Abstract The 2D Dirac oscillator can be viewed as the Dirac operator for a charged particle in a constant magnetic field. This system admits a non-abelian magnetic translation symmetry. Under a cocycle condition with respect to this symmetry, the 2D Dirac oscillator Hamiltonian is made into a Dirac Hamiltonian defined on the two-torus as a magnetic unit cell. A remarkable feature of this system is the existence of an edge-state (or extremal) eigenvalue which is attributed to different bands, according to the range to which the value of the control parameter belongs. The other eigenvalues, which are called bulk-state eigenvalues, are attributed to either of two bands for all values of the control parameter. For a sufficiently weak magnetic flux density, the Dirac oscillator Hamiltonian is brought into a semi-quantum Hamiltonian after the Landau–Peierls substitution. Then, a bulk-edge correspondence is shown to hold in terms of spectral flow for the quantum Hamiltonian and change in (formal) Chern number for the semi-quantum Hamiltonian.

中文翻译：

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两个圆环面上狄拉克振荡器的体边缘对应关系作为磁性单位单元

摘要 二维狄拉克振荡器可以看作是恒定磁场中带电粒子的狄拉克算子。该系统承认非阿贝尔磁平移对称性。在关于这种对称性的共循环条件下，二维狄拉克振荡器哈密顿量被制成一个狄拉克哈密顿量，定义在两个圆环上作为磁晶胞。该系统的一个显着特点是存在一个边缘状态（或极值）特征值，根据控制参数值所属的范围，该特征值归属于不同的波段。其他特征值，称为体态特征值，归属于控制参数所有值的两个波段中的任何一个。对于足够弱的磁通密度，在 Landau-Peierls 替换后，狄拉克振荡器哈密顿量被带入半量子哈密顿量。然后，根据量子哈密顿量的谱流和半量子哈密顿量的（形式）陈数的变化，显示了体边缘对应关系。