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##  Appendix A LiH and BeH${}_{\text{2}}$ Hamiltonians

In this section, examples of complete Hamiltonians of molecules LiH (for an
inter-atomic distance of $1.5$ $\mathrm{\mathring{A}}$) and BeH${}_{\text{2}}$
(for an inter-atomic distance of $1.17$ $\mathrm{\mathring{A}}$) obtained with
the method described in Chap. 4 Sec. 4.3.1 are shown.

$\begin{split}H_{\mathrm{LiH}}=&{\bf-0.19975}+{\bf
0.05393}Z_{0}{\bf-0.12836}Z_{1}{\bf-0.31773}Z_{0}Z_{1}{\bf-0.31773}Z_{3}+{\bf
0.0605}Z_{1}Z_{3}\\\ &+{\bf 0.11409}Z_{0}Z_{1}Z_{3}+{\bf 0.05362}Z_{4}+{\bf
0.11434}Z_{2}Z_{4}{\bf-0.03787}Z_{2}Z_{3}Z_{4}+{\bf
0.05362}Z_{1}Z_{2}Z_{3}Z_{4}\\\ &+{\bf
0.0836}Z_{0}Z_{1}Z_{2}Z_{3}Z_{4}{\bf-0.03787}Z_{5}+{\bf
0.05666}Z_{1}Z_{5}+{\bf 0.11434}Z_{0}Z_{1}Z_{5}+{\bf 0.0836}Z_{3}Z_{5}\\\
&+{\bf 0.05666}Z_{2}Z_{3}Z_{5}{\bf-0.12836}Z_{4}Z_{5}+{\bf
0.0847}Z_{1}Z_{4}Z_{5}+{\bf 0.0605}Z_{0}Z_{1}Z_{4}Z_{5}+{\bf
0.05393}Z_{3}Z_{4}Z_{5}\\\ &+{\bf 0.12357}Z_{2}Z_{3}Z_{4}Z_{5}+{\bf
0.01522}X_{1}{\bf-0.01522}Z_{0}X_{1}+{\bf
0.01089}X_{1}Z_{3}{\bf-0.01089}Z_{0}X_{1}Z_{3}\\\ &+{\bf
0.00436}X_{1}Z_{2}Z_{3}Z_{4}{\bf-0.00436}Z_{0}X_{1}Z_{2}Z_{3}Z_{4}+{\bf
0.01273}X_{1}Z_{5}{\bf-0.01273}Z_{0}X_{1}Z_{5}{\bf-0.00901}X_{1}Z_{4}Z_{5}\\\
&+{\bf 0.00901}Z_{0}X_{1}Z_{4}Z_{5}+{\bf
0.00448}X_{0}X_{2}{\bf-0.00479}X_{0}Z_{1}X_{2}{\bf-0.03512}X_{0}Z_{1}X_{2}Z_{3}{\bf-0.03512}Y_{0}Y_{2}Z_{4}\\\
&{\bf-0.00479}Y_{0}Y_{2}Z_{3}Z_{4}+{\bf
0.00448}Y_{0}Z_{1}Y_{2}Z_{3}Z_{4}{\bf-0.03306}X_{0}Z_{1}X_{2}Z_{5}+{\bf
0.00237}Y_{0}Y_{2}Z_{3}Z_{5}+{\bf 0.00237}X_{0}Z_{1}X_{2}Z_{4}Z_{5}\\\
&{\bf-0.03306}Y_{0}Y_{2}Z_{3}Z_{4}Z_{5}-{\bf(4\times
10^{-5})}X_{0}X_{1}X_{2}{\bf-0.00277}Y_{0}Y_{1}X_{2}+{\bf
0.01054}X_{0}X_{1}X_{2}Z_{3}+{\bf 0.01054}X_{0}Y_{1}Y_{2}Z_{4}\\\ &+{\bf
0.00277}Y_{0}X_{1}Y_{2}Z_{3}Z_{4}-{\bf(4\times
10^{-5})}X_{0}Y_{1}Y_{2}Z_{3}Z_{4}+{\bf
0.01173}X_{0}X_{1}X_{2}Z_{5}{\bf-0.00154}X_{0}Y_{1}Y_{2}Z_{3}Z_{5}\\\
&{\bf-0.00154}X_{0}X_{1}X_{2}Z_{4}Z_{5}+{\bf
0.01173}X_{0}Y_{1}Y_{2}Z_{3}Z_{4}Z_{5}+{\bf
0.01522}X_{3}X_{4}{\bf-0.00901}Z_{1}X_{3}X_{4}+{\bf
0.01089}Z_{0}Z_{1}X_{3}X_{4}\\\ &+{\bf
0.00436}Y_{3}Y_{4}{\bf-0.01273}Z_{2}Y_{3}Y_{4}+{\bf
0.00436}X_{3}X_{4}Z_{5}{\bf-0.01273}Z_{2}X_{3}X_{4}Z_{5}+{\bf
0.01522}Y_{3}Y_{4}Z_{5}{\bf-0.00901}Z_{1}Y_{3}Y_{4}Z_{5}\\\ &+{\bf
0.01089}Z_{0}Z_{1}Y_{3}Y_{4}Z_{5}+{\bf
0.00658}X_{1}X_{3}X_{4}{\bf-0.00658}Z_{0}X_{1}X_{3}X_{4}+{\bf
0.00658}X_{1}Y_{3}Y_{4}Z_{5}{\bf-0.00658}Z_{0}X_{1}Y_{3}Y_{4}Z_{5}\\\
&{\bf-0.00776}X_{0}Z_{1}X_{2}X_{3}X_{4}+{\bf 0.00776}Y_{0}Y_{2}Y_{3}Y_{4}+{\bf
0.00776}Y_{0}Y_{2}X_{3}X_{4}Z_{5}{\bf-0.00776}X_{0}Z_{1}X_{2}Y_{3}Y_{4}Z_{5}\\\
&+{\bf
0.00211}X_{0}X_{1}X_{2}X_{3}X_{4}{\bf-0.00211}X_{0}Y_{1}Y_{2}Y_{3}Y_{4}{\bf-0.00211}X_{0}Y_{1}Y_{2}X_{3}X_{4}Z_{5}+{\bf
0.00211}X_{0}X_{1}X_{2}Y_{3}Y_{4}Z_{5}\\\ &+{\bf
0.00004}X_{5}+{\bf-0.00154}Z_{1}X_{5}+{\bf 0.01054}Z_{0}Z_{1}X_{5}+{\bf
0.00277}Z_{3}X_{5}{\bf-0.01173}Z_{2}Z_{3}X_{5}+{\bf 0.00004}Z_{4}X_{5}\\\
&+{\bf
0.00154}Z_{1}Z_{4}X_{5}+{\bf-0.01054}Z_{0}Z_{1}Z_{4}X_{5}{\bf-0.00277}Z_{3}Z_{4}X_{5}+{\bf
0.01173}Z_{2}Z_{3}Z_{4}X_{5}\\\ &+{\bf
0.00211}X_{1}X_{5}{\bf-0.00211}Z_{0}X_{1}X_{5}+{\bf-0.00211}X_{1}Z_{4}X_{5}+{\bf
0.00211}Z_{0}X_{1}Z_{4}X_{5}{\bf-0.00837}X_{0}Z_{1}X_{2}X_{5}\\\ &+{\bf
0.00837}Y_{0}Y_{2}Z_{3}X_{5}+{\bf 0.00837}X_{0}Z_{1}X_{2}Z_{4}X_{5}\\\
&{\bf-0.00837}Y_{0}Y_{2}Z_{3}Z_{4}X_{5}+{\bf
0.00303}X_{0}X_{1}X_{2}X_{5}{\bf-0.00303}X_{0}Y_{1}Y_{2}Z_{3}X_{5}{\bf-0.00303}X_{0}X_{1}X_{2}Z_{4}X_{5}\\\
&+{\bf 0.00303}X_{0}Y_{1}Y_{2}Z_{3}Z_{4}X_{5}+{\bf
0.00448}X_{3}X_{4}X_{5}+{\bf
0.03306}Z_{2}X_{3}X_{4}X_{5}{\bf-0.00479}Y_{3}Y_{4}X_{5}+{\bf
0.00237}Z_{1}Y_{3}Y_{4}X_{5}\\\ &{\bf-0.03512}Z_{0}Z_{1}Y_{3}Y_{4}X_{5}+{\bf
0.00448}Y_{3}X_{4}Y_{5}+{\bf 0.03306}Z_{2}Y_{3}X_{4}Y_{5}+{\bf
0.00479}X_{3}Y_{4}Y_{5}{\bf-0.00237}Z_{1}X_{3}Y_{4}Y_{5}\\\ &+{\bf
0.03512}Z_{0}Z_{1}X_{3}Y_{4}Y_{5}{\bf-0.00776}X_{1}Y_{3}Y_{4}X_{5}+{\bf
0.00776}Z_{0}X_{1}Y_{3}Y_{4}X_{5}+{\bf
0.00776}X_{1}X_{3}Y_{4}Y_{5}{\bf-0.00776}Z_{0}X_{1}X_{3}Y_{4}Y_{5}\\\
&{\bf-0.03074}Y_{0}Y_{2}X_{3}X_{4}X_{5}+{\bf
0.03074}X_{0}Z_{1}X_{2}Y_{3}Y_{4}X_{5}{\bf-0.03074}Y_{0}Y_{2}Y_{3}X_{4}Y_{5}{\bf-0.03074}X_{0}Z_{1}X_{2}X_{3}Y_{4}Y_{5}\\\
&+{\bf
0.00837}X_{0}Y_{1}Y_{2}X_{3}X_{4}X_{5}{\bf-0.00837}X_{0}X_{1}X_{2}Y_{3}Y_{4}X_{5}+{\bf
0.00837}X_{0}Y_{1}Y_{2}Y_{3}X_{4}Y_{5}+{\bf
0.00837}X_{0}X_{1}X_{2}X_{3}Y_{4}Y_{5}\end{split}$ (A.1)

