Higher order and infinite Trotter-number extrapolations in path integral Monte Carlo

Abstract

Improvements beyond the primitive approximation in the path integral Monte Carlo method are explored both in a model problem and in real systems. Two different strategies are studied: The Richardson extrapolation on top of the path integral Monte Carlo data and the Takahashi-Imada action. The Richardson extrapolation, mainly combined with the primitive action, always reduces the number-of-beads dependence, helps in determining the approach to the dominant power law behavior, and all without additional computational cost. The Takahashi-Imada action has been tested in two hard-core interacting quantum liquids at low temperature. The results obtained show that the fourth-order behavior near the asymptote is conserved, and that the use of this improved action reduces the computing time with respect to the primitive approximation.

High order Chin actions in path integral Monte Carlo

“…Working directly on the exponential of the Hamiltonian, Takahashi and Imada 10 introduced in the action the double commutator [[V ,K],V ] and showed that the new algorithm (TIA) was of fourth order. As showed in previous work, 24 the TIA reduces significantly the number of beads to reach the asymptotic limit and therefore it can be very useful in quantum systems if the temperature is not very small. However, if one is interested on achieving lower temperatures, deep in the quantum regime, the TIA is still not accurate enough since the number of beads required is yet too large.…”

“…As in our previous work, 24 we have studied the accuracy of the CA in a fully many-body calculation, deep in the quantum regime, as it is liquid 4 He. We consider a bulk system at a density ρ = 0.02186Å −3 and at two temperatures, T = 5.1 and 0.8 K. The calculation is performed within a simulation box of 64 atoms with periodic boundary conditions and with an accurate Aziz potential.…”

“…From previous work 24 it is known that δ = 2 and 4 for PA and TIA, respectively. In the case of the CA approximation the departure from E 0 is effectively of sixth order δ = 6 in spite of the fact that CA is rigorously a fourth-order action.…”

“…The potential energy can be calculated from the difference between the total energy and the kinetic one (24) or by means of the general relation (23) with identical result,…”

Lattice stability and high-pressure melting mechanism of dense hydrogen up to 1.5 TPa

“…In the primitive approximation of path integral, the dependence of the integrated quantities such as the energy on the number of beads , is scaled as [41] = 0 + 2 −2 + 4 −4 + ⋯…”