Numerical RG-time integration of the effective potential: Analysis and Benchmark

Abstract

We investigate the RG-time integration of the effective potential in the functional renormalization group in the presence of spontaneous symmetry breaking and its subsequent convexity restoration on the example of a scalar theory in $d=3$. The features of this setup are common to many physical models and our results are, therefore, directly applicable to a variety of situations. We provide exhaustive work-precision benchmarks and numerical stability analyses by considering the combination of different discrete formulations of the flow equation and a large collection of different algorithms. The results are explained by using the different components entering the RG-time integration process and the eigenvalue structure of the discrete system. Particularly, the combination of Rosenbrock methods, implicit multistep methods or certain (diagonally) implicit Runge-Kutta methods with exact or autodiff Jacobians proves to be very potent. Furthermore, a reformulation in a logarithmic variable circumvents issues related to the singularity bound in the flat regime of the potential.

arXiv comment22 pages, 13 figures

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• Conformal invariance and composite operators: A strategy for improving the derivative expansion of the nonperturbative renormalization group
• Universal and non-universal large deviations in critical systems
• Tunneling with physics-informed RG flows in the anharmonic oscillator
• Physics-informed operator flows and observables