$\begin{split}H_{\text{BeH}_{\text{2}}}=&{\bf-1.90305}{\bf-0.48894}Z_{0}+{\bf
0.14357}Z_{1}{\bf-0.18803}Z_{0}Z_{1}+{\bf 0.12314}Z_{2}+{\bf
0.18326}Z_{0}Z_{2}\\\ &+{\bf 0.10964}Z_{1}Z_{2}+{\bf
0.18222}Z_{0}Z_{1}Z_{2}{\bf-0.48894}Z_{3}+{\bf 0.1288}Z_{0}Z_{3}+{\bf
0.1136}Z_{0}Z_{1}Z_{3}\\\ &+{\bf 0.11249}Z_{2}Z_{3}+{\bf
0.11746}Z_{1}Z_{2}Z_{3}+{\bf 0.14357}Z_{4}{\bf-0.18803}Z_{3}Z_{4}+{\bf
0.1136}Z_{0}Z_{3}Z_{4}\\\ &+{\bf 0.10602}Z_{0}Z_{1}Z_{3}Z_{4}+{\bf
0.10306}Z_{2}Z_{3}Z_{4}+{\bf 0.10577}Z_{1}Z_{2}Z_{3}Z_{4}+{\bf
0.12314}Z_{5}+{\bf 0.11249}Z_{0}Z_{5}\\\ &+{\bf 0.10306}Z_{0}Z_{1}Z_{5}+{\bf
0.10451}Z_{2}Z_{5}+{\bf 0.10785}Z_{1}Z_{2}Z_{5}+{\bf 0.18326}Z_{3}Z_{5}+{\bf
0.10964}Z_{4}Z_{5}\\\ &+{\bf 0.11746}Z_{0}Z_{4}Z_{5}+{\bf
0.10577}Z_{0}Z_{1}Z_{4}Z_{5}+{\bf 0.10785}Z_{2}Z_{4}Z_{5}+{\bf
0.11352}Z_{1}Z_{2}Z_{4}Z_{5}+{\bf 0.18222}Z_{3}Z_{4}Z_{5}\\\
&{\bf-0.00743}X_{0}X_{1}{\bf-0.00229}Y_{0}Y_{1}{\bf-0.00229}X_{0}X_{1}Z_{2}{\bf-0.00743}Y_{0}Y_{1}Z_{2}{\bf-0.00711}Y_{0}Y_{1}Z_{3}\\\
&{\bf-0.00711}X_{0}X_{1}Z_{2}Z_{3}{\bf-0.00875}Y_{0}Y_{1}Z_{3}Z_{4}{\bf-0.00875}X_{0}X_{1}Z_{2}Z_{3}Z_{4}{\bf-0.00352}Y_{0}Y_{1}Z_{5}{\bf-0.00352}X_{0}X_{1}Z_{2}Z_{5}\\\
&{\bf-0.0072}Y_{0}Y_{1}Z_{4}Z_{5}{\bf-0.0072}X_{0}X_{1}Z_{2}Z_{4}Z_{5}{\bf-0.04165}X_{0}X_{2}+{\bf
0.03769}X_{0}Z_{1}X_{2}{\bf-0.00396}Y_{0}Y_{2}\\\ &+{\bf
0.00839}X_{1}X_{2}{\bf-0.01015}Z_{0}X_{1}X_{2}{\bf-0.01015}Y_{1}Y_{2}+{\bf
0.00839}Z_{0}Y_{1}Y_{2}+{\bf 0.01355}Z_{0}X_{1}X_{2}Z_{3}\\\ &+{\bf
0.01355}Y_{1}Y_{2}Z_{3}+{\bf 0.01082}Z_{0}X_{1}X_{2}Z_{3}Z_{4}+{\bf
0.01082}Y_{1}Y_{2}Z_{3}Z_{4}+{\bf 0.00854}Z_{0}X_{1}X_{2}Z_{5}+{\bf
0.00854}Y_{1}Y_{2}Z_{5}\\\ &+{\bf 0.01408}Z_{0}X_{1}X_{2}Z_{4}Z_{5}+{\bf
0.01408}Y_{1}Y_{2}Z_{4}Z_{5}+{\bf
0.03611}X_{0}X_{3}{\bf-0.03611}X_{0}Z_{1}X_{3}{\bf-0.03611}X_{0}X_{3}Z_{4}\\\
&+{\bf 0.03611}X_{0}Z_{1}X_{3}Z_{4}{\bf-0.02498}X_{1}X_{3}+{\bf
0.02498}Z_{0}X_{1}Z_{2}X_{3}+{\bf
0.02498}X_{1}X_{3}Z_{4}{\bf-0.02498}Z_{0}X_{1}Z_{2}X_{3}Z_{4}\\\
&{\bf-0.03615}X_{2}X_{3}+{\bf 0.03615}Z_{1}X_{2}X_{3}+{\bf
0.03615}X_{2}X_{3}Z_{4}{\bf-0.03615}Z_{1}X_{2}X_{3}Z_{4}{\bf-0.01573}X_{0}X_{1}X_{2}X_{3}\\\
&{\bf-0.01573}Y_{0}X_{1}Y_{2}X_{3}+{\bf 0.01573}X_{0}X_{1}X_{2}X_{3}Z_{4}+{\bf
0.01573}Y_{0}X_{1}Y_{2}X_{3}Z_{4}{\bf-0.02498}X_{0}X_{4}+{\bf
0.02498}X_{0}Z_{1}X_{4}\\\ &+{\bf
0.02498}X_{0}Z_{3}X_{4}Z_{5}{\bf-0.02498}X_{0}Z_{1}Z_{3}X_{4}Z_{5}+{\bf
0.02085}X_{1}X_{4}{\bf-0.02085}Z_{0}X_{1}Z_{2}X_{4}{\bf-0.02085}X_{1}Z_{3}X_{4}Z_{5}\\\
&+{\bf 0.02085}Z_{0}X_{1}Z_{2}Z_{3}X_{4}Z_{5}+{\bf
0.02464}X_{2}X_{4}{\bf-0.02464}Z_{1}X_{2}X_{4}{\bf-0.02464}X_{2}Z_{3}X_{4}Z_{5}+{\bf
0.02464}Z_{1}X_{2}Z_{3}X_{4}Z_{5}\\\ &+{\bf 0.01532}X_{0}X_{1}X_{2}X_{4}+{\bf
