Electrons in Strong Magnetic Fields Reveal Competition Between Ordered States and Superconductivity

Researchers are increasingly focused on understanding the behaviour of three-dimensional electron gases subjected to strong magnetic fields, where partially flat bands emerge and Landau-level quantization occurs. Nandagopal Manoj, Valerio Peri, and Jason Alicea, all affiliated with the California Institute of Technology and working with colleagues at Hexagon Innovation Hub GmbH, have revisited this complex problem, incorporating realistic factors such as generalized interactions, higher Landau level bands, spin restoration, and broken spatial symmetries to explore the competition between charge density waves and superconductivity. Their work demonstrates that generic local interactions can stabilise a nematic charge density wave with an unconventional Hall response, and crucially, establishes the stability of a non-Fermi liquid state under specific symmetry conditions. Furthermore, the team discovered that breaking translational symmetry can induce a novel layered superconductor hosting Weyl nodes, offering potential insights into the design of high-field superconductivity in materials with low carrier densities.

Scientists are uncovering novel states of matter within three-dimensional electron gases subjected to intense magnetic fields, potentially paving the way for new materials with enhanced superconducting properties. This work revisits a decades-old problem, understanding how electrons interact in these extreme conditions, and reveals a surprising interplay between competing electronic orders. Early research indicated that strong repulsion between electrons in these systems leads to the formation of charge density waves, where electrons arrange themselves into distinct layers exhibiting integer quantum Hall states. Conversely, attractive interactions were predicted to yield a non-Fermi liquid state, a peculiar phase of matter lacking conventional metallic behaviour. Numerical calculations demonstrate that generic local interactions can stabilise a nematic charge density wave, characterised by tilted layers and an unconventional Hall effect. Furthermore, the research establishes that the previously predicted non-Fermi liquid state remains robust against certain types of disturbances, provided that fundamental dipole conservation symmetries are maintained. Most strikingly, researchers discovered that by explicitly breaking the translational symmetry of the electron gas, attraction can induce a novel layered superconductor. This superconductor exhibits Weyl nodes, unique points in its electronic structure, and demonstrates superconductivity within each layer while remaining insulating between layers. These findings not only deepen our understanding of interacting electrons in strong magnetic fields but also offer potential guidance for designing materials that maintain superconductivity even in the presence of strong external fields, a crucial goal for advanced technological applications. The work expands the known range of phenomena in high-field quantum matter and may inform the development of field-resistant superconductivity in materials with low electron densities. A detailed analysis of the condensate order parameter, ΨGL(r) = ⟨ψL(r)ψR(r)⟩, underpins the methodological approach employed in this work. The research begins by expressing the local particle creation operator in terms of Fourier transformed fields, ψR,qx,qy,qz and ψL,qx,qy,qz, facilitating calculations within a defined reciprocal space. Working with units of lB = 1 simplifies the notation and allows for a more streamlined analysis of the electron gas. The field operators are then expanded using a series of integrations and summations over wavevectors, ky, qz, and crucially, Ky, which is quantized in units of λ/2. This expansion leverages the properties of the lowest Landau level, allowing the researchers to express the condensate order parameter as an integral over momentum space, incorporating the influence of the deformed interactions. To avoid computationally intensive integrations, the study utilizes the delta function arising from the Bogoliubov-de Gennes (BdG) Hamiltonian, a standard technique for describing superconducting systems, to directly relate the order parameter to the Fourier transform of the induced pairing amplitude, ∆′(qx, qy). Symmetry considerations are central to the analysis, with the researchers meticulously examining the behaviour of ∆′(qx, qy) under translations in momentum space. By applying specific transformations to the integer indices, n and n′, they identify points where the order parameter vanishes, revealing the underlying lattice structure of the charge density wave. This approach, combined with an examination of the order parameter at translated points, establishes the presence of a triangular lattice with translation vectors (0, 2πl2 B/λ) and (λ/2, πl2 B/λ), providing insight into the spatial arrangement of the electron gas. The nematic charge density wave exhibits a spontaneous tilting of the integer Hall layers, a phenomenon not previously observed in this system. This tilting alters the conventional Hall response, indicating a modification of the electronic transport properties. The stability of the non-Fermi liquid phase is confirmed by its resilience to perturbations that maintain the dipole conservation symmetries, suggesting a robust character to this exotic state of matter. The layered superconductor, formed by breaking translational symmetry, presents a unique architecture with superconducting regions separated by insulating layers. The Weyl nodes within the superconducting layers are a key feature of this novel state, potentially leading to unusual surface states and topological properties. Superconductivity is strictly confined to the individual layers, while the material behaves as an insulator in the direction perpendicular to these layers. This anisotropic superconducting behaviour is a direct consequence of the broken translational symmetry and the resulting layered structure. The calculations reveal a complex interplay between the various instabilities, leading to a rich phase diagram dependent on the strength of interactions and the degree of symmetry breaking. The persistent challenge of achieving high-temperature superconductivity has long focused attention on unconventional pairing mechanisms, but this work suggests that geometric frustration may be equally fruitful. Researchers have demonstrated that in two-dimensional electron systems subjected to intense magnetic fields, the interplay between electron interactions and the peculiar quantum states that emerge, Landau levels, can give rise not simply to superconductivity, but to a layered superconducting state with striking topological properties. What distinguishes this finding is the explicit connection made between the tilting of charge density waves and the emergence of Weyl nodes within the superconducting layers, a combination not previously anticipated. This isn’t merely a refinement of existing theories; it proposes a new route to superconductivity predicated on manipulating the spatial arrangement of electrons. For decades, the quest for robust superconductivity has been hampered by the difficulty of controlling electron correlations and suppressing competing orders. This research indicates that carefully tuning these interactions within a high-magnetic-field environment can not only stabilise superconductivity but also engineer its properties, potentially leading to materials with enhanced critical temperatures and unusual electronic behaviour. However, translating these findings from theoretical models to real materials will be far from straightforward. The extremely high magnetic fields required represent a significant practical hurdle, and the precise control over interactions assumed in the calculations may prove difficult to achieve. Furthermore, the stability of this layered superconducting state against disorder and other imperfections remains an open question. Future work will likely focus on identifying material systems where these conditions can be approximated, and on exploring the potential for exploiting these topological features in novel electronic devices, perhaps even realising dissipationless transport at relatively high temperatures.