Lattice QCD on nonorientable manifolds

Abstract

A common problem in lattice QCD simulations on the torus is the extremely long autocorrelation time of the topological charge when one approaches the continuum limit. The reason is the suppressed tunneling between topological sectors. The problem can be circumvented by replacing the torus with a different manifold, so that the connectivity of the configuration space is changed. This can be achieved by using open boundary conditions on the fields, as proposed earlier. It has the side effect of breaking translational invariance strongly. Here we propose to use a nonorientable manifold and show how to define and simulate lattice QCD on it. We demonstrate in quenched simulations that this leads to a drastic reduction of the autocorrelation time. A feature of the new proposal is that translational invariance is preserved up to exponentially small corrections. A Dirac fermion on a nonorientable manifold poses a challenge to numerical simulations: the fermion determinant becomes complex. We propose two approaches to circumvent this problem.

FLAG Review 2019

“…In the absence of any knowledge on the ¯ i one would assume λ 0 0 λ 2 0 , and with this input Eqs. (108,109) suggest that the NLO contribution to |a 0 0 | is by a factor ∼ 9 larger than the NLO contribution to |a 2 0 |. The experimental numbers quoted before clearly support this view.…”

“…A solution to the problem consists in using open boundary conditions in time [105], instead of the more common antiperiodic ones. More recently two other approaches have been proposed, one based on a multiscale thermalization algorithm [106,107] and another based on defining QCD on a nonorientable manifold [108]. The problem is also touched upon in Sect.…”

“…In passing we note that an alternative version of Eqs. (108,109) is used in the literature, too. For instance Refs.…”

Ergodic sampling of the topological charge using the density of states

“…(19) So the N i can be analytically evaluated in terms of truncated Gaussian integrals or error functions. Using this expression in (17) and (18), we obtain the reweighting formula for fixed separation between Gaussians:…”

“…This dramatic slowing down of Q and the resulting very large autocorrelation times pose serious problems of ergodicity in simulations and cause a very slow convergence of physical observables such as hadron masses to their thermodynamic limit values [13,14]. Since the toroidal topology of the domain plays a crucial role in the freezeout, other setups have been proposed based for example on open boundary conditions in time [15] (used at large N in [16,17]) or on non-orientable manifolds [18]. While these setups are very promising, their properties and potential drawbacks such as the loss of translational invariance with open boundary conditions in time, need further study.…”