0.01532}Y_{0}X_{1}Y_{2}X_{4}{\bf-0.01532}X_{0}X_{1}X_{2}Z_{3}X_{4}Z_{5}{\bf-0.01532}Y_{0}X_{1}Y_{2}Z_{3}X_{4}Z_{5}\\\
&{\bf-0.00743}X_{3}X_{4}{\bf-0.00229}Y_{3}Y_{4}{\bf-0.00711}Z_{0}Y_{3}Y_{4}{\bf-0.00875}Z_{0}Z_{1}Y_{3}Y_{4}{\bf-0.00352}Z_{2}Y_{3}Y_{4}{\bf-0.0072}Z_{1}Z_{2}Y_{3}Y_{4}\\\
&{\bf-0.00229}X_{3}X_{4}Z_{5}{\bf-0.00711}Z_{0}X_{3}X_{4}Z_{5}{\bf-0.00875}Z_{0}Z_{1}X_{3}X_{4}Z_{5}{\bf-0.00352}Z_{2}X_{3}X_{4}Z_{5}{\bf-0.0072}Z_{1}Z_{2}X_{3}X_{4}Z_{5}\\\
&{\bf-0.00743}Y_{3}Y_{4}Z_{5}+{\bf 0.01972}Y_{0}Y_{1}Y_{3}Y_{4}+{\bf
0.01972}X_{0}X_{1}Z_{2}Y_{3}Y_{4}+{\bf 0.01972}Y_{0}Y_{1}X_{3}X_{4}Z_{5}+{\bf
0.01972}X_{0}X_{1}Z_{2}X_{3}X_{4}Z_{5}\\\
&{\bf-0.0173}Z_{0}X_{1}X_{2}Y_{3}Y_{4}{\bf-0.0173}Y_{1}Y_{2}Y_{3}Y_{4}{\bf-0.0173}Z_{0}X_{1}X_{2}X_{3}X_{4}Z_{5}{\bf-0.0173}Y_{1}Y_{2}X_{3}X_{4}Z_{5}{\bf-0.03615}X_{0}X_{5}\\\
&+{\bf 0.03615}X_{0}Z_{1}X_{5}+{\bf
0.03615}X_{0}Z_{4}X_{5}{\bf-0.03615}X_{0}Z_{1}Z_{4}X_{5}+{\bf
0.02464}X_{1}X_{5}{\bf-0.02464}Z_{0}X_{1}Z_{2}X_{5}\\\
&{\bf-0.02464}X_{1}Z_{4}X_{5}+{\bf 0.02464}Z_{0}X_{1}Z_{2}Z_{4}X_{5}+{\bf
0.04177}X_{2}X_{5}{\bf-0.04177}Z_{1}X_{2}X_{5}{\bf-0.04177}X_{2}Z_{4}X_{5}\\\
&+{\bf 0.04177}Z_{1}X_{2}Z_{4}X_{5}+{\bf 0.01232}X_{0}X_{1}X_{2}X_{5}+{\bf
0.01232}Y_{0}X_{1}Y_{2}X_{5}{\bf-0.01232}X_{0}X_{1}X_{2}Z_{4}X_{5}{\bf-0.01232}Y_{0}X_{1}Y_{2}Z_{4}X_{5}\\\
&{\bf-0.04165}X_{3}X_{5}+{\bf
0.03769}X_{3}Z_{4}X_{5}{\bf-0.00396}Y_{3}Y_{5}+{\bf
0.00839}X_{4}X_{5}{\bf-0.01015}Z_{3}X_{4}X_{5}\\\ &+{\bf
0.01355}Z_{0}Z_{3}X_{4}X_{5}+{\bf 0.01082}Z_{0}Z_{1}Z_{3}X_{4}X_{5}+{\bf
0.00854}Z_{2}Z_{3}X_{4}X_{5}+{\bf
0.01408}Z_{1}Z_{2}Z_{3}X_{4}X_{5}{\bf-0.01015}Y_{4}Y_{5}\\\ &+{\bf
0.01355}Z_{0}Y_{4}Y_{5}+{\bf 0.01082}Z_{0}Z_{1}Y_{4}Y_{5}+{\bf
0.00854}Z_{2}Y_{4}Y_{5}+{\bf 0.01408}Z_{1}Z_{2}Y_{4}Y_{5}+{\bf
0.00839}Z_{3}Y_{4}Y_{5}\\\
&{\bf-0.0173}Y_{0}Y_{1}Z_{3}X_{4}X_{5}{\bf-0.0173}X_{0}X_{1}Z_{2}Z_{3}X_{4}X_{5}{\bf-0.0173}Y_{0}Y_{1}Y_{4}Y_{5}{\bf-0.0173}X_{0}X_{1}Z_{2}Y_{4}Y_{5}\\\
&+{\bf 0.01858}Z_{0}X_{1}X_{2}Z_{3}X_{4}X_{5}+{\bf
0.01858}Y_{1}Y_{2}Z_{3}X_{4}X_{5}+{\bf 0.01858}Z_{0}X_{1}X_{2}Y_{4}Y_{5}+{\bf
0.01858}Y_{1}Y_{2}Y_{4}Y_{5}\\\ &{\bf-0.01573}X_{0}X_{3}X_{4}X_{5}+{\bf
0.01573}X_{0}Z_{1}X_{3}X_{4}X_{5}{\bf-0.01573}X_{0}Y_{3}X_{4}Y_{5}+{\bf
0.01573}X_{0}Z_{1}Y_{3}X_{4}Y_{5}\\\ &+{\bf
0.01532}X_{1}X_{3}X_{4}X_{5}{\bf-0.01532}Z_{0}X_{1}Z_{2}X_{3}X_{4}X_{5}\\\
&+{\bf
0.01532}X_{1}Y_{3}X_{4}Y_{5}{\bf-0.01532}Z_{0}X_{1}Z_{2}Y_{3}X_{4}Y_{5}+{\bf
0.01232}X_{2}X_{3}X_{4}X_{5}{\bf-0.01232}Z_{1}X_{2}X_{3}X_{4}X_{5}\\\ &+{\bf
0.01232}X_{2}Y_{3}X_{4}Y_{5}{\bf-0.01232}Z_{1}X_{2}Y_{3}X_{4}Y_{5}+{\bf
0.01415}X_{0}X_{1}X_{2}X_{3}X_{4}X_{5}+{\bf
0.01415}Y_{0}X_{1}Y_{2}X_{3}X_{4}X_{5}\\\ &+{\bf
0.01415}X_{0}X_{1}X_{2}Y_{3}X_{4}Y_{5}+{\bf
0.01415}Y_{0}X_{1}Y_{2}Y_{3}X_{4}Y_{5}\end{split}$ (A.2)

