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Anomalously sharp (delta-function-like) $n=0$ Landau level in the presence ofdisorder is usually considered to be a manifestation of the massless Diracfermions in magnetic fields. This property persists even when the Dirac cone istilted, which has been shown by Kawarabayashi et al. [Phys. Rev. B {\bf 83},153414 (2011)] to be a consequence of a "generalized chiral symmetry". Here wepose a question whether this property will be washed out when the tilted Diracfermion becomes massive. Surprisingly, the levels persist to bedelta-function-like, although the mass term that splits $n=0$ Landau levels mayseem to degrade the anomalous sharpness. This has been shown both numericallyfor a tight-binding model, and analytically in terms of the Aharonov-Casherargument extended to the massive tilted Dirac fermions. A key observation isthat, while the generalized chiral symmetry is broken by the mass term, the$n=0$ Landau level remains to accommodate eigenstates of the generalized chiraloperator, resulting in the robustness against chiral-symmetric disorders.Mathematically, the conventional and generalized chiral operators are relatedwith each other via a non-unitary transformation, with which the split,nonzero-energy $n=0$ wave functions of the massive system are justgauge-transformed zero-mode wave functions of the massless system. A message isthat the chiral symmetry, rather than a simpler notion of the sublatticesymmetry, is essential for the robustness of the $n=0$ Landau level.

Survival of sharp $n=0$ Landau levels in massive tilted Dirac fermions: Protection by generalized chiral operator