Computational Physics

350 Transfer matrix and diagonalisation of spin chains

The real parametertrepresents the hopping rate, andU(which is also real) describes
the Coulomb interaction. The fermion creation and annihilation operatorsci,σand
c†i,σhave a site indexiand a spin indexσ=± 1 /2 (these two values are denoted
as↑and↓in the Coulomb term). They satisfy the usual fermion anticommutation
relations:

{c†i,σ,cj,σ′}=δijδσσ′. (11.32)

The other anticommutators vanish. Finally, the number-operatorsni,σare given by

ni,σ=c†i,σci,σ. (11.33)

Other models, such as thet−Jmodel, can be related to the Hubbard model,
and this model reduces in a particular limit to the Heisenberg chain; we shall not
go into details but refer to the literature [15].
In the following, we shall mainly restrict ourselves to the Heisenberg chain,
which serves to illustrate the numerical methods suitable for studying quantum
chains; we shall briefly mention how these can be adapted to Hubbard-like models
where appropriate.

11.5.1 Lanczos diagonalisation of the Heisenberg model

It is quite straightforward to diagonalise the Hamiltonian for the Heisenberg model –
this is done using the Lanczos method (see Appendix A8.2). The main problem is
writing a procedure for multiplying the Hamiltonian by some given vector|ψ〉.
The basis vectors of anL-site chain can be chosen as

| 〉=|s 1 ,s 2 ,...,sL〉, (11.34)

where thesiare the spins. For a spin-1/2 chain these spins assume the values± 1 /2,
and for theS=1 chain 1, 0 and−1. The dimension of the Hilbert space is( 2 S+ 1 )L.
We shall not use this spin representation in our program, but use a mapping of each
basis state to an integer just as in the transfer matrix program. First, we change
notation and letsirun from 0 to 2Sinstead of from−StoS. We write

|K〉=|s 1 ,s 2 ,...,sL〉, (11.35)

where the integerKis given as

K=

∑L

i= 1

siMiS−^1 ; (11.36)

MS= 2 S+1. It is instructive to calculate the numbersKfor all possible states of
aL=4 spin-1/2 chain, for whichsi=0, 1, and the reader is invited to do this.