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If one wants to evaluate hadron correlation functions, one of the biggest problems is the inversion of the Dirac-matrix that appears in the fermionic action. Computing all elements is most of the times impossible. For propagators from one source point to all other points, there are methods to reduce computational cost. As mentioned above, for multi hadron states and disconnected diagrams propagators from all spatial sites to all spatial sites have to be computed, which makes the situation more complicated. One way to handle the prob- lem is to stochastically estimate the inverse of this large matrix using random noise vectors and combine this approach with Distillation to facilitate all-to-all propagation and amelio- rate the volume scaling [53].

3.3. Stochastic LapH method 27

Since the path integrals are being evaluated using a Monte Carlo based method, the statis- tical errors for the hadron correlators are bound by the statistical fluctuations arising from sampling the gauge-field. This means that the propagators only have to be estimated to a comparable accuracy, exact treatment is not necessary and can even be wasteful. The prop- agator can also be approximated using the Monte Carlo Method.

Random noise vectors η can be used for stochastically estimating the inverse of the large Dirac matrix. The vectorsηsatisfy the following properties

E(ηi) = 0

E(ηiηj∗) =δij,

(3.32)

where E()denotes the expectation value over the random noise sources. For each noise vector, the following system of linear equations can be solved:

ΩXr=ηr (3.33)

forXr_{, whererlabels the noise vectorsr = 1,2,· · ·, N}

R. ThenXr = Ω−1ηr and the expec-

tation value is given by

E(Xiη∗j) =E( X k Ω−1_ik ηkηj∗) = X k Ω−1_ik E(ηkηj∗) = X k Ω−1_ik δkj = Ω−1ij . (3.34)

The left-hand-side can be estimated using the Monte Carlo method, which provides an esti- mate ofΩ−1_ij given by Ω−1_ij ≈N_R−1 NR X r=1 X_irηr∗_j . (3.35)

The problem with this expression is that the variances of the stochastic estimates are usu- ally much too large and hence, the noisy estimates need variance reduction techniques to separate signal from noise. This is possible through dilution of the noise vectors [54, 55]. Dilution

For every noise vector, one can define:

ηr_j =

N

X

b=1

η_jr[b], η_jr[b]=η_jrδjb, where j is not summed over. (3.36)

28 Chapter 3. Methodology

A dilution scheme amounts to the application of a complete set of projection operatorsP(b)_,

which ensure exact zeros for many of the E(ηiηj∗) elements instead of estimates that are

only statistically zero. This reduces the variance dramatically. The dilution projectorsP(b)

are products of time dilution, spin dilution, and LapH eigenvector dilution projectors. The diluted sources are defined as

ηr[b]=P(b)ηr, (3.37)

where the matricesP(b)_satisfy

P(a)P(b)=δabP(a), Na

X

a=1

P(a) = 1, P(a)† =P(a). (3.38)

Xr[b]_{is the solution of}

ΩXr[b]=ηr[b]. (3.39)

Now the Monte Carlo estimate ofΩ−1_ij given in equation (3.35) can be rewritten as

Ω−1_ij ≈ 1 NR NR X r=1 Nb X b=1 X_ir[b]ηr[b]∗_j . (3.40)

Although the expectation value is the same, the variance ofP

aη [a] k η

[a]∗

j is smaller than the

variance of ηkηj∗. The effectiveness of the variance reduction depends on the projectors

chosen. The use of ZN noise ensures zero variance in the estimates of diagonal elements

E(ηiη∗i)[56].

It is now possible to introduce the noise on the entire lattice [57], but distillation offers a more effective way. Noise vectorsρare being introduced only in the smaller LapH-subspace, which have spin, time and eigenmode as their indices. The Dilution projectors are matrices in the subspace and each component of ρ is a random ZN variable with E(ρ) = 0 and

E(ρρ†_{) =I}

d. Note thatρρ†is an outer product. Idis the identity matrix. A quark line as in

equation (3.30) can now be written down in the following way D = SΩ−1S, = SΩ−1V V† = P b SΩ−1_{V P}(b)_P(b)†_V† = P b SΩ−1V P(b)_E(ρρ†_)P(b)†_V† = P b ESΩ−1_{V P}(b)_{ρ(V P}(b)_ρ)†_{. (3.41)}

3.3. Stochastic LapH method 29 The estimate of a quark line on a given gauge configuration therefore reads

D(ij)_uv (y, t;x, t0)≈ 1 NR δij NR P r=1 Nb P b=1 ϕr[b]u (y, t)% r[b]∗ v (x, t0). (3.42)

A quark line is a matrix in space, time, spin and color space. The subscriptsu,v are com- pound indices indicating color and spin,i,jdenote the flavor of the source and sink field.

