This is the official documentation for **meankondo**, version 1.4. The aim of this document is not to give a technical description of how to use the various programs bundled with **meankondo**, nor is it to explain where hierarchical models come from and what their meaning is, but rather a conceptual overview of how **meankondo** approaches the computation of flow equations, and how its programs can be made to interact with one another to compute various quantities. For a more technical description, see the man pages included with the **meankondo** source code. For a more theoretical discussion of Fermionic hierarchical models, see [G.Benfatto, G.Gallavotti, I.Jauslin, 2015].

**meankondo** is a collection of tools to compute and manipulate the renormalization group flow in Fermionic hierarchical models. The programs included in **meankondo** are the following:

**meankondo**: computes the flow equation.**numkondo**: iterate the flow equation numerically.**meantools**: tools to exponentiate, derive and evaluate a flow equation.**meantools-convert**: python script to convert a flow equation to C, javascript or LaTeX code.

**kondo_preprocess**: hierarchical Kondo model.

**Remark**: The name "meankondo" comes from the fact that it was originally developed for the hierarchical Kondo model, though the tools in **meankondo** are quite versatile and can be used for a wide variety of Fermionic hierarchical models.

We first discuss the more elementary commands that can be used to compute and iterate flow equations. The rest of this document is dedicated to what flow equations are and how **meankondo** represents and manipulates them.

Given a configuration file 'config', the flow equation can be computed by
```
meankondo config
```

and it can be iterated for, say, 100 steps starting from \(\ell_0^{[m]}=-0.01\) using
```
meankondo -C config | numkondo -N 100 -I "0:-0.01"
```

In this section, we discuss how the models that **meankondo** can process are defined. A model is specified by a collection of *fields*, and a *propagator* between pairs of fields.

Fields are the elementary objects in terms of which a model is defined. Fields can be one of

**internal**: which are organized in pairs, and are denoted by \((\psi_i^+,\psi_i^-)\) for \(i\in\{1,\cdots,I\}\).**external**: which are organized in pairs, and are denoted by \((\Psi_i^+,\Psi_i^-)\) for \(i\in\{1,\cdots,E\}\).**super-external**: which denoted by \(H_i\) for \(i\in\{1,\cdots,X\}\) (the only difference with external fields is that super-external fields are not in pairs, which is a seemingly innocuous difference; but super-external fields are meant to be used for different purposes as external fields (see Definition below)).

`#!fields`

entry in the configuration file.
In the configuration file of the **meankondo** program, the fields are specified in the `#!fields`

entry.

Given a collection of fields, a *model* is specified as a recipe for computing the *average* of a polynomial of fields. We will use the following notation: given a polynomial \(F(\psi,\Psi,H)\) in the fields, its average will be denoted by \(\langle F(\psi,\Psi,H)\rangle\). The average is a linear operation, and, as indicated by the name "internal", only acts on internal fields, so that the external and super-external fields are viewed as constants for the averaging operation and can be factored out. It therefore suffices to define the average of a monomial of internal fields.

First of all, a monomial of internal fields which does not contain the same number of \(\psi^+_i\) as \(\psi^-_j\) is \(0\). The average of a monomial of the form \(\psi_{i_1}^+\psi_{j_1}^-\cdots\psi_{i_n}^+\psi_{j_n}^-\) in which \(n\in\mathbb N\), \((i_1,\cdots,i_n)\in\{1,\cdots,I\}^n\) and \((j_1,\cdots,j_n)\in\{1,\cdots,I\}^n\), is computed using the *Wick rule*:
$$
\langle\psi_{i_1}^+\psi_{j_1}^-\cdots\psi_{i_n}^+\psi_{j_n}^-\rangle=
\sum_{\pi\in\mathcal S_n}(-1)^\pi\langle\psi_{i_1}^+\psi_{j_{\pi(1)}}^-\rangle\cdots\langle\psi_{i_n}^+\psi_{j_{\pi(N)}}^-\rangle
$$
in which \(\mathcal S_n\) denotes the set of permutations of \(\{1,\cdots,n\}\) and \((-1)^\pi\) denotes the signature of \(\pi\). Using the Wick rule, we can specify any average by defining the quadratic moments of the model, similarly to the moments of a 0-mean Gaussian measure. The collection of all quadratic moments of the form
$$\langle\psi_i^+\psi_j^-\rangle$$
with \((i,j)\in\{1,\cdots,I\}^2\) is called the *propagator* of the model.

