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.


1 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 $q = q^{N^0}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.