NRandNb are the number of independent stochastic noise sources and the number of di-

lution projectors corresponding to the chosen dilution scheme, respectively. The smeared- diluted quark source and quark sink vectors are defined as

%r[b]_{(x, t) = (V P}(b)_ρr_{)(x, t), ϕ}r[b]_{(x, t) = [SΩ}−1_{V P}(b)_ρr_{](x, t). (3.43)}

It is now apparent that a quark line factorizes into an outer product of a source vector and a sink vector. This allows for separate construction of the source and sink hadrons. Source and sink operators can be correlated after all elements ofϕ[b]_{(ρ)have been computed and}

stored once. For an unbiased estimation, it is necessary that each quark line in a hadron cor- relator has independent noise. For a meson correlator, this means at least two independent noises per configuration.

As mentioned above, the dilution projectorsP(b)_{used for this work are direct products of}

time dilution, spin dilution and LapH eigenvector dilution projectors. Sob = (bT, bS, bL)is

a triplet of indices, wherebT is the time projector index,bSis the spin projector index andbL

is the LapH eigenvector projector index. The noise-dilution projectors have the form

P_tαn;(b) _t0_α0_n0 =P (bT) t;t0 P (bS) α;α0 P (bL) n;n0, (3.44)

wheret, t0 refer to time slices,α, α0 are Dirac spin indices, and n, n0 are LapH eigenvector indices. We use projectors which are diagonal with some or all of the diagonal elements set to unity and all other elements vanishing. N denotes the dimension of the space of the dilution type of interest.

P_ij(b) =δij, b= 0, (no dilution)

P_ij(b) =δij δbi, b= 0, . . . , N−1 (full dilution)

P_ij(b) =δij δb, imodJ b= 0, . . . , J −1, (interlace-J)

wherei, j= 0, . . . , N−1. A triplet(T, S, L)specifies a dilution scheme, whereT,SandLde- note time, spin, and LapH eigenvector dilution, respectively. F stands for full dilution and IJ

for interlace-J. For example, full time and spin dilution with interlace-8 LapH eigenvector dilution is denoted by (TF, SF, LI8). If full dilution (TF,SF,LF) is used, the exact propagator

30 Chapter 3. Methodology

could be computed with using one noise source only. This so called homeopathic limit cor- responds to distillation.

It is important to note that full spin dilution is needed, as the intended quantum numbers are only recovered up to stochastic mixing. The interlacing of eigenvectors on the other hand is very beneficial, it helps to ameliorate the volume scaling of the computational cost of inversions. Using eigenvector dilution, the required number of inversions is no longer proportional toNv and correspondingly the volume.

In [53], different dilution schemes have been tested using different spatial lattice sizes and pion masses. It was found that (TF, SF, LI8) produces variances near that of the gauge noise limit for correlatorsD(y, tF;x, t0)withtF 6=t0. For these, inversions are usually computed

for a handful of source times only using full time dilution, as correlation functions extracted using nearby source times tend to be highly correlated. The interlacing in time enables the evaluation of correlators which involve propagators that originate and terminate on the same time slice tF = t0, where full dilution in time would not be feasible. The dilution

scheme has to be chosen and tested for each ensemble individually to ensure that the vari- ances reach the gauge noise limit.

Using the stochastic LapH method, the number of inversions is equal toNρNP, whereNρis

the number of noises used andNP is the number of dilution projectors. The two key features

of the method are the use of noise dilution projectors that interlace in time and the introduc- tion of noise in the LapH subspace instead of the entire lattice. Even though the number of eigenvectors required to span the LapH subspace rises with the Volume, the number of inversions of the Dirac matrix can be kept almost constant if eigenvectors are interlaced. Another advantage of stochastic LapH lies in the complete factorization of hadron sources and sinks, which facilitates the construction of correlation functions a posteriori. For the calculation of the string breaking diagrams, this is no longer true. The usual workflow is broken because we do not stochastically estimate the static quark propagators. However, the method still allows for an effective calculation of the required correlation matrix elements, which is described in detail in chapter 4.5.