Table 4-2.
Test results for ID Ising system. Here we have exact results to compare to the
MC numbers. The E-fit is obtained by fitting the correlation to an exponential
and reading off the exponent as shown on graph 4-2. We can see that the
overall error decreases with increasing number of sweeps, but once the error is
pretty small this decrease is not very consistent. This is an inherent property of
a stochastic method such as this one.
Numerical Results
Coupling g Exact E MC E-fit
104 sweeps 4 104 sweeps 1.6 105 sweeps
0.1 0.20067 0.25724 0.22188 0.21900
0.2 0.40547 0.38471 0.41673 0.41903
0.3 0.61904 0.64785 0.61391 0.61466
wholly local Ising interaction is given by
=- (- A\sis]r+ msi)
(4-11)
with A a matrix with only nearest neighbor interaction. For simplicity of the test we took
m 0 and Aij I= ./ i-1.
All the resemblance almost drives us to test the software and methods on just a
ID Ising spin system, which we did. The MC implementation involved using simply the
software for the Lightcone World Sheet with a simplified interaction. The exact results are
very simple to obtain for the case described here. For a given time the state of the system
is represented by the vector of spins s= (si, S2,... SM). Since the interaction is so simple
then the transition matrix is given by
(4-12)
i4J 9 ( ,j+l + 6J+l,j)
so that s(t + dt) = Ts(t). Taking t A= t/N and s(t) TN(0) then finite time
propagation corresponds to N -> +oo. The eigenvalues of T are = 1 g and for
large enough N then E = Int and therefore the splitting is AE ln((1 + g)/(1 g'
The results for the exact-vs-MC comparison are shown in Table 4-2 and the fitting is
exemplified by the plot in Figure 4-2. Again, the relative agreement with known exact
answers gives the computer code a "pass" for this test as well. Having passed both of