A question about duality for Potts model on square lattice

Question

Consider the Potts model on the square lattice: The Hamiltonian reads: $$-\beta H=K \sum_{<ij>}\delta_{s_i, s_j}$$ where $s_i$ takes values from the set $\{1,2,...,q\}$

The partition function reads: $$Z= \sum_{s_1,s_2,...,s_N}\exp(-\beta H)= \sum_{s_1, s_2,...s_N} \prod_{<ij>} \exp(K\delta_{s_i, s_j})$$

Follow the problem 7.2 and 7.5 of chapter 7 in Mehran Kardar's book "Statistical physics of fields", we can obtain the low-temperature series and high-temperature series as follows:

$$Z = q e^{2NK} f(e^{-K}) = q^{-N} (e^K +q-1)^{2N} f(\frac{e^K-1}{e^K +q-1})$$

Then we can obtain the critical point by requiring that:

$$e^K = \frac{e^K -1}{ e^K+q-1}$$

which gives us the critical coupling $e^{K_c} = 1+\sqrt{q}$.

But then, there is problem, the prefactor for the two series are not the same: $$qe^{2 N K_c} \neq q^{-N} (e^{K_c}+q-1)^{2N}$$.

They differ by a factor of q.

I carefully check the derivation but cannot find what's wrong? Maybe somebody could help me out. Thanks in advance.

Answer 1

I think the answer is that we obtain the low-temperature and high-temperature series for different boundary conditions. Hence two partition functions are not equal exactly. The difference can be by multipliers of this kind: $A^{N^a}$, where $a < 1$. This is the case in you solution as $q = q^{N^0}$.