Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory

Abstract

Lattice QCD with chiral fermions are extremely computationally expensive, but on the other hand provides an accurate tool for studying the physics of strong interactions. Since the truncated overlap variant of domain wall fermions are equivalent to overlap fermions in four dimensions at any lattice spacing, in this paper we have used domain wall fermions for our simulations. The physical information of lattice QCD theory is contained in quark propagators. In practice computing quark propagator in lattice is an inversion problem of the Dirac operator matrix representing this quarks. In order to develop fast inversion algorithms we have used overlap solvers in two dimensions. Lattice QED theory with U(1) group symmetry in two dimensional space-times dimensions has always been a testing ground for algorithms. By the other side, motivated by our previews work that the two-grid algorithm converge faster than the standard iterative methods for overlap inversion but not for all quark masses, we thought to test this idea in less dimensions such as U(1) gauge theory. Our main objective of this paper it is to implement and develop the idea of a two level algorithm in a new algorithm coded in QCDLAB. This implementation is presented in the preconditioned GMRESR algorithm, as our new contribution in QCDLAB package. The preconditioned part of our algorithm, different from the one of [18], is the approximation of the overlap operator with the truncated overlap operator with finite N3 dimension. We have tested it for 100 statistically independent configurations on 32 x 32 lattice background U(1) field at coupling constant β=1 and for different bare quark masses mq = [0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1]. We have compared the convergence history of the preconditioned GMRESR residual norm with another overlap inverter of QCDLAB as an optimal one, such as SHUMR. We have shown that our algorithm converges faster than SHUMR for different quark masses. Also, we have demonstrated that it saves more time for light quarks compared to SHUMR algorithm. Our algorithm is approximately independent from the quark mass. This is a key result in simulations with chiral fermions in lattice theories. By the other side, if we compare the results of [18] for quark mass 0.1 in SU(3), results that our chosen preconditioned saves a factor of 2 but in U(1). Our next step is to test this algorithm in SU(3) and to adopt it in parallel.

Fast algorithms for chiral fermions in 2 dimensions

“…We first proposed this algorithm in [15]. In this paper we tested it for different coupling constant on 64x64 lattice volume.…”

“…Thus, in order to develop fast algorithms, we use the truncated overlap variant of domain wall fermions in 2 + 1 dimensions with the extra finite dimension N 3 . Using this idea we have developed an algorithm called the preconditioned GMRESR in [15] which converges faster than optimal algorithms used in simulations with chiral fermions.In essence the preconditioned GMRESR follows two levels;…”

“…In [15] this algorithm is developed for a single coupling constant of the gauge field in 32x32 lattice volume. In this paper we test it for larger lattice volume 64x64.…”

Chiral Fermions Algorithms In Lattice QCD

“…The QED it is a good "environment" for testing new algorithms of QCD. In [13] we have developed a faster inversion algorithm used for chiral fermions, called the preconditioned GMRESR (Generalized Minimal Residual -Recursive) algorithm. The feature of our preconditioned part is that we used the relation between the overlap operator and the truncated overlap operator with finite extra dimension.…”