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# Self-Consistent Field Theory

## Main Idea

Energy of Ising system is

It can be written as

with molecular field

We substitute by its average value

and choose from consistency condition!

## One-Particle Hamiltonian

Let us write

The product:

Approximation: neglect . Result:

with

 (4)
We separated energy into independent contributions from each spin!

## Free energy

Partition function:

Partition function for one spin

Free energy:
 (5)
This depends on the magnetization M!

## Calculation of M

There are two ways to calculate M:

### Self-Consistency Equation

By definition, . Since only Ei depends on , we have:

or, from (4)
 (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:

or

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

## Critical Point and Phase Diagram

We can rewrite equation (7) as
 (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 we see, that this means phase separation

3.
At critical temperature Tc, we have only one point, where :

At this temperature

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