Resta R, Ceresoli Davide, Thonhauser T, Vanderbilt David
INFM Democritos National Simulation Center and Dipartimento di Fisica Teorica, Università di Trieste, Strada Costiera 11, 34014 Trieste, Italy.
Chemphyschem. 2005 Sep 5;6(9):1815-9. doi: 10.1002/cphc.200400641.
While the orbital magnetic dipole moment of any finite sample is well-defined, it becomes ill-defined in the thermodynamic limit as a result of the unboundedness of the position operator. Effects due to surface currents and to bulk magnetization are not easily disentangled. The corresponding electrical problem, where surface charges and bulk polarization appear as entangled, was solved about a decade ago by the modern theory of polarization, based on a Berry phase. We follow a similar path here, making progress toward a bulk expression for the orbital magnetization in an insulator represented by a lattice-periodic Hamiltonian with broken time-reversal symmetry. We therefore limit ourselves to the case where the macroscopic (i.e. cell-averaged) magnetic field vanishes. We derive an expression for the contribution to the magnetization arising from the circulating currents internal to the bulk Wannier functions, and then transform to obtain a Brillouin zone integral involving the occupied Bloch orbitals. A version suitable for practical implementation in discretized reciprocal space is also derived, and the gauge invariance of both versions is explicitly shown. However, tests on a tight-binding model indicate the presence of additional edge currents, and it remains to be determined whether these can be related to the bulk band structure.
虽然任何有限样品的轨道磁偶极矩都是明确的，但由于位置算符的无界性，在热力学极限下它会变得不明确。表面电流和体磁化所产生的效应不易区分。大约十年前，基于贝里相位的现代极化理论解决了相应的电学问题，其中表面电荷和体极化表现为相互纠缠的情况。我们在此遵循类似的路径，朝着用具有破时间反演对称性的晶格周期哈密顿量表示的绝缘体中轨道磁化的体表达式迈进。因此，我们将自己限制在宏观（即晶胞平均）磁场为零的情况。我们推导了一个表达式，用于表示体万尼尔函数内部循环电流对磁化强度的贡献，然后进行变换以得到一个涉及占据布洛赫轨道的布里渊区积分。还推导了一个适用于离散倒易空间实际实现的版本，并明确展示了两个版本的规范不变性。然而，对紧束缚模型的测试表明存在额外的边缘电流，这些电流是否与体带结构有关仍有待确定。
