(3g2+22 2 x2+g2
3y2 2 +2 x2 2 x Ix2 + 2 y2g2)
a1
a2
f-I
4 (2 y2 2 + +x2 2 + 2y22)
1
4
(3 y2 2+2x2+2x x2+2y2g2)
(4-33)
(4-34)
(4-35)
9y4g4
a4 =- y (4-36)
4 (2 y2 2 + X2 2 + 2 y2g2) 2y2 2 + X2 + x X2 + 2 y2g2)
as = y2 (4-37)
2
a6 -2g (4-38)
4 2 y2 + 2 2 + 2 y22)
e -, /TO--1 2/4TO e 52/STO
where x C 2 T and y = v2
We can now compare the results of the Monte Carlo simulation with the exact results
given by (4-30) as functions of j for given values of g,To and p2. In figure (4-8) and (4-9)
we see the results of a sample calculation using the Monte Carlo simulation for M = 2 and
N = 1000 in conjunction with the exact results obtained above. Although the results are
disp] y.I without proper error analysis the reader should be convinced that the simulation
is in qualitative agreement with the exact results. The idea is that in more complicated
cases where it is intractable to do the exact calculation one should be able to fit the
statistical curve for (Ri) with an exponential and read off the energy gap. Doing this for
this simple case gives:
A very similar procedure can be done for M = 3 in which case there is an eight state
system at each time-slice, namely | TTT), I ITT), I TIT), I TTI), I tIT), I TI), I ITI), III).
The 73 matrix can also be diagonalized (although this time it was done numerically) and
exact results can be obtained and compared to the Monte Carlo simulation as shown in
Figure (4-10). Clearly the size of the matrix TM increases exponentially with M which
4 (2 y2 2+ X2 2 + 2 y2g2 2