general discussion of Monte Carlo simulations and -..,ilications of Markov chains, it is not
a required reading to continue the chapter. In fact, the discussion of spin correlations in
the next section are instructive before reading this section. At the end of the day however,
the considerations here apply for any observable and not just ':: correlations.
When .i1 ':. Monte Carlo simulations to study a physical 'm we sample an
observable in various states of the system. We will work extensively with the correlations
between ':: on various sites of the lattice. ': data which we will have available is
the full spin configuration of the lattice for each sweep. Let us denote by n the sweep
number and assume we have performed a total of K We then have the data: s (T)
for i L [1, MJ, j [ LI, N] and n e [1, K], where each value s'(n) is an up or down
spin, represented by ss (n) = 1 or s'(n) 0 r(e ''ely. Here [ki, k2j means the set of
all integers between ki and k2. We will use the expectation notation ( '), when we are
taking averages over sweeps n, i.e., (s,),,, .,() r, where the surn on n runs over
various subsets of [1, K]. When there is no risk of confusion, we omit the i)t n
and write simply ( '). We shall denote by s(n) the entire configuration of spins at sweep
n, s(n) is in some sense a large matrix of spins. We are interested in the correlations
Corr (s, s') = ,) (s{)( ) and mostly their dependence on the action n j j'. So
tackling the issue top-down we can break the data analysis into two parts:
1. Obtain from the raw data (s(n)) another set of data points of the form (, )
such that x ~ j j' and y ~ Corr (s, s') up to additive constants. Because of
the statistical nature of MC simulations the data will have an inherent distribution,
so that along with the data itself we need the variance and covaariance of the data
>ints. In other words, we need the crror-mfalrix: F = cov ( ..)
2. T. new data ( ,. ) is fitted to an onential-t ;.e function fa(x) where A
(A, 2... ) are fitting parameters. 'i fit is then performed by simply minimizing
the function F(A) where
F(A) r (f -,A(. ))( f-( )). (4-13)
k.1
Recall that since E is symmetric, then F will be non-negative