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Transversal Propagation Behaviors of Helical Edge States In a QSH System

The transversal propagation of the edge states in a two-dimensional quantum spin Hall system are classified by decay characteristic quantity $λ$. Two different modes of the helical edge states exhibit distinct behaviors. The penetration depth is momentum dependent in normal edge states. The finite size gap decays monotonously with sample width, leading to the normal size effect. In contrast, the penetration depth maintains a uniform minimal value in the special edge states with much shorter length. The finite size gap decays non-monotonously with width, leading to the anomalous finite size effect. Real materials are compared in the phase diagram, which explicitly demonstrates their differences. We also propose an intuitive way to search the special edge states in two-dimensional quantum spin Hall system.