Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 5 XY Model Li,Zimeng PB06203182
we can see Hubbard Model takes tremendous simplifications. It only takes into
interacting of the nearest neighbours, and neglects multiband effect. However, it still
works fine with MIT transitions and the Mott insulating phase.As we have stated before
(see Sec. 2.1), the Mott insulating phase appears only at half filling sites.For the nearestneighbour
Hubbard model on a hypercubic lattice, the band structure of the
noninteracting part is described as (see to (2.2.1))
(2.2.2)
We have used Fourier transform in the above derivation.
Therefore, the parameters in the Hamiltonian depend on the direction of i-j. We
compare (1.1) with (2.2.2), and thus we conclude that Mott-insulator can be described
by XY model.
Note the Hubbard Model here has ignored degeneracy of both obitals' and spins'.It
comes into degenerating Hubbard Model when introducing in obital and spin freedoms.
To introduce in the spin interaction, we just need to multiply the original Hubbard
Hamiltonian with
the spin exchange also easily occurs.
. It should be noted here that XY type or Ising type anisotropy for
2.3 Superfluid-insulator transition
The situation for bosonic superfluid-insulator transition is simpler than the fermionic
Mott-insulator transition, similar to the Hubbard Model, we can write the Hamiltonian
for the interacting bosons as follows (with boson operators
this time)
b
Similar to Fig 2.2.1, the system undergoes bandwidth-control phase transition from
Mott insulator to superfluid when U/t is changed. Also, when U is large, the Mott