Conceptual ideas of this package

Flexible implementation

In most MaxEnt codes (and all publicly available codes that we are aware of), the expressions for \(\chi^2\) and \(S\) are fixed and hard-coded in the program. Often, simplifications are performed that are only possible for the usual expressions for these quantities. In this code, in principle any (doubly derivable) expression for \(\chi^2\) and \(S\) can be used. The usual choices as well as choices suited for off-diagonal elements of matrix-valued spectral functions are already implemented, and it is very easy to swap out the functions that are used.

The implementation of the whole framework is so that it is highly flexible, allowing the user to change the individual building blocks (e.g., but not at all limited to, as mentioned above, the expressions for \(\chi^2\) and \(S\)). But even implementing new functions is possible for the user.

Nevertheless, there are sensible defaults for everything and the most common tasks can be carried out in a user-friendly way in a few lines of python.

Different ways of choosing alpha

Furthermore, the TRIQS/maxent provides the spectral functions for different ways of choosing \(\alpha\) at the same time, which provides the user valuable information when assessing the quality of the continuation.

The procedure is done in two steps:

• The first step is to perform the analytic continuation for a range of \(\alpha\) values. If selected by the user also the probability for each \(\alpha\) is calculated. This step is the computationally more demanding part.

• In the second step the solution for each way of choosing \(\alpha\) is then obtained by analyzing the full data set of the first step. The code ships with a variety of Analyzers, which perform this task. Of course, the user can also write their own analyzer for their preferred way of selecting the optimal \(\alpha\).

Continuation of off-diagonal elements

A main feature of this package is the continuation of off-diagonal elements which correspond to spectral functions which are not non-negative. The normal entropy term is not defined for negative spectral functions. To circumvent this we use the so-called PosNeg entropy to continue these elements of matrix-valued spectral functions. The flexibility of the implementation allows us to just swap out the expression for the entropy.

In principle, it is necessary to ensure a hermitian and positive semi-definite spectral function. However, this is only possible when all matrix elements are treated at the same time. The code only supports an elementwise continuation, but the full matrix-version will be released eventually.

Quality control

Along with the desired output (i.e., the spectral function), other quantities are returned by the program. Using these, it is possible to assess the quality and correctness of the result. Due to the ill-posed nature of the problem, it is not always straightforward to decide whether the features of the obtained spectral function are real or artefacts. Investigating this extra information helps to come to a conclusion.

The package also offers tools to visualize these quantities to encourage the user to actually have a look at them.