Wigner Crystals: Magnetic Fields Boost Interactions

The behaviour of electrons squeezed into two dimensions can give rise to exotic states of matter, including the Wigner crystal, where electrons arrange themselves in a regular pattern. Kyung-Su Kim and Anthony J. Leggett, at the University of Illinois Urbana-Champaign, along with their colleagues, investigate how magnetic fields and a property called Berry curvature influence the magnetic interactions within these crystals. Their work reveals that even weak magnetic fields cause electrons to tunnel along complex paths, altering the strength of interactions between them and introducing quantum mechanical phases dependent on the enclosed magnetic flux. Furthermore, the presence of Berry curvature necessitates considering the simultaneous tunneling of multiple electrons, adding another layer of complexity and potentially renormalising the interactions through effective mass corrections, which may explain recent observations in layered materials.

The semi-classical large-r expansion describes behaviour in physical systems, and when only a magnetic field is present, various ring-exchange interactions arise from electron tunneling along complex trajectories. These trajectories constitute complex instanton solutions of the coordinate-space path integral, and to leading order in the magnetic field strength, each ring-exchange constant acquires an Aharonov-Bohm (AB) phase, equal to the magnetic flux enclosed by the real tunneling trajectory of the B = 0 problem, an effect directly relevant to two-dimensional electron systems with a small g-factor. In the presence of Berry curvature, however, multi-particle tunneling must be considered within a complexified phase space, and to leading order, the exchange constants are affected.

Wigner Crystals, Berry Curvature and Quantum Effects

This research provides a detailed investigation into the interplay of Wigner crystals, Berry curvature, and many-body effects in two-dimensional electron systems. Wigner crystals form when strong interactions between electrons cause them to arrange themselves in a regular pattern, and this study explores how this order is affected by quantum phenomena. The research demonstrates that Berry curvature and quantum fluctuations can destabilize the Wigner crystal, leading to novel phases of matter. The authors employ a sophisticated theoretical framework based on complex instantons to calculate the exchange constants between electrons, capturing the non-perturbative effects of electron interactions and Berry curvature.

This reveals a rich phase diagram with various Wigner crystal phases, including those with chiral order and potentially quantum spin liquid behaviour. A particularly interesting finding is the prediction of a microemulsion phase, where the Wigner crystal breaks up into droplets surrounded by a fluid of electrons, stabilized by the interplay of electron repulsion and Berry curvature. The research connects these theoretical predictions to experimental systems like graphene/hBN moiré superlattices, suggesting potential avenues for experimental verification. The theoretical rigor, comprehensive treatment of the topic, clear presentation, and connection to experimental systems are key strengths of this work.

Future research could investigate the effects of disorder, finite-size effects, dynamical properties, higher-order interactions, and specific material systems. Exploring valley polarization and the role of electron-phonon interactions could also provide further insights. Overall, this is a significant contribution to our understanding of Wigner crystals and Berry curvature, likely to stimulate further research in this exciting field.

Magnetic Fields Tune Electron Exchange Interactions

Researchers have investigated how magnetic fields and Berry curvature influence the interactions between electrons in a two-dimensional Wigner crystal. The study reveals that even weak magnetic fields and Berry curvature significantly modify the exchange interactions, the quantum mechanical forces, between these electrons. These modifications arise from the electrons following complex paths as they tunnel between different locations within the crystal. The research demonstrates that when a magnetic field is applied, the electrons’ tunneling paths enclose areas, leading to an Aharonov-Bohm phase shift in the exchange interactions.

This phase shift alters the strength of the interactions, and its magnitude depends on the size of the enclosed area and the strength of the magnetic field. Interestingly, the calculations show that the area enclosed by these paths can be determined solely by the properties of the electron crystal, independent of the specific arrangement of the electrons or the field strength. Furthermore, the study extends to consider the impact of Berry curvature, a property related to the momentum of electrons in materials. When Berry curvature is present, the electrons’ tunneling occurs along imaginary paths in momentum space, again leading to a phase shift in the exchange interactions.

This Berry phase is distinct from the Aharonov-Bohm phase and arises from the unique way electrons experience forces due to their momentum. The calculations reveal that the Berry phase has a negative sign, indicating a different type of interaction compared to the magnetic field case. When both a magnetic field and Berry curvature are present simultaneously, the effects combine additively, influencing the overall exchange interactions. Crucially, the research also shows that these combined effects can subtly alter the effective mass of the electrons, modifying the electron crystal’s properties and influencing the tunneling rates and exchange interactions. These findings provide a deeper understanding of electron interactions in two-dimensional systems and could be relevant to the design of novel electronic devices.

Magnetic Coupling, Berry Curvature, and Wigner Crystals

This research investigates how magnetic fields and Berry curvature influence the magnetic interactions within a two-dimensional Wigner crystal. The findings demonstrate that both a perpendicular magnetic field and Berry curvature introduce phase shifts to the exchange interactions between electrons, effectively altering the strength of magnetic coupling. When both are present, an additional renormalization of the exchange constant occurs due to the coupling of electron orbital magnetic moments to the magnetic field. These effects are particularly relevant to understanding Wigner crystals in layered graphene systems, where recent experiments have revealed an extended region of this phase.

The research highlights that geometric phases, such as the Berry phase, manifest differently in correlated electron crystals compared to simpler systems. The authors acknowledge that a related study appeared concurrently, with largely consistent results where overlap occurred. Future work could focus on systematically investigating the phase diagram of the complex magnetic interactions identified, potentially revealing novel correlated phases, and exploring the broader implications of geometric phases in correlated electron systems. The study is limited to an asymptotic expansion, and further investigation may be needed to fully understand the behavior of the system across a wider range of parameters.