Computational Physics

12.5 Quantum Monte Carlo on a lattice 413

for each plaquette (remember thesioccurring in this equations are the eigenvalues of
the correspondingszioperators). Therefore a single spin flip will never be accepted
as it does not respect this requirement. This was already noted inSection 11.5:
letting a chain evolve under the Hamiltonian time evolution leaves the system in
the ‘sector’ where it started off. Simple changes in the spin configuration which
conserve the total spin from one row to another are spin flips of all the spins at the
corners of a nonshaded plaquette.
In the boson and fermion models, where we have particle numbersniminstead
of spins, the requirement(12.97)is to be replaced by

nim+ni+1,m=ni,m+ 1 +ni+1,m+ 1. (12.98)

In this case the simplest change in the spin configuration consists of an increase
(decrease) by one of the numbers at the two left corners of a nonshaded plaquette and
a decrease (increase) by one of the numbers at the right hand corners (obviously, the
particle numbers must obeynim≥0 (bosons) ornim=0, 1 (fermions)). Such a step
is equivalent to having one particle moving one lattice position to the left (right). The
overall particle number along the time direction is conserved in this procedure. The
particles can be represented byworld lines, as depicted inFigure 12.7. The changes
presented here preserve particle numbers from row to row, so for a simulation of
the full system, one should consider also removals and additions of entire world
lines as possible Monte Carlo moves.
Returning to the Heisenberg model, we note that the operator exp(−
τHi)
couples only spins at the bottom of a shaded plaquette to those at the top. This
means that we can represent this operator as a 4×4 matrix, where the four pos-
sible states|++〉,|+−〉,|−+〉and|−−〉label the rows and columns. For the
Heisenberg model one finds after some calculation

exp

[

−

τ

J

4

σσσi·σσσi+ 1

]

=e−^ J/^4









e

τJ/^2000

0 cosh(

τJ/ 2 ) sinh(

τJ/ 2 ) 0

0 sinh(

τJ/ 2 ) cosh(

τJ/ 2 ) 0

00 0e

τJ/^2







 (12.99)

(σσσis the vector of Pauli matrices(σx,σy,σz)–wehaves=
σσσ/2;
≡1). This
matrix can be diagonalised (only a diagonalisation of the inner 2×2 block is
necessary) and the model can be solved trivially. Some matrix elements become
negative whenJ<0 (Heisenberg antiferromagnet). This minus-sign problem turns
out not to be fundamental, as it can be transformed away by a redefinition of the
spins on alternating sites [ 5 , 28 ].