library(knitr) library(hadron)
cf
for Correlation Functionshadron
provides a data type or better class for correlation
functions and correlation matrices, which is called cf
. There
is a whole list of input
routines available to import data from HDF5, text or binary formats
into the cf
container. The most important ones are compiled
in the following table:
Hadron Function | Correlator Format --- | --- readtextcf | text readbinarycf | binary, HDF5 readbinarysamples | binary more? | ...
For even more flexibility there is the raw_cf
container, for
which we refer to the documentation.
In oder to solve the generalised eigenvalue problem (GEVP) one has to
read several correlation functions into one cf
correlator
matrix. For this purpose the combine operation c
is defined
for the class cf
. Thus, for instance the following code
snipped can be used:
Time <- 48 correlatormatrix <- cf() for(i in c(1:4)) { tmp <- readbinarycf(files=paste0("corr", i, ".dat"), T=Time) correlatormatrix <- c(correlatormatrix, tmp) } rm(tmp)
This code snippet reads a correlator matrix with four correlation functions from four files. The read functions can also directly read from a list of files. File lists can be created conveniently using the following routines
getorderedfilelist <- function(path="./", basename="onlinemeas", last.digits=4, ending="") getconfignumbers <- function(ofiles, basename="onlinemeas", last.digits=4, ending="") getorderedconfigindices <- function(path="./", basename="onlinemeas", last.digits=4, ending="")
for which we refer to the documentation.
Once the bare data is available as a cf
, one has to decide
for an error analysis strategy. This can be either the bootstrap or
the jackknife. To demonstrate this we first load the sample
correlation matrix provided by hadron
data(correlatormatrix)
which corresponds to a $2\times 2$ local-fuzzed correlator matrix with quantum numbers of the pion. First the resampling needs to be performed, for instance for the (blocked) bootstrap
boot.R <- 150 boot.l <- 1 seed <- 1433567 correlatormatrix <- bootstrap.cf(cf=correlatormatrix, boot.R=boot.R, boot.l=boot.l, seed=seed)
Analogously, jackknife.cf
initiates the jackknife
resampling. boot.R
is the number of bootstrap replicates,
boot.l
the block lentgh. Now, it is also possible to plot the
data with errors
plot(correlatormatrix, log="y", xlab=c("t/a"), ylab="C(t)")
Let us denote the correlator matrix by $C(t)$. Now we are going to solve the generalised eigenvalue problem [ C(t)\, v_i(t, t_0)\ =\ \lambda_i(t, t_0)\, C(t_0)\, v_i(t, t_0) ] with some reference time value $t_0$. One can show that the so-called principal correlators $\lambda(t, t_0)$ follow for large $t$-values the following behaviour [ \lambda_i(t, t_0)\ \propto\ e^{-E_i(t-t_0)} + e^{-E_i(T-t+t_0)}\,. ] Here, $T$ is the time extent and we focus on a symmetric correlation matrix in time. However, analogously one can show this with a minus sign for anti-symmetric correlation matrices in time. Of course, we also have $\lambda(t_0, t_0) = 1$. We re-write the generalised eigenvalue problem by defining [ w_i\ =\ \sqrt{C(t_0)} v_i ] and solve the simple eigenvalue problem [ \sqrt{C(t_0)}^{-1}\,C(t)\,\sqrt{C(t_0)}^{-1}\, w_i\ =\ A\, w_i\ =\ \lambda_i(t, t_0)\, w_i ] instead.
In hadron
this task is performed as follows on the bootstrap
correlator matrix in the most simple case
t0 <- 4 correlatormatrix.gevp <- bootstrap.gevp(cf=correlatormatrix, t0=t0, element.order=c(1,2,3,4), sort.type="values")
Next, the principal correlators $\lambda_i$ are obtained as follows, where in this case we have $i=1,2$
pc1 <- gevp2cf(gevp=correlatormatrix.gevp, id=1) pc2 <- gevp2cf(gevp=correlatormatrix.gevp, id=2) plot(pc1, col="red", pch=21, log="y", xlab="t", ylab="C(t)") plot(pc2, rep=TRUE, col="blue", pch=22)
These principal correlators can be analysed as every object of type
cf
, see below.
bootstrap.gevp
has some additional options which are worth
mentioning.
During the bootstrap procedure for the GEVP, eigenvalues have to be
sorted for every $t$-value. This can be either done by
values
, vectors
or det
passed via the
parameter sort.type
. When vectors
is chosen,
scalar products of eigenvectors are computed
[
v(t', t_0) \cdot v(t, t_0)
]
and the overlap maximised. When sort.t0
is set to
TRUE
, the comparison time is chosen constant as
$t'=t_0+1$. Otherwise, $t'=t-1$ is set in dependence of $t$.
With parameter element.order
the correlation functions in
the input correlator matrix are specified for use in the GEVP. This
can be a sub-set of all the correlation functions in the
matrix. Double usage is allowed as well.
First, a fit directly to the (principal) correlator can be
performed. The corresponding functionality is provided in
hadron
by the function matrixfit
and, more modern,
new_matrixfit
. Let us discuss here the former in its
application to
pc1.matrixfit <- matrixfit(cf=pc1, t1=6, t2=21, useCov=TRUE, parlist=array(c(1,1), dim=c(2,1)), sym.vec=c("cosh"), fit.method="lm") plot(pc1.matrixfit, do.qqplot=FALSE, xlab="t", ylab="C(t)")
An extended overview is provided by the overloaded summary
function
summary(pc1.matrixfit)
This yields an energy level with error of
$E =r tex.catwitherror(x=pc1.matrixfit$t0[1], dx=pc1.matrixfit$se[1], digits=2, with.dollar=FALSE)
$.
As we know that $\lambda(t_0, t_0)=1$, we can fit more than a single
exponential to the principal correlator. For this matrixfit
knows the model pc
. The corresponding fit model reads
[
f(t; E, \Delta E, A)\ =\ \exp(-E(t-t_0))(A + (1-A)\exp(-\Delta E(t-t_0))
]
involving three fit parameters. Of course, the fit must be started at
earlier time slices in order to be sensitive to excited states.
pc1.matrixfit <- matrixfit(cf=pc1, t1=3, t2=20, useCov=TRUE, parlist=array(c(1,1), dim=c(2,1)), sym.vec=c("cosh"), fit.method="lm", model="pc") plot(pc1.matrixfit, do.qqplot=FALSE, xlab="t", ylab="C(t)")
A useful crosscheck is to not plot the raw correlator, but the correlator with the leading exponential divided out
plot(pc1.matrixfit, do.qqplot=FALSE, xlab="t", ylab="C(t)", plot.raw=FALSE) abline(h=1, lty=2)
In such a plot all the data points should fluctuate around one.
This matrixfit gives as a result
$E =r tex.catwitherror(x=pc1.matrixfit$t0[1], dx=pc1.matrixfit$se[1], digits=2, with.dollar=FALSE)
$.
Similaryly, effective masses [ M_\mathrm{eff}\ =\ -\log\frac{C(t)}{C(t+1)} ] can be computed and bootstrapped as follows
pc1.effectivemass <- fit.effectivemass(cf=bootstrap.effectivemass(cf=pc1), t1=5, t2=20) plot(pc1.effectivemass, col="red", pch=21, ylim=c(0,1), xlab="t", ylab="Meff")
From the fit to the effective masses we obtain in this case
$E =r tex.catwitherror(x=pc1.effectivemass$effmassfit$t0[1], dx=pc1.effectivemass$effmassfit$se[1], digits=2, with.dollar=FALSE)
$.
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