| kolbe2018hubbard_EQ0001_p007(2.2) | \begin{aligned} c(|\phi\rangle+|\psi\rangle) & =c|\phi\rangle+c|\psi\rangle \\ (c+d)|\phi\rangle & =c|\phi\rangle+d|\phi\rangle \\ c(d|\phi\rangle) & =c d|\phi\rangle \end{aligned} | |  |
| kolbe2018hubbard_EQ0002_p008(2.4) | \vec{J}=\vec{r} \times \vec{p} | |  |
| kolbe2018hubbard_EQ0003_p008(\2.5\) | \hat{J}=\frac{\mathrm{i}}{\hbar} \hat{r} \times \nabla | |  |
| kolbe2018hubbard_EQ0004_p008(\2.5\) | \hat{J}_{x}=\frac{\mathrm{i}}{\hbar}\left(y \frac{\partial}{\partial z}-z \frac{\partial}{\partial y}\right) \quad ; \quad \hat{J}_{y}=\frac{\mathrm{i}}{\hbar}\left(z \frac{\partial}{\partial x}-x \frac{\partial}{\partial z}\right) \quad ; \quad \hat{J}_{z}=\frac{\mathrm{i}}{\hbar}\left(x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right) | |  |
| kolbe2018hubbard_EQ0005_p008 | \left[\hat{J}_{x}, \hat{J}_{y}\right]=\mathrm{i} \hbar \hat{J}_{z} \quad ; \quad\left[\hat{J}_{y}, \hat{J}_{z}\right]=\mathrm{i} \hbar \hat{J}_{x} \quad ; \quad\left[\hat{J}_{z}, \hat{J}_{x}\right]=\mathrm{i} \hbar \hat{J}_{y} | |  |
| kolbe2018hubbard_EQ0006_p009(2.6) | \epsilon_{i j k}=\left\{\begin{aligned} +1, & \text { wenn ijk eine gerade Permutation von } 123 \text { ist } \\ -1, & \text { wen ijk eine ungerade Permutation von } 123 \text { ist } \\ 0, & \text { wenn mindestens zwei Indizes gleich sind } \end{aligned}\right. | |  |
| kolbe2018hubbard_EQ0007_p009(\2.7\) | \left[\hat{J}_{i}, \hat{J}_{j}\right]=\mathrm{i} \hbar \sum_{k} \hat{J}_{k} \epsilon_{i j k} | |  |
| kolbe2018hubbard_EQ0008_p009(\2.8\) | \left[\hat{J}_{i}, \hat{J}^{2}\right]=0 \quad i=x, y, z | |  |
| kolbe2018hubbard_EQ0009_p009(2.10) | \begin{aligned} \hat{\mathbf{J}}^{2}|j, m\rangle & =\hbar^{2} j(j+1)|j, m\rangle \\ \hat{J}_{z}|j, m\rangle & =\hbar m|j, m\rangle \end{aligned} | |  |
| kolbe2018hubbard_EQ0010_p009(2.11) | \begin{aligned} \hat{J}^{+} & =\left(\hat{J}_{x}+\mathrm{i} \hat{J}_{y}\right) \\ \hat{J}^{-} & =\left(\hat{J}_{x}-\mathrm{i} \hat{J}_{y}\right) \end{aligned} | |  |
| kolbe2018hubbard_EQ0011_p009(2.13) | \left[\hat{J}_{z}, \hat{J}^{+}\right]=\hbar \hat{J}^{+} \quad, \quad\left[\hat{J}_{z}, \hat{J}^{-}\right]=-\hbar \hat{J}^{-} \quad, \quad\left[\hat{J}^{+}, \hat{J}^{-}\right]=2 \hbar \hat{J}_{z} | |  |
| kolbe2018hubbard_EQ0012_p010(2.14) | |s\rangle=\alpha|\uparrow\rangle+\beta|\downarrow\rangle=\binom{\alpha}{\beta} | |  |
| kolbe2018hubbard_EQ0013_p010(\2.15\) | \hat{\sigma}_{x}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \quad ; \quad \hat{\sigma}_{y}=\left(\begin{array}{cc} 0 & -\mathrm{i} \\ \mathrm{i} & 0 \end{array}\right) \quad ; \quad \hat{\sigma}_{z}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right) \quad ; \quad \hat{\sigma}_{0}=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right) | |  |
