Geometry of High-dimensional Landau Levels

Reimagining Landau quantization beyond two dimensions -- Two-dimensional Landau levels provide one of the fundamental structures of condensed matter physics. A charged particle moving in a magnetic field develops highly degenerate quantum levels; after interactions are included, this simple structure underlies the integer and fractional quantum Hall effects.

A natural but surprisingly interesting question is: Is there a fundamental Landau-level structure in three dimensions? My early work with collaborators developed a family of higher-dimensional Landau-level systems in which ordinary Abelian magnetic fields are replaced or supplemented by spin-orbit and non-Abelian gauge structures. This program is now being revisited from a many-body perspective.

Quaternionic analytic Landau levels

In High-Dimensional Topological Insulators with Quaternionic Analytic Landau Levels, with Congjun Wu, we uncovered a three-dimensional Landau level model with wavefunctions of its lowest Landau level exhibiting mathematical structure based on quaternionic analyticity, a higher-dimensional analogue of the complex-analytic structure familiar from ordinary two-dimensional Landau levels.

This connection between quantum mechanics, topology, and quaternionic geometry provides a natural language for constructing highly degenerate topological states in dimensions where ordinary complex analysis is no longer sufficient.

Isotropic Landau levels in higher dimensions

In Isotropic Landau Levels of Dirac Fermions in High Dimensions, with Kenneth Intriligator, Yue Yu, and Congjun Wu, we further constructed higher-dimensional Landau-level structures for relativistic fermions.

SU(2) Landau levels and topological insulators

In Topological Insulators with SU(2) Landau Levels, with Shou-Cheng Zhang and Congjun Wu, we showed how non-Abelian SU(2) gauge structure can generate Landau-level physics and associated topological states in higher dimensions. The work is part of a sequence connecting Landau quantization, spin-orbit coupling, and topological-insulator physics.

From continuum Landau levels to a three-dimensional SU(2) Hofstadter problem

A natural next question was whether this higher-dimensional Landau-level physics survives when continuous space is replaced by a lattice. Yi Li addressed this in the single-author work Time-Reversal-Invariant SU(2) Hofstadter Problem in Three-Dimensional Lattices, published in Physical Review B 91, 195133 (2015). The work formulated the lattice version of the three-dimensional SU(2) Landau-level problem while preserving time-reversal symmetry.

The ordinary Hofstadter problem provides a lattice counterpart of two-dimensional Landau quantization: magnetic flux competes with lattice periodicity and produces an intricate topological band structure. The 2015 work generalized this idea from the familiar Abelian (U(1)) setting to a three-dimensional non-Abelian SU(2) gauge field, providing a lattice realization of the continuum SU(2) Landau-level construction developed in the earlier work. Using an SU(2) analogue of the Landau gauge, the three-dimensional lattice problem can be reduced to a one-dimensional SU(2) Harper equation with a periodic, spin-dependent gauge potential. The surface spectrum exhibits spatially separated helical boundary states associated with opposite eigenvalues of a lattice helicity operator. The bulk topology was characterized independently through the boundary helical Fermi surfaces and through calculation of the bulk (Z_2) indices.

This work also points naturally toward synthetic quantum systems. The paper discussed ultracold-atom implementations based on extending techniques used to engineer Abelian Hofstadter Hamiltonians to spin-dependent hopping and synthetic SU(2) gauge structures.

Current direction: Guiding center and many-body geometry in three-dimensional Landau levels

Current work studies SU(2) Runge–Lenz symmetry and semiclassical orbits in three-dimensional Landau levels, extending the geometric understanding of these systems, as well as investigations of the guiding-center algebra that controls interacting Landau-level physics.

The motion of a particle in three-dimensional Landau levels.

The longer-term question is whether the extraordinary many-body physics generated by two-dimensional Landau levels -- especially fractionalization and collective quantum geometry --has fundamentally new analogues in three dimensions.

Representative works

Current work: SU(2) Runge–Lenz symmetry, semiclassical dynamics, guiding-center algebra, and interacting physics in three-dimensional Landau levels.

Time-Reversal-Invariant SU(2) Hofstadter Problem in Three-Dimensional Lattices

Yi Li

Physical Review B 91, 195133 (2015)

arXiv:1410.6189

Topological Insulators with SU(2) Landau Levels

Yi Li, Shou-Cheng Zhang, Congjun Wu

Physical Review Letters 111, 186803 (2013)

High-Dimensional Topological Insulators with Quaternionic Analytic Landau Levels

Yi Li, Congjun Wu

Physical Review Letters 110, 216802 (2013)

Isotropic Landau Levels of Dirac Fermions in High Dimensions

Yi Li, Kenneth Intriligator, Yue Yu, Congjun Wu

Physical Review B 85, 085132 (2012)