Stochastic many-body perturbation theory for high-order calculations

Abstract

High-order perturbative $\textit{ab initio}$ calculations are challenging due to the rapidly growing configuration space and the difficulty of assessing convergence. In this work, we introduce perturbation theory quantum Monte Carlo (PTQMC), a stochastic approach designed to compute high-order many-body perturbative corrections. By representing the perturbative wave function with random walkers in configuration space, PTQMC avoids the exponential scaling inherent to conventional constructions of high-rank excitation operators. Benchmark calculations for the Richardson pairing model demonstrate that PTQMC accurately reproduces exact many-body perturbation theory (MBPT) coefficients up to 16th order, even in strongly divergent regimes. We further show that combining PTQMC with series resummation techniques yields stable and precise energy estimates in cases where the straightforward perturbative series fails. Finally, we propose the effective number of configurations, $e^{S}$, as a global measure of perturbative wave-function complexity that can be directly extracted within PTQMC. We demonstrate that the saturation behavior of $e^{S}$ provides a more reliable indicator of the validity of perturbative expansions than energy convergence alone.

Figures (4)

• Figure 1: High-order MBPT calculations for Richardson pairing model.(a) The area corresponding to $-1.3<g<0.75$. (b) The area corresponding to $1.5<g<3.0$.
• Figure 2: Benchmarking PTQMC against exact MBPT calculations. (a) Eighth-order PTQMC correlation energy $E_{\mathrm{corr}}$ for the Richardson pairing model at $g=-1.2$, compared with the exact MBPT result. (b) Relative error of the PTQMC energy with respect to the exact MBPT value as a function of the number of walkers, demonstrating the expected statistical scaling $\propto N_\mathrm{w}^{-1/2}$. (c) PTQMC correlation energies computed up to 16th order over the coupling range $-1.2 < g < -0.76$, a regime where the conventional MBPT series is strongly divergent.
• Figure 3: Resummation of high-order PTQMC data using Padé approximation and comparison with exact value. (a) Correlation energy as a function of approximation order at $g=-1.1$. (b) Same as (a) but for $g=+3.0$. (c) Correlation energy as a function of coupling strength $g$ obtained from the resummed PTQMC results, compared with different kinds of non-perturbative calculations. The lower panel shows the deviation of each method from the exact FCI result.
• Figure 4: Effective number of configurations $e^{S}$ as a function of the coupling strength $g$ for different perturbative orders. The dark-shaded regions mark coupling intervals where $e^{S}$ exhibits divergence.