| kolbe2018hubbard_EQ0014_p010(2.16) | \begin{aligned} \vec{S} & =\frac{\hbar}{2} \vec{\sigma} \\ \hat{S}_{i} & =\frac{\hbar}{2} \hat{\sigma}_{i} \quad i=x, y, z \end{aligned} | |  |
| kolbe2018hubbard_EQ0015_p010(2.19) | \begin{aligned} \mathbf{1}_{2} & =\hat{\sigma}_{x}^{2}+\hat{\sigma}_{y}^{2}+\hat{\sigma}_{z}^{2} \\ 2 \mathrm{i} \hat{\sigma}_{z} & =\left[\hat{\sigma}_{x}, \hat{\sigma}_{y}\right] \quad \text { und zyklisch } \\ \left\{\hat{\sigma}_{x}, \hat{\sigma}_{y}\right\} & =\left\{\hat{\sigma}_{y}, \hat{\sigma}_{z}\right\}=\left\{\hat{\sigma}_{z}, \hat{\sigma}_{x}\right\}=0 \end{aligned} | |  |
| kolbe2018hubbard_EQ0016_p011(2.21) | \begin{aligned} & \hat{S}_{x}=\frac{1}{2}\left(\hat{S}^{+}+\hat{S}^{-}\right) \\ & \hat{S}_{y}=\frac{1}{2 \mathrm{i}}\left(\hat{S}^{+}-\hat{S}^{-}\right) \end{aligned} | |  |
| kolbe2018hubbard_EQ0017_p011(2.23) | \hat{S}^{+}=\hbar\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right) \quad ; \quad \hat{S}^{-}=\hbar\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right) | |  |
| kolbe2018hubbard_EQ0018_p011(2.25) | \begin{aligned} \hat{S}^{+}|\downarrow\rangle & =\hbar|\uparrow\rangle & \hat{S}^{+}|\uparrow\rangle & =0 \\ \hat{S}^{-}|\uparrow\rangle & =\hbar|\downarrow\rangle & \hat{S}^{-}|\downarrow\rangle & =0 \\ \hat{S}_{z}|\uparrow\rangle & =\frac{\hbar}{2}|\uparrow\rangle & \hat{S}_{z}|\downarrow\rangle & =-\frac{\hbar}{2}|\downarrow\rangle \end{aligned} | |  |
| kolbe2018hubbard_EQ0019_p011(2.28) | \begin{aligned} -1 & <\left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}}<0 & & \text { antikorreliert. } \\ 0 & <\left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}}<1 & & \text { korreliert. } \\ 0 & =\left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}} & & \text { unkorreliert. } \\ 1 & =\left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}} & & \text { maximal korreliert. } \end{aligned} | |  |
| kolbe2018hubbard_EQ0020_p011(2.31) | \left\langle\vec{S}_{i} \vec{S}_{j}\right\rangle_{T}=\frac{1}{Z} \sum_{n} e^{\beta E_{n}} \sum_{l=1}^{g_{n}}\left\langle\Psi_{E_{n}, l}\right| \vec{S}_{i} \vec{S}_{j}\left|\Psi_{E_{n}, l}\right\rangle | |  |
| kolbe2018hubbard_EQ0021_p011(2.32) | \left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}}=\frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \hat{S}_{i} \hat{S}_{j}\left|\psi^{0, l}\right\rangle | |  |
| kolbe2018hubbard_EQ0022_p012(2.33) | \mathcal{H}=\mathcal{H}_{1} \otimes \mathcal{H}_{2} \otimes \ldots \otimes \mathcal{H}_{N-1} \otimes \mathcal{H}_{N} | |  |
| kolbe2018hubbard_EQ0023_p012(2.34) | |\psi\rangle=|\psi\rangle_{1} \otimes|\psi\rangle_{2} \otimes \ldots \otimes|\psi\rangle_{N-1} \otimes|\psi\rangle_{N} | |  |
| kolbe2018hubbard_EQ0024_p012(2.35) | \hat{\mathrm{A}}_{i}: \mathcal{H}_{i} \rightarrow \mathcal{H}_{i} | |  |
