Auxiliary field Monte Carlo

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).

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. This causes that the many-body problem is reduced 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.

Reweighting procedure and numerical sign problem

It is here, as in other related methods, that Monte Carlo enters the game in the guise ofimportance sampling: the large sum over auxiliary field configurations is performed by sampling over the most important ones, with a certain probability. Since such fieldtheories generally possess a complex or non-positive semidefinite weight function, onehas to resort to a reweighting procedure, to get a positive definite reference distributionsuitable 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

* Quantum Monte Carlo

References

* cite journal
last = Blankenbecler
first = R.
coauthors = Scalapino, D. J. and Sugar, R. L.
title = Monte Carlo calculations of coupled boson-fermion systems. I
journal = Phys. Rev. D.
volume = 24
pages = 2278
year = 1981
url = http://link.aps.org/abstract/PRD/v24/p2278

* cite journal
url = http://link.aps.org/abstract/PRB/v16/p3081
last = Ceperley
first = D.
coauthors = Chester, G.V. and Kalos, M.H.
title = Monte Carlo simulation of a many-fermion study
journal = Phys. Rev. B
volume = 16
pages = 3081
year = 1977

* cite journal
url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ5-4BHK1JN-4&_user=616165&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=fe7a7f52aadb5375994112ee03cebefc
last = Baeurle
first = S.A.
title = Grand canonical auxiliary field Monte Carlo: a new technique for simulating open systems at high density
journal = Comput. Phys. Commun.
volume = 157
pages = 201
year = 2004

* cite journal
url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-47MKK8M-2&_user=616165&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=60e4b8712970cf7e07a14b05c5f0e609
last = Baeurle
first = S.A.
title = Computation within the auxiliary field approach
journal = J. Comput. Phys.
volume = 184
pages = 540
year = 2003

* cite journal |
url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000117000007003027000001&idtype=cvips&gifs=yes
last = Baeurle
first = S.A.
coauthors = Martonak, R.; Parrinello, M.
title = A field-theoretical approach to simulation in the classical canonical and grand canonical ensemble
journal = J. Chem. Phys.
volume = 117
pages = 3027
year = 2002a

* cite journal
url = http://prola.aps.org/abstract/PRL/v89/i8/e080602
last = Baeurle
first = S.A.
title = Method of Gaussian Equivalent Representation: A New Technique for Reducing the Sign Problem of Functional Integral Methods
journal = Phys. Rev. Lett.
volume = 89
pages = 080602
year = 2002

* cite journal |
url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ5-48XCNNY-3&_user=616165&_coverDate=08%2F01%2F2003&_alid=745709877&_rdoc=6&_fmt=high&_orig=search&_cdi=5301&_sort=d&_docanchor=&view=c&_ct=8&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=c5a0e9b03b74325c0dc7bf7d88406ecf
last = Baeurle
first = S.A.
title = The stationary phase auxiliary field Monte Carlo method: a new strategy for reducing the sign problem of auxiliary field methodologies
journal = Comput. Phys. Commun.
volume = 154
pages = 111
year = 2003a

External links

* [http://www-dick.chemie.uni-regensburg.de/group/stephan_baeurle/index.html Particle and Polymer Field Theory Group]