In this document, we discuss the meankondo configuration files and scripts used to study the hierarchical s-d model, as defined in [G.Gallavotti, I.Jauslin, 2015]
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 's-d' directory, which contains the 'configs' directory.
meankondo configs/sd/sd.mk
meankondo configs/sd_susc/sd_susc.mk
./scripts/sd_susceptibility configs/sd_susc/sd_susc.mk 200 '-0.01' '1e-6' -X
sd_susceptibility
sd_susceptibility is a python script to compute the magnetization and susceptibility for the hierarchical s-d model. Its basic syntax is
sd_susceptibility <config_file> <nbeta> <l0> <h> [-X] [-F]
We specify the hierarchical s-d model by listing its internal and external fields and giving expressions for the propagator between them. The model is quite similar to the hierarchical Kondo model, though the \(\mathbf B\) polynomials are replaced by Pauli operators \(\tau\). 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 s-d model is defined as the map relating \(\ell^{[m-1]}\) to \(\ell^{[m]}\) in $$ C^{[m]}\left(1+\sum_{n}\ell^{[m-1]}_{n}O_{n}^{[\le m-1]}\right)= \left<\prod_{\eta=\pm}\left(1+\sum_{n}\ell_n^{[m]}O_{n,\eta}^{[\le m]}\right)\right>_m $$ where $$ O^{[\le m]}_{0,\eta}:=\frac12\mathbf A^{[\le m]}_\eta\cdot\tau,\quad O^{[\le m]}_{1,\eta}:=\frac12\mathbf B^{[\le m]}_\eta\cdot\tau,\quad O^{[\le m]}_{4,\eta}:=\frac12\mathbf A^{[\le m]}_\eta\cdot\omega, $$ $$ O^{[\le m]}_{5,\eta}:=\frac12\tau\cdot\omega,\quad O^{[\le m]}_{6,\eta}:=\frac12\left(\mathbf A^{[\le m]}_\eta\cdot\omega\right)\left(\tau\cdot\omega\right),\quad O^{[\le m]}_{7,\eta}:=\frac12\left(\mathbf A^{[\le m]}_\eta\cdot\mathbf A^{[\le m]}_\eta\right)\left(\tau\cdot\omega\right) $$ in which \(\omega\in\mathbb R^3\) is of norm 1, \(\tau=(\tau_1,\tau_2,\tau_3)\) where \(\tau_j\) denotes the \(j\)-th Pauli matrix, and \(\mathbf A_\eta^{[\le m]}\) is a vector whose \(j\)-th component, for \(j\in\{1,2,3\}\), is $$ A^{[\le m]j}_\eta:=\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}\psi_{\alpha,\eta}^{[\le m]+}\sigma^j_{\alpha,\alpha'}\psi_{\alpha',\eta}^{[\le m]-} $$ in which \(\sigma^j\) is the \(j\)-th Pauli matrix and $$ \psi_{\alpha,\eta}^{[\le m]\pm}:=\frac1{\sqrt2}\psi_\alpha^{[\le m-1]\pm}+\psi_{\alpha,\eta}^{[m]\pm}; $$ \(O_n^{[\le m-1]}\) is defined like \(O_{n,\eta}^{[\le m]}\) with \(\psi_{\alpha,\eta}^{[\le m]\pm}\) replaced by \(\psi_\alpha^{[\le m-1]\pm}\); and \(\left<\cdot\right>_m\) is the average over \(\psi_{\alpha,\eta}^{[m]\pm}\) and \(\varphi_{\alpha,\eta}^{[m]\pm}\).
There are therefore 8 external fields:
$$
\psi_{\alpha,\eta}^{[m]\pm},\qquad
\alpha\in\{\downarrow,\uparrow\},\ \eta\in\{-,+\};
$$
4 external fields:
$$
\psi_{\alpha}^{[\le m-1]\pm},\qquad
\alpha\in\{\downarrow,\uparrow\};
$$
and 6 super-external fields:
$$
\omega_j,\quad\tau_j,\qquad j\in\{1,2\}
$$
in which \(\tau_j\) are non-commuting, and the commutation relations as well as the condition that \(\omega\) is of norm 1 are implemented using the #!identities entry.
The propagator is defined as $$ \left<\psi_{\alpha',\eta'}^{[m]-}\psi_{\alpha,\eta}^{[m]+}\right>=\delta_{\alpha,\alpha'}\delta_{\eta,-\eta'}\eta. $$