| kolbe2018hubbard_EQ0025_p012(2.36) | \overline{\mathrm{A}}_{1}+\ldots+\overline{\mathrm{A}}_{N}=\hat{\mathrm{A}}_{1} \otimes \hat{\mathrm{I}} \otimes \ldots \otimes \hat{\mathrm{I}}+\hat{\mathrm{I}} \otimes \hat{\mathrm{~A}}_{2} \otimes \ldots \otimes \hat{\mathrm{I}}+\ldots+\hat{\mathrm{I}} \otimes \ldots \otimes \hat{\mathrm{~A}_{N}} | |  |
| kolbe2018hubbard_EQ0026_p013(2.37) | \mathcal{H}=\mathcal{H}_{1}^{(1)} \otimes \mathcal{H}_{1}^{(2)} \otimes \ldots \otimes \mathcal{H}_{1}^{(N)} | |  |
| kolbe2018hubbard_EQ0027_p013(2.38) | \mathcal{V}\left(x_{1}, x_{2}, \ldots, x_{N}\right)=\sum_{\left\{\alpha_{i}\right\}} a_{\alpha_{1} \alpha_{2} \ldots \alpha_{N}} \nu_{\alpha_{1}}\left(x_{1}\right) \nu_{\alpha_{2}}\left(x_{2}\right) \ldots \nu_{\alpha_{N}}\left(x_{N}\right), \quad a_{\alpha_{i}} \in \mathbb{C} | |  |
| kolbe2018hubbard_EQ0028_p013(2.40) | \begin{array}{lr} \Psi\left(x_{1}, x_{2}, \ldots, x_{N}\right)=\operatorname{sign}(\pi) \Psi\left(x_{\pi(1)}, x_{\pi(2)}, \ldots, x_{\pi(N)}\right) & \text { für Fermionen } \\ \Phi\left(x_{1}, x_{2}, \ldots, x_{N}\right)=\Phi\left(x_{\pi(1)}, x_{\pi(2)}, \ldots, x_{\pi(N)}\right) & \text { für Bosonen } \end{array} | |  |
| kolbe2018hubbard_EQ0029_p013(2.41) | \Psi\left(x_{1}, x_{2}, \ldots, x_{N}\right)=\sum_{\left\{\alpha_{i}\right\}} a_{\alpha_{1} \alpha_{2} \ldots \alpha_{N}} \psi_{\alpha_{1} \alpha_{2} \ldots \alpha_{N}}\left(x_{1}, x_{2}, \ldots, x_{N}\right), \quad a_{\alpha_{1} \alpha_{2} \ldots \alpha_{N}} \in \mathbb{C} | |  |
| kolbe2018hubbard_EQ0030_p013(2.42) | \psi_{\alpha_{1} \alpha_{2} \ldots \alpha_{N}}\left(x_{1}, x_{2}, \ldots, x_{N}\right)=\frac{1}{\sqrt{N!}} \operatorname{det}\left(\begin{array}{ccc} \psi_{\alpha_{1}}\left(x_{1}\right) & \ldots & \psi_{\alpha_{N}}\left(x_{1}\right) \\ \vdots & \ddots & \vdots \\ \psi_{\alpha_{1}}\left(x_{N}\right) & \ldots & \psi_{\alpha_{N}}\left(x_{N}\right) \end{array}\right) | |  |
| kolbe2018hubbard_EQ0031_p014(2.43) | \left\langle n_{1 \sigma}, n_{2 \sigma}, \ldots \mid n_{1 \sigma}^{\prime}, n_{2 \sigma}^{\prime}, \ldots\right\rangle=\prod_{i, \sigma} \delta_{n_{i \sigma}, n_{i \sigma}^{\prime}} | |  |
| kolbe2018hubbard_EQ0032_p014(2.44) | |\Psi\rangle=\sum_{n_{1 \sigma}, n_{2 \sigma}, \ldots} a_{n_{1 \sigma}, n_{2 \sigma}, \ldots}\left|n_{1 \sigma}, n_{2 \sigma}, \ldots\right\rangle, \quad \text { wobei } 2 N=\sum_{i, \sigma} n_{i, \sigma} | |  |
| kolbe2018hubbard_EQ0033_p014(\2.45\) | \left.F^{(2 N)}=\mathcal{H}^{(+)}=\{|\Psi\rangle \in \mathcal{H}|\hat{\pi}| \Psi\rangle=(-1)^{\pi}|\Psi\rangle, \forall \pi \in S_{n}\right\} | |  |
| kolbe2018hubbard_EQ0034_p014(2.46) | c_{i \sigma}^{\dagger}\left|n_{1 \sigma}, \ldots, n_{i \sigma}, \ldots\right\rangle=\left(1-n_{i \sigma}(-1)^{\sum_{\sigma, j<i} n_{j \sigma}}\left|n_{1 \sigma}, \ldots, n_{i \sigma}+1, \ldots\right\rangle\right. | |  |
