Potts Model in statistical physics

[CAF123]

Homework Statement

The 3 state Potts model is defined by $$-\beta \mathcal H = J \sum_{r,r'} (3 \delta_{\sigma(r), \sigma(r')} - 1) + h\sum_r \delta_{\sigma(r),1},$$ with J > 0 to encourage neighbouring Potts spins to have same value and h orienting field. The spin like variables can take values 1,2 or 3. The spins are on a d dim hypercubic lattice so that each spin has z=2d nearest neighbours.

Consider the easier hamiltonian $$-\beta \mathcal H_o = H \sum_r \delta_{\sigma(r),1}$$ where H is an external field.

a) Compute the quantity $\langle -\beta(\mathcal H- \mathcal H_o)\rangle$ wrt easy hamiltonian. Use the fact that, wrt the easy hamiltonian, $$\langle \delta_{\sigma, \sigma'} = \langle \delta_{\sigma,1}\rangle^2 + \langle \delta_{\sigma,2}\rangle^2 + \langle \delta_{\sigma,3}\rangle^2 = \langle \delta_{\sigma,1}\rangle^2 + \frac{(1-\langle \delta_{\sigma,1}\rangle)^2}{2}$$

b) Use variational mean field theory to find the best lower bound for the original partition function using the easy hamiltonian above. Show that the resulting mean field equation is $$m = \frac{e^{h+3Jzm} - 1}{e^{h+3Jzm} + 2}$$

Homework Equations

All in section 1, and $\sum_{\sigma} e^{-\beta \mathcal H_o}$ is partition function associated with system governed by easy hamiltonian

The Attempt at a Solution

I found the partition function associated with the easy hamiltonian and the object for part a) is: $$\langle (h-H) \sum_r \delta_{\sigma(r),1} + J \sum_{\langle r,r'\rangle} (3 \delta_{\sigma(r), \sigma(r')} - 1) \rangle_{0,H}$$ There are N nodes with a spin and at each node the average of $\delta_{\sigma(r),1}$ is (1+0+0)/3 = 1/3, so for N spins, the first term average is N/3. For the second term, I get a 2dN multiplied by average of $\langle \delta_{\sigma(r),\sigma(r')}\rangle$. I can use the suggested formula and I get this to also evaluate to 1/3.

I am just wondering how this equation they gave comes about. I can write $$\langle \delta_{\sigma,\sigma'}\rangle = \sum_{\sigma, \sigma'} \delta_{\sigma, \sigma'}e^{-\beta \mathcal H_o}$$ which is written like $$\langle \delta_{1,\sigma'}\rangle \langle \delta_{\sigma,1}\rangle + \dots$$ I guess?

Thanks!

 

[CAF123]

Can anyone help me understand the formula in a)? part b) should be fine thereafter. Thanks