**Auxiliary field Monte Carlo** is a method that allows the calculation, by use of

Monte Carlo technique, of averages of operators in many-body

quantum mechanical (Blankenbecler 1981, Ceperley 1977) or classical problems (Baeurle 2004, Baeurle 2003, Baeurle 2002a).

## Reweighting procedure and numerical sign problem

The distinctive ingredient of "auxiliary field Monte Carlo" is the fact that the interactions are decoupled by means of the application of the

Hubbard-Stratonovich transformation, which permits the reformulation of

many-body theory in terms of a scalar auxiliary

field representation. This reduces the

many-body problem to the calculation of a sum or integral over all possible

auxiliary field configurations. In this sense, there is a trade-off: instead of dealing with one very complicated many-body problem, one faces the calculation of an infinite number of simple external-field problems.

It is here, as in other related methods, that Monte Carlo enters the game in the guise of

importance sampling: the large sum over auxiliary field configurations is performed by sampling over the most important ones, with a certain

probability. Since such field theories generally possess a complex or non-positive semidefinite weight function, one has to resort to a reweighting procedure, to get a strictly positive reference distribution suitable for Monte Carlo sampling. However, it is well-known that, in specific parameter ranges of the model under consideration, the...

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