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Himaldrmann (talk | contribs) →Methods for reducing the sign problem: final run thru |
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In systems with a moderate sign problem, such as field theories at a sufficiently high temperature or in a sufficiently small volume, the sign problem is not too severe and useful results can be obtained by various methods, such as more carefully tuned reweighting, [[analytic continuation]] from imaginary <math>\mu</math> to real <math>\mu</math>, or [[Taylor series|Taylor expansion]] in powers of <math>\mu</math>.<ref name='Philipsen'/><ref>{{Cite journal |arxiv=hep-lat/0610116 |last1=Schmidt |first1=Christian |title=Lattice QCD at Finite Density |journal=Pos Lat |volume=021 |pages=21.1 |year=2006|doi=10.22323/1.032.0021 |bibcode=2006slft.confE..21S |s2cid=14890549 |doi-access=free }}</ref>
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There are various proposals for solving systems with a severe sign problem:
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* ''Contour deformation:'' The field space is complexified and the [[Contour integration|path integral contour]] is deformed from <math>R^N</math> to another <math>N</math>-dimensional manifold embedded in complex <math>C^N</math> space.<ref>{{Cite journal |last1=Alexandru |first1=Andrei |last2=Basar |first2=Gokce |last3=Bedaque |first3=Paulo |last4=Warrington |first4=Neill |year=2022 |title=Complex paths around the sign problem |journal=Reviews of Modern Physics |volume=94 |pages=015006 |arxiv=2007.05436 |doi=10.1103/RevModPhys.94.015006}}</ref>
* ''[[Meron (physics)|Meron]]-cluster algorithms:'' These achieve an exponential speed-up by decomposing the fermion world lines into clusters that contribute independently. Cluster algorithms have been developed for certain theories,<ref name='Wiese-cluster'>{{cite journal |doi=10.1103/PhysRevLett.83.3116 |arxiv=cond-mat/9902128 |bibcode=1999PhRvL..83.3116C |title=Meron-Cluster Solution of Fermion Sign Problems |journal=Physical Review Letters |volume=83 |issue=16 |pages=3116–3119 |year=1999 |last1=Chandrasekharan |first1=Shailesh |last2=Wiese |first2=Uwe-Jens|s2cid=119061060 }}</ref> but not for the [[Hubbard model]] of electrons, nor for [[Quantum chromodynamics|QCD]] ''i.e.'' the theory of quarks.
* ''[[Stochastic quantization]]:'' The sum over configurations is obtained as the equilibrium distribution of states explored by a complex [[Langevin equation]]. So far, the algorithm has been found to evade the sign problem in test models that have a sign problem but do not involve fermions.<ref>{{cite journal |doi=10.1103/PhysRevLett.102.131601 |pmid=19392346 |arxiv=0810.2089 |bibcode=2009PhRvL.102m1601A |title=Can Stochastic Quantization Evade the Sign Problem? The Relativistic Bose Gas at Finite Chemical Potential |journal=Physical Review Letters |volume=102 |issue=13 |pages=131601 |year=2009 |last1=Aarts |first1=Gert|s2cid=12719451 }}</ref>
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