| kolbe2018hubbard_EQ0035_p014(2.47) | \left|n_{1 \uparrow} n_{1 \downarrow} \ldots\right\rangle=\prod_{i, \sigma} \frac{1}{\sqrt{n_{i \sigma!}}}\left(c_{i, \sigma}^{\dagger}\right)^{n_{i \sigma}}|0\rangle | |  |
| kolbe2018hubbard_EQ0036_p015(2.48) | c_{i \sigma}\left|n_{1 \sigma}, \ldots, n_{i \sigma}, \ldots\right\rangle=n_{i \sigma}(-1)^{\sum_{\sigma, j<i} 2 n_{j \sigma}}\left|n_{1 \sigma}, \ldots, n_{i \sigma}-1, \ldots\right\rangle | |  |
| kolbe2018hubbard_EQ0037_p015(2.50) | \begin{aligned} {\left[c_{i \sigma}, c_{j \sigma^{\prime}}^{\dagger}\right] } & =\delta_{i j} \delta_{\sigma \sigma^{\prime}} \\ {\left[c_{i \sigma}, c_{j \sigma^{\prime}}\right] } & =0 \\ {\left[c_{i \sigma}^{\dagger}, c_{j \sigma^{\prime}}^{\dagger}\right] } & =0 \end{aligned} | |  |
| kolbe2018hubbard_EQ0038_p015(2.52) | \hat{n}_{i \sigma}=c_{i \sigma}^{\dagger} c_{i \sigma} | |  |
| kolbe2018hubbard_EQ0039_p015(2.53) | \hat{n}_{i \sigma}\left|n_{1 \sigma}, \ldots, n_{i \sigma}, \ldots\right\rangle=n_{i \sigma}\left|n_{1 \sigma}, \ldots, n_{i \sigma}, \ldots\right\rangle | |  |
| kolbe2018hubbard_EQ0040_p017(3.1) | \mathbf{H}_{\mathrm{Heis}}=-J \sum_{\langle i, j\rangle} \vec{S}_{i} \cdot \vec{S}_{j} | |  |
| kolbe2018hubbard_EQ0041_p017(3.2) | \vec{S}_{i} \cdot \vec{S}_{j}=S_{i}^{x} S_{j}^{x}+S_{i}^{y} S_{j}^{y}+S_{i}^{z} S_{j}^{z} | |  |
| kolbe2018hubbard_EQ0042_p017(3.3) | \mathbf{H}_{\mathrm{Heis}}=-J \sum_{\langle i, j\rangle} \frac{\hat{S}_{i}^{+} \hat{S}_{j}^{-}}{2}+\frac{\hat{S}_{i}^{-} \hat{S}_{j}^{+}}{2}+\hat{S}_{i}^{z} \hat{S}_{j}^{z} | |  |
| kolbe2018hubbard_EQ0043_p019(3.4) | \mathbf{H}_{\mathrm{Hub}}=\mathbf{H}_{U}+\mathbf{H}_{\epsilon}+\mathbf{H}_{t} | |  |
| kolbe2018hubbard_EQ0044_p019(3.5) | \mathbf{H}_{\mathrm{Hub}}=U \sum_{i=1}^{N} \hat{c}_{i, \uparrow}^{\dagger} \hat{c}_{i, \uparrow} \hat{c}_{i, \downarrow}^{\dagger} \hat{c}_{i, \downarrow}+\sum_{\sigma}\left(\sum_{i=1}^{N} \epsilon_{i} \hat{c}_{i, \sigma}^{\dagger} \hat{c}_{i, \sigma}-t \sum_{\langle i, j\rangle}\left(\hat{c}_{i, \sigma}^{\dagger} \hat{c}_{j, \sigma}+\hat{c}_{j, \sigma}^{\dagger} \hat{c}_{i, \sigma}\right)\right) | |  |
| kolbe2018hubbard_EQ0045_p021(4.1) | \hat{S}_{i}^{+} \hat{S}_{j}^{-}|Z u s t a n d\rangle \equiv \operatorname{SpinPlus}(i, \operatorname{SpinMinus}(j, Z u s t a n d)) | |  |
| kolbe2018hubbard_EQ0046_p021 | \left(\begin{array}{lll} 0 & \mathrm{~J} & 0 \\ 0 & 0 & \mathrm{~J} \\ \mathrm{~J} & 0 & 0 \end{array}\right) | |  |
| kolbe2018hubbard_EQ0047_p021(4.2) | \left\langle\hat{S}_{1} \hat{S}_{j}\right\rangle_{\psi_{g}}=\frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \frac{\hat{S}_{1}^{+} \hat{S}_{j}^{-}}{2}+\frac{\hat{S}_{1}^{-} \hat{S}_{j}^{+}}{2}+\hat{S}_{1}^{z} \hat{S}_{j}^{z}\left|\psi^{0, l}\right\rangle | |  |
