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arXiv:2501.17618 · The Journal of Chemical Physics

Periodic boundary conditions for bosonic path integral molecular dynamics

Jacob Higer, Yotam M. Y. Feldman, Barak Hirshberg

Published Jul 8, 2025• 3 citations
arXivHTMLPDF

Abstract

We develop an algorithm for bosonic path integral molecular dynamics (PIMD) simulations with periodic boundary conditions (PBC) that scales quadratically with the number of particles. Path integral methods are powerful tools for simulating bosonic condensed phases, which exhibit fundamental physical phenomena such as Bose-Einstein condensation and superfluidity. Recently, we developed a quadratic-scaling algorithm for bosonic PIMD but employed an ad hoc treatment of PBC. Here, we rigorously enforce PBC in bosonic PIMD. This requires summing over the spring energies of all periodic images in the partition function, and a naïve implementation scales exponentially with system size. We present an algorithm for bosonic PIMD simulations of periodic systems that scales only quadratically. We benchmark our implementation on the free Bose gas and a model system of cold atoms in optical lattices. We also study an approximate treatment of PBC based on the minimum-image convention and derive a numerical criterion to determine when it is valid.

DOI
10.1063/5.0265906

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We develop an algorithm for bosonic path integral molecular dynamics (PIMD) simulations with periodic boundary conditions (PBC) that scales quadratically with the number of particles. Path integral methods are powerful tools for simulating bosonic condensed phases, which exhibit fundamental physical phenomena such as Bose-Einstein condensation and superfluidity. Recently, we developed a quadratic-scaling algorithm for bosonic PIMD but employed an ad hoc treatment of PBC. Here, we rigorously enforce PBC in bosonic PIMD. This requires summing over the spring energies of all periodic images in the partition function, and a naïve implementation scales exponentially with system size. We present an algorithm for bosonic PIMD simulations of periodic systems that scales only quadratically. We benchmark our implementation on the free Bose gas and a model system of cold atoms in optical lattices. We also study an approximate treatment of PBC based on the minimum-image convention and derive a numerical criterion to determine when it is valid.
Recently, fictitious identical particles have provided a promising way to overcome the fermion sign problem and have been used in path integral Monte Carlo to accurately simulate warm dense matter with up to 1000 electrons [T. Dornheim et al., J. Phys. Chem. Lett. 15, 1305 (2024)1948-718510.1021/acs.jpclett.3c03193]. The inclusion of fictitious identical particles in path integral molecular dynamics (PIMD) can provide another way to simulate fermion systems. In a recent paper [Y. M. Y. Feldman et al., J. Chem. Phys. 159, 154107 (2023)0021-960610.1063/5.0173749], Feldman and Hirshberg improved the recursive formula for PIMD of identical bosons, significantly reducing the computational complexity. In this paper, we extend this latest recursive formula for bosons to PIMD of fictitious identical particles to improve the efficiency of simulating fermion systems. We also provide the virial estimator for calculating energy of fictitious identical particles by using the new recursive technique, which can help suppress fluctuations in energy calculation. As an example, we use the quadratic scaling PIMD for fictitious identical particles to study the simulation of hundreds and even thousands of fermions in a two-dimensional periodic potential, in the hope of providing a simulation tool for the large-scale Fermi-Hubbard model and other strongly correlated fermion systems, such as the simulation of ultracold fermionic gases in optical lattices and the extended Fermi-Hubbard model using quantum dots.
We present a method for performing path integral molecular dynamics (PIMD) simulations for fermions and address its sign problem. PIMD simulations are widely used for studying many-body quantum systems at thermal equilibrium. However, they assume that the particles are distinguishable and neglect bosonic and fermionic exchange effects. Interacting fermions play a key role in many chemical and physical systems, such as electrons in quantum dots and ultracold trapped atoms. A direct sampling of the fermionic partition function is impossible using PIMD since its integrand is not positive definite. We show that PIMD simulations for fermions are feasible by employing our recently developed method for bosonic PIMD and reweighting the results to obtain fermionic expectation values. The approach is tested against path integral Monte Carlo (PIMC) simulations for up to seven electrons in a two-dimensional quantum dot for a range of interaction strengths. However, like PIMC, the method suffers from the sign problem at low temperatures. We propose a simple approach for alleviating it by simulating an auxiliary system with a larger average sign and obtaining an upper bound to the energy of the original system using the Bogoliubov inequality. This allows fermions to be studied at temperatures lower than would otherwise have been feasible using PIMD, as demonstrated in the case of a three-electron quantum dot. Our results extend the boundaries of PIMD simulations of fermions and will hopefully stimulate the development of new approaches for tackling the sign problem.