##  Appendix B Slave-spin theory for the Mott transition in the triangular
lattice

###  B.1 Forewords

In the main chapter (Chap. 5), we focus on the square lattice because of it is
a bipartite lattice and therefore the constraint of the slave-spin method is
useless. In addition, it is the most studied lattice. At first, we did not
focus on this lattice but on the triangular lattice, (111) plane of the
crystal structure of face-centered cubic materials like iron. The two main
differences are the condition for half-filling and the constraint fulfillment.
In order to keep the system at half-filling, we use an optimizer to find the
good chemical potential for every values of $U/t$. Yet, the constraint in
never satisfied in the following. Despite this strong approximation, I think
it is interesting to see the result as the geometry is totally different and
the cluster mean-field is not defined in the same way. The simulation are
performed for cluster of 6 and 10 sites.

The out of equilibrium case is not studied here. The content of this chapter
is extracted from a first version of the article on the Slave-Spin theory.

###  B.2 Solving the spin Hamiltonian $H_{\mathrm{S}}$ for $Q$: cluster mean-
field

We now focus on the computation of $Q_{i,j}$. It requires the computations the
spin-spin correlation function $\langle S_{i}^{z}S_{j}^{z}\rangle$. Many
strategies are available. Most authors [452, 323] handle $H_{\mathrm{S}}$ at
the single-site mean-field level. This yields local observables, but neglects
important spatial correlation effects, thereby limiting the scope of the
method. In particular, spatial (or orbital) resolution in quasi-particle
weights is a key factor to investigate e.g hot and cold spots observed in
angle-resolved photoemission spectroscopy [318, 298, 363].

One must thus go beyond single-site mean-field. A straightforward approach
would be to directly diagonalize a finite-site version of $H_{\mathrm{S}}$ and
compute the finite-size ground-state. However, finite-size effects would
prevent us from observing the phase transition we are looking for, namely the
transition from a Fermi liquid to a Mott insulator.

We thus tackle $H_{\mathrm{S}}$ (Eq. (5.2b)) with a cluster mean-field
approach, as done in e.g [365]. Finite-size effects are thus reduced thanks to
the influence of a self-consistent mean field $m=\langle S^{z}\rangle$ at the
boundary of the cluster, at the same numerical price as the exact
diagonalization (except the procedure needs to be repeated in a self-
consistent fashion until convergence of the mean field).

The cluster mean-field approximation leads to

$S_{i}^{z}S_{j}^{z}\approx\langle S_{i}^{z}\rangle S_{j}^{z}+\langle
S_{j}^{z}\rangle S_{i}^{z}-\langle S_{i}^{z}\rangle\langle S_{j}^{z}\rangle,$
(B.1)

where $i$ ($j$) is inside the cluster and $j$ ($i$) is not. The mean-field
parameter $\langle S^{z}\rangle$ is the same for all sites in the
thermodynamic limit. As we consider finite-size systems, we numerically
compute

$m=\frac{1}{N_{a}}\sum_{i}^{N_{a}}\langle S_{i}^{z}\rangle$ (B.2)

With $N_{a}$ being the number of sites. Eq. (5.2b) can be rewritten as

$H_{\mathrm{S}}^{(C)}(J,m)=H_{\mathrm{S}}(J)+\sum_{i\in\mathrm{C}}h_{i}S_{i}^{z}$
(B.3)

where $C$ is the cluster and $h_{i}=\sum_{j}mJ_{i,j}$

The shape of the cluster and the mean field are illustrated in Fig. B.1 for
clusters of size $6$ and $10$.

