Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 4 Quantum Ising model Li,Zimeng PB06203182
We define the ferromagnetic moment =-
At g=0 we have
, and even in the thermodynamic limit, this ground state still
survives for a small range of g ( ), but with .This can be proved by
perturbation theory, and is also similar to the extension of first order line in the
tricritical Ising model.(see Homework Blume-Capel Model Sec.2) Therefore the two
ferromagnetic ground state remains and are still 2-fold degenerate.We notice that the
symmetry is broken as the state is divided into two independent state and
Now consider the ground state for g ,when g= , we have a single nondegenerate
ground state which mix and (thus preserving all symmetries), the state is
written as
We can verify that this state has no ferromagnetic moment (which is in z direction):
Similar to the case of , the ground state is preserved for a finite range of large g (g
),we can view this ground state as a result of strong quantum fluctuations, as the
mixed state shows quantum tunneling between spin up and spin down.
Therefore, the very different ground states of and indicates that the
ground state cannot evolve smoothly as a function of g. There must be singularity at
some point as a function of g for the quantum Hamiltonian, and g=1 is the nonanlytical
point.
We thereby conclude the critical point for the second order quantum phase transition at
g=
2.Derive the mapping: a d dimensional quantum Ising model can be mapped onto a
d+1 dimensional classical Ising model.
The mapping is a bridge between quantum field theory and classical statistic physics.We
shall see below how quantum quantities is connected with statistic quantities.