Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 7 Renormalization Group Li,Zimeng PB06203182
The second term of (1.1.1) is rewritten as
Therefore Z can be rewritten as
The free energy is F=
Because and B(k) (see (1.2.1))
Therefore the singularity can only occur when b=B(0)=2Kd=-2d
, which can be used
to identify critical temperature
Since only k=0 has a significant impact on the free energy, we thus expand the cosine in
(1.1.2) at k=0 to , thus
in effective Hamiltonian is calculated as
The effective Hamiltonian is thus +
We change the sum to integration and get +
(1.2.2)
where
1.3 Renormalization Group in Momentum Space
Two steps in RG method: Coarse Grain Transformation and Rescaling process.
[1]Coarse Grain Transformation