Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 5 XY Model Li,Zimeng PB06203182
insulating phase appears similarly to the fermionic case.
We can use path integral method to solve the criticality. First we write down the action
Similar to the derivation of Hubbard model, the above action can be mapped into d+1
dimensional XY model. So the universality of superfluid-insulator transition can also be
described by XY model.
3.Mermin-Wagner theorem - 2D XY Model has no long-ranged ordering at non-zero
temperature
3.1 Spin Correlation
To demonstrate the above theorem, we first look at the correlation of the spins
between two sites:
and we can demonstrate that this equals to
Proof:
In order to demonstrate it, first we can draw from the second part of equ (1.2) that
expand
=
So the Hamiltonian would have the common form of binomial
We introduce a generate function
, and
Here < >=0 can be easily drawn, so when we derivate and from the above equation,
and set
, we get,
, and so ,
the above equation has set ,