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arXiv:1905.09053 · Proceedings of the National Academy of Sciences

Path integral molecular dynamics for bosons

Barak Hirshberg, Valerio Rizzi, Michele Parrinello

Published Oct 7, 2019• 86 citations
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Abstract

Trapped bosons exhibit fundamental physical phenomena and are at the core of emerging quantum technologies. We present a method for simulating bosons using path integral molecular dynamics. The main difficulty in performing such simulations is enumerating all ring-polymer configurations, which arise due to permutations of identical particles. We show that the potential and forces at each time step can be evaluated by using a recurrence relation which avoids enumerating all permutations, while providing the correct thermal expectation values. The resulting algorithm scales cubically with system size. The method is tested and applied to bosons in a 2-dimensional (2D) trap and agrees with analytical results and numerical diagonalization of the many-body Hamiltonian. An analysis of the role of exchange effects at different temperatures, through the relative probability of different ring-polymer configurations, is also presented.

DOI
10.1073/pnas.1913365116

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Trapped bosons exhibit fundamental physical phenomena and are at the core of emerging quantum technologies. We present a method for simulating bosons using path integral molecular dynamics. The main difficulty in performing such simulations is enumerating all ring-polymer configurations, which arise due to permutations of identical particles. We show that the potential and forces at each time step can be evaluated by using a recurrence relation which avoids enumerating all permutations, while providing the correct thermal expectation values. The resulting algorithm scales cubically with system size. The method is tested and applied to bosons in a 2-dimensional (2D) trap and agrees with analytical results and numerical diagonalization of the many-body Hamiltonian. An analysis of the role of exchange effects at different temperatures, through the relative probability of different ring-polymer configurations, is also presented.
In this article we develop a general method to numerically calculate physical properties for a system of anyons with path integral molecular dynamics. We provide a unified method to calculate the thermodynamics of identical bosons, fermions, and anyons. Our method is tested and applied to systems of anyons, bosons, and fermions in a two-dimensional harmonic trap. We also consider a method to calculate the energy for fermions as an application of the path integral molecular dynamics to simulate the anyon model.
Using imaginary time, quantum statistical mechanics of a finite-temperature system in thermodynamical equilibrium turns into a problem of classical ring polymers made of particle paths. We need to consider all possible ring polymers at a given temperature, which again is where Monte Carlo methods are effective, especially using a so-called worm algorithm. We show how paths can be sampled using Monte Carlo, and how the physical properties, such as energy and radial distribution, are computed from the paths. Identical particles can have their paths exchanged at will, and we see how boson superfluidity comes out of this picture. Finally, the stochastic series expansion method is introduced to show finite temperature Hubbard model can be simulated.
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We present a method for approximating real-time correlation functions and quantum transport coefficients of bosonic condensed phases. The direct evaluation of quantum real-time correlation functions in the path integral formulation is impossible due to a severe dynamical sign problem. Evaluating imaginary-time correlation functions and inverting them using analytic continuation is famously ill-posed. Ring polymer molecular dynamics (RPMD) provides an alternative approach resulting in accurate approximations for real-time correlation functions of condensed phases but neglects exchange effects. In this work, we develop a bosonic RPMD method, which is exact in the harmonic, short time, and high temperature limits. A critical enabling observation is that the Kubo-transformed correlation function for bosons is real only when the correlated observables are symmetric under exchange. We benchmark the method on harmonic and anharmonic model systems and then use it to directly obtain the Lieb-Liniger gas density-density real-time correlation functions at various temperatures and momentum transfers. This work enables the direct simulation of finite-temperature, real-time dynamics of large bosonic condensed phases using RPMD for the first time.
In this article we develop a general method to numerically calculate physical properties for a system of anyons with path integral molecular dynamics. We provide a unified method to calculate the thermodynamics of identical bosons, fermions and anyons. Our method is tested and applied to systems of anyons, bosons and fermions in a two-dimensional harmonic trap. We also consider a method to calculate the energy for fermions as an application of the path integral molecular dynamics to simulate the anyon model.
We build on the framework for studying systems containing spin-phonon coupling using molecular dynamics as the method for sampling the positions of atoms carrying spins, whose interaction is described by the Heisenberg model, and which we presented in a previous paper Srpak et al. [Phys. Rev. B 112, 205132 (2025)]. In applying the real-space path integral framework to the spin-phonon problem, neighboring imaginary time slices become coupled via overlap of eigenvector matrices propagating along the path integral, leading to interactions that go beyond the pure harmonic interactions between neighboring beads of the ring-polymers in Born-Oppenheimer-based path integral molecular dynamics (PIMD) simulations of quantum nuclei. We show how the use of a spin-force, defined as the negative gradient of the spin free energy, accounts for this type of coupling, valid for systems in thermal equilibrium. The spin force is computed using either ring-polymer exact diagonalization (for small systems) or stochastic series expansion (for large systems). We provide the technical details for these implementations. We present a neural network approach for this type of PIMD simulation, where we compute the spin force based on training data obtained from the ring-polymer stochastic series expansion method. Utilizing these methods, we compare them with the literature and demonstrate how one can obtain phase diagrams clearly showing how the thermal and nuclear quantum effects influence the stability of the spin-Peierls dimerized phase. We also show how machine learning makes it possible to apply the method to two dimensional systems.