I have been trying to work out a practice problem after reading about Transfer Matrices method for solving 1D Ising Model. Please, if you are able to, tell me whether the way I introduced transfer matrices and dealt with matrix multiplication is correct, I do not care much about whether I made any algebraic errors but rather if I got the concept of transfer matrices correctly.
Consider a modified 1D Ising Model with N sites, where N is odd, and magnetic field applied on every even site of strength $2\cdot h$. The interaction between neighbours has strength $J$. The Hamiltonian is given by:
$ H = -J \sum_{j=1}^{N-1} \sigma_j \cdot \sigma_{j+1} + 2\cdot h \sum_{j=1}^{\frac{N-1}{2}}\sigma_{2j}$
The objective is first to write a solution in a form:
$ \vec{v}^T \hat{T}^{\frac{N-1}{2}}\vec{u}$
Where $\vec{u}, \vec{v}$ are two dimensional vectors and $T$ is appropriately constructed transfer matrix.
My attempt:
Consider the energy when $N=3$:
$E = -J\sigma_1\sigma_2+h\sigma_2 -J\sigma_2\sigma_3 + h\sigma_2$
My idea was to write:
$E = E'(\sigma_1, \sigma_2) + E^*(\sigma_2,\sigma_3)$
Using this notation, we can write the partition function as:
$Z = \sum_{{\sigma_j}} e^{-\beta\cdot E'(\sigma_1, \sigma_2)}e^{-\beta\cdot E^*(\sigma_2, \sigma_3)}...e^{-\beta\cdot E^*(\sigma_{N-1}, \sigma_N)} $
Now the part where my mistake could be:
I introduce:
$T' = e^{-\beta\cdot E'(\sigma, \sigma')}$
$T^*= e^{-\beta\cdot E^*(\sigma, \sigma')}$
As $\sigma$s are independent this can be viewed as consecutive matrix multiplication of $T'$ and $T^*$. By introducing:
$ T(\sigma, \sigma') = \sum_{\sigma^*}T'(\sigma, \sigma^*)T^*(\sigma^*, \sigma')$
which can be computed to be:
$T = 2 \begin{bmatrix} cosh(2\beta(J-h)) & cosh(2\beta\cdot h) \\ cosh(2\beta\cdot h) & cosh(2\beta(J+h)) \end{bmatrix}$
Here the first matrix field corresponds to both spin up.
Using this matrix, I suppose that partition function could be written as:
$ Z = \sum_{\sigma_1}\sum_{\sigma_N} T^{\frac{N-1}{2}}(\sigma_1, \sigma_N)$
I understand this expression as sum of all the elements so I concluded that $\vec{u}^T=\vec{v}^T= (1,1)$.
The problem arises when I try to compute $Z$ when $h=0$. I get the following expression:
$Z = 2^{\frac{N+1}{2}} cosh^{\frac{N-1}{2}}(2\beta J)$
However, the well known expression should be:
$Z = 2(2 cosh(\beta J))^{N-1}$