research confirmed that the patterns within the Hofstadter butterfly accurately reflect the Hall conductivity of interacting electron systems. This is significant because understanding electron behaviour in magnetic fields is important for the development of electronic materials. Scientists proved that the ‘colours’ within the butterfly correspond to constant, quantized values of Hall conductivity, even when electrons interact with each other. The study focused on weakly interacting electrons and the authors suggest further investigation is needed to understand how these findings apply when interactions are stronger.