Computational Physics

416 Quantum Monte Carlo methods

Table 12.3.Largest eigenvalues of the

transfer matrix of the Ising model on a strip

with helical boundary conditions

(Figure 12.8) versus strip width L.

L lnλ 0 (MCTM) lnλ 0 (Lanczos)

6 0.9368(2) 0.9369

7 0.9348(2) 0.9350

8 0.9337(2) 0.9338

9 0.9328(2) 0.9329

10 0.9321(2) 0.9323

11 0.9316(2) 0.9318

The target number of walkers is equal to 5000, and

they performed 10 000 diffusion steps. The third

column gives the eigenvalues obtained by diagon-

alising the full transfer matrix using the Lanczos

method. These values are determined with high

accuracy and are rounded to four significant digits.

The walkers are ‘columns’ of lattice spins,(s 0 ,...,sL− 1 ), as represented in
Figure 12.8. In the diffusion step, a new spin is added to the system, and its value
is thes 0 -component of the new configuration of the walker. The spin components
1toL−1 of the new configuration are filled with the components 0 toL−2of
the old walker – the walker is shifted one position over the cylinder. To sample the
news′ 0 -value, we use the ‘shooting method’ in which the interval[0, 1]is divided
up intopsegments corresponding to the conditional probabilityP(s′ 0 |S)which is
proportional to the Boltzmann factor for adding a spins′ 0 =0,...,p−1tothe
existing columnS. In our clock model example, we have

P(s 0 ′|S)=eJcos(s

′ 0 −s 0 )+Jcos(s′ 0 −sL− 1 )

/D(S), (12.104)

with normalisation factor

D(S)=

∑

s′ 0

eJcos(s

′ 0 −s 0 )+Jcos(s′ 0 −sL− 1 )

. (12.105)

A random number between 0 and 1 is then generated and the new spin value
corresponds to the index of the segment in which the random number falls.
The next step is then the assignment of the weightD(S′)to the walker withD
given in (12.105). Branching is then carried out exactly as in the DMC method. In
fact, the weights are also multiplied by a factor exp(Etrial), whereEtrialis the same
for all walkers but varies in time. It is updated as in the DMC method according to

Etrial=E 0 −αln(N/N 0 ), (12.106)