Variations on PIGS: Non-standard approaches for imaginary-time path integrals
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The second Rényi entropy has been used as a measure of entanglement in various model systems, including those on a lattice and in the continuum. The present work focuses on extending the existing ideas to measurement of entanglement in physically relevant systems, such as molecular clusters. We show that using the simple estimator with the regular Path Integral Ground State (PIGS) distribution is not effective, but a superior estimator exists so long as one has access to other configuration sectors. To this end, we implement the ability to explore different sectors in the Molecular Modelling Toolkit (MMTK) and use it to obtain the entanglement entropy for a test system of coupled harmonic oscillators. The Semiclassical Initial Value Representation (SC-IVR) method for real-time dynamics using the Herman-Kluk propagator is known to be an effective semiclassical method. In the present work, we combine this approximate real-time propagator with exact and approximate ground state wavefunctions in order to find ground state survival amplitudes. The necessary integrals are first performed numerically on a grid (which is feasible for only low-dimensional systems) and then stochastically using MMTK (which has applicability to high-dimensional systems). The stochastic approach is used to compare two estimators, and it is again demonstrated that better results are obtained in a specialized configuration sector.