Improvements in the Nodal Action

In the previous section, it has been shown how a nodal action can be used to predict the probability that paths cross the nodes between two time slices. The procedure has the advantage that it allows to employ a large time step, which can be crucial for simulation at low temperature with a high computational demand. However, there exists still an upper bound on the time step because the assumption of a planar nodal surface between two slices breaks down. The effects on the thermodynamic variables calculated from a simulation with too large a time step are especially drastic in attractive systems like hydrogen. There, the nodes realize the Pauli exclusion principle, which make matter stable. If it is not guaranteed the system will inevitably collapse at some point in time during a simulation as shown by Theilhaber and Alder (1991).

An example of node violations is shown in Fig. 2.3. All determinants at a time slice have a positive sign and the predicted distance to the nodes has a reasonable value greater than 0. However, if one interpolates the coordinates linearly between two slices and then calculates the determinant and the distance to the node one finds some points where the path crosses the node. This could not happen if the nodes were planar. This analysis shows the break down of the nodal action procedure described in the previous section for too large time steps. In a simulation of hydrogen, one finds that the pressure becomes unphysically low and even negative. Simultaneously, the system partially collapses.

Fig. 2.3 also reveals ways to improve the nodal action. One possibility is to study the classical path that connects the two slices ${\bf R}_i$ and ${\bf R}_{i+1}$ in order to predict violations of the nodal surfaces. We propose to use the function,

$$f({\bf R}(t);t) = \frac{ \rho({\bf R}(t),{\bf R}^*;t)}{\rho({\bf R}^*,{\bf R}^*;t)}$$ (112)

and to determine its value and its gradient at the two slices. Those are fit a third order polynomial and it will be checked if it goes through zero. The reason for dividing by the term $\rho ({\bf R}^*,{\bf R}^*;t)$ is that the magnitude of the density matrix changes considerably even with a small time interval. Checking for node violations on the classical path gives rise to an additional restriction for a proposed configuration. In order to derive a nodal action one needs to study an ensemble of the paths. Due to the lack of analytical solutions of the diffusion equation for this problem we suggest to use a set of randomly sampled semi-classical paths,

$${\bf R}(t) = (1-x) \: {\bf R}_i + x \: {\bf R}_{i+1} + \sum_... ... k} \quad\quad{\rm with} \quad\quad x=\frac{t}{\tau}-i \quad,$$ (113)

where $Q_k$ are $DN$ dimensional normal-mode vectors that have a Gaussian distribution. The simplest way is to use only the first mode and to determine the width of the Gaussian from the free particle density matrix. Practically, one would sample a number $Q_k$, constructed the corresponding semi-classical paths, perform the fit of $f$ to the polynomial along each paths and check if the node would be crossed. The fraction of node avoiding paths would then be used in a metropolis rejection step. This analysis does predict some of the node violations that could not be detected with previous method. However, a detailed analysis if it actually improves the efficiency compare to a simulation with a smaller time step remains to be done.

Figure 2.3: Demonstration of violations of the nodal surfaces for one configuration of a hydrogen simulation with too large a time step. The $\circ$ correspond to the time slice, at which the sign of the trial density matrix is checked. The solid lines display same properties on a classical path connecting the slices. The middle graphs shows the trial density matrix $\rho({\bf R}(t),{\bf R}^*;t)$ divided by $\rho({\bf R}^*,{\bf R}^*;t)$ vs. imaginary time $t$. The upper graph exhibits sign of $\rho$ and in the lower graph, the distance to the node from Eq. 2.94. The sign of this distance indicates, which side of the node the paths is on. The classical paths exhibits four node crossings the could not be predict using the nodal action from Eq. 2.93. This fact as well the middle graph demonstrates the nodes are not sufficiently planar in the time interval $\tau$.