find . -name \*.txt -type f -exec /Users/ruess/Downloads/pandoc-2.7.2/bin/pandoc -f mediawiki -t markdown -o {}.md {} \;
then replace all .txt.md files by .txt
and finally run git mv on all .txt files to convert to .md files

fermi_surface/theory.md

+###### Perform Fermi surfaces calculations using pkkr code {#perform_fermi_surfaces_calculations_using_pkkr_code}
+
+The computation of Fermi surfaces, or band structures can be easily done
+setting the *RUNOPTION* *qdos* (see [Qdos
+tutorial](jumu:qdos "wikilink")). The \"qdos approach\" is based on the
+scan of the k-space for a given energy (or different energy values if we
+are interested on band-structure calculations) by the spectral function
+: \$\\rho(E+i\\eta,k) = \\frac{\\eta}{\\pi}\\frac{1}{(E-\\epsilon\_k)\^2
++ \\eta\^2}\$. \\\\ So the density is written as a Lorentzian which
+reaches its maximum \$\\frac{1}{\\eta\\pi}\$ at
+\$E=\\epsilon\_{\\vec{k},\\sigma}\$. All the maximums of this spectral
+function constitute the Fermi surface (at \$E=E\_f\$).
+
+Even if this method is straightforward, it has some inconveniences.
+First in order to obtain accurate dispersions fine \$(\\vec{k},E)\$
+meshes must be chosen, which increases computation time. Also a precise
+Fermi surface computation requires a small \$\\eta\$ which increases as
+well the computational effort. Finally because of the spreading around
+\$E=\\epsilon\_{\\vec{k},\\sigma}\$ the exact value of \$\\vec{k}\$ is
+not known which is a necessary input data for further calculations,
+particularly transport ones. However, the \'qdos method\' allows to have
+a first glance on the Fermi surface, before launching more accurate
+computations with the pkkr code.
+
+##### Fermi surface calculation {#fermi_surface_calculation}
+
+The goal is to solve numerically the secular equation of KKR : \\\\
+\$\|\\underline{\\underline{M}}(\\vec{k},E)\|=0 \$ where
+\$\\underline{\\underline{M}}(\\vec{k},E)=\\delta\_{\\Lambda\\Lambda\'}-\\sum\\limits\_{\\Lambda*}
+g\_{\\Lambda\\Lambda*}(\\vec{k};E) t\_{\\Lambda\'\'\\Lambda\'}(E)\$ and
+\$\\ \\Lambda=(L,\\sigma)\$.
+
+In most cases this equation can\'t be solved analytically so a \"trick\"
+is done. This trick consists on rather solving:\\\\
+\$\\underline{\\underline{M}}(\\vec{k},E) \\underline{c\_\\nu} =
+\\lambda\_\\nu(\\vec{k},E) \\underline{c\_\\nu}\$\\\\ and find the
+eigenvalue which satisfies the following condition
+\$\\lambda\_\\nu(\\vec{k},E)=0\$.\\\\ Thus the objective is to scan the
+k-space, for a fixed energy \$E\$, and find
+\$\\min\\limits\_{\\vec{k},\\nu}(\|\\lambda\_\\nu(\\vec{k},E)\|)\$.
+
+##### Qualitative explanation of the iterative method {#qualitative_explanation_of_the_iterative_method}
+
+The main feature of this method is that the Brillouin zone, will be
+divided into cubes and into each cube the secular equation
+\$\\underline{\\underline{M}}(\\vec{k},E) \\underline{c\_\\nu} =
+\\lambda\_\\nu(\\vec{k},E) \\underline{c\_\\nu}\$ for
+\$\\lambda\_\\nu(\\vec{k},E)=0\$ will be solved numerically. However,
+scanning the k-space can be a heavy computational task, especially for
+3D Brillouin zones. That\'s why an iterative method is done in order to
+refine the mesh and keep in memory only the zones crossed by a band. So
+in a first step the algorithm will quad the Brillouin zone, into
+tetrahedras (triangles in 2D), and then evaluate if along the edges we
+can find a solution satisfying \$\|\\lambda\_\\nu(\\vec{k},E)\|\<a\_n\$,
+where \$a\_n\$ is a given accuracy.
+
+If a solution is found along a tetrahedra edges, the mesh is refined in
+this region. Otherwise, we don\'t take any more into account this
+spatial region. This step is repeated until we reach the final number of
+desired cubes in the Brillouin zone ((Total cubes number = cubes with
+intersect the bands and cubes which doesn\'t intersect the bands)).
+
+Once the mesh is refined to a certain accuracy, the research of
+\$\\vec{k}\$ that minimize \$\\lambda\_\\nu(\\vec{k},E)\$ start in the
+cubes which are crossed by a band (found in the previous step). For
+doing so a dichotomy procedure is employed:
+
+`  - The spectrum of $\underline{\underline{M}}(\vec{k},E)$ is calculated for discrete k-points in the tetrahedra edges.`\
+`  - The local $\Omega_k$ region where the transition $\lambda_\nu(\vec{k},E) > 0$ -> $\lambda_\nu(\vec{k},E) < 0$ is kept in memory. So by the Bijection Theorem,  $\exists \ \vec{k}_0 \in \Omega_k \ , \  \lambda_\nu(\vec{k}_0,E)=0$.`\
+`  - The k-mesh in $\Omega_k$ is refined.`\
+`  - The same steps are repeated until the given accuracy is reached, or the maximal number of steps is exhausted.`
+
+```{=mediawiki}
+{{:jumu:pkkr1.png?300|}}
+```
+```{=mediawiki}
+{{:jumu:pkkr2.png?300|}}
+```
+```{=mediawiki}
+{{:jumu:pkkr3.png?300|}}
+```
+```{=mediawiki}
+{{:jumu:pkkr4.png?300|}}
+```