模型 | Hamiltonian | relevant models |
| 1D Ising model | ||
| 2D Ising model | ||
| 3D Ising model | ||
| \(Z_{N}\) model |
模型 | Hamiltonian | relevant models |
| Classical XY model | \(H(\mathbf{s})=-\sum_{i \neq j} J_{i j} \mathbf{s}_{i} \cdot \mathbf{s}_{j}-\sum_{j} \mathbf{h}_{j} \cdot \mathbf{s}_{j}=-\sum_{i \neq j} J_{i j} \cos \left(\theta_{i}-\theta_{j}\right)-\sum_{j} h_{j} \cos \theta_{j}\) | |
| 2D Ising model | ||
| 3D Ising model | ||
| \(Z_{N}\) model |
模型 | Hamiltonian | relevant models |
| Quantum Heisenberg model | \(\hat{H}=-J \sum_{j=1}^{N} \sigma_{j} \sigma_{j+1}-h \sum_{j=1}^{N} \sigma_{j}\) | |
| XXX class | \(\hat{H}=-J \sum_{j=1}^{N} \sigma_{j}^{z} \sigma_{j+1}^{z}-g J \sum_{j=1}^{N} \sigma_{j}^{x}\) | |
| Quantum rotor model | \(H_{R}=\frac{J \bar{g}}{2} \sum_{i} \mathbf{L}_{i}^{2}-J \sum_{\langle i j\rangle} \mathbf{n}_{i} \cdot \mathbf{n}_{j}\) | |
| \(J_{1}-J_{2}\) model | \(\hat{H}=J_{1} \sum_{\langle i j\rangle} \vec{S}_{i} \cdot \vec{S}_{j}+J_{2} \sum_{\langle\langle i j\rangle\rangle} \vec{S}_{i} \cdot \vec{S}_{j}\) with \(J_{1}\)-nearest-neighbor interaction, \(J_{2}\)-next nearest-neighbor | |
| Majumdar–Ghosh model | \(J_{1}=2J_{2}\) \(J_{1}-J_{2}\) model \(\hat{H}=J \sum_{j=1}^{N} \vec{S}_{j} \cdot \vec{S}_{j+1}+\frac{J}{2} \sum_{j=1}^{N} \vec{S}_{j} \cdot \vec{S}_{j+2}\) | |
| AKLT model | \(\hat{H}=\sum_{j} \vec{S}_{j} \cdot \vec{S}_{j+1}+\frac{1}{3}\left(\vec{S}_{j} \cdot \vec{S}_{j+1}\right)^{2}\) where the \(\overrightarrow{S_{i}}\) are spin-1 operators. | |
| \(t–J\) model |
模型 | Hamiltonian | relevant models |
| \(n\)-vector model/O(\(n\)) model | \(H=-J \sum_{\langle i, j\rangle} \mathbf{s}_{i} \cdot \mathbf{s}_{j}\) |
模型 | Hamiltonian | relevant models |
| Potts model | \(H_{p}=-J_{p} \sum_{(i, j)} \delta\left(s_{i}, s_{j}\right)\) where \(\delta\left(s_{i}, s_{j}\right)\) is the Kronecker delta, which equals one whenever \(s_{i}=s_{j}\) and zero otherwise | |
| Ashkin–Teller model | ||
| Kac model (The infinite-range Potts model) | ||
| cellular Potts model |
模型 | Hamiltonian | relevant models |
| Lieb-Liniger model(Boson) | ||
| Yang-Gaudin model(spin-1/2 fermion) | ||
| Sutherland model(SU(N) fermion) | ||
| Lieb model(Bose-Fermi mixture) |
模型 | Hamiltonian | relevant models |
| Bose-Hubbard model | \(H=-t \sum_{\langle i, j\rangle} \hat{b}_{i}^{\dagger} \hat{b}_{j}+\frac{U}{2} \sum_{i} \hat{n}_{i}\left(\hat{n}_{i}-1\right)-\mu \sum_{i} \hat{n}_{i}\) \(\hat{n}_{i}=\hat{b}_{i}^{\dagger} \hat{b}_{i}\) on-site interaction: attractive\(U<0\), repulsive\(U<0\) chemical potential \(\mu\) sets the total number of particles. | |
