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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).
Reweighting procedure and numerical sign problemedit
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. In classical statistical physics, this probability is usually given by the (positive semi-definite) Boltzmann factor. 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 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).
Blankenbecler, R.; Scalapino, D. J.; 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.; 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): 201–206. 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): 540–558. 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): 111–120. Bibcode:2003CoPhC.154..111B. doi:10.1016/S0010-4655(03)00284-4.
Baer, R.; Head-Gordon, M.; 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.
Implementationsedit
ALF
QUEST
QMCPACK
External linksedit
Theory and Computation of Advanced Materials and Sensors Group
April 13, 2024
auxiliary, field, monte, carlo, this, article, relies, largely, entirely, single, source, relevant, discussion, found, talk, page, please, help, improve, this, article, introducing, citations, additional, sources, find, sources, news, newspapers, books, schola. This article relies largely or entirely on a single source Relevant discussion may be found on the talk page Please help improve this article by introducing citations to additional sources Find sources Auxiliary field Monte Carlo news newspapers books scholar JSTOR September 2010 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 Contents 1 Reweighting procedure and numerical sign problem 2 See also 3 References 4 Implementations 5 External linksReweighting procedure and numerical sign problem editThe 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 In classical statistical physics this probability is usually given by the positive semi definite Boltzmann factor 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 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 See also editQuantum Monte CarloThis article includes a list of references related reading or external links but its sources remain unclear because it lacks inline citations Please help improve this article by introducing more precise citations November 2010 Learn how and when to remove this template message References editBlankenbecler R Scalapino D J 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 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 3 201 206 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 2 540 558 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 2 111 120 Bibcode 2003CoPhC 154 111B doi 10 1016 S0010 4655 03 00284 4 Baer R Head Gordon M 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 Implementations editALF QUEST QMCPACKExternal links editTheory and Computation of Advanced Materials and Sensors Group Retrieved from https en wikipedia org w index php title Auxiliary field Monte Carlo amp oldid 1113902893, wikipedia, wiki, book, books, library,