| kolbe2018hubbard_EQ0048_p022(4.3) | \left|\psi_{g}\right\rangle=\sum_{k=1}^{2^{N}} \alpha_{k}|k\rangle \quad \text { Heisenberg-Modell } | |  |
| kolbe2018hubbard_EQ0049_p023(4.4) | \hat{n}_{\uparrow}|Z u s t a n d\rangle=\hat{c}_{i \uparrow}^{\dagger} \hat{c}_{j \uparrow}|Z u s t a n d\rangle \Leftrightarrow \operatorname{CupDagger}(i, \operatorname{Cup}(j, \text { Zustand })) | |  |
| kolbe2018hubbard_EQ0050_p023 | \left(\begin{array}{lll} 0 & \mathrm{t} & 0 \\ 0 & 0 & \mathrm{t} \\ 0 & 0 & 0 \end{array}\right) | |  |
| kolbe2018hubbard_EQ0051_p023(4.5) | \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}^{\dagger}|0\rangle_{j}=|\uparrow \downarrow\rangle_{j} \quad \quad \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}^{\dagger}|0\rangle_{j}=-\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}^{\dagger}|0\rangle_{j}=-|\uparrow \downarrow\rangle_{j} | |  |
| kolbe2018hubbard_EQ0052_p023(4.6) | |0,0\rangle=[0,0] \quad|\downarrow, \downarrow\rangle=[1,1] \quad|\uparrow, \uparrow\rangle=[2,2] \quad|\uparrow \downarrow, \uparrow \downarrow\rangle=[3,3] | |  |
| kolbe2018hubbard_EQ0053_p023(4.7) | \begin{array}{rll} |0, \downarrow\rangle=[0,1] & \rightarrow & |\downarrow, 0\rangle=[1,0] \\ |\downarrow, 0\rangle=[1,0] & \rightarrow & |0, \downarrow\rangle=[0,1] \\ |0, \uparrow\rangle=[0,2] & \rightarrow & |\uparrow, 0\rangle=[2,0] \\ |\uparrow, 0\rangle=[2,0] & \rightarrow & |0, \uparrow\rangle=[0,2] \end{array} | |  |
| kolbe2018hubbard_EQ0054_p024(4.8) | \begin{array}{lll} |\uparrow, \uparrow \downarrow\rangle=[1,3] & \rightarrow & |\uparrow \downarrow, \uparrow\rangle=[3,1] \\ |\downarrow, \downarrow \uparrow\rangle=[1,3] & \rightarrow & |\uparrow \downarrow, \downarrow\rangle=[3,2] \\ |\uparrow \downarrow, \uparrow\rangle=[3,1] & \rightarrow & |\uparrow, \uparrow \downarrow\rangle=[1,3] \\ |\uparrow \downarrow, \downarrow\rangle=[3,2] & \rightarrow & |\downarrow, \uparrow \downarrow\rangle=[2,3] \end{array} | |  |
| kolbe2018hubbard_EQ0055_p024(4.9) | \begin{aligned} |\uparrow, \downarrow\rangle=[1,2] & \rightarrow & |\uparrow \downarrow, 0\rangle+|0, \uparrow \downarrow\rangle=[3,0]+[0,3] \\ |\downarrow, \uparrow\rangle=[2,1] & \rightarrow & -|\uparrow \downarrow, 0\rangle-|0, \uparrow \downarrow\rangle=-[3,0]-[0,3] \\ |\uparrow \downarrow, 0\rangle=[3,0] & \rightarrow & |\uparrow, \downarrow\rangle-|\downarrow, \uparrow\rangle=[2,1]-[1,2] \\ |0, \uparrow \downarrow\rangle=[0,3] & \rightarrow & |\uparrow, \downarrow\rangle-|\downarrow, \uparrow\rangle=[1,2]-[2,1] \end{aligned} | |  |
| kolbe2018hubbard_EQ0056_p024(4.10) | \mathbf{H}_{\epsilon}+\mathbf{H}_{U}=\sum_{i=1}^{N}\left(\epsilon_{i} \cdot\left(\hat{n}_{i, \uparrow}+\hat{n}_{i, \downarrow}\right)+U_{i} \cdot \hat{n}_{i, \uparrow} \hat{n}_{i, \downarrow}\right) | |  |
