Non-renormalizability of the HMC algorithm

TL;DR

This paper demonstrates that the Hybrid Monte Carlo (HMC) algorithm, based on molecular dynamics, exhibits non-renormalizability due to non-local ultraviolet singularities, contrasting with Langevin-based algorithms.

Contribution

It reveals that the HMC algorithm's molecular dynamics component leads to non-renormalizable behavior in phi^4 theory, highlighting fundamental limitations.

Findings

HMC's molecular dynamics equations are hyperbolic.

Non-local ultraviolet singularities appear at one-loop order.

HMC differs from Langevin algorithms in renormalizability.

Abstract

In lattice field theory, renormalizable simulation algorithms are attractive, because their scaling behaviour as a function of the lattice spacing is predictable. Algorithms implementing the Langevin equation, for example, are known to be renormalizable if the simulated theory is. In this paper we show that the situation is different in the case of the molecular-dynamics evolution on which the HMC algorithm is based. More precisely, studying the phi^4 theory, we find that the hyperbolic character of the molecular-dynamics equations leads to non-local (and thus non-removable) ultraviolet singularities already at one-loop order of perturbation theory.