`cf`

for Correlation Functions`hadron`

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")
```

```
## Loading required namespace: minpack.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)
```

```
##
## ** Result of one state exponential fit **
##
## based on 541 measurements
## time range from 6 to 21
##
## ground state energy:
## E = 0.4020216
## dE = 0.05814395
##
## Amplitudes:
## P 1 = 2.997469
## dP 1 = 0.4713082
##
## boot.R = 150 (bootstrap samples)
## boot.l = 1 (block length)
## useCov = TRUE
## chisqr = 4.182661
## dof = 14
## chisqr/dof= 0.2987615
## Quality of the fit (p-value): 0.9942633
```

This yields an energy level with error of \(E =0.402(58)\).

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 \(E =0.334(60)\).

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 \(E =0.365(33)\).