QuantumPhysics.dvi

Here, we have dropped the′onm 1 andm 2. The initial condition on this recursion relation

is the highest weight matrix element

〈j 1 ,j 2 ;j,j|j 1 ,j 2 ;j 1 ,j 2 〉= 1 j=j 1 +j 2 (8.67)

A detailed discussion on how the above recursion relation fixes all theClebsch Gordan

coefficients may be found in Sakurai.

8.11 Spin Models

A spin model is a statistical mechanical lattice model for which each lattice site has a two-

dimensional Hilbert space of states. (More generally, one could consider spin models with

higher angular momentum representations, and thus with higher dimensional Hilbert spaces

at each lattice point.) A natural way of realizing this two-dimensionalHilbert space is by a

spin 1/2 degree of freedom at each lattice site. The lattice Λ is usuallya square lattice in

ddimensions, and the interactions are usually limited to nearest neighbor interactions only.

Spin often being responsible for the presence of a magnetic moment, one often also includes

a uniform magnetic fieldB which acts as an external source. Schematically, the general

Hamiltonian is of the form,

H=−J

∑

〈i,j〉,i,j∈Λ

SiSj−B

∑

i∈Λ

Si (8.68)

Here the notation〈i,j〉stands for the inclusions of nearest neighbor pairs only, andiandj

run over all points of a latticeL. One such Hamiltonian is given by theHeisenberg model,

for which all three components of spin are retained,

H=−J

∑

〈i,j〉

Si·Sj−B

∑

i

Siz (8.69)

where theBfield has been taken to be in thezdirection.

8.12 The Ising Model

Henceforth, we shall concentrate on the simplest Hamiltonian, namely that of the Ising

model. Here, the spinsSiare taken to be thez-componentSzi of the spin operatorS. The

Ising model Hamiltonian is given by

H=−J

∑

〈i,j〉

SizSjz−B

∑

i

Siz (8.70)