Tunneling with physics-informed RG flows in the anharmonic oscillator

We solve the anharmonic oscillator using physics-informed renormalisation group (PIRG) flows, focusing on the weak coupling regime dominated by instanton-induced tunnelling processes. We demonstrate that the instanton physics underlying the exponential suppression of the energy gap is already captured at the first order of the derivative expansion within the PIRG framework. The key advances of this work are the implementation of a ground-state expansion in the PIRG flow and the adoption of high-precision Galerkin-based numerics. Our determination of the decay constant, $ a_{} = 1.910(2) $, agrees quantitatively with the analytic value $ a_{} = 1.886 $, differing by only 1\% 1 % . While the present formulation relies on a heuristic mapping that will be systematically refined in future extensions of the PIRG approach, these results already highlight the remarkable ability of the method to capture non-perturbative quantum phenomena within relatively simple truncations.