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 (Blankenbecler 1981, Ceperley 1977) or classical problems (Baeurle 2004, Baeurle 2003, Baeurle 2002a).
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[edit] 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 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).
[edit] See also
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This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations. (November 2010) |
[edit] References
- Blankenbecler, R.; Scalapino, D. J. and Sugar, R. L. (1981). "Monte Carlo calculations of coupled boson-fermion systems. I". Physical Review D 24 (8): 2278. Bibcode 1981PhRvD..24.2278B. doi:10.1103/PhysRevD.24.2278.
- Ceperley, D.; Chester, G.V. and Kalos, M.H. (1977). "Monte Carlo simulation of a many-fermion study". Physical Review B 16 (7): 3081. Bibcode 1977PhRvB..16.3081C. doi:10.1103/PhysRevB.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 (3): 201. Bibcode 2004CoPhC.157..201B. doi:10.1016/j.comphy.2003.11.001.
- Baeurle, S.A. (2003). "Computation within the auxiliary field approach". J. Comput. Phys. 184 (2): 540. Bibcode 2003JCoPh.184..540B. doi:10.1016/S0021-9991(02)00036-0.
- 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 (7): 3027. Bibcode 2002JChPh.117.3027B. doi:10.1063/1.1488587.
- 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 (8): 080602. Bibcode 2002PhRvL..89h0602B. doi:10.1103/PhysRevLett.89.080602. PMID 12190451.
- 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 (2): 111. Bibcode 2003CoPhC.154..111B. doi:10.1016/S0010-4655(03)00284-4.
- Baer, R.; Head-Gordon, M. and Neuhauser, D. (1998). "Shifted-contour auxiliary field Monte Carlo for ab initio electronic structure: Straddling the sign problem". Journal of Chemical Physics 109 (15): 6219. Bibcode 1998JChPh.109.6219B. doi:10.1063/1.477300.
