Phase transition and critical phenomena

Ising model

Lattice model

Describe spontaneous magnetization of uniaxial magnet. - Demension:

• State persite:

• , means summing over the nearest-neighbor site.

J>0: ferromagnetic

J<0: anti-ferromagnetic

Partition function

This is an interacting system.

Mean field theory:

Approximation: assume a uniform average magnetization:

: number of nearest-neighbors per particle.

We define : 1. :

• :

• : 3 solutions,

• :

Free energy

- : Expand close to the critical point Thermal equilibrium saddle point: minimize : Spontaneous Symmetry Breaking (SSB) : - :

Critical exponents

Exact solutions of Ising model