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arXiv:2506.14977 · The Journal of Chemical Physics

Path integral Monte Carlo in the angular momentum basis for a chain of planar rotors

Estêvão V. B. de Oliveira, Muhammad Shaeer Moeed, Pierre–Nicholas Roy

Published Oct 9, 2025• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0284612

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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.
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.
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.
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.
In this work, we develop a theoretical description of the collective behavior of interacting dipolar planar rotors by using time independent perturbation theory and a small angle quadratic approximation. The ground state properties for both the ordered and disordered quantum phases of the system are directly calculated and analyzed. Time-independent perturbation theory is shown to be appropriate for the disordered phase. For the ordered phase, we construct a quadratic approximation based on the stable equilibrium configurations of the dipolar ordering; we show that the inclusion of the quartic terms from the expansion of the potential energy are essential to correct the shift in the energy spectrum due to quantization ambiguities. Numerical techniques such as Exact Diagonalization and Density Matrix Renormalization Group are used for the benchmark the quality of both approximations.
The ground state properties of the quantum anisotropic planar rotor (QAPR) model, which was constructed to describe the orientational ordering of homonuclear diatomic molecules on inert surfaces, are investigated theoretically using diffusion Monte Carlo. The implementation of the descendant weighing (DW) technique due to Casulleras and Boronat [Phys. Rev. B 52, 3654 (1995)] is used, for which an alternate derivation is presented, based on the path-integral representation of the imaginary time propagator. We calculate the order parameter and then perform finite size scaling in order to search for a critical reduced rotational constant Bc* at zero temperature. Our simulation results indicate that a critical rotational constant is at Bc*≈0.25. The behavior of the kinetic and potential energies show strong evidence for local, single-rotor tunneling as the driving mechanism for the phase transition. A Gaussian mean-field treatment is also presented, in which the most important mechanism is local, single-rotor tunneling. While quantitatively the mean-field phase transition is not in agreement with the simulation results, the energy curves show qualitative similarities. In both cases, the phase transition occurs at the point where the kinetic energy reaches a maximum as a function of the reduced rotational constant B*.
The orientational behavior of the quantum anisotropic planar rotor model is studied by means of the path-integral Monte Carlo method combined with finite size scaling. We compare a system where all angular momentum states are allowed, and one where the angular momentum is restricted to have even values. We find that a reentrant phase transition is present in the unrestricted case, where the system reenters a disordered phase upon cooling. Cumulants that are sensitive to the order of the transition indicate that the phase transition is second order. In the parameter range investigated in this paper, the even angular momentum system does not show reentrance.