标题： 各向异性与低维导体中的Zeeman、自旋-轨道和量子自旋霍尔相互作用 显示英文标题

标题： The Zeeman, Spin-Orbit, and Quantum Spin-Hall Interactions in Anisotropic and Low-Dimensional Conductors

摘要： 当电子或空穴处于晶体的导带中时，它可能与2有很大不同，这取决于晶体的各向异性和施加的磁感应方向 ${\bf B}$。 事实上，它甚至可以是0！ 为了定量地证明这一点，狄拉克方程被扩展用于正交各向异性导带中的相对论性电子或空穴，其有效质量为 $m_j$，对于 $j=1,2,3$，几何平均值为 $m_g=(m_1m_2m_3)^{1/3}$。 适当的Foldy-Wouthuysen变换被扩展以计算非相对论哈密顿量到 $O({\rm m}c^2)^{-4}$，其中 ${\rm m}c^2$是粒子的爱因斯坦静止能量。 对于${\bf B}||\hat{\bf e}_{\mu}$，Zeeman$g_{\mu}$因子是$2{\rm m}\sqrt{m_{\mu}}/m_g^{3/2} + O({\rm m}c^2)^{-2}$。 在二维（2D）导带中传播时，$m_3\gg m_1,m_2$，$g_{||}<<2$，与最近对超导单层NbSe$_2$以及扭曲双层石墨烯中平行上临界感应强度$B_{c2,||}(T)$的温度$T$依赖性的测量结果一致。 当粒子处于沿$\hat{\bf e}_{\mu}$的原子薄一维金属链的导带中时，$g<<2$对所有${\bf B}={\bf\nabla}\times{\bf A}$方向都存在，而对${\bf B}||\hat{\bf e}_{\mu}$则几乎为零。 二维金属具有$m_1=m_2=m_{||}$的量子自旋霍尔哈密顿量是$K[{\bf E}\times({\bf p}-q{\bf A})]_{\perp}\sigma_{\perp}+O({\rm m}c^2)^{-4}$，其中${\bf E}$和${\bf p}-q{\bf A}$是平面电场和规范不变动量，$q=\mp|e|$是粒子的电荷，$\sigma_{\perp}$是垂直于层的泡利矩阵，$K=\pm\mu_B/(2m_{||}c^2)$和$\mu_B$是玻尔磁子。 显示英文摘要

摘要： When an electron or hole is in a conduction band of a crystal, it can be very different from 2, depending upon the crystalline anisotropy and the direction of the applied magnetic induction ${\bf B}$. In fact, it can even be 0! To demonstrate this quantitatively, the Dirac equation is extended for a relativistic electron or hole in an orthorhombically-anisotropic conduction band with effective masses $m_j$ for $j=1,2,3$ with geometric mean $m_g=(m_1m_2m_3)^{1/3}$. The appropriate Foldy-Wouthuysen transformations are extended to evaluate the non-relativistic Hamiltonian to $O({\rm m}c^2)^{-4}$, where ${\rm m}c^2$ is the particle's Einstein rest energy. For ${\bf B}||\hat{\bf e}_{\mu}$, the Zeeman $g_{\mu}$ factor is $2{\rm m}\sqrt{m_{\mu}}/m_g^{3/2} + O({\rm m}c^2)^{-2}$. While propagating in a two-dimensional (2D) conduction band with $m_3\gg m_1,m_2$, $g_{||}<<2$, consistent with recent measurements of the temperature $T$ dependence of the parallel upper critical induction $B_{c2,||}(T)$ in superconducting monolayer NbSe$_2$ and in twisted bilayer graphene. While a particle is in its conduction band of an atomically thin one-dimensional metallic chain along $\hat{\bf e}_{\mu}$, $g<<2$ for all ${\bf B}={\bf\nabla}\times{\bf A}$ directions and vanishingly small for ${\bf B}||\hat{\bf e}_{\mu}$. The quantum spin Hall Hamiltonian for 2D metals with $m_1=m_2=m_{||}$ is $K[{\bf E}\times({\bf p}-q{\bf A})]_{\perp}\sigma_{\perp}+O({\rm m}c^2)^{-4}$, where ${\bf E}$ and ${\bf p}-q{\bf A}$ are the planar electric field and gauge-invariant momentum, $q=\mp|e|$ is the particle's charge, $\sigma_{\perp}$ is the Pauli matrix normal to the layer, $K=\pm\mu_B/(2m_{||}c^2)$, and $\mu_B$ is the Bohr magneton.