| Jaynes–Cumming model | \(\hat{H}_{\mathrm{JC}}=\hbar \omega_{c} \hat{a}^{\dagger} \hat{a}+\hbar \omega_{a} \frac{\hat{\sigma}_{z}}{2}+\frac{\hbar \Omega}{2}\left(\hat{a} \hat{\sigma}_{+}+\hat{a}^{\dagger} \hat{\sigma}_{-}\right)\) | |
| Jaynes–Cummings–Hubbard model | \(H=\sum_{n=1}^{N} \omega_{c} a_{n}^{\dagger} a_{n}+\sum_{n=1}^{N} \omega_{a} \sigma_{n}^{+} \sigma_{n}^{-}+\kappa \sum_{n=1}^{N}\left(a_{n+1}^{\dagger} a_{n}+a_{n}^{\dagger} a_{n+1}\right)+\eta \sum_{n=1}^{N}\left(a_{n} \sigma_{n}^{+}+a_{n}^{\dagger} \sigma_{n}^{-}\right)\) \(\hat{N}_{c} \equiv \sum_{n=1}^{N} a_{n}^{\dagger} a_{n}\) and \(\hat{N}_{a} \equiv \sum_{n=1}^{N} \sigma_{n}^{+} \sigma_{n}^{-}\) | |
| Fermi-Hubbard model | \(\hat{H}=-t \sum_{i, \sigma}\left(\hat{c}_{i, \sigma}^{\dagger} \hat{c}_{i+1, \sigma}+\hat{c}_{i+1, \sigma}^{\dagger} \hat{c}_{i, \sigma}\right)+U \sum_{i} \hat{n}_{i \uparrow} \hat{n}_{i \downarrow}\) | |
| \(t-J\) model | ||
| Heisenberg model | ||
| Gaussian class | ||
| Gaussian class |
模型 | Lagrangian density | relevant models |
| Schwinger model | \(\mathcal{L}=-\frac{1}{4 g^{2}} F_{\mu \nu} F^{\mu \nu}+\bar{\psi}\left(i \gamma^{\mu} D_{\mu}-m\right) \psi\) | |
| Soler model | \(\mathcal{L}=\bar{\psi}(i \not \partial-m) \psi+\frac{g}{2}(\bar{\psi} \psi)^{2}\) | |
| chiral model | ||
| Non-linear \(\sigma\) model | ||
| Gross–Neveu model | \(\mathcal{L}=\bar{\psi}_{a}(i \not \partial-m) \psi^{a}+\frac{g^{2}}{2 N}\left[\bar{\psi}_{a} \psi^{a}\right]^{2}\) \(\mathcal{L}=\bar{\psi}_{a}(i \not \partial-m) \psi^{a}+\frac{g^{2}}{2 N}\left(\left[\bar{\psi}_{a} \psi^{a}\right]^{2}-\left[\bar{\psi}_{a} \gamma_{5} \psi^{a}\right]^{2}\right)\) | |
| Nambu–Jona-Lasinio model | ||
| Thirring model | \(\mathcal{L}=\bar{\psi}(i \not \partial-m) \psi-\frac{g}{2}\left(\bar{\psi} \gamma^{\mu} \psi\right)\left(\bar{\psi} \gamma_{\mu} \psi\right)\) | |
| Thirring–Wess model | ||
| Sine-Gordon model | \(\mathcal{L}_{\mathrm{SG}}(\varphi)=\frac{1}{2}\left(\varphi_{t}^{2}-\varphi_{x}^{2}\right)-1+\cos \varphi\) | |
| Wess–Zumino model | \(\mathcal{L}=-\frac{1}{2}(\partial S)^{2}-\frac{1}{2}(\partial P)^{2}-\frac{1}{2} \bar{\psi} \not \partial \psi\) with \(S\) a scalar field, \(P\) a pseudoscalar field and \(\psi\) a Majorana spinor field | |
| Thirring–Wess model | ||
| Standard Model |