Subsections

• Zero Field. Curie Point
• Non-Zero Field. Susceptibility
• Phase Diagram

Phase Transitions of Ising Model

Zero Field. Curie Point

Let H=0 (symmetric case). Entropy wants spins to be disoriented, energy--to be parallel

• At large T entropy wins: M=0
• At small T energy wins: $M=\pm1$
• In the Curie point T_c--phase transition. This is a second order phase transition (why?).

Non-Zero Field. Susceptibility

If $H\ne0$, magnetization M is parallel to H. There is a function M(H,T). What happens at $H\to0$?

• If T>T_c, the system is disordered at H=0:
  
  

  $$\lim_{H\to0} M(H,T)=0,\quad T\gt T_c$$

• If T<T_c, the system is ordered at H=0:
  
  

  $$\lim_{H\to\pm0} M(H,T)=\pm M_0(T),\quad T<T_c$$

• If T=T_c, the points M₀ and -M₀ coincide--the tangent is vertical
  
  

  $$\lim_{H\to\pm0} M(H,T)=0,\quad \lim_{H\to0} \frac{\partial M(H,T)}{\partial H} = \infty, \quad T=T_c$$

In critical point susceptibility is infinity!

Since amplitude of fluctuations is

$$\left\langle (\Delta M)^2\right\rangle = \frac{\chi kT}{V}$$

the fluctuations infinitely grow at T_c!

Phase Diagram