Subsections

• Main Idea
• One-Particle Hamiltonian
• Free energy
• Calculation of M
  • Self-Consistency Equation
  • Minimization of Free Energy
• Critical Point and Phase Diagram
• Flory Theory and Self-Consistent Field Theory

Self-Consistent Field Theory

Main Idea

Energy of Ising system is

$$E = -H\sum_i \sigma_i -\frac{J}{2}\sum_{(i,j)}\sigma_i\sigma_j$$

It can be written as

$$E = -\sum_i \sigma_i H_{\text{eff}}(i)$$

with molecular field

$$H_{\text{eff}}(i) = H + \frac{J}{2}\sum_{\text{neighbors of $i$}} \sigma_j$$

We substitute $H_{\text{eff}}(i)$ by its average value

$$H_{\text{eff}}=\left\langle H_{\text{eff}}(i)\right\rangle= H + Jz\left\langle \sigma\right\rangle$$

and choose $\left\langle \sigma\right\rangle$ from consistency condition!

One-Particle Hamiltonian

Let us write

$$\sigma_i = M + \delta_i$$

The product:

$$\sigma_i\sigma_j = (M+\delta_i)(M+\delta_j) = M^2 + M\delta_i + M\delta_j + \delta_i\delta_j$$

Approximation: neglect $\delta_i\delta_j$. Result:

$$E=\sum_i E_i$$

with

 

$$\begin{split} E_i(\sigma_i) &= -MH-\delta_i(H + JzM) - \fr... ...^2 \ &= -\sigma_i(H + JzM) + \frac{1}{2}JzM^2 \ \end{split}$$ (4)

We separated energy into independent contributions from each spin!

Free energy

Partition function:

$$Q = \prod_i Q_i = Q_1^N$$

Partition function for one spin

$$\begin{split} Q_i &= \exp\bigl(-\beta E_i(-1)\bigr) + \exp\... ...= 2\exp(-JzM^2/2kT)\cosh\bigl(\beta(H + JzM)\bigr) \end{split}$$

Free energy:  

$$\begin{split} A &= -NkT\ln Q_i \ &= \frac{JzM^2N}{2} - NkT\ln\Bigl(2\cosh\bigl(\beta(H + JzM)\bigr)\Bigr) \end{split}$$ (5)

This depends on the magnetization M!

Calculation of M

There are two ways to calculate M:

Self-Consistency Equation

By definition, $M=\left\langle \sigma_i\right\rangle$. Since only E_i depends on $\sigma_i$, we have:

$$M = \frac{1}{Q_i}\left[\exp\bigl(-\beta E_i(1)\bigr) - \exp\bigl(-\beta E_i(-1)\bigr) \right]$$

or, from (4)  

$$M = \tanh\bigl(\beta(H+JzM)\bigr)$$ (6)

This is a transcendent equation for M.

Minimization of Free Energy

Free energy (5) depends on M. It should be minimal as function of M:

$$\frac{\partial A}{\partial M} = 0$$

or

$$JzMN - NJz\frac{\sinh\bigl(\beta(H + JzM)\bigr)}{\cosh\bigl(\beta(H + JzM)\bigr)}$$

This gives  

$$M = \tanh\bigl(\beta(H+JzM)\bigr)$$ (7)

The results (6) and (7) coincide! Our theory is self-consistent.

Critical Point and Phase Diagram

We can rewrite equation (7) as  

$$H = kT\tanh^{-1} M - JzM$$ (8)

1.

At large T, this is monotonic function:

This means one-phase regime

2.

At low T, this is non-monotonic function:

From stability condition $(\partial M/\partial H)\gt$ we see, that this means phase separation

3.

At critical temperature T_c, we have only one point, where $(\partial H/\partial M)=0$:

At this temperature

$$\left.\frac{\partial H}{\partial M}\right\rvert_{M=0} = 0$$

or, from (8)

kT=Jz

Phase diagram:

Flory Theory and Self-Consistent Field Theory

• Give same critical point
• Expansions near critical point coincide up to quadratic terms
• Give different results farther from critical point

Advantages of SCF:

1.

We know how to extend it--just make a better approximation for $\delta_i\delta_j$.

2.

Works better farther from critical point