For instance, for the 6-site cluster, sites $0$, $3$ and $5$ have 2 nearest-
neighbor whereas sites $1$, $2$ and $4$ are connected to 4 sites respectively.
If we consider a hopping term of value $-1$ for connected sites and $0$ for
the others, the matrix $t$ can be written:

$t=-\begin{pmatrix}0&1&1&0&0&0\\\ 1&0&1&1&1&0\\\ 1&1&0&0&1&1\\\ 0&1&0&0&1&0\\\
0&1&1&1&0&1\\\ 0&0&1&0&1&0\\\ \end{pmatrix}$ (B.4)

This embedding will be the one studied in the rest of the article.

Figure B.1: Cluster mean field and parameterized geometry of Rydberg atoms.(a)
Geometry of a 6-site cluster. After the slave spin mean field approach
combined with the cluster mean field approach, all spins $S^{z}_{i}$ are
immersed in a mean-field bath $h_{i}$ and undergo a transverse field $\propto
U$. (b) Optimization of the geometry with parameters $\alpha$ and $\beta$ to
fulfill Eq.(B.12). (c-d) Same as (a-b), for a 10-site cluster.

The self-consistent procedure to solve for the approximate ground state of
$H_{\mathrm{S}}$, $|\psi_{0}\rangle_{C}$, and hence $Q_{i,j}$, is illustrated
in Fig. 5.1. The solution of the cluster Hamiltonian for a fixed mean field
$h$, $H_{\mathrm{S}}^{(C)}$, is described in section B.3.1.

###  B.3 Self-consistent outer cycle

We are now ready to describe the outer self-consistent loop. It is illustrated
in Fig. 5.1.

In the regime of cluster mean-field theory, there is a difference between the
quasi-particle weight $Z$ and the effective mass renormalization
$\frac{m}{m*}$. across the Mott insulator phase, $Z$ vanishes to $0$ beside
$g=\frac{m}{m*}$ stays finite [452, 365, 344]. They are the order parameters
of the Fermi-liquid/Mott insulator phase transition. The final output of our
method is then the quasi-particle weight defined as

$Z=h^{2}=\langle S^{z}\rangle^{2}.$ (B.5)

It is the order parameter for the phase transition : it is $\neq 0$ before
transition and should be $=0$ after the phase transition. The effective mass

$g=\frac{m}{m^{*}}=\langle S_{i}^{z}S_{i+1}^{z}\rangle$ (B.6)

on the other hand, does not reach $0$ after the transition. These definitions
are true (i.e do not depend on the site we choose) if we consider a system
with an infinite number of size. Numerically, these values are computed as
mean values over all spins:

$\displaystyle Z=h^{2}=\Big{(}\frac{1}{N_{a}}\sum_{i}^{N_{a}}\langle
S_{i}^{z}\rangle\Big{)}^{2}$ (B.7a) $\displaystyle
g=\frac{1}{(N_{a}-1)}\sum_{i}^{N_{a}-1}\langle S_{i}^{z}S_{i+1}^{z}\rangle.$
(B.7b)

In the following, the whole procedure to find these two values is named SSMF-
CMFT for Slave Spin Mean Field - Cluster Mean Field Theory.

####  B.3.1 Solving the embedded model

For a given value of the mean field $h$, we now need to compute $\langle
S_{i}^{z}S_{j}^{z}\rangle$ and $\langle S_{i}^{z}\rangle$ with
$\langle\dots\rangle={}_{\mathrm{S}}\bra{\psi_{0}}\dots\ket{\psi_{0}}_{\mathrm{S}}$
to obtain $g$, $Q$ and $Z$ at the end of the loop. It is possible to compute
these values without knowing the exact groundstate of the system but in our
case, considering an implementation on a Rydberg quantum processor, it is
simpler to get the groundstate of $H_{\mathrm{S}}^{(C)}(J,h)$.

Figure B.2: Quasiparticle weight and mass renormalization as a function of
$U/t$ with the SSMF-CMFT method. Systems plotted here 10 sites triangular
lattices. $H_{\mathrm{S}}^{(C)}(J,h)$ is solved by exact diagonalization
method.

Exact diagonalization of the cluster Hamiltonian Eq. (B.3) is possible only
for a few sites because of the exponential size of the corresponding Hilbert
space. We therefore turn to our Rydberg platform, characterized by the
Hamiltonian given in Eq. (3.6), to find the solution to our spin model Eq.
(B.3).

These two Hamiltonians have a major difference: while the Rydberg Hamiltonian
has an antiferromagnetic interaction, the auxiliary spin Hamiltonian has a
ferromagnetic interaction as long as $t_{i,j}>0$. We can nevertheless make use
of the Rydberg platform: instead of looking for its ground state, we are going
to be looking for its most excited state.

To this aim, we resort to an annealing procedure. The atomic register is
prepared in a state

$\ket{\psi_{\mathrm{start}}}=\ket{g}^{\otimes N_{\mathrm{a}}}$ (B.8)

where $N_{a}$ is the number of atoms.

The adiabatic theorem states that a physical system remains in its
instantaneous eigenstate if the Hamiltonian is driven slowly enough compare to
the inverse square gap between the eigenvalue and the rest of the
Hamiltonian’s spectrum. In particular, this is true for the most excited
state. We can rewrite (B.3) as:

$H_{\mathrm{target}}=\sum_{i,j,\sigma}(-J_{i,j})S^{z}_{i}S^{z}_{j}-\sum_{i\in
C}h_{i}S_{i}^{z}-\frac{U}{4}\sum_{i}S_{i}^{x}$ (B.9)

that is the target Hamiltonian. The first step is to calculate which
experimental values should be obtained to go from Eq. (3.6) to Eq. (B.9). The
Rabi frequency, detuning and the interaction matrix of the atoms have to
fulfil the following relations at the end of the annealing procedure:

$\displaystyle\Omega(t_{\mathrm{max}})$ $\displaystyle=\frac{U}{2},$ (B.10)
$\displaystyle\delta_{i}(t_{\mathrm{max}})$
$\displaystyle=\frac{1}{2}\sum_{j\neq
i}\frac{C_{6}}{r_{i,j}^{6}}+2h\sum_{j\neq i}J_{i,j},$ (B.11)
$\displaystyle\frac{C_{6}}{r_{i,j}^{6}}$ $\displaystyle=-4J_{i,j}.$ (B.12)

In our embedding, sites can be described in two categories: those which have
two nearest neighbours and those which have four nearest neighbours. It leads
to a matrix $J_{i,j}$ of the form

$\begin{pmatrix}0.&a&a&0.&0.&0.\\\ a&0.&b&a&b&0.\\\ a&b&0.&0.&b&a\\\
0.&a&0.&0.&a&0.\\\ 0.&b&b&a&0.&a\\\ 0.&0.&a&0.&a&0.\\\ \end{pmatrix}.$ (B.13)

