QuantumPhysics.dvi

Since all operators in the Hamiltonian mutually commute, the quantumsystem actually

behaves classically. Note that, forB= 0, the Hamiltonian is invariant under the unitary

operationRof simultaneous reversal of all spins

RSizR†=−Siz (8.71)

A basis for all the quantum states is given by the tensor product ofthe states|i, σi〉where

iruns through the lattice Λ and for eachi, the variableσi=±1, namely the eigenvalues of

Siz. The eigenvalue of the Hamiltonian on such a state is

H|σi, i∈Λ〉=E{σi}|σi, i∈Λ〉 |σi, i∈Λ〉=

⊗

i∈Λ

|i, σi〉 (8.72)

and the energy eigenvalue is

E{σi}=−J

∑

〈i,j〉

σiσj−B

∑

i

σi (8.73)

Under the operation of spin reversalR, the eigenvalues behave asR(σi) =−σi.

There are two important cases to distinguish between depending onthe sign ofJ. If

J >0 andB= 0, the ground (or minimum energy) state of the system is attainedwhen all

spins are lined up in the same direction, either all upσi= +1 or all downσi=−1. This

interaction is referred to asferromagnetic. IfJ <0 andB= 0, the configuration with all

spins aligned is actually maximum energy, so the minimum energy configuration will have

alternation between spins up and down. This interaction is referredto asanti-ferromagnetic.

8.13 Solution of the 1-dimensional Ising Model

The simplest Ising model is in one dimension, in which case the Hamiltonianmay be written

down even more explicitly,

H=−J

∑N

i=1

SizSi+1z−B

∑N

i=1

Siz (8.74)

and we use the periodicity conventionSN+1z=S 1 z. To computeZ, we write it as a sequential

product,

Z = tr

(

E 1 E 2 ···EN− 1 EN

)

Ei = exp

{

βJSizSi+1z+

1

2

βB(Siz+Si+1z)

}

(8.75)

We define the identity operator

Ii=

∑

σi=± 1

|i, σi〉〈i, σi| (8.76)