Numerical sign problem: Difference between revisions

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* [[Condensed matter physics]] — It prevents the numerical solution of systems with a high density of strongly correlated electrons, such as the [[Hubbard model]].<ref>{{cite journal |doi=10.1103/PhysRevB.41.9301 |pmid=9993272 |bibcode=1990PhRvB..41.9301L |title=Sign problem in the numerical simulation of many-electron systems |journal=Physical Review B |volume=41 |issue=13 |pages=9301–9307 |year=1990 |last1=Loh |first1=E. Y. |last2=Gubernatis |first2=J. E. |last3=Scalettar |first3=R. T. |last4=White |first4=S. R. |last5=Scalapino |first5=D. J. |last6=Sugar |first6=R. L.}}</ref>

* [[Nuclear physics]] — It prevents the ''[[ab initio]]'' calculation of properties of [[nuclear matter]] and hence limits our understanding of [[atomic nucleus|nuclei]] and [[neutron star]]s.

* [[Quantum field theory]] — It prevents the use of [[lattice QCD]]<ref>{{Cite journal |author=de Forcrand, Philippe |title=Simulating QCD at finite density |journal=Pos Lat |volume=010 |pages=010 |year=2010 |arxiv=1005.0539 |bibcode=2010arXiv1005.0539D}}</ref> to predict the phases and properties of [[quark matter]].<ref name='Philipsen'>{{cite journalbook |last=Philipsen |first=O. |yeartitle=2008Proceedings of VIIIth Conference Quark Confinement and the Hadron Spectrum — PoS(ConfinementVIII) |titlechapter=Lattice calculations at non- zero chemical potential: The QCD phase diagram |journalyear=Proceedings of Science2008 |volume=77 |pages=011 |doi=10.22323/1.077.0011|doi-access=free }}</ref> (In [[lattice field theory]], the problem is also known as the '''complex action problem'''<!--boldface per WP:R#PLA-->.)<ref>{{cite journal |doi=10.1103/PhysRevD.66.106008 |arxiv=hep-th/0108041 |bibcode=2002PhRvD..66j6008A |title=New approach to the complex-action problem and its application to a nonperturbative study of superstring theory |journal=Physical Review D |volume=66 |issue=10 |pages=106008 |year=2002 |last1=Anagnostopoulos |first1=K. N. |last2=Nishimura |first2=J.|s2cid=119384615 }}</ref>

==The sign problem in field theory==

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The sign problem is [[NP-hard]], implying that a full and generic solution of the sign problem would also solve all problems in the complexity class NP in polynomial time.<ref>{{Cite journal |arxiv=cond-mat/0408370 |doi=10.1103/PhysRevLett.94.170201 |pmid=15904269 |bibcode=2005PhRvL..94q0201T |title=Computational Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations |journal=Physical Review Letters |volume=94 |issue=17 |pages=170201 |year=2005 |last1=Troyer |first1=Matthias |last2=Wiese |first2=Uwe-Jens |s2cid=11394699 }}</ref> If (as is generally suspected) there are no polynomial-time solutions to NP problems (see [[P versus NP problem]]), then there is no ''generic'' solution to the sign problem. This leaves open the possibility that there may be solutions that work in specific cases, where the oscillations of the integrand have a structure that can be exploited to reduce the numerical errors.

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 journalbook |arxiv=hep-lat/0610116 |last1=Schmidt |first1=Christian |title=Proceedings of XXIVth International Symposium on Lattice QCDField atTheory Finite— DensityPoS(LAT2006) |journalchapter=PosLattice QCD at finite Latdensity |volume=021 |pages=21.1 |year=2006|doi=10.22323/1.032.0021 |bibcode=2006slft.confE..21S |s2cid=14890549 |doi-access=free }}</ref>

=== List: current approaches ===

There are various proposals for solving systems with a severe sign problem:

* ''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 |issue=1 |pages=015006 |arxiv=2007.05436 |doi=10.1103/RevModPhys.94.015006|bibcode=2022RvMP...94a5006A }}</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>