Metadynamics surfing on topology barriers: the CP N−1 case

Abstract

As one approaches the continuum limit, QCD systems, investigated via numerical simulations, remain trapped in sectors of field space with fixed topological charge. As a consequence the numerical studies of physical quantities may give biased results. The same is true in the case of two dimensional CP N −1 models. In this paper we show that metadynamics, when used to simulate CP N −1 , allows to address efficiently this problem. By studying CP 20 we show that we are able to reconstruct the free energy of the topological charge F (Q) and compute the topological susceptibility as a function of the coupling and of the volume. This is a very important physical quantity in studies of the dynamics of the θ vacuum and of the axion. This method can in principle be extended to QCD applications.

Axion phenomenology and θ-dependence from N f = 2 + 1 lattice QCD

“…Such an obstruction could be overcome by the development of new MonteCarlo algorithms. Proposals in this respect have been made in the past [77,78] and some new strategies have been put forward quite recently [79,80]. In view of the exploration of higher temperatures, one should also consider the inclusion of dynamical charm quarks.…”

Fighting topological freezing in the two-dimensional CPN−1 model

“…For N = 21 they went up to a correlation length ξ 2nd ≈ 4.2, where they find an integrated autocorrelation time of τ int,χt = 19000(3000) for the topological susceptibility. In the recent work [9], the metadynamics method was tested. As basic update algorithm the Hybrid-Monte-Carlo (HMC) algorithm [30] was used.…”

“…[7] with moderate success. More recently, "trivializing maps in the Hybrid Monte Carlo algorithm" [8] or the "Metadynamics" method [9] have been tested.…”

The topological properties of QCD at high temperature: problems and perspectives

“…A possibility that can be thought as a dynamical implementation of the multicanonical idea is metadynamics [43], in which an history dependent biasing potential is added to the system and updated in such a way that, as the simulation goes on, the biasing potential converges to the logarithm of the density of states. This idea has been successfully applied to the case of CP N models in [44] and we will discuss some more details of metadinamics in the next section, where the application to the QCD case will be analyzed. A more radical approach is to precisely estimate the density of states [45,46] and then compute observables without the need of a Monte Carlo.…”