The intrinsic spin Hall conductivity and the anomalous Hall conductivity of ferromagnetic L1₀-CoPt are studied using first principles calculations. We identify the symmetry of the Berry curvature and the spin Berry curvature to reveal the correlation between the anomalous Hall and the spin Hall conductivities. Whereas the Berry curvature preserves the C_4v crystal rotation symmetry along the c-axis, in accordance with the crystal symmetry, the symmetry of the spin Berry curvature is reduced to C_2v. Such symmetry reduction is most notable at band crossing points where the Bloch states have opposite spin characters. We show that the symmetry reduction of the spin Berry curvature originates from the difference in the matrix elements of the velocity and spin current operators at such crossing point. The symmetry reduction implies that the anomalous Hall conductivity does not simply scale with the spin Hall conductivity, with a scaling factor being the spin polarization at the Fermi level, as band crossing points with opposite spin characters are ubiquitous in many systems.

中文翻译：

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铁磁CoPt的异常霍尔和自旋霍尔效应的第一性原理研究。

利用第一性原理计算研究了铁磁L1 ₀ -CoPt的本征自旋霍尔电导率和反常霍尔电导率。我们确定了贝里曲率和自旋贝里曲率的对称性，以揭示异常霍尔和自旋霍尔电导率之间的相关性。贝里曲率保持沿c轴的C _{4 v}晶体旋转对称性，但根据晶体对称性，自旋贝里曲率的对称性减小为C _{2 v}。这种对称性降低在布洛赫状态具有相反的自旋特性的带交叉点处最为明显。我们表明，自旋贝里曲率的对称性减小源于在该交叉点处速度和自旋电流算子的矩阵元素的差异。对称性的降低意味着异常的霍尔电导率不仅仅随自旋霍尔电导率成比例，缩放因数是费米能级的自旋极化，因为在许多系统中普遍存在具有相反自旋特性的带交点。