Hubbard-Stratonovich transformation
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The Hubbard-Stratonovich (HS) transformation is an exact mathematical transformation, which allows to convert a particle theory into its respective field theory by linearizing the density operator in the many-body interaction term of the Hamiltonian and introducing a scalar auxiliary field. It is defined as (Baeurle 2002, Baeurle 2003)
![\exp \left\{ - \frac{a}{2} x^2 \right\} =
\sqrt{\frac{1}{2 \pi a}} \; \int\limits_{-\infty}^{\infty}
\exp \left[ - \frac{y^2}{2 a} - i x y \right] d y,](../../../../math/3/1/1/311c476206dded34a20d49f3c1ff904b.png)
where the real constant a > 0. The basic idea of the HS transformation is to reformulate a system of particles interacting through two-body potentials into a system of independent particles interacting with a fluctuating field. The procedure is widely used in polymer physics (Baeurle 2007, Schmid 1998, Matsen 2002, Fredrickson 2002), classical particle physics (Baeurle 2002a, Baeurle 2004) and electronic structure theory (Rom 1997, Baer 1998).
[edit] Calculation of resulting field theories
The resulting field theories are well-suited for the application of effective approximation techniques, like the mean field approximation (Matsen 2002, Fredrickson 2002) or beyond mean field approximation procedures (Baeurle 2007, Baeurle 2006, Baeurle 2007a). A major difficulty arising in the simulation with such field theories is their highly oscillatory nature in case of strong interactions, which leads to the well-known numerical sign problem (Baeurle 2002, Baeurle 2003). The problem originates from the repulsive part of the interaction potential, which implicates the introduction of the complex factor via the HS transformation. Several analytical and numerical techniques have been developed recently to alleviate the sign problem in Monte Carlo simulation in an efficient way (Baeurle 2002, Baeurle 2002a, Baeurle 2003a).
[edit] References
- 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. (2003). "Computation within the auxiliary field approach". J. Comput. Phys. 184: 540.
- Baeurle, S.A.; Nogovitsin, E.A. (2007). "Challenging scaling laws of flexible polyelectrolyte solutions with effective renormalization concepts". Polymer 48: 4883.
- Schmid, F. (1998). "Self-consistent-field theories for complex fluids". J. Phys.: Condens. Matter 10: 8105.
- Matsen, M.W. (2002). "The standard Gaussian model for block copolymer melts". J. Phys.: Condens. Matter 14: R21.
- Fredrickson, G.H.; Ganesan, V.; Drolet, F. (2002). "Field-Theoretic Computer Simulation Methods for Polymers and Complex Fluids". Macromolecules 35: 16.
- 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. (2004). "Grand canonical auxiliary field Monte Carlo: a new technique for simulating open systems at high density". Comput. Phys. Commun. 157: 201.
- Rom, N.; Charutz, D.M.; Neuhauser, D. (1997). "Shifted-contour auxiliary-field Monte Carlo: circumventing the sign difficulty for electronic-structure calculations". Chem. Phys. Lett. 270: 382.
- Baer, R.; Head-Gordon, M.; Neuhauser, D. (1998). "Shifted-contour auxiliary field Monte Carlo for ab initio electronic structure: Straddling the sign problem". J. Chem. Phys. 109: 6219.
- Baeurle, S.A.; Efimov, G.V.; Nogovitsin, E.A. (2006). "Calculating field theories beyond the mean-field level". Europhys. Lett. 75: 378.
- Baeurle, S.A.; Efimov, G.V.; Nogovitsin, E.A. (2007a). "Grand canonical investigations of prototypical polyelectrolyte models beyond the mean field level of approximation". Phys. Rev. E. 75: 011804.
- 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.
