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--- technical:ising1d [2022/10/29 13:53] – [Characteristic polynomial] chunchung
+++ technical:ising1d [2022/10/29 15:10] (current) – [Simple case 1: $J = 1$, $h = 0$] chunchung
@@ Line 24: / Line 24: @@
 \begin{equation} \lambda^2 - \left(e^{\beta (J+h)}+e^{\beta (J-h)}\right) \lambda + e^{2\beta J} - e^{-2\beta J}. \end{equation}
 Since the discriminant \begin{align}
-&   \left(e^{\beta (J+h)}+e^{\beta (J-h)}\right)^2 - 4 \left(e^{2\beta J} - e^{-2\beta J}\right) \notag \\
+\mathrm{Disc}_\lambda & = \left(e^{\beta (J+h)}+e^{\beta (J-h)}\right)^2 - 4 \left(e^{2\beta J} - e^{-2\beta J}\right) \notag \\
-& =  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) \notag \\
+& = 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) \notag \\
-& =  \left(e^{\beta (J+h)}-e^{\beta (J-h)}\right)^2 + 4 e^{-2\beta J} \notag \\
+& = \left(e^{\beta (J+h)}-e^{\beta (J-h)}\right)^2 + 4 e^{-2\beta J} \notag \\
-& >  0 .
+& >  0,
 \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====
 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] \notag \\
-& = & e^{\beta J}\left[\cosh(\beta h)\pm\sqrt{\sinh^2(\beta h)+e^{-4\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
-$$ Z = \lambda_+^N+\lambda_-^N . $$
+\begin{equation} Z = \lambda_+^N+\lambda_-^N . \end{equation}
 ====Simple case 1: $J = 1$, $h = 0$====
 The eigenvalues become
-$$ \lambda_\pm = e^{\beta}\pm e^{-\beta}. $$
+\begin{equation} \lambda_\pm = e^{\beta}\pm e^{-\beta}. \end{equation}
 The partition function is
-$$ Z = \left(e^\beta+e^{-\beta}\right)^N+\left(e^\beta-e^{-\beta}\right)^N . $$
+\begin{equation} Z = \left(e^\beta+e^{-\beta}\right)^N+\left(e^\beta-e^{-\beta}\right)^N . \end{equation}
 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 \notag \\
-& = & 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 $r = \left(e^\beta-e^{-\beta}\right)/\left(e^\beta+e^{-\beta}\right)<1$ is the ratio between the two eigenvalues.