Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 5 XY Model Li,Zimeng PB06203182
XY Model
Edited by Li,Zimeng
Contents:
1.Definition
2.XY Model is frequently used to describe the universal behavior of Mott-insulator and
superfluidity phase transition. Why it is possible?
2.1 Mott-insulator
2.2 Hubbard Model
2.3 Superfluid-insulator transition
3.Mermin-Wagner theorem - 2D XY Model has no long-ranged ordering at non-zero
temperature
3.1 Spin Correlation
3.2
4.Summarize the phase transition of Classical XY Model in 1D,2D,3D and more
4.1 1D Ising Model
4.2 2D XY Model - KT phase transition
4.3 3D - Heisenberg Model
1.Definition
XY Model is a continuous spin system with its spin s=(sx,sy), so it is a two-dimensional
spin system with its two order parameters (n=2). This type of system (d=2, n=2) is called
KT phase transition.
First we define , and here is the phase angle. Thus we can conclude the
Hamiltonian for XY Model:
(1.1)
Here the sum is counted so that i,j is not repeated, otherwise we have to multiply a
factor ahead of (1.1)
Because the sum is calculated within the nearest neighbours, therefore we can expand