At filling ν=1, intervalley-coherent states in opposite-Chern Landau level models are ground states only for reduced intravalley interactions, and their gapless spin mode rules them out as the sole explanation of the fractional quantum spin Hall effect.
The presence of a magnetic field modifies the translation operators and the way in which periodic boundary conditions are applied to finite-size systems
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Valley Order in Moir\'e Topological Insulators
At filling ν=1, intervalley-coherent states in opposite-Chern Landau level models are ground states only for reduced intravalley interactions, and their gapless spin mode rules them out as the sole explanation of the fractional quantum spin Hall effect.