Non-Fermi liquid and Weyl superconductivity from the weakly interacting 3D electron gas at high magnetic fields

Three-dimensional electron gases in strong magnetic fields host partially flat bands that disperse along the field direction yet exhibit Landau-level quantization in the transverse dimensions. Early work established that for spin-polarized electrons confined to the lowest Landau level band, repulsion triggers a charge density wave (CDW) in which electrons ‘self-layer’ into integer quantum Hall states, while attraction generates a non-Fermi liquid (rather than a superconductor). We revisit this problem with physically motivated deformations – including generalized local interactions, higher Landau level bands, restoration of spin, and explicit breaking of spatial symmetries – paying particular attention to the competition between CDWs and superconductivity. Our main findings are: (1) Generic local interactions can stabilize a nematic CDW in which integer quantum Hall layers spontaneously ’tilt’, yielding unconventional Hall response. (2) We numerically establish that the non-Fermi liquid appears stable to perturbations that preserve effective dipole conservation symmetries that emerge within a Landau level band. (3) Upon explicitly breaking translation symmetry, attraction catalyzes a novel layered superconductor that hosts Weyl nodes, superconducts within each layer, and insulates transverse to the layers. These results expand the rich phenomenology of interacting bulk electrons in the high-field regime and potentially inform the design of field-resistant superconductivity in low-carrier-density materials.

💡 Research Summary

The paper investigates a three‑dimensional electron gas subjected to a strong magnetic field, a regime in which the kinetic energy is quenched in the transverse (x‑y) directions and the electronic dispersion remains only along the field (z) direction. In this “ultra‑quantum” limit the system occupies a small number of Landau‑level bands that are flat in k_y and dispersive in k_z, producing a pair of parallel Fermi sheets at ±k_F. The authors start from a spin‑polarized, lowest‑Landau‑level (LLL) model and introduce the most general local four‑fermion interaction consistent with the microscopic symmetries: U(1) charge conservation, magnetic translation and rotation in the xy‑plane, translation along z, and combined time‑reversal–mirror symmetries. Because the LLL wavefunctions are exponentially localized in x, the interaction kernel h(k_y1,k_y2) decays exponentially with the transverse momentum transfer, but otherwise remains an arbitrary function of the two transverse momenta.

Using a functional renormalization‑group (FRG) approach based on a one‑loop Wilsonian scheme, the authors derive a flow equation for the entire coupling function h(k_y1,k_y2). At tree level every such interaction is marginal (scaling dimension two) because only the longitudinal momentum and frequency are rescaled; the transverse momenta are not. The FRG analysis reveals three distinct asymptotic behaviors. First, most initial couplings flow to strong coupling, indicating an instability toward an ordered phase. Second, when the emergent dipole‑conservation symmetries—one for each transverse direction—are strictly preserved, the flow is arrested and the system settles into a non‑Fermi‑liquid (NFL) fixed point. This NFL is the three‑dimensional analogue of a Luttinger liquid: the charge sector remains gapless, but the coupling function does not diverge, reflecting a delicate balance between particle‑hole (CDW) and particle‑particle (BCS) channels. The authors provide numerical evidence that the NFL is robust for a wide range of initial h(k_y1,k_y2) as long as both dipole symmetries are intact.

Third, the authors explicitly break translation symmetry along the field direction by adding a weak periodic potential V(z)=V_0 cos(Qz). This perturbation violates one of the dipole‑conservation laws, suppresses the CDW channel, and allows the Cooper‑pair channel to dominate even for weak attractive interactions. The resulting ground state is a layered superconductor: each “layer” (a region of constant V) hosts a two‑dimensional integer quantum Hall state intertwined with a superconducting order parameter. Solving the Bogoliubov‑de Gennes equations shows that the quasiparticle spectrum contains Weyl nodes at momenta where the superconducting gap changes sign, i.e., a Weyl superconducting phase. The Josephson coupling between neighboring layers inherits the residual dipole symmetry, leading to an unconventional, coordinate‑dependent inter‑layer coupling that prevents phase stiffness in the transverse direction. Consequently the bulk behaves as a “Bose‑Einstein insulator” (superconducting within each layer but insulating across layers), while surface boundaries, which break the remaining dipole symmetry, can support transverse supercurrents.

Beyond the LLL, the paper explores higher Landau‑level bands and restores spin degrees of freedom. Inclusion of spin modifies the stability criteria for the NFL, establishing a quantitative correspondence with a strictly one‑dimensional spinful Luttinger liquid. For higher Landau levels the authors find that a nematic charge‑density‑wave (CDW) can become energetically favorable: the integer‑quantum‑Hall layers tilt relative to the magnetic field, producing an anisotropic Hall response that deviates from the conventional quantized value.

The work concludes with a discussion of experimental relevance. Materials with low carrier density—such as Weyl semimetals, narrow‑gap semiconductors, or moiré heterostructures—can reach the ultra‑quantum limit at experimentally accessible magnetic fields. By engineering a weak periodic modulation (e.g., via a superlattice or strain pattern) one could realize the layered Weyl superconducting state, while preserving dipole‑conservation symmetries would be essential for stabilizing the NFL phase. The authors suggest that their findings open a pathway toward designing field‑resistant superconductors and exploring exotic non‑Fermi‑liquid behavior in three‑dimensional flat‑band systems.