12.6 The Monte Carlo transfer matrix method 415s' 0s' 1 (= s 0 )s' 2 (= s 1 )s'L- 1 (= sL- 2 )sL- 1Figure 12.8. Helical boundary conditions for the spin model with nearest neigh-
bour interactions on a strip. A step of the algorithm consists of evolving the ‘old’
walkerSinto a new one calledS′. This is done by first adding a new ‘head’s′ 0 of
S′according to a probability distribution like(12.104). Then the ‘old’ components
sL− 2 tos 0 are copied ontosL′− 1 tos′ 1.The ground state will be represented by a collection of random walkers{Sk}which
diffuse in configuration space according to the transition probabilityP. Each dif-
fusion step is followed by a branching step in which the walkers are eliminated or
multiplied, i.e. split into a collection of identical walkers, depending on the value
of the weight factorD(S′k).
Let us describe the procedure for ap-state clock modelwith stochastic variables
(spins) which assume values
θ=2 πn
p,n=0,...,p− 1 (12.102)and a nearest neighbour coupling
−H
kBT=
∑
〈ij〉Jcos(θi−θj). (12.103)Forp=2 this is equivalent to the Ising model (with zero magnetic field), with
Jbeing exactly the same coupling constant as in the standard formulation of this
model (Chapter 7). For largepthe model is equivalent to theXYmodel. TheXY
model will be discussed inChapter 15– at this moment it is sufficient to know that
this model is critical for all temperatures between 0 andTKT, which corresponds to
βJ≈1.1 (the subscript KT denotes the Kosterlitz–Thouless phase transition; see
Chapter 15). The central chargec(seeSection 11.3) is equal to 1 on this critical line.