The distribution of permutation cycles has significant effects on the thermodynamics properties of the studied system since they represent the fermionic character. Here, we will discuss how it changes with increasing degeneracy.
In Fig. 2.4, the probability distribution of permutation cycles of different length from PIMC simulations are shown. The normalization is given . We found that the fraction of -cycles is a good candidate to discuss the degree degeneracy. Beginning at in a non-degenerate system, it decreases with increasing degeneracy. Simultaneously, states with higher cycle lengths are populated. At first, odd cycles have a higher probability, while one finds an almost uniform distribution at high degeneracy. The reason is that the nodal surfaces prohibit even permutations, which means an even number of even cycles must occur simultaneously and more importantly within the distance of the order of the thermal de Broglie wave length. It can be shown from the determinant that isolated even cycles would violate the nodes. This is also the reason why systems with even numbers of particles cannot form one long chain ( in Fig. 2.4). At high degeneracy where the thermal de Broglie wave length is larger than the inter-particle spacing, the nodal constraint does not discriminate between even and odd cycles.
This observed cycle distribution with fermion nodes is very different
from what one expects from direct fermion methods, where one considers
the signs explicitly and does not use nodal surfaces, e.g. the cycle
distribution of a system of non-interacting fermions can be calculated
in the grand canonical ensemble (Feynman, 1972). One has to differentiate
between odd and even cycle lengths leading to positive and negative
contributions to the partition function. Incorporating the sign into
, it reads