| kolbe2018hubbard_EQ0057_p024(4.11) | \langle l| \mathbf{H}_{\epsilon}+\mathbf{H}_{U}|m\rangle= \begin{cases}\epsilon_{l m}+U_{l m} & \text { wenn } l=m \text { ist. } \\ 0 & \text { wenn } l \neq m \text { ist. }\end{cases} | |  |
| kolbe2018hubbard_EQ0058_p025(4.12) | \hat{S}_{i}^{+}=\hat{c}_{i, \uparrow}^{\dagger} \hat{c}_{i, \downarrow} \quad ; \quad \hat{S}_{i}^{-}=\hat{c}_{i, \downarrow}^{\dagger} \hat{c}_{i, \uparrow} \quad ; \quad \hat{S}_{i}^{z}=\frac{1}{2}\left(\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}-\hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}\right) | |  |
| kolbe2018hubbard_EQ0059_p025(4.12) | \begin{aligned} C_{1 j}= & \frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \frac{1}{2}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \downarrow} \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \uparrow} \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}\right) \\ & +\frac{1}{4}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \uparrow}-\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \downarrow}\right)\left(\hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}-\hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}\right)\left|\psi^{0, l}\right\rangle \\ = & \frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \frac{1}{2}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \downarrow} \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \uparrow} \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}\right) \\ & +\frac{1}{4}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \uparrow} \cdot \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}-\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \uparrow} \cdot \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}-\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \downarrow} \cdot \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \downarrow} \cdot \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \downarrow}\right)\left|\psi^{0, l}\right\rangle \end{aligned} | |  |
| kolbe2018hubbard_EQ0060_p025(4.13) | \begin{aligned} \left\langle\hat{S}_{i} \hat{S}_{j}\right\rangle_{\psi_{g}}= & \frac{1}{g_{0}} \sum_{l=0}^{g_{0}}\left\langle\psi^{0, l}\right| \frac{1}{2}\left(\hat{c}_{1, \uparrow}^{\dagger} \hat{c}_{1, \downarrow} \hat{c}_{j, \downarrow}^{\dagger} \hat{c}_{j, \uparrow}+\hat{c}_{1, \downarrow}^{\dagger} \hat{c}_{1, \uparrow} \hat{c}_{j, \uparrow}^{\dagger} \hat{c}_{j, \downarrow}\right) \\ & +\frac{1}{4}\left(\hat{n}_{1, \uparrow} \hat{n}_{j, \uparrow}-\hat{n}_{1, \uparrow} \hat{n}_{j, \downarrow}-\hat{n}_{1, \downarrow} \hat{n}_{j, \uparrow}+\hat{n}_{1, \downarrow} \hat{n}_{j, \downarrow}\right)\left|\psi^{0, l}\right\rangle \end{aligned} | |  |
| kolbe2018hubbard_EQ0061_p025(4.14) | \left|\psi_{g}\right\rangle=\sum_{k=1}^{4^{N}} \alpha_{k}|k\rangle \quad \text { Hubbard-Modell } | |  |
