Computational Physics

Exercises 367

This leads to the following code for the multiplication routine involving the first

(leftmost) vertex:

FORn=0TO2M−^1 −1DO{nruns over half the number of

row states }

l= 2 n;{lis a number with least significant

bit equal to 0}

ψ′(l,0,0)=ω 1 ψ(n);

ψ′(l,1,1)=ω 3 ψ(n);

ψ′(l+1, 1, 0)=ω 6 ψ(n);

l=l+1{lis now a number with least

significant bit equal to 1}

ψ′(l,0,0)=ω 2 ψ(n);

ψ′(l,1,1)=ω 4 ψ(n);

ψ′(l−1, 0, 1)=ω 6 ψ(n);

END FOR

Now the second vertex must be treated. We start from the vectorψ′(n,μ,μ′)and

calculate the next vectorψ′′(n,μ,μ′). The indexμremains the leftmost horizontal

arrow (we need this to ensure PBC in the end), and the second corresponds to the

link connecting sites 1 and 2. The procedure is analogous to that for the first vertex.

After completing the sequence through the full row, we calculate the resulting

vectorφ(n)from the last vectorψlast(n,μ,μ′)obtained in the foregoing procedure

by taking the trace:

FORN=0TO2M−^1

φ(n)=ψlast(n,0,0)+ψlast(n,1,1);

END FOR

The results can be analysed in a similar fashion to the Ising model. For the
weights of the F-model, the central charge should be equal to 1.
11.3 The Hamiltonian of the periodic Heisenberg chain can be written as

H=J

∑L

l= 1

SlSl+ 1

withSL+ 1 ≡S 1.

(a) Show that this can be transformed into

H=J

∑L

l= 1

[

1

2

(S+,lS−,l+ 1 +S−,lS+,l+ 1 )+Sz,lSz,l+ 1

]

,

whereS±=Sx±iSy. Give the matrix forms ofS±fors= 1 /2 ands=1.