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arXiv:2605.14687

Generalized Suzuki-Chin Factorization in Bosonic Path Integral Molecular Dynamics

Jacob Higer, Barak Hirshberg

Published May 14, 2026• 0 citations
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Abstract

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 $\sim 2-4$ across a wide temperature range, at only a modest computational cost.

DOI
10.48550/arxiv.2605.14687

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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.
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.
Accurately simulating long-time dynamics of many-body systems is a challenge in both classical and quantum computing due to the accumulation of Trotter errors. While low-order Trotter-Suzuki decompositions are straightforward to implement, their rapidly growing error limits access to long-time observables. We present a framework for constructing efficient high-order Trotter-Suzuki schemes by identifying their structure and directly optimizing their parameters over a high-dimensional space. This method enables the discovery of new schemes with significantly improved efficiency compared to traditional constructions, such as those by Suzuki and Yoshida. Based on the theoretical efficiency and practical performance, we recommend two novel highly efficient schemes at 4th4^{\textrm{th}}4th and 6th6^{\textrm{th}}6th order. We also demonstrate the effectiveness of these decompositions on the Heisenberg model and the quantum harmonic oscillator, and find that for a fixed final time they perform better across the computational cost. Even when using large time steps, they surpass established low-order schemes like the Leapfrog. Finally, we investigate the in-practice performance of different Trotter schemes and find the decompositions with more uniform coefficients tend to feature improved error accumulation over long times. We have included this observation into our choice of recommended schemes.
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.
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.