Nodal Restriction for Open Paths

The restricted PIMC method leads to some additional questions on how to constrain the open paths, which will be addressed in this section. First of all, one has to decide where to put the open ends with respect to the reference point . Since we are going to use the double reference point method from section 2.6.4, the open ends consequently have to be located in slice , which is illustrated in Fig. 5.1. This method has the advantage that the trial density matrix is only needed up .

The most significant difference between fermionic PIMC simulations with
closed and open paths lies in the fact that *for open paths, odd
permutations do not necessarily cross the nodes*. They will be
included in the sampling and lead to *negative signs*. As pointed
out in section
2.6.2, the nodal constraint only
prevents negative contributions to diagonal density matrix
elements. For open paths, the sign
comes into a PIMC
simulation as an additional factor when the averages are computed.

The nodes are taken from the trial density matrix in
Eq. 2.82. They are enforced by the condition
where the reference point is
. This is used for closed as well as for open
paths. For closed paths, one can simply check if the signs of all the
determinants are positive. For open paths, the situation is more
complicated because one needs to be able to move the point of
permutation
described in section
2.5.4 to any time slice.
*If
is at the slice with the open ends,
, all
determinants must be positive as in the case of closed paths.* If
is moved to a different time slice some rows in the
determinants
on the way are
switched because a permutation is applied to the coordinates
. For the slices between
and , the determinants
must have the sign
in order to fulfill the nodal condition.
This is illustrated in Fig. 5.1. This reasoning
still holds if
is moved across the reference point slice
because then the columns in change as well.
These rules have consequences for the ways odd permutations can be introduced
into a system. There are two required conditions for such a move
in order to have at least the chance not to violate the nodes:

- It is impossible to permute an even number of closed path while keeping all other particle coordinates fixed.
- It is impossible to permute an open and a closed path in a move that does not change the slice with the open ends .