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 $\rho({\bf R},{\bf R}^*;\beta)$ rather than from $\rho({\bf R},{\bf R}^*;2\beta)$.

The density matrix $\rho({\bf R}_\beta,{\bf R}'_\beta;2\beta)$ can be expressed in terms of the convolution equation,

$$\rho_F({\bf R}_\beta,{\bf R}'_\beta;2\beta)= \int \! {\bf d... ...f R}^*;\beta) \; \rho_F({\bf R}'_\beta,{\bf R}^*;\beta) \quad,$$ (93)

which can be interpreted as an integral over all pairs of paths, one going from ${\bf R}^*$ to ${\bf R}_\beta$ and a second one from ${\bf R}^*$ to ${\bf R}'_\beta$. Both fermion density matrices can evaluated using a restricted path integral with the same reference point ${\bf R}^*$. This requires the time argument to be zero at ${\bf R}^*$ and to increase in both directions up to $\beta$ at ${\bf R}_\beta$ and ${\bf R}'_\beta$. Using the explicit form of $\rho_F$ in Eq. 2.77 the above equation becomes,

$$\rho({\bf R}_\beta,{\bf R}'_\beta;2\beta)\!\! = \frac{1}{{N!}^{\,2}} \int {\bf d}{\bf R}^* \sum_{{\mathcal{P}}{\m... ...\! \! \! \! \! \! \! \! \! \! \! \! \! {\bf d}{\bf R}_t \;\; e^{-S[{\bf R}_t] }$$ (94)

$$= \frac{1}{{N!}^{\,2}} \; \sum_{{\mathcal{P}}{\mathcal{P}}'} \; (-1... ...\! \! \! \! \! \! \! \! \! \! \! \! \! {\bf d}{\bf R}_t \;\; e^{-S[{\bf R}_t] }$$ (95)

$$= \frac{1}{{N!}} \; \sum_{{\mathcal{P}}} \; (-1)^{{\mathcal{P}}} \;... ...\! \! \! \! \! \! \! %\! \! \! {\bf d}{\bf R}_t \;\; e^{-S[{\bf R}_t] } \quad,$$ (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 $\rho_F({\bf R}_\beta,{\bf R}'_\beta;2 \beta)$. The paths start at ${\bf R}_\beta$, goes through the reference ${\bf R}^*$ at the middle of the path, and ends at ${\mathcal{P}}{\bf R}'_\beta$. The time argument to check the nodes gets chosen according to,

$$t_{\rm ref} = \left\{ \begin{array}{cl} t & {\rm {for}}~~... ...{\rm {for}}~~ \beta/2 \le t \le \beta\quad, \end{array}\right.$$ (97)

which means one only needs to evaluate the trial density matrix up to $\beta/2$. This time doubling procedure cannot be applied further without reintroducing the sign problem.