Computational Physics

11.5 ‘Exact’ diagonalisation of quantum chains 349

11.5 ‘Exact’ diagonalisation of quantum chains

So far in this chapter, we have studied methods for diagonalising the transfer
matrices of two-dimensional models. This is equivalent to diagonalising Hamiltoni-
ans of one-dimensional quantum systems. In the next few sections we shall consider
the analysis of one-dimensional quantum chains more systematically.
You may be sceptical about the use of studying one-dimensional quantum chains.
There are, however, good reasons for considering such systems. First, they are
equivalent to classical systems in two dimensions, as we have seen extensively in
the first half of this chapter. Second, there exist experimental realisations of quasi-
onedimensionalsystemsinsomeparticularcrystals;seeforexampleRef. [13]. Last
but not least, the fact that one-dimensional chains can be studied successfully using
analytical and computational tools makes them useful as testing grounds for new
methods which may be successful in higher dimensions too. A nice introduction to
thematerialcoveredinthisandthefollowingsectionsisRef. [14].
The quantum chains which are studied most intensively are spin chains and
Hubbard-like systems. Spin chains consist of quantum spins located on a one-
dimensional lattice. The magnetic spin quantum numbers can assume values−S,
−S+1uptoS, where the maximal spinSis a positive, (half-)integer number. Note
that from now on we shall simply call ‘spin’ the eigenvalue of thez-component
of the spin-angular momentum operator (without the factor
). A famous example
is the Heisenberg (anti-)ferromagnet, which is described by the Hamiltonian:

H=J

L∑− 1

i= 1

SiSi+ 1. (11.30)

This is a chain withopen endsas the first and last spins couple only to a single
neighbour. If we connect the first and last spin by puttingSL+ 1 ≡S 1 ,wehavea
periodic chain. For positive values ofJthe chain is antiferromagnetic.
The one-dimensional Hubbard model describes fermions^4 on a chain. The fermi-
ons are spin-1/2 particles and because of the Pauli principle there are on each site
four possibilities: zero particles, one spin-up or one spin-down particle, or both.
The motion of the particles along the chain is represented by a hopping term. There
is also a Coulomb interaction, which is assumed to be strongly local: only particles
on the same site can feel it. For particles on different sites, the Coulomb interaction
is neglected. The Hamiltonian is

H=−t

L∑− 1

σ;i= 1

(c†i,σci+1,σ+c†i+1,σci,σ)+U

∑L

i= 1

ni,↑ni,↓. (11.31)

(^4) Bose–Hubbard models are presently also the subject of research, but the fermion version has been studied
most intensively.