In the configuration file of the **meankondo** program, the propagator is specified in the `#!propagator`

entry. Note that **meankondo** recognizes numeric propagators as well as symbolic ones.

In this section, we discuss what flow equations are, and how **meankondo** computes them.

We first discuss how a flow equation is defined from a renormalization group map. The discussion below is not, in any sense, precise, and is meant as a guiding idea to understand why **meankondo** does what it does in the way it does.

Consider a map, which we will call the *renormalization group* flow, of the form:
$$
V^{[m]}(\Phi^{[m]},H)\longmapsto V^{[m-1]}(\Phi^{[m-1]},H)=\frac1{C^{[m]}}\left(\left\langle\prod_{\nu=1}^k\left(1+V^{[m]}(\Phi_\nu^{[m]},H)\right)\right\rangle-1\right)
$$
where \(V^{[m]}\) anf \(V^{[m-1]}\) are polynomials with no constant term, \(C^{[m]}\in\mathbb R\setminus\{0\}\) is a constant, \(\langle\cdot\rangle\) is the average defined in Propagator, \(H\) is the collection of super-external fields, \(k=I/E\) which we assume to be an integer, \(\{\Phi^{[m]\pm}_{\nu,i}\}_{i\in\{1,\cdots,E\},\ k\in\{1,\cdots,k\}}\) and \(\{\Phi^{[m-1]\pm}_{i}\}_{i\in\{1,\cdots,E\}}\) are collections of combinations of the internal and external fields, of the form
$$
\Phi^{[m]\pm}_{\nu,i}=\alpha_i\Psi_i^\pm+\psi_{j_\nu^{(i)}}^\pm,\quad
\Phi^{[m-1]\pm}_{i}=\Psi_i^\pm
$$
in which \(\alpha_i\in\mathbb R\setminus\{0\}\) is some *re-scaling* factor, and
$$
\bigcup_{i\in\{1,\cdots,E\}}^\circ\{j_1^{(i)},\cdots,j_k^{(i)}\}=\{1,\cdots,I\}
$$
in which the \(\circ\) over the union means it is disjoint.

If \(V^{[m-1]}\) can be written in the same form as \(V^{[m]}\), then the renormalization group map can be written as a finite system of equations: if there exist \(p\in\mathbb N\), \((\ell_1^{[m]},\cdots,\ell_p^{[m]})\in\mathbb C^p\), \((\ell_1^{[m-1]},\cdots,\ell_p^{[m-1]})\in\mathbb C^p\), and polynomials \((O_1(\Phi,H),\cdots,O_l(\Phi,H))\) such that
$$
V^{[m]}(\Phi_\nu^{[m]},H)=\sum_{n=1}^p\ell_n^{[m]}O_n(\Phi_\nu^{[m]},H),\quad
V^{[m-1]}(\Phi^{[m-1]},H)=\sum_{n=1}^p\ell_n^{[m-1]}O_n(\Phi^{[m-1]},H)
$$
then the renormalization group map can be written as a finite system of equations, called the *flow equation*, which maps
$$
(\ell_1^{[m]},\cdots,\ell_p^{[m]})\longmapsto(C^{[m]},\ell_1^{[m-1]},\cdots,\ell_1^{[m-1]})
$$
(we added the constant \(C^{[m]}\) which plays an important role). The collection \(\underline\ell\) is called the collection of *running coupling constants*.

We now discuss how **meankondo** computes flow equations. The two relevant configuration file entries are `#!input_polynomial`

and `#!id_table`

.

The entry `#!input_polynomial`

specifies a polynomial \(F(\underline\ell,\psi,\Psi,H)\) in the fields, as well as in a family of complex symbolic variables \(\underline\ell=(\ell_1,\cdots,\ell_p)\). In the context of the computation in Definition, it would be of the form
$$
\prod_{\nu=1}^k\left(1+\sum_{n=1}^p\ell_nO_n(\Phi_{\nu}^{[m]},H)\right).
$$

The entry `#!id_table`

specifies a sequence of \(p\) polynomials \((O_1(\Psi,H),\cdots,O_p(\Psi,H))\) in the external fields. In the context of the computation in Definition the \(n\)-th polynomial would be
$$
O_n(\Phi_{\nu}^{[m]},H)
$$
which specifies that \(O_n\) corresponds to \(\ell_n\).

