# Hierarchical s-d model

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]

## Portability

The configuration files are compatible with meankondo version 1.3.1.

## Quickstart

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.

• Compute the flow equation at $$h=0$$:  kondo_preprocess -d 1 configs/sd/sd.pre | meankondo 
• Compute the flow equation at $$h\neq0$$  kondo_preprocess -d 1 configs/sd_susc/sd_susc.pre | meankondo 
• Compute the susceptibility at $$h\neq0$$ after $$200$$ iterations at interaction strength $$\lambda_0=-0.01$$  ./scripts/sd_susceptibility configs/sd_susc/sd_susc.pre 200 '-0.01' 

## The model

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 $$\left<\psi_{\alpha',\eta'}^{[m]-}\psi_{\alpha,\eta}^{[m]+}\right>=\delta_{\alpha,\alpha'}\delta_{\eta,-\eta'}\eta.$$

## Preprocessing

The kondo_preprocess tool was written to simplify the writing of configuration files for this model, which is indeed quite intricate. Essentially, it defines the $$\mathbf A$$, $$\omega$$ and $$\tau$$ variables, in terms of which the model can be easily specified.

More precisely, kondo_preprocess defines $${\tt A1}\equiv\mathbf A_+^{[\le m]},\quad {\tt A2}\equiv\mathbf A_-^{[\le m]},\quad {\tt a}\equiv\mathbf A^{[\le m-1]},\qquad {\tt t}\equiv \tau,\qquad {\tt h}\equiv \omega.$$ The #!input_polynomial and #!id_table entries of the configuration file can then make use of these variables as well as their scalar product, which greatly simplifies them (see the kondo_preprocessor(1) man page for details). The #!propagator entry is simplified as well by allowing the use of $$\tt A1$$ as a shorthand for $$\psi_{\alpha,+}^{[m]}$$ for any $$\alpha$$ and similarly for the other variables, which reduces the number of entries of the propagator by a factor 2.

In addition, kondo_preprocessor generates the appropriate #!fields, #!symbols, #!identities and #!groups entries of the configuration file.

The configuration file therefore reduces to reduced #!propagator and #!input_polynomial entries, as well as a #!id_table entry that contains the expressions of $$O_{n}^{[\le m-1]}$$ (remark: since $$\tt a$$ was defined without a rescaling factor, the entries of the #!id_table must contain the appropriate rescaling factor (1, 2 or 4 depending on the number of $$\tt a$$)).