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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,
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
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Burkhard Militzer
20030115