Widely-used models

$$H = J \sum_{\langle ij\rangle} \mathbf{S}_i\cdot\mathbf{S}_j$$

$$H = J \sum_i \left( S^x_i S^x_{i+1} + S^y_i S^y_{i+1} + \Delta\, S^z_i S^z_{i+1} \right)$$

$$H = \sum_{\langle ij\rangle}\left[ S_i^x S_j^x + S_i^y S_j^y + \Delta\, S_i^z S_j^z \right] + D \sum_i (S_i^z)^2$$

$$H = -J \sum_{\langle ij\rangle} \sigma^z_i \sigma^z_j - \Gamma \sum_i \sigma^x_i$$

$$H = \sum_i \left[ \mathbf{S}_i\cdot\mathbf{S}_{i+1} + \tfrac{1}{3}\left(\mathbf{S}_i\cdot\mathbf{S}_{i+1}\right)^2 \right]$$

$$H = -\sum_{\langle ij\rangle}\left(X_i X_j^\dagger + X_i^\dagger X_j\right) - h\sum_i\left(Z_i + Z_i^\dagger\right)$$

$$H = J_1 \sum_{\langle ij\rangle} \mathbf{S}_i\cdot\mathbf{S}_j + J_2 \sum_{\langle\langle ij\rangle\rangle} \mathbf{S}_i\cdot\mathbf{S}_j$$

$$H = J \sum_{\langle ij\rangle_{NN}} \mathbf{S}_i\cdot\mathbf{S}_j \;+\; J' \sum_{\langle ij\rangle_{\text{dimer}}} \mathbf{S}_i\cdot\mathbf{S}_j$$

$$H = -J_x \sum_{\langle ij\rangle_x}\sigma^x_i\sigma^x_j \;-\; J_y \sum_{\langle ij\rangle_y}\sigma^y_i\sigma^y_j \;-\; J_z \sum_{\langle ij\rangle_z}\sigma^z_i\sigma^z_j$$

$$H = J_{\text{eff}} \sum_{\langle ij\rangle} S^{z_i}_i S^{z_j}_j \;\;\Big(\text{equivalently } H = -J \sum_{\langle ij\rangle} \mathbf{S}_i\cdot\mathbf{S}_j \text{ with } \mathbf{S}_i \parallel \hat{z}_i\Big) \;+\; D \sum_{i<j} \frac{\hat{z}_i\cdot\hat{z}_j - 3(\hat{z}_i\cdot\hat{r}_{ij})(\hat{z}_j\cdot\hat{r}_{ij})}{|r_{ij}|^3}$$

$$H = -\sum_v A_v \;-\; \sum_p B_p, \qquad A_v=\prod_{i\in v}\sigma^x_i,\quad B_p=\prod_{i\in p}\sigma^z_i$$

$$H = -t \sum_{\langle ij\rangle,\sigma} \left( c^\dagger_{i\sigma} c_{j\sigma} + \text{h.c.} \right) + U \sum_i n_{i\uparrow} n_{i\downarrow} - \mu \sum_i n_i$$

$$H = -t \sum_{\langle ij\rangle,\sigma} \left( c^\dagger_{i\sigma} c_{j\sigma} + \text{h.c.} \right) \;-\; |U| \sum_i n_{i\uparrow} n_{i\downarrow} \;-\; \mu \sum_i n_i$$

$$H = -t \sum_{\langle ij\rangle,m,\sigma} \left( c^\dagger_{im\sigma} c_{jm\sigma} + \text{h.c.} \right) + H_{\text{Kanamori}}$$

$$H = -t \sum_{\langle ij\rangle,\sigma} P \left( c^\dagger_{i\sigma} c_{j\sigma} + \text{h.c.} \right) P + J \sum_{\langle ij\rangle} \left( \mathbf{S}_i \cdot \mathbf{S}_j - \tfrac{1}{4} n_i n_j \right)$$

$$H = -t \sum_{\langle ij\rangle} \left( c^\dagger_i c_j + \text{h.c.} \right) + V \sum_{\langle ij\rangle} n_i n_j$$

$$H = -t \sum_{\langle ij\rangle} \left( c^\dagger_i c_j + \text{h.c.} \right) + U \sum_i n^c_i\, n^f_i$$

$$H = -t \sum_{\langle ij\rangle,\sigma} \left( c^\dagger_{i\sigma} c_{j\sigma} + \text{h.c.} \right) + J_K \sum_i \mathbf{S}_i \cdot \mathbf{s}_i$$

$$H = \varepsilon_d \sum_\sigma n_{d\sigma} + U\, n_{d\uparrow} n_{d\downarrow} + \sum_{k\sigma} \varepsilon_k c^\dagger_{k\sigma} c_{k\sigma} + \sum_{k\sigma} \left( V_k\, d^\dagger_\sigma c_{k\sigma} + \text{h.c.} \right)$$

$$H = \sum_{n} \left[ t_1\, c^\dagger_{A,n} c_{B,n} + t_2\, c^\dagger_{B,n} c_{A,n+1} + \text{h.c.} \right]$$

$$H = t_1 \sum_{\langle ij\rangle} c^\dagger_i c_j \;+\; t_2 \sum_{\langle\langle ij\rangle\rangle} e^{i\phi_{ij}}\, c^\dagger_i c_j \;+\; M \sum_i \xi_i\, n_i$$

$$H = -t \sum_{\langle ij\rangle} e^{i\theta_{ij}}\, c^\dagger_i c_j + \text{h.c.}, \qquad \sum_{\square} \theta_{ij} = 2\pi\alpha,\quad \alpha = \frac{p}{q}$$

$$H = \sum_i \left[ -t\left(c^\dagger_i c_{i+1} + \text{h.c.}\right) - \mu\left(n_i - \tfrac12\right) + \Delta\left(c_i c_{i+1} + \text{h.c.}\right) \right]$$

$$H = -t \sum_{\langle ij\rangle}\bigl(c^\dagger_i c_j + \text{h.c.}\bigr) \;+\; \sum_i \varepsilon_i\, n_i, \qquad \varepsilon_i \in [-W/2,\,+W/2]\ \text{i.i.d. uniform}$$

$$H = J \sum_i \mathbf{S}_i\cdot\mathbf{S}_{i+1} \;+\; \sum_i h_i\, S^z_i, \qquad h_i \in [-W,\,+W]\ \text{i.i.d. uniform}$$

$$H = \sum_{i<j<k<l} J_{ijkl}\, \chi_i \chi_j \chi_k \chi_l, \qquad \overline{J_{ijkl}} = 0,\quad \overline{J_{ijkl}^2} = \frac{3!\,J^2}{N^3}$$

$$\frac{d\rho}{dt} = -i[H,\rho] + \sum_k\Bigl(L_k\,\rho\,L_k^\dagger - \tfrac12\{L_k^\dagger L_k,\,\rho\}\Bigr) \equiv \mathcal{L}\rho$$

$$H = -t \sum_{\langle ij\rangle} \left( b^\dagger_i b_j + \text{h.c.} \right) + \frac{U}{2} \sum_i n_i (n_i - 1) - \mu \sum_i n_i$$

$$H = \Omega \sum_i P_{i-1}\, \sigma^x_i\, P_{i+1} \;-\; \Delta \sum_i n_i$$