Auxiliary field Monte Carlo

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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