Subsections

• Inhomogeneous systems
• Gradient expansion
• Conclusion

Self-Consistent Field Theory

Inhomogeneous systems

In the previous lecture we obtained for Ising lattice:  

$$A = \frac{JzM^2N}{2} - NkT\ln\Bigl(2\cosh\bigl(\beta(H + JzM)\bigr)\Bigr)$$ (1)

This is for homogeneous systems. Now M depends on x. We introduce layers normal to x axis:

and M becomes M(l).

Consider a simple cubic lattice. Each spin in the layer l has z=6 neighbors:

• One neighbor in the layer l+1
• One neighbor in the layer l-1
• z-2 neighbors in the layer l

Changes to equation (1): substitute N by sum over layers. Then:

1.

The first term describes mean interaction of a given spin with z neighbors. Substitute zM² for

$$M(l)\bigl[M(l+1)+M(l-1)+(z-2)M(l)\bigr]$$

2.

The second term describes molecular field acting on a given spin (in the layer l). Substitute zM by

$$\bigl[M(l+1)+M(l-1)+(z-2)M(l)\bigr]$$

Gradient expansion

Assumption:

Magnetization M(l) changes at distances much larger than the lattice size a₀

This means that the difference between neighboring layers is small:

$$M(l+1)-M(l)\ll M(l)$$

We substitute M(l) by a smooth function M(x) and use the expansion

$$\begin{aligned} M(x+a_0) &= M(x) + a_0\frac{\partial M(x)}{... ...2}{2}\frac{\partial^2 M(x)}{\partial x^2}+\dots \end{aligned}$$

and

$$\begin{split} \bigl[M(x+a_0)+M(x-a_0)+(z-2)M(x)\bigr] &= \\ zM(x) + a_0^2\frac{\partial^2 M(x)}{\partial x^2} \end{split}$$

Conclusion

We see that in SCF A can be represented as

$$A = \sum_{\text{layers}}A_l$$

with

$$A_l = A_l\left(M(x),a_0^2\frac{\partial^2 M(x)}{\partial x^2} \right)$$

We will not solve the resulting non-linear equations, but rather use this in the Landau theory.