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 (Blankenbecler 1981, Ceperley 1977) or classical problems (Baeurle 2004, Baeurle 2003, Baeurle 2002a). Monte Carlo methods are a class of Computational Algorithms that rely on repeated Random sampling to compute their results Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons
The distinctive ingredient of this method is the fact that the interactions are decoupled by means of the application of the Hubbard-Stratonovich transformation, which permits to reformulate the many-body theory in terms of a scalar auxiliary field representation. The Hubbard-Stratonovich (HS transformation is an exact mathematical transformation, which allows to convert a particle theory into its respective In Physics, a field is a Physical quantity associated to each point of Spacetime. This causes that the many-body problem is reduced to the calculation of a sum or integral over all possible auxiliary field configurations. In Physics, and especially Quantum field theory, an auxiliary field is one whose equations of motion admit a single solution 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.
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Reweighting procedure and numerical sign problem
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. Probability is the likelihood or chance that something is the case or will happen 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 positive definite 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). In Physics, the numerical sign problem refers to the slow statistical convergence of a numerical integration procedure like the Metropolis Monte Carlo algorithm
See also
References
- Blankenbecler, R. Quantum Monte Carlo is a large class of computer algorithms that simulate Quantum systems with the idea of solving the Many-body problem. ; Scalapino, D. J. and Sugar, R. L. (1981). "Monte Carlo calculations of coupled boson-fermion systems. I". Phys. Rev. D. 24: 2278.
- Ceperley, D. ; Chester, G. V. and Kalos, M. H. (1977). "Monte Carlo simulation of a many-fermion study". Phys. Rev. B 16: 3081.
- Baeurle, S. A. (2004). "Grand canonical auxiliary field Monte Carlo: a new technique for simulating open systems at high density". Comput. Phys. Commun. 157: 201.
- Baeurle, S. A. (2003). "Computation within the auxiliary field approach". J. Comput. Phys. 184: 540.
- Baeurle, S. A. ; Martonak, R. ; Parrinello, M. (2002a). "A field-theoretical approach to simulation in the classical canonical and grand canonical ensemble". J. Chem. Phys. 117: 3027.
- Baeurle, S. A. (2002). "Method of Gaussian Equivalent Representation: A New Technique for Reducing the Sign Problem of Functional Integral Methods". Phys. Rev. Lett. 89: 080602.
- Baeurle, S. A. (2003a). "The stationary phase auxiliary field Monte Carlo method: a new strategy for reducing the sign problem of auxiliary field methodologies". Comput. Phys. Commun. 154: 111.
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