fermi_surface/theory.md

-###### Perform Fermi surfaces calculations using pkkr code
+###### Perform Fermi surfaces calculations using PKKprime 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\$).
+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)\$
+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
+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
+$`E=\epsilon\_{\vec{k},\sigma}`$ 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.
+computations with the PKKprime code.
 
 ##### 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)\$.
+The goal is to solve numerically the secular equation of KKR :
 
-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
+$`\|\underline{\underline{M}}(\vec{k},E)\|=0`$ 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)\|)\$.
+$`\lambda\_\nu(\vec{k},E)=0`$.
+ 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
 
 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,
+$`\underline{\underline{M}}(\vec{k},E) \underline{c\_\nu} =
+\lambda\_\nu(\vec{k},E) \underline{c\_\nu}`$ 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.
+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
@@ -61,13 +61,13 @@ 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
+$`\vec{k}`$ 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 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.`
 
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