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Homework Statement:

Consider the 1domensional Ising model given by the Hamiltonian
$$
H = J \sum_{i=1}^{N} s_{i} s_{i+1}
$$
where $s_{i} = 1, 0, 1$
Calculate the partition function $Z$ and the phase transition temperature.
Relevant Equations:

$$
Z = Tr(T^{n})
$$
I did the first part using the transfer matrix method:
$$
Z = Tr(T^{N})
$$
In this case, the transfer matrix will be
$$
T(i,i') =
\begin{pmatrix}
e^{\beta J} & 1 & e^{\beta J}\\
1 &1 &1 \\
e^{\beta J} & 1 & e^{\beta J}
\end{pmatrix}
$$
To get the trace of $T^N$, you find the eigenvalues of T:
$$
\lambda_{1} = 2 sinh(\beta J)
$$
$$
\lambda_{2,3} = \frac{1}{2} (2cosh(\beta J) + 1 + \sqrt{ ( 2cosh^2(\beta J) + 2sinh^2(\beta J) )^2  cosh(\beta J) + e^{2\beta J} +9})
$$
With this, the partition function is:
$$
Z = \lambda_{1}^{N}+\lambda_{2}^N + \lambda_{3}^N
$$
Now, to find the transition temperature, I'd have to find the zeros of the partition function or the singular points of the Helmholtz free energy, but I'm not quite sure on how to proceed from here. Any ideas? Thanks in advance.
$$
Z = Tr(T^{N})
$$
In this case, the transfer matrix will be
$$
T(i,i') =
\begin{pmatrix}
e^{\beta J} & 1 & e^{\beta J}\\
1 &1 &1 \\
e^{\beta J} & 1 & e^{\beta J}
\end{pmatrix}
$$
To get the trace of $T^N$, you find the eigenvalues of T:
$$
\lambda_{1} = 2 sinh(\beta J)
$$
$$
\lambda_{2,3} = \frac{1}{2} (2cosh(\beta J) + 1 + \sqrt{ ( 2cosh^2(\beta J) + 2sinh^2(\beta J) )^2  cosh(\beta J) + e^{2\beta J} +9})
$$
With this, the partition function is:
$$
Z = \lambda_{1}^{N}+\lambda_{2}^N + \lambda_{3}^N
$$
Now, to find the transition temperature, I'd have to find the zeros of the partition function or the singular points of the Helmholtz free energy, but I'm not quite sure on how to proceed from here. Any ideas? Thanks in advance.