**meankondo** first computes the average
$$
F'(\underline\ell,\Psi,H):=\langle F(\underline\ell,\psi,\Psi,H)\rangle
$$
and then *groups* \(F'\) according to the id table, that is it tries to put it in the form
$$
F'(\underline\ell,\Psi,H)=C(\underline\ell)\left(1+\sum_{n=1}^p\ell'_n(\underline\ell)O_n(\Psi,H)\right)
$$
in which \(C(\underline\ell)\) is a complex constant term and \(\ell'_n(\underline\ell)\) is some complex coefficient, which are both functions of \(\underline\ell\). Note that for Fermionic hierarchical models, \(\ell'_n\) is a rational function of \(\underline\ell\), and \(C\) is a polynomial in \(\underline\ell\). In addition, \(\ell'_n\) can be expressed as a polynomial in \((\underline\ell,C^{-1}(\underline\ell))\). The flow equation is then defined as
$$
\mathcal R:\underline\ell\longmapsto(C(\underline\ell),\ell'_1(\underline\ell),\cdots,\ell'_p(\underline\ell))
$$
If \(F'\) cannot be grouped, then **meankondo** exits with error code `-1`

.

In this section we describe the various operations on flow equations that the tools bundled with **meankondo** support.

Once a flow equation \(\mathcal R\) has been computed, it can be numerically evaluated by passing a vector \(\underline\ell\) of double precision floating point numbers to **meantools eval** or **numkondo** which computes \(\mathcal R(\underline\ell)\) in the former case, and \(\mathcal R^m(\underline\ell)\) for any \(m\in\mathbb N\) in the latter.

Numerical evaluation is handled in a straightforward manner, but for the following consideration. As was mentioned in Computation, \(\mathcal R\) is a polynomial in \((\underline\ell,C^{-1}(\underline\ell))\), and when evaluating \(\mathcal R(\underline\ell)\), **meankondo** first evaluates \(C\) and the computes \(\ell'_n(\underline\ell)\).

Oftentimes the renormalization group flow is expressed in terms of an exponential of an effective potential \(\exp(W)\), in which case the exponential must be computed before it can be processed by **meankondo**:
$$
\exp(W)=1+V.
$$
This is handled by **meantools exp**, which computes the running coupling constants appearing in \(V\) in terms of those in \(W\).

This feature was introduced to compute the susceptibility in the hierarchical Kondo model. In that case, some of the running coupling constants depend on the field, \(h\), and the susceptibility is expressed as a derivative of \(C(\underline\ell(h))\) with respect to \(h\). To that end, we wrote **meantools derive** to compute the derivatives of a flow equation with respect to an external variable.

The input of **meantools derive** consists in a flow equation and a collection of variables \(X\subset\{1,\cdots,p\}\). Each running coupling constant \(\ell_i\) for \(i\in X\) is assumed to depend on an external parameter, \(h\). The flow equation is then derived with respect to \(h\): for every \(n\in\{1,\cdots,p\}\), the derivative of \(\ell_n'(\underline\ell)\) with respect to \(h\) in terms of \(\partial_h\ell_i\) for \(i\in X\) is computed. It is then appended to the input flow equation.

The computation of the flow equation, as well as its exponentiation and derivation, are *exact* in the sense that they only involve operations on integers and are not subject to truncations. The coefficients appearing in the flow equation are therefore *exact*. This statement has one major caveat: integer operations are only correct as long as the integers involved are not too large. The precise meaning of "not too large" is system dependent. In the source code, integers relating to flow equation coefficients are declared with the `long int`

type, which, at least using the C library **meankondo** was tested with (that is `glibc 2.21`

), means integers are encoded on 64 bits on 64-bit systems and 32 bits on 32-bit systems. All operations are therefore exact as long as all integers are in \([-2^{31},2^{31}-1]\) on 64-bit systems and \([-2^{15},2^{15}-1]\) on 32-bit systems.

Numerical evaluations are not exact. The numbers manipulated **meankondo** are either "long doubles" or "MPFR floats", depending on the options passed to **numkondo** (see `man numkondo`

).

- Long doubles: when compiled for x86 processors, have a precision of 64 bits, which implies they are accurate to 19 decimal places; and the absolute value of doubles is bounded above by \(2^{16384}-2^{16384-64}\) (that is the number whose binary expansion has \(16383\) digits and whose \(64\) left-most digits are \(1\) whereas the others are \(0\)) and below by \(2^{-16382}\).
- MPFR floats: the precision and size of the exponent can be specified as options on the command line. The maximal precision and maximal value of the exponent are, on 64 bit systems, \(2^{63}\) bits and \(2^{62}\) respectively.

**meankondo** was written by Ian Jauslin, in the context of a project in collaboration with Giuseppe Benfatto and Giovanni Gallavotti.