Auxiliary field Monte Carlo
Encyclopedia
Auxiliary field Monte Carlo is a method that allows the calculation, by use of Monte Carlo technique
s, of averages of operators in many-body quantum mechanical
(Blankenbecler 1981, Ceperley 1977) or classical problems (Baeurle 2004, Baeurle 2003, Baeurle 2002a).
, 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 oscillatory nature of the weight function can lead to a bad statistical convergence
of the numerical integration
procedure. The problem is known as the numerical sign problem
and can be alleviated with analytical and numerical convergence acceleration procedures (Baeurle 2002, Baeurle 2003a).
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
s, of averages of operators in many-body quantum mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
(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 transformationHubbard-Stratonovich transformation
The Hubbard–Stratonovich transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard...
, which permits the reformulation of many-body theory
Many-body theory
The many-body theory is an area of physics which provides the framework for understanding the collective behavior of vast assemblies of interacting particles. In general terms, the many-body theory deals with effects that manifest themselves only in systems containing large numbers of constituents...
in terms of a scalar auxiliary field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
representation. This reduces the many-body problem
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 a large number of interacting particles. Microscopic here implies that quantum mechanics has to be used to provide an accurate description of the system...
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
Importance sampling
In statistics, importance sampling is a general technique for estimating properties of a particular distribution, while only having samples generated from a different distribution rather than the distribution of interest. It is related to Umbrella sampling in computational physics...
: the large sum over auxiliary field configurations is performed by sampling over the most important ones, with a certain probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
. 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 oscillatory nature of the weight function can lead to a bad statistical convergence
Convergence of random variables
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes...
of the numerical integration
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
procedure. The problem is known as the numerical sign problem
Numerical sign problem
The numerical sign problem refers to the difficulty 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 negative contributions to the integral...
and can be alleviated with analytical and numerical convergence acceleration procedures (Baeurle 2002, Baeurle 2003a).