Commit c6aae37c authored by Florian Goth's avatar Florian Goth
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write sth. for the documentation

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......@@ -371,3 +371,17 @@ eprint = {https://doi.org/10.1137/17M1117732}
timestamp = {2018.10.18},
url = {https://arxiv.org/abs/1810.06595},
}
@Article{Lee2013,
Title = {Minimal Split Checkerboard Method for Exponentiating Sparse Matrices and Its Applications in Quantum Statistical Mechanics},
Author = {Lee, Che-Rung.},
Journal = {SIAM Journal on Scientific Computing},
Year = {2013},
Number = {2},
Pages = {C143-C171},
Volume = {35},
Doi = {10.1137/110838716},
Owner = {sifff},
Timestamp = {2020.01.05}
}
......@@ -12,7 +12,34 @@
ALF has a module for the automatic decomposition of arbitrary graphs/ hermitian matrices into
an almost minimal number of so-called strictly sparse matrices. We call a hermitian matrix $A$
strictly sparse if it has the property
\begin{equation*}
A_{i,j} =0 \lrarrow \left( e^A \right)_{i,j} = 0 \forall i,j
\end{equation*}
\begin{equation}
A_{i,j} = 0 \lrarrow \left( e^A \right)_{i,j} = 0 \forall i,j
\label{eq:strictsparse}
\end{equation}
To exploit that property for a mtrix, \eg the hopping matrix $T$
we need a decomposition of $T$ in the form
\begin{equation}
T=\sum_i^N T^i
\label{eq:sep}
\end{equation}
with each $T^i$ having the property \eqref{strictsparse}, because then by an application
of the Lie-Trotter formula a first order approximation of $e^{\Delta \tau T}$ can be constructed.
Of course the question remains how to find a decomposition \eqref{eq:sep}.
For some lattices you have seen an example of this type of decomposition in Figure \ref{fig_predefined_lattices}.
As you might notice from the figure, all edges that are attached to a vertex have a different color.
And indeed, as pointed out by \cite{Lee2013} a color decomposition will give a decomposition of $T$
with the required properties. More generally the problem of coloring the edges of a graph
is known as the graph-coloring problem and the number of colors required for a graph $G$ is known as
its chromaticity index $\chi(G)$. An important structural result is Vizing's theorem
which states that a graph $G$ of degree $g$ (meaning the maximum of the number of edges over $G$) the following holds
\begin{equation}
g \leq \chi(G) \leq g+1
\end{equation}
thereby enabling a classification of graphs into two classes.
Determining the chromaticity index for an arbitrary $G$ is an NP-hard problem.
ALF does not try to solve this challenge but tries to find a decomposition with the help of the
Misra-van-Gries graph coloring algorithm.
This algorithm allows itself the freedom to utilize $g+1$ colors, from which it can derive the important
knowledge that at every vertex it encounters during the process it has always one free color.
In ALF we allow to utilize higher-order checkerboard (as e.g. demonstrated in the appendix of \cite{goth2020})
via a config switch.
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