Multi-particle quantum systems within the Worldline Monte Carlo formalism

TL;DR

The paper tackles the challenge of computing ground-state properties for interacting multi-particle quantum systems by extending Worldline Monte Carlo (WMC) to the non-relativistic multi-particle path integral. It develops a general framework where each particle samples free Brownian worldlines and is coupled through a local interaction via a Wilson line, allowing ground-state energies to be extracted from the late-time behaviour of the propagator $K_n$. The authors validate the method on separable two-particle setups (e.g., harmonic oscillator) and non-separable cases (soft-Coulomb with external potentials), as well as 3D excition-like models and three-particle trions, and benchmark against analytical results or direct diagonalisation, reporting favorable scaling with dimensionality and particle number. They also discuss smoothing for singular potentials, computational efficiency, and the potential for extensions to relativistic systems and quantum field theory contexts, highlighting practical impact for exciton/trion physics and beyond.

Abstract

We extend the Worldline Monte Carlo approach to computationally simulating the Feynman path integral of non-relativistic multi-particle quantum-mechanical systems. We show how to generate an arbitrary number of worldlines distributed according to the (free) kinetic part of the multi-particle quantum dynamics and how to simulate interactions between worldlines in the ensemble. We test this formalism with two- and three-particle quantum mechanical systems, with both long range Coulomb-like interactions between the particles and external fields acting separately on the particles, in various spatial dimensionality. We extract accurate estimations of the ground state energy of these systems using the late-time behaviour of the propagator, validating our approach with numerically exact solutions obtained via straightforward diagonalisation of the Hamiltonian. Systematic benchmarking of the new approach, presented here for the first time, shows that the computational complexity of Wordline Monte Carlo scales more favourably with respect to standard numerical alternatives. The method, which is general, numerically exact, and computationally not intensive, can easily be generalised to relativistic systems.

Figures (14)

• Figure 1: WMC estimation for the logarithm of the kernel for two particles with harmonic interaction in 1D and 2D for two values of the parameter $d$ (see equation \ref{['eq:propHO']}). Calculations performed with $N_L=1000^2$ , $N_p=5000$, $\omega=1$ . Dots: WMC estimates with statistical error bars. Lines: analytic results (Eq. \ref{['eq:propHO']}).
• Figure 2: Ground state energy estimates for two particles in 1D and 2D with harmonic oscillator interaction with $\omega=1$ . Parameters of the WMC calculation are $N_L=500^2$ , $N_p=10000$. Parameters for diagonalisation: $N_{T}=300$, $N_{T}=300^{2}$ in 1D, 2D, respectively. Inset: zoom at $d=0.25$ in 2D.
• Figure 3: Ground state energy of two particles in 1D and 2D interacting via a soft-Coulomb interaction vs the cut-off parameter $d$. For WMC: $N_L=500^2$ , $N_p=10000$. For diagonalisation: $N_{T}=300$, $N_{T}=300^{2}$ in 1D, 2D, respectively. Insets: zoom at $d=1.5\;\&\;d=2.5$ in 2D.
• Figure 4: WMC estimates for the logarithm of the kernel for two particles with soft Coulomb interaction and external square potentials vs time $T$, in 1D and 2D, with their respective linear fit. Potential parameters: $L_{x}=L_{y}=1$, $V_{e}=-1$ and $V_{h}=1$. Parameters of the calculation: $N_L=5000^2$ , $N_p=20000$.
• Figure 5: Ground state energy of two particles in 1D with soft-Coulomb interaction and square or Gaussian external potentials vs cut-off parameter $d$. Potential parameters: $V_e=-1$, $V_h=1$, $L=1$ , $\lambda_e=\lambda_{h}=1$ . Numerical calculation parameters: $N_L=5000^2$, $N_{T}=900$ for square potential; $N_L=4480^2$, $N_{T}=300$ for Gaussian potential. $N_p=20000$ . Inset: zoom at $d=1.25$ for the 1D Gaussian potentials case.