Numerical sign problem: Difference between revisions

Revision as of 05:52, 15 August 2018

Revision as of 05:52, 15 August 2018

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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|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|year=2010|arxiv=1005.0539|bibcode=2010arXiv1005.0539D}}</ref> to predict the phases and properties of [[quark matter]].<ref name='Philipsen'>{{cite journal |last=Philipsen |first=O. |year=2008 |title=Lattice calculations at non-zero chemical potential: The QCD phase diagram |url=http://pos.sissa.it//archive/conferences/077/011/Confinement8_011.pdf PoS Confinement8 |journal=Proceedings of Science|pages=011 |doi=10.22323/1.077.0011}}</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|year=2002|last1=Anagnostopoulos|first1=K. N|last2=Nishimura|first2=J}}</ref>

==The sign problem in field theory==

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* 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}}</ref> but not for the Hubbard model of electrons, nor for [[Quantum chromodynamics|QCD]], 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}}</ref>

* Fixed node method. One fixes the ___location of nodes (zeros) of the multiparticle wavefunction, and uses Monte Carlo methods to obtain an estimate of the energy of the ground state, subject to that constraint.<ref>{{cite journal |doi=10.1103/PhysRevLett.72.2442|pmid=10055881|bibcode=1994PhRvL..72.2442V|title=Fixed-Node Quantum Monte Carlo Method for Lattice Fermions|journal=Physical Review Letters|volume=72|issue=15|pages=2442–2445|year=1994|last1=Van Bemmel|first1=H. J. M|last2=Ten Haaf|first2=D. F. B|last3=Van Saarloos|first3=W|last4=Van Leeuwen|first4=J. M. J|last5=An|first5=G|hdl=1887/5478|url=http://hdl.handle.net/1887/5478|format=Submitted manuscript}}</ref>

* Majorana algorithms. Using Majorana fermion representation to perform Hubbard-Stratenovich transformations can help to solve the fermion sign problem of a class of fermionic many-body models.<ref>{{cite journal |doi=10.1103/PhysRevB.91.241117 |arxiv=1408.2269|bibcode=2015PhRvB..91x1117L|title=Solving the fermion sign problem in quantum Monte Carlo simulations by Majorana representation|journal=Physical Review B|volume=91|issue=24|year=2015|last1=Li|first1=Zi-Xiang|last2=Jiang|first2=Yi-Fan|last3=Yao|first3=Hong}}</ref><ref>{{Cite journal |doi=10.1103/PhysRevLett.117.267002 |pmid=28059531|arxiv=1601.05780|bibcode=2016PhRvL.117z7002L|title=Majorana-Time-Reversal Symmetries: A Fundamental Principle for Sign-Problem-Free Quantum Monte&nbsp;Carlo Simulations|journal=Physical Review Letters|volume=117|issue=26|pages=267002|year=2016|last1=Li|first1=Zi-Xiang|last2=Jiang|first2=Yi-Fan|last3=Yao|first3=Hong}}</ref>