414 Quantum Monte Carlo methods
In the case where, instead of spin-1/2 degrees of freedom, we have (boson)
numbers on the sites, the matrixH 1 becomes infinite-dimensional. In that case we
must expand exp(−
τHi)in a Taylor series expansion in
τ. We shall not go into
details but refer to the literature[5].
If we have fermions, there is again a minus-sign problem. This turns out to
be removable for a one-dimensional chain, but not for two and three dimensions.
In these cases one uses fixed-node and transient estimator methods as described
above[29].
12.6 The Monte Carlo transfer matrix method
InChapter 11we have seen that it is possible to calculate the free energy of a
discrete lattice spin model on a strip by solving the largest eigenvalue of the transfer
matrix. The size of the transfer matrix increases rapidly with the strip width and the
calculation soon becomes unfeasible, in particular for models in which the spins
can assume more than two different values. The QMC techniques which have been
presented in the previous sections can be used to tackle the problem of finding
the largest eigenvalues of the very large matrices arising in such models. Here
we discuss such a method. It goes by the name of ‘Monte Carlo transfer matrix’
(MCTM) method and it was pioneered by Nightingale and Blöte[30].
Let us briefly recall the transfer matrix theory. The elementsT(S′,S)=〈S′|T|S〉
of the transfer matrixTare the Boltzmann weights for adding new spins to a semi-
infinite system. For example, the transfer matrix might contain the Boltzmann
weights for adding an entire row of spins to a semi-infinite lattice model, or a single
spin, in which case we take helical boundary conditions so that the transfer matrix
is the same for each spin addition (see Figure 12.8). The free energy is given in
terms of the largest eigenvalueλ 0 of the transfer matrix:
F=−kBTln(λ 0 ). (12.100)From discussions inChapter 11andSection 12.4, it is clear that the transfer matrix
of a lattice spin model is the analogue of the time-evolution operator in quantum
mechanics.
We now apply a technique analogous to diffusion Monte Carlo to sample the
eigenvector corresponding to the largest eigenvalue. In the following we use the
terms ‘ground state’ for this eigenvector, because the transfer matrix can be written
in the formT=exp(−τH), so that the ground state ofHgives the largest eigenvalue
of the transfer matrix. We write the transfer matrix as a product of a normalised
transition probabilityPand a weight factorD. In Dirac notation:
〈S′|T|S〉=D(S′)〈S′|P|S〉. (12.101)