Computational Physics

344 Transfer matrix and diagonalisation of spin chains

s'i

ssi

s'i– 1

i+ 1

Figure 11.1. A step in the multiplication of a vector with the transfer matrix for

the Ising case.

There is, however, a way to perform matrix–vector multiplications efficiently,
using the fact thatTfactorises into a product of sparse matrices:

T=TMTM− 1 ···T 1 (11.23)

where the matrixTiis associated with site numberi. To understand the explicit form
of the submatricesTi, it is useful to look atFigure 11.1. Let us callφthe state with
matrix elements corresponding to the bottom row in that figure. Calling the bottom
row configurations|S〉, and the top row|S′〉, we want to calculate the elements of
the vector|ψ〉=T|φ〉:

〈S′|ψ〉=〈S′|T|φ〉=

∑

S

〈S′|T|S〉〈S|φ〉 (11.24)

for each top row configurationS′. The full matrixTcontains couplings within the
two horizontal rows and on the vertical links between them. The submatricesTi
contain only the couplings on the thick lines inFigure 11.1. We could therefore say
that the application of a matrixTireplaces the lower spinsiby the upper spins′i,
leaving the remaining spins unchanged.
We can now give the form of the matricesTi. It acts between two states which are
somehow intermediate betweenSandS′: the state ′on the left ofTirepresents the
spinss′ 1 ,s 2 ′,...,s′i,si+ 1 ,...,sM, and the state on the right represents the spins
s′ 1 ,s 2 ′,...,s′i− 1 ,si,...,sM. In terms of these states, which have elementsσjandσj′
respectively, we have

〈 ′|Ti| 〉=〈σ 1 ′...σi′− 1 σi′σi′+ 1 ...σM′|Ti|σ 1 ...σi− 1 σiσi+ 1 ...σM〉

=exp

{

βJ

[

1

2

(σiσi+ 1 +σi′σi′− 1 )+σiσi′

]}

×δσ 1 σ 1 ′δσ 2 σ 2 ′...δσi− 1 σi′− 1 δσi+ 1 σi′+ 1 ...δσMσM′. (11.25)

The horizontal couplings have a factor 1/2 because they are taken into account
twice: once when the transfer matrix couples the previous row to the current row,
and once when the transfer matrix couples the present row to the next one. The form