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| technical:ising1d [2022/10/29 14:09] – [Characteristic polynomial] chunchung | technical:ising1d [2022/10/29 15:10] (current) – [Simple case 1: $J = 1$, $h = 0$] chunchung | ||
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| ====Simple case 1: $J = 1$, $h = 0$==== | ====Simple case 1: $J = 1$, $h = 0$==== | ||
| The eigenvalues become | The eigenvalues become | ||
| - | $$ \lambda_\pm = e^{\beta}\pm e^{-\beta}. | + | \begin{equation} |
| The partition function is | The partition function is | ||
| - | $$ Z = \left(e^\beta+e^{-\beta}\right)^N+\left(e^\beta-e^{-\beta}\right)^N . $$ | + | \begin{equation} |
| Generally, the average energy $U = \langle E\rangle$ can be calculated from the $\beta$-derivative of $-\log Z$: | Generally, the average energy $U = \langle E\rangle$ can be calculated from the $\beta$-derivative of $-\log Z$: | ||
| - | \begin{eqnarray*} | + | \begin{align} |
| - | U & = & \langle E\rangle \\ | + | U &= \langle E\rangle |
| - | & = & Z^{-1} \sum_{\mathbf{s}} E e^{-\beta E} \\ | + | &= Z^{-1} \sum_{\mathbf{s}} E e^{-\beta E} \notag |
| - | & = & - Z^{-1} \frac{\partial}{\partial\beta} Z \\ | + | &= - Z^{-1} \frac{\partial}{\partial\beta} Z \notag |
| - | & = & - \frac{\partial}{\partial\beta}\log Z | + | &= - \frac{\partial}{\partial\beta}\log Z \\ |
| - | \end{eqnarray*} | + | &= \frac{N\left[\left(e^\beta+e^{-\beta}\right)^{N-1}\left(e^\beta-e^{-\beta}\right)+\left(e^\beta+e^{-\beta}\right)\left(e^\beta-e^{-\beta}\right)^{N-1}\right]}{\left(e^\beta+e^{-\beta}\right)^N+\left(e^\beta-e^{-\beta}\right)^N} \notag \\ |
| + | &= N\frac{r+r^{N-1}}{1+r^N} | ||
| + | \end{align} | ||
| + | where $r = \left(e^\beta-e^{-\beta}\right)/ | ||