Numerical methods for the QCD overlap operator IV: Hybrid Monte Carlo

Abstract

The computational costs of calculating the matrix sign function of the overlap operator together with fundamental numerical problems related to the discontinuity of the sign function in the kernel eigenvalues are the major obstacles towards simulations with dynamical overlap fermions using the Hybrid Monte Carlo algorithm. In a previous paper of the present series we introduced optimal numerical approximation of the sign function and have developed highly advanced preconditioning and relaxation techniques which speed up the the inversion of the overlap operator by nearly an order of magnitude.In this fourth paper of the series we construct an HMC algorithm for overlap fermions. We approximate the matrix sign function using the Zolotarev rational approximation, treating the smallest eigenvalues of the Wilson operator exactly within the fermionic force. Based on this we derive the fermionic force for the overlap operator. We explicitly solve the problem of the Dirac delta-function terms arising through zero crossings of eigenvalues of the Wilson operator. The main advantage of this scheme is that its energy violations scale better than O(∆τ 2 ) and thus are comparable with the violations of the standard leapfrog algorithm over the course of a trajectory. We explicitly prove that our algorithm satisfies reversibility and area conservation. We present test results from our algorithm on 4 4 , 6 4 , and 8 4 lattices.

Physics results from dynamical overlap fermion simulations

“…Dynamical overlap fermions have been attempted by several authors [16,17,18], in which the treatment of the topology tunneling is one of the main issues. Among them, our simulations have reached the level to produce broad physics results for the first time.…”

Mixed action simulations: approaching physical quark masses

“…[2,3,4,5]). One way to circumvent this problem is the mixed action approach: the ensemble production is performed with a relatively cheap fermion discretization and GW fermions are only used for calculations in the valence sector.…”

Avoiding ergodicity problems in lattice discretizations of the Hubbard model

“…As a result, algorithms which combine BSS with HMC updates are in general not ergodic. Note that similar ergodicity problems appear for overlap fermions in Lattice QCD, which has prompted the development of a class of HMC algorithms which can reflect from or refract through boundaries where det M [φ] changes sign [30][31][32]. However, such algorithms are rather costly computationally, which suggests that a more amenable approach is to find a lattice action free from ergodicity problems.…”