Path integral molecular dynamics (PIMD) has been successfully applied to perform simulations of large bosonic systems in a recent study [Hirshberg et al., Proc. Natl. Acad. Sci. U. S. A. 116, 21445 (2019)]. In this work, we extend PIMD techniques to study Green's function for bosonic systems. We demonstrate that the development of the original PIMD method enables us to calculate Green's function and extract momentum distribution from our simulations. We also apply our method to systems of identical interacting bosons to study Berezinskii-Kosterlitz-Thouless transition around its critical temperature.
The Lindblad equation determines the time evolution of the density operator of open quantum systems. While valid for any system size, its use is, in practice, restricted to prototype/surrogate models with the aim of tackling specific aspects of the overall quantum complexity of a multi-atomic system. Path integral molecular dynamics (PIMD) instead provides static and dynamical quantum statistical averages of physical observables for systems in equilibrium composed of up to thousands of atoms over timescales up to nanoseconds, under the condition that short-time quantum coherence is not relevant for the properties of interest. PIMD relies on the well-established technique of molecular dynamics (MD) with its associated classical trajectories. However, it cannot describe a direct time evolution of a system and its convergence to a stationary state in situations out of equilibrium. In this work, we analyze the link between the Lindblad equation and PIMD; specifically, we will discuss how PIMD can actually be used to calculate the time evolution of ensemble-averaged physical observables and their convergence to a stationary state for situations out of equilibrium, bypassing the need of explicitly solving the Lindblad equation. Yet, at the same time, the Lindblad equation and PIMD are linked to one another through a formal relation of equivalence, which provides an argument for the consistency of PIMD results, namely the positivity of the density operator at any time. A numerical study of a prototype system, which is of interest in chemical physics, will be used to showcase the method.
We provide a detailed description of the path-integral Monte Carlo worm algorithm used to exactly calculate the thermodynamics of Bose systems in the canonical ensemble. The algorithm is fully consistent with periodic boundary conditions, which are applied to simulate homogeneous phases of bulk systems, and it does not require any limitation in the length of the Monte Carlo moves realizing the sampling of the probability distribution function in the space of path configurations. The result is achieved by adopting a representation of the path coordinates where only the initial point of each path is inside the simulation box, the remaining ones being free to span the entire space. Detailed balance can thereby be ensured for any update of the path configurations without the ambiguity of the selection of the periodic image of the particles involved. We benchmark the algorithm using the non-interacting Bose gas model for which exact results for the partition function at finite number of particles can be derived. Convergence issues and the approach to the thermodynamic limit are also addressed for interacting systems of hard spheres in the regime of high density.
Modern implementations of path integral molecular dynamics (PIMD) simulations of distinguishable particles frequently make use of high order factorization schemes for the Boltzmann operator to expedite convergence of equilibrium averages. Among these methods is the generalized Suzuki-Chin factorization (GSF), which is accurate up to fourth order in the imaginary-time step. In this work, we show that the GSF decomposition of the Boltzmann operator is applicable to bosonic PIMD, and results in an improved convergence of estimators. In particular, we show that the recently developed quadratic scaling bosonic PIMD need not change when using the GSF. The GSF scheme is implemented as a re-weighting factor for observables, without affecting the sampling generated by the standard, second-order, primitive factorization. We study the effect of this factorization for bosons in a harmonic trap and a sinusoidal potential. We also assess the effectiveness of GSF in calculating fermionic expectation values for harmonically-trapped atoms. In all these cases, we find that the GSF speeds up convergence with the Trotter number by a factor of ∼2−4\sim 2-4∼2−4 across a wide temperature range, at only a modest computational cost.