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 ${\bf R}^*$. 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 $t_O=\frac{\beta}{2}$, which is illustrated in Fig. 5.1. This method has the advantage that the trial density matrix is only needed up $\frac{\beta}{2}$.

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 $(-1)^{\mathcal{P}}$ 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 $\rho_T({\bf R}(t),{\bf R}^*;t)>0$ where the reference point is ${\bf R}^*\equiv{\bf R}(0)$. 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 $t_{\mathcal{P}}$ described in section 2.5.4 to any time slice. If $t_{\mathcal{P}}$ is at the slice with the open ends, $t_O=t_{\mathcal{P}}$, all determinants must be positive as in the case of closed paths. If $t_{\mathcal{P}}$ is moved to a different time slice some rows in the determinants $\rho_{ij} = \rho_1({\bf r}_i,{\bf r}^*_j;\beta)$ on the way are switched because a permutation ${\mathcal{P}}$ is applied to the coordinates ${\bf r}_i$. For the slices between $t_{\mathcal{P}}$ and $t_O$, the determinants must have the sign $(-1)^{\mathcal{P}}$ in order to fulfill the nodal condition. This is illustrated in Fig. 5.1. This reasoning still holds if $t_{\mathcal{P}}$ is moved across the reference point slice because then the columns in $\rho_{ij}$ 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 $t_O$.

This puts a restriction on the trial permutations entering the permutation table from section 2.5.4 because certain permutations would inevitably be rejected when the nodes are checked.