Non-renormalizability of the HMC algorithm

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 φ 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.

Fourier acceleration, the HMC algorithm and renormalizability

“…This will join pairs of directed lines ending in a square into what, following the conventions of Ref. [7], we would represent as a single, directionless line, as suggested in Fig. 2.…”

Instanton dominance over αs at low momenta from lattice QCD simulations at Nf = 0, Nf = 2 + 1 and Nf = 2 + 1 + 1

“…While Wilson flow (WF) shares the capacity of eliminating the short-distance fluctuations with many other techniques such as cooling or smearing, it has solid theoretical foundations analyzed in [19]. We refer the reader to the original literature for all the details of the WF.…”

Lattice QCD on nonorientable manifolds

“…For comparison we simulate three streams at every set of parameters: one with periodic, one with open, and one with P-periodic boundaries. Our main observables are the topological charge Q and time slice averages of the topological charge and action densities QðtÞ and EðtÞ as defined in [4]. All are evaluated along the Wilson flow [27] at a flow time of w 2 0 .…”

“…The slow modes can be removed from the theory by changing the topology of the manifold M. The authors of [4] proposed to introduce an open boundary in one of the directions. This change in the topology of space-time also changes the topology of the gauge field configuration space, which becomes connected and Q is not restricted to an integer value anymore.…”

“…This eliminates the slow modes from the theory. Indeed, in [4] a drastic reduction of the autocorrelation time of the topological charge was observed. A disadvantage of this approach is the lack of the translational invariance in the open direction, which introduces boundary effects and decreases the effectively available space-time volume.…”