Memo

Memo#

Bloch sphere#

Consider a spinor \(\psi_{\uparrow}(\mathbf{r})\ket{\uparrow} + \psi_{\downarrow}(\mathbf{r})\ket{\downarrow}\) with \(|\psi_{\uparrow}(\mathbf{r})|^{2} + |\psi_{\downarrow}(\mathbf{r})|^{2} = 1\).

The following construction maps the spinor to a point \(\left( m_{x}(\mathbf{r}), m_{y}(\mathbf{r}), m_{z}(\mathbf{r}) \right) \in S^{2}\):

\[\begin{split} m_{x}(\mathbf{r}) &:= \begin{pmatrix} \psi_{\uparrow}(\mathbf{r})^{\ast} & \psi_{\downarrow}(\mathbf{r})^{\ast} \end{pmatrix} \mathbf{\sigma}_{x} \begin{pmatrix} \psi_{\uparrow}(\mathbf{r}) \\ \psi_{\downarrow}(\mathbf{r}) \end{pmatrix} \\ &= 2 \,\mathrm{Re} \left( \psi_{\uparrow}(\mathbf{r})^{\ast} \psi_{\downarrow}(\mathbf{r}) \right) \in \mathbb{R} \\ m_{y}(\mathbf{r}) &:= \begin{pmatrix} \psi_{\uparrow}(\mathbf{r})^{\ast} & \psi_{\downarrow}(\mathbf{r})^{\ast} \end{pmatrix} \mathbf{\sigma}_{y} \begin{pmatrix} \psi_{\uparrow}(\mathbf{r}) \\ \psi_{\downarrow}(\mathbf{r}) \end{pmatrix} \\ &= 2 \,\mathrm{Im} \left( \psi_{\uparrow}(\mathbf{r})^{\ast} \psi_{\downarrow}(\mathbf{r}) \right) \in \mathbb{R} \\ m_{z}(\mathbf{r}) &:= \begin{pmatrix} \psi_{\uparrow}(\mathbf{r})^{\ast} & \psi_{\downarrow}(\mathbf{r})^{\ast} \end{pmatrix} \mathbf{\sigma}_{z} \begin{pmatrix} \psi_{\uparrow}(\mathbf{r}) \\ \psi_{\downarrow}(\mathbf{r}) \end{pmatrix} \\ &= |\psi_{\uparrow}(\mathbf{r})|^{2} - |\psi_{\downarrow}(\mathbf{r})|^{2} \in \mathbb{R} \\ \end{split}\]

\[ m_{x}(\mathbf{r})^{2} + m_{y}(\mathbf{r})^{2} + m_{z}(\mathbf{r})^{2} = 1 \]

Symmetry-adapted tensor with intrinsic symmetry#

Ref. [EBW08]

Consider vector space \(V\) and its symmetry adapted basis \(\{ \mathbf{f}^{\alpha m}_{i} \}\) w.r.t. group \(G\).

\[\begin{split} V &= \bigoplus_{\alpha} \bigoplus_{m=1}^{m_{\alpha}} V^{\alpha m} \\ V^{\alpha m} &= \bigoplus_{i=1}^{d_{\alpha}} K \mathbf{f}^{\alpha m}_{i} \\ g \mathbf{f}^{\alpha m}_{j} &= \sum_{i=1} ^{d_{\alpha}} \mathbf{f}^{\alpha m}_{i} \Gamma^{\alpha}_{ij}(g) \quad (g \in G, j = 1, \dots, d_{\alpha}), \end{split}\]
where \(K\) is \(\mathbb{C}\) or \(\mathbb{R}\), and \(\Gamma^{\alpha}\) is irrep over \(K\).

Action of \(G\) on rank-\(p\) tensor \(\mathsf{T}: V^{\ast \otimes p}\) is defined as

\[ (g \mathsf{T})(\mathbf{v}_{1}, \dots, \mathbf{v}_{p}) := \mathsf{T}(g^{-1} \mathbf{v}_{1}, \dots, g^{-1} \mathbf{v}_{p}) \quad (g \in G). \]
We also consider intrinsic symmetry \(\Sigma\) of \(\mathsf{T}\) as [1]
\[ (\sigma \mathsf{T})(\mathbf{v}_{1}, \dots, \mathbf{v}_{p}) := \mathsf{T}(\mathbf{v}_{\sigma^{-1}(1)}, \dots, \mathbf{v}_{\sigma^{-1}(p)}) \quad (\sigma \in \Sigma). \]

References#

[Alt05]

Simon L Altmann. Rotations, quaternions, and double groups. Courier Corporation, 2005.

[EBW08]

Michael El-Batanouny and Frederick Wooten. Symmetry and condensed matter physics: a computational approach. Cambridge University Press, 2008.

[HKH00]

D. Hobbs, G. Kresse, and J. Hafner. Fully unconstrained noncollinear magnetism within the projector augmented-wave method. Phys. Rev. B, 62:11556–11570, Nov 2000. URL: https://link.aps.org/doi/10.1103/PhysRevB.62.11556, doi:10.1103/PhysRevB.62.11556.

[SKMK16]

Soner Steiner, Sergii Khmelevskyi, Martijn Marsmann, and Georg Kresse. Calculation of the magnetic anisotropy with projected-augmented-wave methodology and the case study of disordered $\mathrm Fe_1\ensuremath -x\mathrm Co_x$ alloys. Phys. Rev. B, 93:224425, Jun 2016. URL: https://link.aps.org/doi/10.1103/PhysRevB.93.224425, doi:10.1103/PhysRevB.93.224425.

[Wes]

D.B. Westra. SU(2) and SO(3). https://www.mat.univie.ac.at/ westra/so3su2.pdf.

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