It means that only two parameters are needed to satisfy Eq. (B.12). The
parameters $\alpha$ and $\beta$ described in Fig. B.1 are then optimized in
each outer loop of the algorithm (Fig. 5.1). The optimization of the geometry
starts with an initial guess for parameters $\alpha$ and $\beta$. Then, a
Nelder-Mead algorithm is used to minimize the value

$\sum_{i,j}\left|\frac{C_{6}}{r_{i,j}^{6}}+4J_{i,j}\right|$ (B.14)

It usually takes less than $50$ steps to find the final parameters with very
good results (see Sec.B.5.2). In the following, we describe the experimental
procedure to solve (B.3). First, the atoms are prepared in the state (B.8) and
in the optimised geometry. The following Hamiltonian is the one applied at
$t=0$

$H_{\mathrm{start}}=\sum_{i\neq
j}\frac{C_{6}}{|\textbf{r}_{i}-\textbf{r}_{j}|^{6}}\hat{n}_{i}\hat{n}_{j}-\hbar\delta_{\mathrm{start}}\sum_{i}n_{i}$
(B.15)

where $\delta_{\mathrm{start}}$ is set so that $\ket{\psi_{\text{start}}}$ is
the most excited state of Eq. (B.15). The Rabi frequency and the detuning are
then driven during a time $t_{max}$ to reach the Hamiltonian Eq. (B.9). A
global addressing is performed for the Rabi frequency whereas a local one is
used for the detuning. Following Eq. (B.10), the Rabi frequency starts at $0$
MHz and is driven linearly to $\frac{U}{2}$. Similarly, the detunings are all
prepared at a value $\delta_{\mathrm{start}}$ and are driven separately to
values Eq. (B.11). $h\in[0,1]$ and Eq. (B.12) ensures that
$\delta_{i}(t_{\mathrm{max}})$ is always positive. Observables ($Q$, $Z$, $g$)
are measured with a sample of shots obtained from the state of the atoms.

###  B.4 Discussion on units

Experimentally, the Rydberg states used impose a van der Waals coefficient
$\frac{C_{6}}{h}=1947.10^{3}$ MHz.$\mu$m6 [464]. This leads to fix all
parameters in our protocol. The Rabi frequency (Eq. (B.10)) can be driven up
to a maximal value of $\frac{\Omega}{2\pi}\approx 2.5$ MHz. Therefore,
considering $U\in[0,5]$ is tailored to our device. In addition, It is guessed
that the phase transition is for $\frac{U}{t}\in[10,15]$ (see Sec. B.5.1). We
consider then $t=\frac{1}{3}$ such as all the interval is span. This yields to
value distance between atoms $\approx 11\mu$m for the closest ones and
$\approx 23\mu$m for unintended interactions. Interaction values are then
between $10^{-2}$ MHz and $\approx 0.8$ MHz. Frequencies of local detuning are
then imposed by Eq. (B.11). It is important to note that for $h$ values close
to $1$ (i.e. $\frac{U}{t}\ll\frac{U_{c}}{t}$), all
$\delta_{i}(t_{\mathrm{max}})$ are expected to reach $0$ MHz111We aim at
$\frac{C_{6}}{r_{i,j}^{6}}=-4J_{i,j}$. If we consider this perfectly
fulfilled, we can replace in Eq. (B.11) and we obtain
$\delta_{i}(t_{\mathrm{max}})=\frac{1}{2}(-4\sum_{i\neq j}J_{i,j})+2\times
1\times\sum_{j\neq i}J_{i,j}=-2\sum_{j\neq i}J_{i,j}+2\sum_{j\neq
i}J_{i,j}=0$. In our numerical simulation, optimization of geometry provides
us $\delta_{i}(t_{\mathrm{max}})\approx 10^{-2}$MHz. For values of
$\frac{U}{t}$ at the phase transition and after, $h\approx 0$ and
$\delta_{i}(t_{\mathrm{max}})=-2\sum_{j\neq i}J_{i,j}\approx 0.8$ MHz.

##### Numerical tools

All numerical simulations are performed with the library qutip [386] (exact
diagonalization) and Quantum Learning Machine. The SPAM error is implemented
with a code from Pulser [476].

During the SSMF-CMFT procedure, the vector state obtained for each
diagonalization of Eq. (B.3) is sampled to measure $Q$ and $Z$. Sampling the
vector state above $100$ samples allows to see the Mott transition. If the
maximum number of iterations is set to $5$ for the SSMF loop and the CMFT
loop, the maximal number of shots is $100\times 25=2500$. The working rate of
the device is $\approx 5$ Hz. It means it would take $500$ s to perform one
complete run for a given $U$.

###  B.5 Results

####  B.5.1 Exact diagonalization method

This method is performed in Fig. B.4. The Mott transition appears for
$U/t\approx 11.3$. The Mott insulating phase occurs between $10.5$ and $16.2$
for other numerical methods like DMFT-exact diagonalization ($\frac{U}{t}=15$,
8 sites), exact diagonalization on 12 sites ($\frac{U}{t}=12$) and CDMFT
($\frac{U}{t}=10.5$) [365]: the difference lies in the number of sites taken
in the cluster and the choice of gauge considered in the slave-spin theory
[323].

####  B.5.2 Impact of experimental parameters

The evolution of $h$ during the SSMF-CMFT procedure is shown Fig. B.3.
Different starting points $h^{0}$ are tested. If $h^{0}=0$ or $h^{0}=1$, the
value of $h$ stays constant, it is therefore needed to choose $h^{0}\in]0,1[$.
In this interval, $h$ always converges towards the same value. Far from the
phase transition, the convergence is relatively quick (less than $35$
iterations in total). Near the phase transition, the convergence takes a lot
more time as $h^{2}=Z$ is an order parameter, but it finally reaches the same
value regardless of initial guess for $h^{0}$. The same protocol can be
applied to check the evolution of $Q$ during loops. The initial guess for $Q$
is always $t$ in all results.

Figure B.3: Evolution of $Z$ as a function of loop iterations for a 6 sites
embedding. Different starting point $h^{0}$ (0.1, 0.5 and 0.9) are considered
for each value of $U/t$ (clockwise starting from upper left: $U/t=2.2$,
$U/t=11.0$, $U/t=20.0$ and $U/t=12.76$). The solving method is exact
diagonalization. The number of allowed iteration is increased to $2500$ and
the error accepted is $\eta=10^{-3}$.

##### Optimized geometry

The result of the optimization of atomic positions is shown Fig. B.4 (6
sites). The difference in $Z$ values is under $0.1\%$. The norm of the
difference Eq. (B.14) after optimization is $\sim 0.18$ for each value of
$\frac{U}{t}$.

