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]

Table of contents

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.

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 \(\tt a\))).