Topics in Statistic Mechanics

Adv. Sta. Phy. Homework 4 Quantum Ising model Li,Zimeng PB06203182
We will now show how the above functional integral is connected with analogous
expressions in the language of statistic mechanics.
We consider a statistical system on a d+1 dimensions hypercubic lattice, and define one
of its dimension as time and the other d dimensions as space.
Consider two times and their time-dependent Ising spins, the transfer matrix
links the two configurations and has the elements:
The partition function Z can thus be written:
Z=
(2.2)
where M is the number of sites in time direction and periodic boundary condition |
is assumed.
Compare the transition amplitude in (2.1) and the partition function in (2.2), we can
write the correspondence between statistic physics and quantum mechanics.
We have
which the quantum Hamiltonian is defined.
Therefore we obtain a map of a d+1 dimensional problem with classical variables onto a
d dimensional problem with quantum variables.
3.Suzuki-Trotter Formula
In the derivation of (2.1), we have used the Suzuki-Trotter Formula, which we will
illustrate below:
The path integral arises from the fact that (3.1)