Auxiliary field Monte Carlo

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Auxiliary field Monte Carlo is a method that allows the calculation, by use of Monte Carlo techniques, of averages of operators in many-body quantum mechanical problems.

The distinctive ingredient of this method is the fact that the interaction is expressed, by means of a Hubbard-Stratonovich transformation, in terms of an auxiliary external field. Once this transformation is performed, the many-body problem is reduced to the calculation of a sum over all possible configurations of such auxiliary field. In this sense, there is a trade off: instead of dealing with one very complicated self-interacting quantum mechanical 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 external field configurations is performed by sampling over the most important ones, with certain probability. A central question at that point is whether the probability measure associated with the field configurations is well defined, i.e., positive (one can always normalize later).