Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 5 XY Model Li,Zimeng PB06203182
(3.2.3)
and there goes
(3.2.4)
The long-range correlation is thus power-law decreasing (see (3.2.4)), and from (3.2.3)
the angle deviation is increasing with r increased.
Therefore there couldn't be long-ranged correlation in 2D XY model.
4.Summarize the phase transition of Classical XY Model in 1D,2D,3D and more
4.1 1D Ising Model
The phase transition of Ising Model is done on another article - Ising Model (Homework
3)
4.2 2D XY Model - KT phase transition
The derivation in Section 3 is based on a continuous function of , and so inconsistent
with situations when vortice state occur.The spin experience mutation in a loop and
so we divide the loop into two parts, one is (describing vortice state) and the other is
(describing spin state).
, and and , we define
Here q is the vortice quantum number, s is the arc length and ds=rd
It can be easily drawn that
Thus (3.2.0) can be rewritten as