Abstract
According to the Onsager’s semiclassical quantization rule, the Landau levels of a band are bounded by its upper and lower band edges at zero magnetic field. However, there are two notable systems where the Landau level spectra violate this expectation, including topological bands and flat bands with singular band crossings, whose wave functions possess some singularities. Here, we introduce a distinct class of flat band systems where anomalous Landau level spreading (LLS) appears outside the zero-field energy bounds, although the relevant wave function is nonsingular. The anomalous LLS of isolated flat bands are governed by the cross-gap Berry connection that measures the wave-function geometry of multi bands. We also find that symmetry puts strong constraints on the LLS of flat bands. Our work demonstrates that an isolated flat band is an ideal system for studying the fundamental role of wave-function geometry in describing magnetic responses of solids.
Highlights
According to the Onsager’s semiclassical quantization rule, the Landau levels of a band are bounded by its upper and lower band edges at zero magnetic field
B linear quantum corrections, we find that an isolated flat band (IFB) generally exhibits anomalous level spreading (LLS), and the upper and lower energy bounds for the LLS are determined by the cross-gap Berry connection defined as ðn ≠ mÞ; ð2Þ
We demonstrate that the LLS is proportional to B2 for a flat-band system with IST symmetry in the zero magnetic fields, while the LLS is forbidden in the presence of chiral symmetry
Summary
According to the Onsager’s semiclassical quantization rule, the Landau levels of a band are bounded by its upper and lower band edges at zero magnetic field. In graphene witph ffirffiffieffiffilativistic energy dispersion, Eq (1) successfully predicts the nB dependence of the Landau levels, where the existence of the zero-energy Landau level is a direct manifestation of the π-Berry phase of massless Dirac particles[6, 7] Later, this semiclassical approach is generalized further to the cases with an arbitrary strength of magnetic field[8] where the zero-field energy dispersion in Eq (1) is replaced by the magnetic band structure with B linear quantum corrections. In usual dispersive bands where B linear quantum corrections are negligible in weak-field limit, Onsager’s semiclassical approach in Eq (1) predicts that the Landau levels are developed in the energy interval bounded by the upper and lower band edges of the zero-field band structure. The total energy spreading of the flat band’s Landau levels, dubbed the Landau level spreading (LLS), is solely determined by a geometric quantity, called the maximum quantum distance which characterizes the singularity of the relevant Bloch wave function[14]
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