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| technical:ising1d [2022/10/29 13:52] – [Partition function] chunchung | technical:ising1d [2022/10/29 15:10] (current) – [Simple case 1: $J = 1$, $h = 0$] chunchung | ||
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| Line 22: | Line 22: | ||
| ====Characteristic polynomial==== | ====Characteristic polynomial==== | ||
| The characteristic polynomial is | The characteristic polynomial is | ||
| - | $$ \lambda^2 - \left(e^{\beta (J+h)}+e^{\beta (J-h)}\right) \lambda + e^{2\beta J} - e^{-2\beta J}. $$ | + | \begin{equation} |
| - | Since the discriminant \begin{eqnarray*} | + | Since the discriminant \begin{align} |
| - | & & | + | \mathrm{Disc}_\lambda |
| - | & = & e^{2\beta(J+h)} + e^{2\beta(J-h)} + 2 e^{2\beta J} - 4 \left(e^{2\beta J} - e^{-2\beta J}\right) \\ | + | & = e^{2\beta(J+h)} + e^{2\beta(J-h)} + 2 e^{2\beta J} - 4 \left(e^{2\beta J} - e^{-2\beta J}\right) |
| - | & = & \left(e^{\beta (J+h)}-e^{\beta (J-h)}\right)^2 + 4 e^{-2\beta J} \\ | + | & = \left(e^{\beta (J+h)}-e^{\beta (J-h)}\right)^2 + 4 e^{-2\beta J} \notag |
| - | & > & 0 . | + | & > 0, |
| - | \end{eqnarray*} | + | \end{align} |
| - | The characteristic polynomial has two real roots, that is, the transfer matrix $T$ has two real eigenvalues. | + | the characteristic polynomial has two real roots, that is, the transfer matrix $T$ has two real eigenvalues. |
| ====Eigenvalues==== | ====Eigenvalues==== | ||
| The two eigenvalues are | The two eigenvalues are | ||
| - | \begin{eqnarray*} | + | \begin{align} |
| - | \lambda_\pm & = & \frac{1}{2} \left[ \left( e^{\beta(J+h)}+e^{\beta(J−h)} \right) \pm \sqrt{ \left(e^{\beta (J+h)}-e^{\beta (J-h)}\right)^2 + 4 e^{-2\beta J} } \right] \\ | + | \lambda_\pm &= \frac{1}{2} \left[ \left( e^{\beta(J+h)}+e^{\beta(J−h)} \right) \pm \sqrt{ \left(e^{\beta (J+h)}-e^{\beta (J-h)}\right)^2 + 4 e^{-2\beta J} } \right] |
| - | & = & e^{\beta J}\left[\cosh(\beta h)\pm\sqrt{\sinh^2(\beta h)+e^{-4\beta J}}\right] | + | & |
| - | \end{eqnarray*} | + | \end{align} |
| and the partition function is given by | and the partition function is given by | ||
| - | $$ Z = \lambda_+^N+\lambda_-^N . $$ | + | \begin{equation} |
| ====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)/ | ||