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5 | 5 |
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6 | 6 | ## Bloch sphere |
7 | 7 |
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8 | | -Consider a spinor $\\psi\_{\\uparrow}(\\mathbf{r})\\ket{\\uparrow} + \\psi\_{\\downarrow}(\\mathbf{r})\\ket{\\downarrow}$ with $|\\psi\_{\\uparrow}(\\mathbf{r})|^{2} + |\\psi\_{\\downarrow}(\\mathbf{r})|^{2} = 1$. |
| 8 | +Consider a spinor $\psi_{\uparrow}(\mathbf{r})\ket{\uparrow} + \psi_{\downarrow}(\mathbf{r})\ket{\downarrow}$ with $|\psi_{\uparrow}(\mathbf{r})|^{2} + |\psi_{\downarrow}(\mathbf{r})|^{2} = 1$. |
9 | 9 |
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10 | | -The following construction maps the spinor to a point $\\left( m\_{x}(\\mathbf{r}), m\_{y}(\\mathbf{r}), m\_{z}(\\mathbf{r}) \\right) \\in S^{2}$: |
| 10 | +The following construction maps the spinor to a point $\left( m_{x}(\mathbf{r}), m_{y}(\mathbf{r}), m_{z}(\mathbf{r}) \right) \in S^{2}$: |
11 | 11 | $$ |
12 | | -m\_{x}(\\mathbf{r}) |
13 | | -&:= |
14 | | -\\begin{pmatrix} \\psi\_{\\uparrow}(\\mathbf{r})^{\\ast} & \\psi\_{\\downarrow}(\\mathbf{r})^{\\ast} \\end{pmatrix} |
15 | | -\\mathbf{\\sigma}_{x} |
16 | | -\\begin{pmatrix} \\psi_{\\uparrow}(\\mathbf{r}) \\ \\psi\_{\\downarrow}(\\mathbf{r}) \\end{pmatrix} \\ |
17 | | -&= |
18 | | -2 ,\\mathrm{Re} \\left( \\psi\_{\\uparrow}(\\mathbf{r})^{\\ast} \\psi\_{\\downarrow}(\\mathbf{r}) \\right) \\in \\mathbb{R} \\ |
19 | | -m\_{y}(\\mathbf{r}) |
20 | | -&:= |
21 | | -\\begin{pmatrix} \\psi\_{\\uparrow}(\\mathbf{r})^{\\ast} & \\psi\_{\\downarrow}(\\mathbf{r})^{\\ast} \\end{pmatrix} |
22 | | -\\mathbf{\\sigma}_{y} |
23 | | -\\begin{pmatrix} \\psi_{\\uparrow}(\\mathbf{r}) \\ \\psi\_{\\downarrow}(\\mathbf{r}) \\end{pmatrix} \\ |
24 | | -&= |
25 | | -2 ,\\mathrm{Im} \\left( \\psi\_{\\uparrow}(\\mathbf{r})^{\\ast} \\psi\_{\\downarrow}(\\mathbf{r}) \\right) \\in \\mathbb{R} \\ |
26 | | -m\_{z}(\\mathbf{r}) |
27 | | -&:= |
28 | | -\\begin{pmatrix} \\psi\_{\\uparrow}(\\mathbf{r})^{\\ast} & \\psi\_{\\downarrow}(\\mathbf{r})^{\\ast} \\end{pmatrix} |
29 | | -\\mathbf{\\sigma}_{z} |
30 | | -\\begin{pmatrix} \\psi_{\\uparrow}(\\mathbf{r}) \\ \\psi\_{\\downarrow}(\\mathbf{r}) \\end{pmatrix} \\ |
31 | | -&= |
32 | | -|\\psi\_{\\uparrow}(\\mathbf{r})|^{2} - |\\psi\_{\\downarrow}(\\mathbf{r})|^{2} \\in \\mathbb{R} \\ |
| 12 | +m_{x}(\mathbf{r}) |
| 13 | + &:= |
| 14 | + \begin{pmatrix} \psi_{\uparrow}(\mathbf{r})^{\ast} & \psi_{\downarrow}(\mathbf{r})^{\ast} \end{pmatrix} |
| 15 | + \mathbf{\sigma}_{x} |
| 16 | + \begin{pmatrix} \psi_{\uparrow}(\mathbf{r}) \\ \psi_{\downarrow}(\mathbf{r}) \end{pmatrix} \\ |
| 17 | + &= |
