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 Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of a ...

(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 The Hubbard–Stratonovich (HS) transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard. It is used to convert a particle theory into its resp ...

, which permits the reformulation of

many-body theory The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. Terminology ''Microscopic'' here implies that quantum mechanics has to be ...

in terms of a scalar auxiliary-

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

representation. This reduces the

many-body problem The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. Terminology ''Microscopic'' here implies that quantum mechanics has to be ...

to the calculation of a sum or integral over all possible

auxiliary-field In physics, and especially quantum field theory, an auxiliary field is one whose equations of motion admit a single solution. Therefore, the Lagrangian describing such a field A contains an algebraic quadratic term and an arbitrary linear term, whil ...

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 Importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally at ...

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

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

. In classical

statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

, this probability is usually given by the (positive semi-definite)

Boltzmann factor Factor (Latin, ) may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, such a factor is a resource used ...

. Similar factors arise also in quantum field theories; however, these can have indefinite sign (especially in the case of Fermions) or even be complex-valued, which precludes their direct interpretation as probabilities. In these cases, one has to resort to a reweighting procedure (i.e., interpret the absolute value as probability and multiply the sign or phase to the observable) 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 oscillatory nature of the weight function can lead to a bad statistical convergence of the

numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...

procedure. The problem is known as the

numerical sign problem In applied mathematics, the numerical sign problem is the problem of numerically evaluating the integral of a highly oscillatory function of a large number of variables. Numerical methods fail because of the near-cancellation of the positive and n ...

and can be alleviated with analytical and numerical

convergence acceleration Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen * "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four Wei ...

procedures (Baeurle 2002, Baeurle 2003a).

References

* * * * * * * *

Implementations

ALF

QUEST

QMCPACK

External links

{{DEFAULTSORT:Auxiliary Field Monte Carlo Monte Carlo methods Quantum Monte Carlo

Reweighting procedure and numerical sign problem

See also

References

Implementations

External links