# 3-state Potts model - probability of finding a site in state 1

Question: Consider the 1d 3-state Potts model of N sites (i.e., each site can be in either state 1, 2 or 3). Find the partition function and the probability of finding a site in state 1, $< \frac{1}{N} \sum_i \delta_{\sigma_i, 1}>$.

My answer so far: The Hamiltonian of the system is:

$$H = J \sum_{<i,j>} \delta_{\sigma_i, \sigma_j} + h \sum_i \delta_{ \sigma_i}$$

where $J$ is the strength of interaction between spins (uniform in this model) and $h$ is magnetic field. The Hamiltonian leads to the transfer matrix:

$$T = \left[ \begin{array}{ccc} e^{\beta(J+h)} & e^{\beta h} & e^{\beta h} \\ e^{\beta h} & e^{\beta J} & 1 \\ e^{\beta h} & 1 & e^{\beta J} \end{array} \right]$$

In the thermodynamic limit, the partition function $Z = \lambda^N$, where $\lambda$ is the largest eigenvalue of $T$. Plugging this into Wolfram Alpha/Mathematica tells me that $\lambda$ is long and messy (9 terms, 6 of them in a square root).

I have no idea how to go about finding the probability of finding a site in state 1, and don't know what to do with $< \frac{1}{N} \sum_i \delta_{\sigma_i, 1}>$. Usually $Z$ is used as some sort of normalisation but otherwise I'm clutching at straws.

• Once you have computed the free energy, just observe that differentiating it with respect to h gives you the density of $1$ in the system. (Just differentiate $N^{-1}\log Z_N$ at finite $N$ to see that.) – Yvan Velenik Mar 22 '16 at 17:28
• Aaaah. It's just clicked that what I'm after is the same thing as finding the average magnetisation in the Ising model. Thanks! – nancy Mar 23 '16 at 10:57