Restricted Path Integrals

In the work by Ceperley (1996,1991), it has been shown that one can evaluate the path integral by restricting the path to only specific positive contributions. One introduces a reference point ${\bf R}^*$ on the path that specifies the nodes of the density matrix, $\rho_F({\bf R},{\bf R}^*,t)=0$. A node-avoiding path for $0 < t \leq \beta$ neither touches nor crosses a node: $\rho_F({\bf R}(t),{\bf R}^*,t) \not= 0$. By restricting the integral to node-avoiding paths,

$$\rho_F({\bf R}^*, {\bf R}_{\beta};\beta) = \int \! {\bf d}{\bf R}_0 \: \rho_F({\bf R}_0, {\bf R}^* ; 0) \! \... ...\! \! \! \! \! \! \! \! \! \! \! \! \! {\bf d}{\bf R}_t \;\; e^{-S[{\bf R}_t] }$$ (85)

$$= \frac{1}{N!}\; \sum_{\mathcal{P}}\; (-1)^{\mathcal{P}} \! \! \! \... ...! \! \! \! \! \! \! \! \! \! \! {\bf d}{\bf R}_t \;\; e^{-S[{\bf R}_t] } \quad,$$ (86)

where $\Upsilon ({\bf R}^*)$ denotes the nodal restriction with respect to the reference point ${\bf R}^*$. The nodal restriction remains the same if any permutation ${\mathcal{P}}$ is applied to the reference point, which leads to $\Upsilon ({\bf R}^*) \equiv \Upsilon ({\mathcal{P}}{\bf R}^*)$ because $\rho_F({\bf R},{\bf R}^*;\beta)=0$ implies $\rho_F({\bf R},{\mathcal{P}} {\bf R}^*;\beta)=0$. Eq. 2.77 can now be written in the following alternative form,

$$\rho_F({\bf R}^*, {\bf R}_{\beta};\beta) = \frac{1}{N!}\; \sum_{\mathcal{P}}\; (-1)^{\mathcal{P}} \! \! \! \... ...\! \! \! \! \! \! \! \! \! \! \! \! \! {\bf d}{\bf R}_t \;\; e^{-S[{\bf R}_t] }$$ (87)

$$= \frac{1}{N!}\; \sum_{\mathcal{P}}\; (-1)^{\mathcal{P}} \! \! \! \... ...! \! \! \! \! \! \! \! \! \! \! {\bf d}{\bf R}_t \;\; e^{-S[{\bf R}_t] } \quad,$$ (88)

where we have applied the permutation ${\mathcal{P}}^{-1}$ to the entire path and changed the summation index using that the sign of ${\mathcal{P}}$ and ${\mathcal{P}}^{-1}$ are equal.

In the case of diagonal density matrix elements, Eq. 2.79 can be simplified because odd permutations inevitably cross a node since $\rho_F({\bf R}^*,{\bf R}^*;0) = -\rho_F({\bf R}^*,{\mathcal{P}}_{\rm odd}{\bf R}^*;0)$. Eq. 2.79 then reads,

$$\rho_F({\bf R}, {\bf R};\beta) = \frac{1}{N!}\; \sum_{{\math... ...! \! \! \! \! {\bf d}{\bf R}_t \;\; e^{-S[{\bf R}_t] }\quad.$$ (89)

For off-diagonal density matrix elements however, odd permutations need to be considered and lead to negative contributions, which will be discussed in chapter 5.

Since all contributions to the diagonal density matrix elements are positive the restricted PIMC technique represents, in principle, a solution to the sign problem. The method is exact if the exact fermionic density matrix is used in the restriction. The proof given by Ceperley (1996) consists of three steps.

(i)

The initial condition for the Bloch equation 2.13 are given by,

$$\rho_F({\bf R},{\bf R}^*;0) = \frac{1}{N!}\; \sum_{{\mathcal... ...\; (-1)^{\mathcal{P}}\; \delta({\bf R}-{\mathcal{P}}{\bf R}^*)$$ (90)

and $\rho({\bf R}^*,{\bf R}^*;0) \ge 0$. ${\bf R}^*$ and therefore the initial conditions are kept fixed for the following arguments. The solution of the Bloch equation is uniquely determined by the boundary conditions, which means $\rho_F({\bf R},{\bf R}';\beta)$ can be derived from the values on a certain boundary $\Upsilon({\bf R};\beta')$ for all $\beta' < \beta$.

(ii)

The nodes of $\rho_F({\bf R},{\bf R}^*;\beta)$ carve the space-time into a finite number ($\le N!$) of nodal cells, that are sets of points in the space-time connected by node-avoiding paths. From (i), it follows that the solution inside each nodal cell can be constructed from the initial condition and the zero boundary condition on the surface, which is determined by the nodes.

(iii)

Enforcing zero boundary conditions at the nodes can be done by introducing a infinite repulsive potential on the nodes, which prevents any paths from crossing and therefore guarantees that the density matrix vanishes at the cell boundaries.