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## The Reference Point

The FP nodes as well as the VDM nodes represent approximations to the exact fermion nodes that become increasingly accurate for high temperatures. There is a simple trick that uses two reference points instead of one and allows to enforce the nodes by taking the sign of the trial density matrix from rather than from .

The density matrix can be expressed in terms of the convolution equation,

 (93)

which can be interpreted as an integral over all pairs of paths, one going from to and a second one from to . Both fermion density matrices can evaluated using a restricted path integral with the same reference point . This requires the time argument to be zero at and to increase in both directions up to at and . Using the explicit form of in Eq. 2.77 the above equation becomes,
 (94) (95) (96)

where we have employed the equivalence of Eq. 2.77 and 2.79 and also the fact that the double sum over permutations be converted into a single one because the following path integral can be treated as two independent factors. This expression can be interpreted as a single path integral of the form . The paths start at , goes through the reference at the middle of the path, and ends at . The time argument to check the nodes gets chosen according to,
 (97)

which means one only needs to evaluate the trial density matrix up to . This time doubling procedure cannot be applied further without reintroducing the sign problem.

Next: Example: Nodes for Two Up: Fermion Nodes Previous: Trial Density Matrix   Contents
Burkhard Militzer 2003-01-15