Figure B.4: Impact of considering a realistic geometry. Comparison of $Z$
values between method with the real matrix $J$ and the optimized one for 6
sites. In both cases, exact diagonalization method is performed to obtain the
groundstate of $H_{\mathrm{S}}^{(C)}$

##### Dephasing noise

The time of annealing is $t_{max}=$ 5 $\mu s$ and the exact state vector is
considered to measure $Z$. The dephasing effect is shown in Fig. B.5. We can
notice that for small $\gamma$ ($0.01$ and $0.3$ MHz), the phase transition is
still present but with smaller values of $\frac{U}{t}$. For larger values, the
phase transition disappears. Experimentally, $\gamma$ is measured around
$0.02$ MHz. We will use this value in our simulations.

Figure B.5: Impact of considering dephasing noise A triangular lattice cluster
of 6 sites is considered.

##### Measurement error

We show the effect of these errors in Fig. B.6. The trivial groundstate of Eq.
(5.3) when $U=0$ is Eq. (B.8). Therefore, a finite $\epsilon$ will have an
impact on the measurement of this state.

Figure B.6: Impact of measurement error. The top panel shows the effect of
$\epsilon$ and the bottom panel shows the effect of $\epsilon^{\prime}$. The
number of shots considered for each measurement is $1000$. The time of
annealing is 4 $\mu$s.

##### Shot noise

The effect of shot noise is shown in Fig. B.7. We clearly see that increasing
the number of shot increases the precision of the result. In order to fit with
experimental limitations and stay in the phase transition regime, we will
impose $n_{\mathrm{shot}}=150$ for numerical simulations.

Figure B.7: Impact of considering a realistic geometry. Comparison of $Z$
values between method with the real matrix $J$ and the optimized one for 6
sites. In both cases, exact diagonalization method is performed to obtain the
groundstate of $H_{\mathrm{S}}^{(C)}$ Figure B.8: Correlations $C_{zz}(l)$ and
correlation length of spins in the final state for a triangular lattice. (a)
correlations $C_{zz}$ vs. distance between atoms for an exact diagonalization
method with an optimized geometry. values are shown for several values of
$U/t$ and a system with 6 sites (circle dashed lines) and 10 sites (dotted
square lines). (b) Correlation lenghts vs. $U/t$ for 6 sites (circle green
line) and 10 sites (circle red line). The exact diagonalization method is used
to solve the Ising Hamiltonian with an optimized geometry. (c) Correlation
length for 6 sites vs. $U/t$ for different dephasing noise (blue line
$\gamma=0.001$ MHz, orange line $\gamma=0.01$ MHz, green line $\gamma=0.01$
MHz and red line $\gamma=0.5$ MHz). The annealing procedure is performed to
obtain correlations. the dashed black line depictes the result without
dephasing ($\gamma=0.001$ MHz). (d) Correlation length as function of
$|U-U_{c}|/t$ on a log-scale for 6 and 10 sites.

###  B.6 Discussion

####  B.6.1 Convergence of the method

Different initial guess of $h$ ($0.1$, $0.5$ and $0.9$) for this method are
tested in Fig. B.3. For $U\ll U_{c}$, $U\approx U_{c}$ and $U\gg U_{c}$, the
convergent point is the same for all initial value, which confirms that this
method is quite unrelated to the choice of the initial value of the mean
field. In addition, when $U\approx U_{c}$, it takes more iteration steps to
converge, which is normal close to the phase transition.

####  B.6.2 Experimental feasibility

The method proposed in this study offers several advantages with regard to
experimental feasibility. First, it avoids non-local terms that occurs when
usual method of solving fermionic systems are used (Jordan-Wigner or Bravyi-
Kitaev for instance). This leads to a relatively simple experimental setup
where only annealing is performed. The scalability of annealing seems better
compare to Variational Quantum Eigensolver algorithms for instance, where the
number of parameters, the number of loop and the intractability of
calculations increases with the system size [303, 341]. Finding the good
optimizer in the classical part of this kind of algorithm is still an active
field of research. In the work presented here, annealing only needs a global
detuning and Rabi frequency, which suits well with experimental device. The
low number of loop needed (set to a maximum value of $25$) leads to a short
number of realizations. In addition, the classical part of the method only
involves diagonalization of a free fermion system being scalable and easy to
tract. For the quantum part of the algorithm, it solves the two-dimensional
transverse-field Ising model that exhibits several interesting properties as
it is equivalent with the anisotropic limit of the three-dimensional lattice
Ising model [327, 292]. In addition, The quantum phase transition of the 2D
Ising transverse system is not quite well understood [459, 364, 274] and some
very recent works show a first-order phase transition for the first excited
state of this model [505]. For all these reasons, Rydberg atoms can help to
solve actual issues in the 2D ferromagnetic transverse Ising model and thus
gained a good understanding of the 2D Hubbard model through SSMF. Yet, the
results above focus on the application of the slave-spin method applied to the
half-filled, single-site Hubbard model and one should not expect them to prove
any quantum advantage with respect to classical methods. Indeed, in this
parameters regime, classical methods like diagrammatic Monte-Carlo are
efficient in the absence of a sign problem and can essentially reach the exact
solution in the thermodynamic limit [466]. This therefore constitutes the
short-term quantitative target for the Rydberg platform, provided a sufficient
quality can be attained.

##  Appendix C Discussion on the correlation length of the system

###  C.1 Forewords

Achieving a quantum advantage strongly relies being able to simulate strongly
interacting state in the quantum simulator. This means that these states can
not be reached by simulating small parts separately but the whole system is
needed (all qubits are entangled). A good quantity to check if our system is
interacting strongly is the correlation length. It describes the distance at
which two spins interact in average. If the correlation length is small one
spin "only sees" its nearest-neigbor and its state only depends of them. On
the other hand, if the correlation length is very large, two very distant
spins interact and it means that the state is very difficult to reach
classically.

I propose therefore a little study of correlation length in the Ising with and
without slave-spin theory and I focus on the difference between the
ferrromagnetic model and the antiferromagnetic model.

###  C.2 Correlation length with the slave-spin method implemented in a RQP

A concrete quantitative yardstick for telling if our problem generates
strongly interacting states, is the correlation length $\xi$ than can be
achieved in the ground states prepared on the Rydberg platform. Essentially,
the size of the problem that is effectively handled by the quantum processor
is of order $O(\xi^{2})$ (for a 2D geometry), say
$N_{c}^{(\mathrm{Q})}=\pi\xi^{2}/4$ for concreteness. If one denotes by
$N_{c}^{(\mathrm{C})}$ the number of spins that can be successfully handled by
a classical algorithm (for solving $H_{\mathrm{S}}$), it means one needs
$\xi=(4N_{c}^{(\mathrm{C})}/\pi)^{1/2}$ to reach quantum advantage. Let us
assess $\xi$ for the platform under consideration. For this, we focus on the
spin-spin correlation length, defined as

$C_{zz}(l)=\langle S_{0}^{x}S_{l}^{x}\rangle-\langle S_{0}^{x}\rangle\langle
S_{l}^{x}\rangle\propto e^{-l/\xi}.$ (C.1)

Correlation length and $C_{zz}(l)$ obtained with the SSMF-CMFT method for a
triangular lattice ( the results shown in this appendix) are shown in Fig.
B.8. A maximum value of the correlation length is observed for different
values of $U/t$ which confirms the emergence of a phase transition. The
dephasing noise has a strong impact on the value of the phase transition as
notices in Fig. B.5. The phase transition value is shifted to small $U/t$
values as the dephasing noise is increase. For very high values of $\gamma$,
correlations become very close to $0$. It appears that the correlation length
is smaller than the minimum distance between atoms ($l\approx 7$ $\mu$m$<11$
$\mu$m).

