It seems that the many-body fermionic system or many-body spin systems generically can have sign problems.

The sign problems occur in the fermionic matter: electrons, quarks, many-body/QFT include fermions.

The sign problems occur in the spin matter: Hubbard model.

(1) This problem seems to originate from the anti-commutative or non-commutative nature of fermionic operators $\{c_i,c_j\}=i \delta_{ij}$ and spin operators $[S_i,S_j]=2i\epsilon S_k$. Is this true?

In contrary, if we consider the bosonic model, in general, should that model have no sign problem by its nature? (Say bosonic degree of freedoms without spins.) Or is there any counter example?

(2) If we are able to absorb the anti-commutative or non-commutative nature of operators in a new formulation of the system with only a commutator or in a commutative nature, then we may be able to get rid of sign problem. Is this true? What are some solutions of sign-problem free models in the fermionic and spin matter? What are the ideas behind these sign-problem-free model?


2 Answers 2


I think it is more complicated than that. Most of the interesting sign problems occur in Fermionic models, but there are bosonic examples as well. Examples include pure gauge QCD at finite theta angle, or the charged Bose gas at finite chemical potential.

Also, there are sign free fermionic theories, like the attractive Hubbard model (or the repulsive model at half-flling).

  • $\begingroup$ Do you have any references for the bosonic examples? $\endgroup$
    – leongz
    Commented May 30, 2017 at 4:02
  • 1
    $\begingroup$ @ leongz arxiv.org/abs/0810.2089v2 $\endgroup$
    – Thomas
    Commented May 30, 2017 at 4:09

Today's article claims progress with the fermion sign problem.


The exchange antisymmetry between identical fermions gives rise to infamous fermion sign problem, e.g. both path integral Monte Carlo and path integral molecular dynamics could only give accurate result for only a few noninteracting fermions at low temperatures. By considering fictitious particles with a real parameter interpolating continuously between bosons and fermions, we use path integral molecular dynamics to propose a general strategy to solve fermion sign problem in polynomial time. We verify that our method can efficiently give accurate energy values for large fermion systems through a series of numerical experiments.


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