| 18 | + 2 \,\mathrm{Re} \left( \psi_{\uparrow}(\mathbf{r})^{\ast} \psi_{\downarrow}(\mathbf{r}) \right) \in \mathbb{R} \\ |
| 19 | +m_{y}(\mathbf{r}) |
| 20 | + &:= |
| 21 | + \begin{pmatrix} \psi_{\uparrow}(\mathbf{r})^{\ast} & \psi_{\downarrow}(\mathbf{r})^{\ast} \end{pmatrix} |
| 22 | + \mathbf{\sigma}_{y} |
| 23 | + \begin{pmatrix} \psi_{\uparrow}(\mathbf{r}) \\ \psi_{\downarrow}(\mathbf{r}) \end{pmatrix} \\ |
| 24 | + &= |
| 25 | + 2 \,\mathrm{Im} \left( \psi_{\uparrow}(\mathbf{r})^{\ast} \psi_{\downarrow}(\mathbf{r}) \right) \in \mathbb{R} \\ |
| 26 | +m_{z}(\mathbf{r}) |
| 27 | + &:= |
| 28 | + \begin{pmatrix} \psi_{\uparrow}(\mathbf{r})^{\ast} & \psi_{\downarrow}(\mathbf{r})^{\ast} \end{pmatrix} |
| 29 | + \mathbf{\sigma}_{z} |
| 30 | + \begin{pmatrix} \psi_{\uparrow}(\mathbf{r}) \\ \psi_{\downarrow}(\mathbf{r}) \end{pmatrix} \\ |
| 31 | + &= |
| 32 | + |\psi_{\uparrow}(\mathbf{r})|^{2} - |\psi_{\downarrow}(\mathbf{r})|^{2} \in \mathbb{R} \\ |
33 | 33 | $$ |
34 | 34 |
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35 | 35 | $$ |
36 | | -m\_{x}(\\mathbf{r})^{2} + m\_{y}(\\mathbf{r})^{2} + m\_{z}(\\mathbf{r})^{2} = 1 |
| 36 | +m_{x}(\mathbf{r})^{2} + m_{y}(\mathbf{r})^{2} + m_{z}(\mathbf{r})^{2} = 1 |
37 | 37 | $$ |
38 | 38 |
|
39 | 39 | ## Symmetry-adapted tensor with intrinsic symmetry |
40 | 40 |
|
41 | 41 | Ref. {cite}`el2008symmetry` |
42 | 42 |
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43 | | -Consider vector space $V$ and its symmetry adapted basis ${ \\mathbf{f}^{\\alpha m}_{i} }$ w.r.t. group $G$. |
| 43 | +Consider vector space $V$ and its symmetry adapted basis $\{ \mathbf{f}^{\alpha m}_{i} \}$ w.r.t. group $G$. |
44 | 44 | $$ |
45 | | -V &= \\bigoplus_{\\alpha} \\bigoplus\_{m=1}^{m\_{\\alpha}} V^{\\alpha m} \\ |
46 | | -V^{\\alpha m} &= \\bigoplus\_{i=1}^{d\_{\\alpha}} K \\mathbf{f}^{\\alpha m}_{i} \\ |
47 | | -g \\mathbf{f}^{\\alpha m}_{j} |
48 | | -&= \\sum\_{i=1} ^{d\_{\\alpha}} \\mathbf{f}^{\\alpha m}_{i} \\Gamma^{\\alpha}_{ij}(g) |
49 | | -\\quad (g \\in G, j = 1, \\dots, d\_{\\alpha}), |
| 45 | + V &= \bigoplus_{\alpha} \bigoplus_{m=1}^{m_{\alpha}} V^{\alpha m} \\ |
| 46 | + V^{\alpha m} &= \bigoplus_{i=1}^{d_{\alpha}} K \mathbf{f}^{\alpha m}_{i} \\ |
| 47 | + g \mathbf{f}^{\alpha m}_{j} |
| 48 | + &= \sum_{i=1} ^{d_{\alpha}} \mathbf{f}^{\alpha m}_{i} \Gamma^{\alpha}_{ij}(g) |
| 49 | + \quad (g \in G, j = 1, \dots, d_{\alpha}), |
50 | 50 | $$ |
51 | | -where $K$ is $\\mathbb{C}$ or $\\mathbb{R}$, and $\\Gamma^{\\alpha}$ is irrep over $K$. |
| 51 | +where $K$ is $\mathbb{C}$ or $\mathbb{R}$, and $\Gamma^{\alpha}$ is irrep over $K$. |
52 | 52 |
