A field-theoretic simulation is a numerical strategy to calculate structure and physical properties of a many-particle system within the framework of a statistical field theory, like e.g. a polymer field theory. A convenient possibility is to use Monte Carlo (MC) algorithms, to sample the full partition function integral expressed in field-theoretic representation. The procedure is then called the auxiliary field Monte Carlo method. However, it is well known that MC sampling in conjunction with the basic field-theoretic representation of the partition function integral, directly obtained via the Hubbard-Stratonovich transformation, is impracticable, due to the so-called numerical sign problem (Baeurle 2002, Fredrickson 2002). The difficulty is related to the complex and oscillatory nature of the resulting distribution function, which causes a bad statistical convergence of the ensemble averages of the desired structural and thermodynamic quantities. In such cases special analytical and numerical techniques are required to accelerate the statistical convergence of the field-theoretic simulation (Baeurle 2003, Baeurle 2003a, Baeurle 2004).
Shifted-contour Monte Carlo technique
Mean field representation
To make the field-theoretic methodology amenable for computation, Baeurle proposed to shift the contour of integration of the partition function integral through the homogeneous mean field (MF) solution using Cauchy's integral theorem, which provides its so-called mean-field representation. This strategy was previously successfully employed in field-theoretic electronic structure calculations (Rom 1997, Baer 1998). Baeurle could demonstrate that this technique provides a significant acceleration of the statistical convergence of the ensemble averages in the MC sampling procedure (Baeurle 2002).
Gaussian equivalent representation
In subsequent works Baeurle et al. (Baeurle 2002, Baeurle 2002a) applied the concept of tadpole renormalization, which originates from quantum field theory and leads to the Gaussian equivalent representation of the partition function integral, in conjunction with advanced MC techniques in the grand canonical ensemble. They could convincingly demonstrate that this strategy provides an additional boost in the statistical convergence of the desired ensemble averages (Baeurle 2002).
Alternative techniques
Other promising field-theoretic simulation techniques have been developed recently, but they either still lack the proof of correct statistical convergence, like e.g. the Complex Langevin method (Ganesan 2001), and/or still need to prove their effectiveness on systems, where multiple saddle points are important (Moreira 2003).
References
Baeurle, S.A. (2002). "Method of Gaussian Equivalent Representation: A New Technique for Reducing the Sign Problem of Functional Integral Methods". Physical Review Letters. 89 (8): 080602. Bibcode:2002PhRvL..89h0602B. doi:10.1103/PhysRevLett.89.080602. PMID 12190451.
Fredrickson, G.H.; Ganesan, V.; Drolet, F. (2002). "Field-Theoretic Computer Simulation Methods for Polymers and Complex Fluids" (PDF). Macromolecules. 35 (1): 16. Bibcode:2002MaMol..35...16F. doi:10.1021/ma011515t. Archived from the original (PDF) on 2005-09-02.
Baeurle, S.A. (2003). "Computation within the auxiliary field approach". Journal of Computational Physics. 184 (2): 540–558. Bibcode:2003JCoPh.184..540B. doi:10.1016/S0021-9991(02)00036-0.
Baeurle, S.A. (2003a). "The stationary phase auxiliary field Monte Carlo method: a new strategy for reducing the sign problem of auxiliary field methodologies". Computer Physics Communications. 154 (2): 111–120. Bibcode:2003CoPhC.154..111B. doi:10.1016/S0010-4655(03)00284-4.
Baeurle, S.A. (2004). "Grand canonical auxiliary field Monte Carlo: a new technique for simulating open systems at high density". Computer Physics Communications. 157 (3): 201–206. Bibcode:2004CoPhC.157..201B. doi:10.1016/j.comphy.2003.11.001.
Rom, N.; Charutz, D.M.; Neuhauser, D. (1997). "Shifted-contour auxiliary-field Monte Carlo: circumventing the sign difficulty for electronic-structure calculations". Chemical Physics Letters. 270 (3–4): 382. Bibcode:1997CPL...270..382R. doi:10.1016/S0009-2614(97)00370-9.
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.
Baeurle, S.A.; Martonak, R.; Parrinello, M. (2002a). "A field-theoretical approach to simulation in the classical canonical and grand canonical ensemble". Journal of Chemical Physics. 117 (7): 3027. Bibcode:2002JChPh.117.3027B. doi:10.1063/1.1488587.
Ganesan, V.; Fredrickson, G.H. (2001). "Field-theoretic polymer simulations". Europhysics Letters. 55 (6): 814. Bibcode:2001EL.....55..814G. doi:10.1209/epl/i2001-00353-8.
Moreira, A.G.; Baeurle, S.A.; Fredrickson, G.H. (2003). "Global Stationary Phase and the Sign Problem". Physical Review Letters. 91 (15): 150201. arXiv:physics/0304086. Bibcode:2003PhRvL..91o0201M. doi:10.1103/PhysRevLett.91.150201. PMID 14611450. S2CID 38324821.
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