fermi_surface/theory.md

@@ -2,17 +2,17 @@
 
 The computation of Fermi surfaces, or band structures can be easily done
 setting the *RUNOPTION* *qdos* (see [Qdos tutorial](jumu/qdos)). 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
-: $`\rho(E+i\eta,k) = \frac{\eta}{\pi}\frac{1}{(E-\epsilon\_k)\^2 + \eta\^2}`$.
+: $`\rho(E+i\eta,k) = \frac{\eta}{\pi}\frac{1}{(E-\epsilon_k)^2 + \eta^2}`$.
 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 $`E=E\_f`$).
+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
+$`E=\epsilon_{\vec{k},\sigma}`$ 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
@@ -23,36 +23,36 @@ computations with the PKKprime code.
 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
+$`\underline{\underline{M}}(\vec{k},E)=\delta_{\Lambda\Lambda\'}-\sum\limits_{\Lambda*}
+g_{\Lambda\Lambda*}(\vec{k};E) t_{\Lambda\'\'\Lambda\'}(E)`$ 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:
 
-$`\underline{\underline{M}}(\vec{k},E) \underline{c\_\nu} =
-\lambda\_\nu(\vec{k},E) \underline{c\_\nu}`$
+$`\underline{\underline{M}}(\vec{k},E) \underline{c_\nu} =
+\lambda_\nu(\vec{k},E) \underline{c_\nu}`$
  and find the
 eigenvalue which satisfies the following condition
-$`\lambda\_\nu(\vec{k},E)=0`$.
+$`\lambda_\nu(\vec{k},E)=0`$.
  Thus the objective is to scan the
 k-space, for a fixed energy $`E`$, and find
-$`\min\limits\_{\vec{k},\nu}(\|\lambda\_\nu(\vec{k},E)\|)`$.
+$`\min\limits_{\vec{k},\nu}(\|\lambda_\nu(\vec{k},E)\|)`$.
 
 ## 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
@@ -61,24 +61,16 @@ 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 same steps are repeated until the given accuracy is reached, or the maximal number of steps is exhausted.`
+- 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 $`\lambda_\nu(\vec{k},E) > 0 \rightarrow \lambda_\nu(\vec{k},E) < 0`$ 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|}}
-```
+![pkkr1](uploads/a9da9acec1072936dfb60b8c7daccdaa/pkkr1.png)
+![pkkr2](uploads/2879acff3105852c7873b2f3df4f4afd/pkkr2.png)
+![pkkr3](uploads/be7bdcee93845372d4131485bb229749/pkkr3.png)
+![pkkr4](uploads/aca21b458af178f369dd9a81287e6597/pkkr4.png)