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53 | | -Action of $G$ on rank-$p$ tensor $\\mathsf{T}: V^{\\ast \\otimes p}$ is defined as |
| 53 | +Action of $G$ on rank-$p$ tensor $\mathsf{T}: V^{\ast \otimes p}$ is defined as |
54 | 54 | $$ |
55 | | -(g \\mathsf{T})(\\mathbf{v}_{1}, \\dots, \\mathbf{v}_{p}) |
56 | | -:= \\mathsf{T}(g^{-1} \\mathbf{v}_{1}, \\dots, g^{-1} \\mathbf{v}_{p}) |
57 | | -\\quad (g \\in G). |
| 55 | + (g \mathsf{T})(\mathbf{v}_{1}, \dots, \mathbf{v}_{p}) |
| 56 | + := \mathsf{T}(g^{-1} \mathbf{v}_{1}, \dots, g^{-1} \mathbf{v}_{p}) |
| 57 | + \quad (g \in G). |
58 | 58 | $$ |
59 | | -We also consider intrinsic symmetry $\\Sigma$ of $\\mathsf{T}$ as [^check_action] |
| 59 | +We also consider intrinsic symmetry $\Sigma$ of $\mathsf{T}$ as [^check_action] |
60 | 60 | $$ |
61 | | -(\\sigma \\mathsf{T})(\\mathbf{v}_{1}, \\dots, \\mathbf{v}_{p}) |
62 | | -:= \\mathsf{T}(\\mathbf{v}_{\\sigma^{-1}(1)}, \\dots, \\mathbf{v}_{\\sigma^{-1}(p)}) |
63 | | -\\quad (\\sigma \\in \\Sigma). |
| 61 | + (\sigma \mathsf{T})(\mathbf{v}_{1}, \dots, \mathbf{v}_{p}) |
| 62 | + := \mathsf{T}(\mathbf{v}_{\sigma^{-1}(1)}, \dots, \mathbf{v}_{\sigma^{-1}(p)}) |
| 63 | + \quad (\sigma \in \Sigma). |
64 | 64 | $$ |
65 | 65 |
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66 | | -\[^check_action\]: These definitions follow the condition of left actions: |
67 | | -`{math} (\sigma (\sigma' \mathsf{T}))(\mathbf{v}_{1}, \dots, \mathbf{v}_{p}) &= (\sigma' \mathsf{T})(\mathbf{v}_{\sigma^{-1}(1)}, \dots, \mathbf{v}_{\sigma^{-1}(p)}) \\ &= \mathsf{T}(\mathbf{v}_{\sigma'^{-1}(\sigma^{-1}(1))}, \dots, \mathbf{v}_{\sigma'^{-1}(\sigma^{-1}(p))}) \\ &= \mathsf{T}(\mathbf{v}_{\sigma\sigma'(1)}, \dots, \mathbf{v}_{\sigma\sigma'(p)}) \\ &= ((\sigma \sigma') \mathsf{T})(\mathbf{v}_{1}, \dots, \mathbf{v}_{p}) \\ \therefore \sigma (\sigma' \mathsf{T}) &= (\sigma \sigma') \mathsf{T} ` |
| 66 | +[^check_action]: These definitions follow the condition of left actions: |
| 67 | + ```{math} |
| 68 | + (\sigma (\sigma' \mathsf{T}))(\mathbf{v}_{1}, \dots, \mathbf{v}_{p}) |
| 69 | + &= (\sigma' \mathsf{T})(\mathbf{v}_{\sigma^{-1}(1)}, \dots, \mathbf{v}_{\sigma^{-1}(p)}) \\ |
| 70 | + &= \mathsf{T}(\mathbf{v}_{\sigma'^{-1}(\sigma^{-1}(1))}, \dots, \mathbf{v}_{\sigma'^{-1}(\sigma^{-1}(p))}) \\ |
| 71 | + &= \mathsf{T}(\mathbf{v}_{\sigma\sigma'(1)}, \dots, \mathbf{v}_{\sigma\sigma'(p)}) \\ |
| 72 | + &= ((\sigma \sigma') \mathsf{T})(\mathbf{v}_{1}, \dots, \mathbf{v}_{p}) \\ |
| 73 | + \therefore \sigma (\sigma' \mathsf{T}) &= (\sigma \sigma') \mathsf{T} |
| 74 | + ``` |
68 | 75 |
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69 | 76 | ## References |
70 | 77 |
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