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arXiv:2410.13633 · The Journal of Chemical Physics

Path integral Monte Carlo in a discrete variable representation with Gibbs sampling: Dipolar planar rotor chain

Wenxue Zhang, Muhammad Shaeer Moeed, Andrew Bright, Tobias Serwatka, Estêvão V. B. de Oliveira, Pierre–Nicholas Roy

Published Jan 2, 2025• 4 citations
arXivHTMLPDF

Abstract

In this work, we propose a path integral Monte Carlo approach based on discretized continuous degrees of freedom and rejection-free Gibbs sampling. The ground state properties of a chain of planar rotors with dipole-dipole interactions are used to illustrate the approach. Energetic and structural properties are computed and compared to exact diagonalization and numerical matrix multiplication for N ≤ 3 to assess the systematic Trotter factorization error convergence. For larger chains with up to N = 100 rotors, Density Matrix Renormalization Group calculations are used as a benchmark. We show that using Gibbs sampling is advantageous compared to traditional Metropolis-Hastings rejection importance sampling. Indeed, Gibbs sampling leads to lower variance and correlation in the computed observables.

DOI
10.1063/5.0244162

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In this work, we propose a path integral Monte Carlo approach based on discretized continuous degrees of freedom and rejection-free Gibbs sampling. The ground state properties of a chain of planar rotors with dipole-dipole interactions are used to illustrate the approach. Energetic and structural properties are computed and compared to exact diagonalization and numerical matrix multiplication for N ≤ 3 to assess the systematic Trotter factorization error convergence. For larger chains with up to N = 100 rotors, Density Matrix Renormalization Group calculations are used as a benchmark. We show that using Gibbs sampling is advantageous compared to traditional Metropolis-Hastings rejection importance sampling. Indeed, Gibbs sampling leads to lower variance and correlation in the computed observables.
We introduce a path integral Monte Carlo approach that uses the angular momentum representation for the description of interacting rotor systems. Such a choice of representation allows the calculation of momentum properties without having to break the paths. The discrete nature of the momentum basis also allows the use of rejection-free Gibbs sampling techniques. To illustrate the method, we study the collective behavior of N confined planar rotors with dipole-dipole interactions, a system known to exhibit a quantum phase transition from a disordered to an ordered state at zero temperature. Ground state properties are obtained using the path integral ground state method. We propose a Bond-Hamiltonian decomposition for the high temperature density matrix factorization of the imaginary time propagator. We show that cluster-loop type moves are necessary to overcome ergodicity issues and to achieve efficient Markov chain updates. Ground state energies and angular momentum properties are computed and compared with density matrix renormalization group benchmark results. In particular, the derivative of the kinetic energy with respect to the interaction strength estimator is presented as a successful order parameter for the detection of the quantum phase transition.
Typical path integral Monte Carlo approaches use the primitive approximation to compute the probability density for a given path. In this work, we develop the pair discrete variable representation (pair-DVR) approach to study molecular rotations. The pair propagator, which was initially introduced to study superfluidity in condensed helium, is naturally well-suited for systems interacting with a pairwise potential. Consequently, paths sampled using the pair action tend to be closer to the exact paths (compared to primitive Trotter paths) for such systems leading to convergence with less imaginary time steps. Our approach relies on using the pair factorization approach in conjunction with a discretized path integral ground state paradigm to study a chain of planar rotors interacting with a pairwise dipole interaction. We first use the Wigner-Kirkwood density expansion to analyze the asymptotics of the pair propagator in imaginary time. Then, we exhibit the utility of the pair factorization scheme via convergence studies comparing the pair and primitive propagators. Finally, we compute energetic and structural properties of this system including the correlation function and Binder ratio as functions of the coupling strength to examine the behavior of the pair-DVR method near criticality. The density matrix renormalization group results are used for benchmarking throughout.
This thesis describes several novel approaches in quantum molecular dynamics for obtaining properties of molecular systems in different regimes. We investigate ground state properties of chains of linear rotors with dipole–dipole interactions via the density matrix renormalization group (DMRG), by deriving the appropriate form of the interaction operator and implementing it in ITensor. This provides us with further evidence of a quantum phase transition in this system. We also improve the sampling of Gaussian mixture distributions for finite temperature path integral Monte Carlo (PIMC) of vibronic Hamiltonians. To do this, we replace random sampling by quasi-random sampling, and improve sampling distributions by optimizing their parameters. Finally, we introduce estimators and integrators for constrained free energy simulations in path integral molecular dynamics (PIMD). This method is applied to the study of a water dimer, for which we obtain a quantum potential of mean force.
We introduce a computationally simple algorithm for sampling open-chain distributions within the framework of imaginary-time Feynman path integration. The present method is based on the staging algorithm introduced by Pollock and Ceperley [Phys. Rev. B 30, 2555 (1984)] originally developed for computing position-dependent observables. Here, we sample off-diagonal elements of the density matrix, formulated as a distribution describing a linear polymer-like chain of beads, each connected via nearest-neighbor springs to calculate momentum-dependent quantities. This is achieved using a Monte Carlo scheme that ensures efficient and unbiased sampling of all beads along the chain from the free-particle distribution via a staging transformation; we refer to this approach as staging open path integral Monte Carlo (OPIMC). The proposed algorithm is straightforward to implement, as it only involves sampling Gaussian distributions through a transformation defined by a set of recursion relations, followed by a standard Metropolis acceptance/rejection step. The staging OPIMC method accurately reproduces end-to-end and momentum distributions for quantum systems ranging from coupled harmonic oscillators to liquid water.
We consider the problem of the extrapolation of path-integral Monte Carlo (PIMC) data to infinite Trotter number P. Finite-P data, being even functions of P, have high-P dependence that is generally well described by a quadratic fit, a0{\mathit{a}}_{0}a0​+a1{\mathit{a}}_{1}a1​${\mathit{P}}^{\mathrm{\ensuremath{-}}2}$, where a0{\mathit{a}}_{0}a0​ is the exact quantum value. However, in order to get convergence it is often necessary to run PIMC codes with rather high P values, which implies long computer times and larger statistical errors of the data. It is well known that also for harmonic systems the finite-P data are not exact; nevertheless, they can be easily calculated by Gaussian quadrature. Starting from this observation, we suggest an easy way to correct PIMC data for anharmonic systems in order to take into account the harmonic part exactly, with strong improvement of the extrapolation to P=\ensuremath{\infty}. Lower Trotter numbers are thus required, with the advantages of computer-time saving and much better accuracy of the extrapolated values, without any change in the PIMC code. In order to demonstrate the effectiveness of the approach, we report finite-P data processing for a single anharmonic particle, whose finite-P data are obtained by the matrix-squaring method, and for a chain of atoms with Morse interaction.
We calculate the second Rényi entanglement entropy for systems of interacting linear rotors in their ground state as a measure of entanglement for continuous rotational degrees of freedom. The entropy is defined in relation to the purity of a subsystem in a bipartite quantum system, and to compute it, we compare two sampling ensembles based on the path integral ground state (PIGS) formalism. This scheme centers on the replica trick and is aided by the ratio trick, both developed in this context by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)]. We study a system composed of linear quantum rotors on a lattice in one dimension, interacting via an anisotropic dipole-dipole potential. The ground state second Rényi entropies estimated by PIGS are benchmarked against those from the density matrix renormalization group for various interaction strengths and system sizes. We find that the entropy grows with an increase in interaction strength, and for large enough systems, it appears to plateau near log(2). We posit that the limiting case of many strongly interacting rotors behaves akin to a lattice of two-level particles in a cat state, in which one naturally finds an entanglement entropy of log(2).