In this document, we discuss the meankondo configuration files and scripts used to study the hierarchical graphene model, as defined in [I.Jauslin, 2022]
The configuration files are compatible with meankondo version 1.5.
We start out by giving the commands needed to compute the flow equation for the hierarchical s-d model. The current directory is assumed to be the 'graphene' directory, which contains the 'configs' directory.
meankondo configs/graphene.mk
We specify the hierarchical graphene model by listing its internal and external fields and giving expressions for the propagator between them. The discussion below may look somewhat daunting, but its main purpose is merely to settle the notations, and may be skipped.
The flow equation of the hierarchical graphene model is defined as the map relating \(\ell^{[m-1]}\) to \(\ell^{[m]}\) in $$ \exp\left(C^{[h]}+\sum_{n}\ell^{[m-1]}_{n}O_n(\psi^{[\le h-1]})\right)= \left<\exp\left(1+\sum_{n}\ell_n^{[h]}O_{n}(\psi^{[\le h]})\right)\right>_h^8 $$ where $$ O_0(\psi):= \sum_{\sigma\in\{\uparrow,\downarrow\}}\left( \psi_{a,\sigma}^{+} \psi_{b,\sigma}^{-} + \psi_{b,\sigma}^{+} \psi_{a,\sigma}^{-} \right) ,\quad O_1(\psi):= \sum_{\alpha\in\{a,b\}} \psi_{\alpha,\uparrow}^{+} \psi_{\alpha,\uparrow}^{-} \psi_{\alpha,\downarrow}^{+} \psi_{\alpha,\downarrow}^{-} $$ $$ O_2(\psi):= \psi_{a,\uparrow}^{+} \psi_{a,\downarrow}^{-} \psi_{b,\downarrow}^{+} \psi_{b,\uparrow}^{-} + \psi_{b,\uparrow}^{+} \psi_{b,\downarrow}^{-} \psi_{a,\downarrow}^{+} \psi_{a,\uparrow}^{-} + \psi_{a,\downarrow}^{+} \psi_{a,\uparrow}^{-} \psi_{b,\uparrow}^{+} \psi_{b,\downarrow}^{-} + \psi_{b,\downarrow}^{+} \psi_{b,\uparrow}^{-} \psi_{a,\uparrow}^{+} \psi_{a,\downarrow}^{-} $$ $$ O_3(\psi):= \sum_{\sigma\in\{\uparrow,\downarrow\}} \psi_{a,\sigma}^+ \psi_{a,\sigma}^- \psi_{b,\sigma}^+ \psi_{b,\sigma}^- ,\quad O_4(\psi):= \psi_{a,\uparrow}^+ \psi_{b,\uparrow}^- \psi_{a,\downarrow}^+ \psi_{b,\downarrow}^- + \psi_{b,\uparrow}^+ \psi_{a,\uparrow}^- \psi_{b,\downarrow}^+ \psi_{a,\downarrow}^- $$ $$ O_5(\psi):= \psi_{a,\uparrow}^+ \psi_{a,\uparrow}^- \psi_{a,\downarrow}^+ \psi_{b,\uparrow}^- \psi_{b,\uparrow}^+ \psi_{b,\downarrow}^- + \psi_{a,\downarrow}^+ \psi_{a,\downarrow}^- \psi_{a,\uparrow}^+ \psi_{b,\downarrow}^- \psi_{b,\downarrow}^+ \psi_{b,\uparrow}^- + \psi_{b,\uparrow}^+ \psi_{b,\uparrow}^- \psi_{b,\downarrow}^+ \psi_{a,\uparrow}^- \psi_{a,\uparrow}^+ \psi_{a,\downarrow}^- + \psi_{b,\downarrow}^+ \psi_{b,\downarrow}^- \psi_{b,\uparrow}^+ \psi_{a,\downarrow}^- \psi_{a,\downarrow}^+ \psi_{a,\uparrow}^- $$ $$ O_6(\psi):= \psi_{a,\uparrow}^+ \psi_{a,\uparrow}^- \psi_{a,\downarrow}^+ \psi_{a,\downarrow}^- \psi_{b,\uparrow}^+ \psi_{b,\uparrow}^- \psi_{b,\downarrow}^+ \psi_{b,\downarrow}^- . $$ in which $$ \psi_{\alpha,\sigma}^{[\le h]\pm}:=\frac12\psi_{\alpha,\sigma}^{[\le h-1]\pm}+\psi_{\alpha,\sigma}^{[h]\pm}; $$
There are therefore 8 external fields: $$ \psi_{\alpha,\sigma}^{[h]\pm},\qquad \alpha\in\{a,b\},\ \sigma\in\{\downarrow,\uparrow\}; $$ and 4 external fields: $$ \psi_{\alpha}^{[\le h-1]\pm},\qquad \alpha\in\{a,b\}; ,\ \sigma\in\{\downarrow,\uparrow\}; $$
The propagator is defined as $$ \left<\psi_{\alpha',\sigma'}^{[h]-}\psi_{\alpha,\sigma}^{[h]+}\right>=\delta_{\sigma,\sigma'}(\delta_{\alpha,a}\delta_{\alpha',b}+\delta_{\alpha,b}\delta_{\alpha',a}) $$