# Eigenvalues of transfer matrix Ising model spin 1 system

I am calculating the partition function of an Ising model with spin 1 ($$\sigma_i \in \{-1,0,1\}$$) for $$n$$ sites. The following Hamiltonian has been used:

$$H = -J \sum_{i=1}^{n} \sigma_i\sigma_{i+1},$$

assuming periodic boundary conditions, i.e. $$\sigma_1 = \sigma_{n+1}$$. From this I derived that $$Z = \mathrm{tr}(P^n)$$ where $$P$$ is the transfer matrix as follows:

$$$$P = \begin{pmatrix} e^K & 1& e^{-K} \\ 1&1&1\\ e^{-K} & 1& e^K \end{pmatrix}$$$$ in the basis $$\{|1\rangle,|0\rangle,|-1\rangle\}$$ where $$K := -\beta J$$.

I know that $$Z \approx \lambda^N$$ for the greatest eigenvalue of $$P$$, however I cannot find the eigenvalues. I ended up with this characteristic equation that I cannot solve:

$$(e^K-\lambda)^2(1-\lambda) - 2(e^{K}-\lambda) + 2e^{-K}(1-\lambda)=0.$$

Could someone clarify how to proceed further?

• Have you tried using Mathematica to solve your cubic equation? – mike stone Sep 5 '19 at 15:41
• No I am doing self-study on this topic and found this file: ocw.mit.edu/courses/physics/…. There it is stated that it should be done by symmetry considerations – Mathphys meister Sep 5 '19 at 15:48
• Well one eigenvector is clearly $(1,0,-1)^T$ with eigenvalue $2\sinh K$ – mike stone Sep 5 '19 at 15:57
• When the matrix is symmetric and has lots of "kinda the same" elements, it's often a good idea to try various vectors made up of $1$, $-1$, and $0$. @mikestone gave you one eigenvector in that style, and it's easy to find another one in that style right away. – Lagerbaer Sep 5 '19 at 16:39
• I get some different terms in the char. equation. Why don't you have an $e^{-2K}$ term? I recommend you redo your equation without looking at your previous attempt. – Bill N Sep 5 '19 at 17:36