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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,

(82) 
Even permutations have positive signs and odd ones have negative
signs. The magnitude of the contributions from nonidentity
permutations depends on the degeneracy of the systems, which can be
discussed in terms of the parameter or equivalently as
the ratio of temperature to Fermi temperature . Here, we
compare to the Fermi energy of an ideal quantum gas in 3
dimensions,

(83) 
where is the density of particles in this particular spin state,
which leads to

(84) 
If the temperature is of the order of the nonidentity
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
, where is the free energy difference per particle of a
corresponding fermionic and bosonic system while is the number of
particles.
Next: Restricted Path Integrals
Up: Fermion Nodes
Previous: Fermion Nodes
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Burkhard Militzer
20030115