This result seems to be well known for the 2D-ferromagnetic Ising system with
a transverse field [446], especially when the longitudinal field is non null.

In order to confirm this result for a square lattice, I have run simulation on
the Ising transverse field model on a square lattice with a simple cluster
mean-field approach.

The considered Hamiltonian is:

$H=\sum_{\langle
i,j\rangle}J_{i,j}S^{z}_{i}S^{z}_{j}+\overline{m}\sum_{i}S^{z}_{i}+U\sum_{i}S^{z}_{i}$
(C.2)

with $J_{i,j}=-1$ for the ferromagnetic phase and $J_{i,j}=1$ for the
antiferromagnetic phase and $\overline{m}=1/N\sum_{i}\langle
S^{z}_{i}\rangle$. The lattice spacing (and the distance) is set to one
between nearest neighbors. Therefore, the distance between points is the
Euclidean distance $\sqrt{|x_{i}-x_{j}|^{2}+|y_{i}-y_{j}|^{2}}$ where
$(x_{i},y_{i})$ and $(x_{j},y_{j})$ are the coordinates of the spins $i$ and
$j$ respectively (see Fig. C.1 and Fig. C.2). The cluster mean-field
resolution starts with a first guess of $\overline{m}$ and the groundstate is
calculated with an exact diagonalization method. Then $\overline{m}$ is
calculated with the system in this groundstate and the loop goes on until
convergence. The last groundstate computed is the one used to calculate all
correlators and then, $\xi$.

The computed values are shown in fig. C.1 for the ferromagnetic phase and in
Fig. C.2 for the antiferromagnetic phase. In the case of ferromagnetic
interactions, the correlation length goes to only one neighbor, in accordance
with our result with the slave-spin theory whereas in the antiferromagnetic
cases, the correlation length is more than $12$ times the distance between
atoms in accordance with experimental results [464]. In both cases, a phase
transition is clearly seen as $\Delta Z/\Delta U$ diverges in both cases. The
difference between this two transitions seems to not be clearly understood in
the literature. The resolution of such a system with a cluster mean-field
approach deserves more research for the slave-spin method to be more deeply
understood.

Figure C.1: Correlations and phase transition for the ferromagnetic Ising
transverse field model.(a) A square lattice of $3\times 3$ spins is used. (b)
Correlation length as a function of $U$. The unit is the lattice spacing. The
correlation length has a pic at $1.1$. (c) Evolution of $Z=\langle
S^{z}\rangle^{2}$ as a function of $U$. (d) Variation of $Z$ with regard to
variation $U$ as a function of $U$. Figure C.2: Correlations and phase
transition for the antiferromagnetic Ising transverse field model.(a) A square
lattice of $3\times 3$ spins is used. (b) Correlation length as a function of
$U$. The unit is the lattice spacing. The correlation length has a pic at
$12.5$. (c) Evolution of $Z=\langle S^{z}\rangle^{2}$ as a function of $U$.
(d) Variation of $Z$ with regard to variation of $U$ as a function of $U$.

##  Appendix D  Quasi-particle weight oscillations

###  D.1 Forewords

In this appendix, I show the data I have obtained with the slave spin method
for a quench $0\rightarrow U_{\mathrm{f}}$ (see Chapter 5) and how we can
extract information from it even when a noisy numerical simulation is
considered.

###  D.2 Data of the signal

Quasi-particle response for different $U_{f}$ is shown in Fig. D.1. The mean
value of the oscillation converges to $0$ when $U_{\mathrm{f}}$ increases.

Figure D.1: Response of the quasi-particle weight after a quench $0\rightarrow
U_{\mathrm{f}}$.

###  D.3 Fourier transform

The frequency of the oscillations of the quasi-particle weight obtained with
the slave-spin method is computed from the Fourier transform of the signal.
The signal is a list of points ($Z(\tau_{k})$) taken during a time
$\tau_{\mathrm{max}}$. The library NUMPY is used to perform the Fourier
transform of the signal version. The result is shown in Fig. D.2. One can
clearly the shift in frequency as $U_{\mathrm{f}}$ increases. In addition we
see two pics at a frequency $U_{\mathrm{f}}$ and $U_{\mathrm{f}}/2$ as
explained in chapter 5.

Figure D.2: Fourier transform of the response of the quasi-particle weight
after a quench.

###  D.4 Effect of noise

The effect of noise is shown in Fig. D.3. When $U_{\mathrm{f}}\ll
U_{\mathrm{c}}$, the oscillations are to small and the noise interferes with
the signal. However, when $U_{\mathrm{f}}\geq U_{\mathrm{c}}$, the Fourier
transform can clearly be recovered even in the presence of noise.

Figure D.3: Fourier transform of the response of the quasi-particle weight
after a quench considering a noiseless (pink) or a noisy numerical simulation
(blue).

##  Appendix E Résumé en français

EDF a pour objectif de délivrer une énergie électrique, stable et à bas prix
quels que soient les défis sociétaux, politiques ou scientifiques rencontrés.
Pour ce faire, il est apparu nécessaire, dans le projet de créer un réseau
énergétique totalement décarboné d’ici 2050, d’augmenter la durée de vie des
centrales nucléaires actuelles (à réacteur à eau pressurisée) au-delà de 60
ans. Une très bonne compréhension du vieillissement des matériaux (batteries,
panneaux photovoltaïques, matériaux de structure dans les centrales
nucléaires) devient alors primordiale pour anticiper l’apparition de défauts,
de gonflement ou de corrosion sous contrainte par exemple dans les différents
éléments de la centrale. Comme il est très difficile d’observer directement le
vieillissement des matériaux dans un milieu fortement irradié, être capable de
simuler numériquement ou expérimentalement correctement les matériaux
d’intérêts dans les conditions réelles est la voie prise aujourd’hui à EDF. À
cet effet, plusieurs méthodes de simulation numérique à l’échelle atomique
sont utilisées et développées au sein d’EDF comme la théorie de la