Fermion Sign Problem

In simulations of fermionic systems, one has to deal with an extra complication emerging from the cancellation of positive and negative contributions to the averages calculated from,

$$\left< {\mathcal{O}}\right> = \frac{ \sum_{\mathcal{P}}(-1)... ...f d}{\bf R}\: \rho({\bf R},{\mathcal{P}}{\bf R};\beta) }\quad.$$ (82)

Even permutations have positive signs and odd ones have negative signs. The magnitude of the contributions from non-identity permutations depends on the degeneracy of the systems, which can be discussed in terms of the parameter $n\Lambda^D$ or equivalently as the ratio of temperature to Fermi temperature $\theta=T/T_F$. Here, we compare to the Fermi energy $E_F=k_B T_F$ of an ideal quantum gas in 3 dimensions,

$$E_F = \lambda \; (6 \pi^2 n)^{2/3},$$ (83)

where $n$ is the density of particles in this particular spin state, which leads to

$$\theta^{-3}=\frac{9\pi}{16} \; (n\Lambda^3)^2 \quad.$$ (84)

If the temperature is of the order of the $T_F$ non-identity permutations or lower are important. Those also lead to a significant fraction of negative contributions to the enumerator as well as to the denominator in Eq. 2.73. The consequence are large fluctuations in the computed averages. This is known as the fermion sign problem. While Eq. 2.73 always leads to the exact answer it becomes numerically increasingly difficult to compute the averages at the point where the interesting fermionic effects start to occur. It was shown by Ceperley (1996) that the efficiency of the straightforward implementation scales like $e^{-2 \beta N f}$, where $f$ is the free energy difference per particle of a corresponding fermionic and bosonic system while $N$ is the number of particles.