The eigSUMR inverter for overlap fermions

Cundy, Nigel,Lee, Weonjong Elsevier 2016 Computer physics communications Vol.203 No.-

<P><B>Abstract</B></P> <P>We discuss the usage and applicability of deflation methods for the overlap lattice Dirac operator, focusing on calculating the eigenvalues using a method similar to the eigCG algorithm used for other Dirac operators. The overlap operator, which contains several theoretical advantages over other formulations of lattice Quantum Chromodynamics, is more computationally expensive because it requires the computation of the matrix sign function. The principal change made compared to deflation methods for other formulations of lattice QCD is that it is necessary for best performance to tune the accuracy of the matrix sign function as the computation proceeds. We present two possible relaxation strategies, one which provides a rigorous bound for the eigenvalues but seems to be too conservative in practice, and a second which is less conservative but, while its stability is not guaranteed, seems to work well in practice.</P> <P>We adapt the original eigCG algorithm for two of the preferred inversion algorithms for overlap fermions, GMRESR(relCG) and GMRESR(relSUMR). Before deflation, the rate of convergence of these routines in terms of iterations is similar, but, since the Shifted Unitary Minimal Residual (SUMR) algorithm only requires one call to the matrix sign function compared to the two calls required for Conjugate Gradient (CG), SUMR is usually preferred for single inversions of the Dirac operator. We construct bounds for the required accuracy of the matrix sign function during the eigenvalue calculation. For the SUMR algorithm, we use a variant of the Galerkin projection to perform the deflation; while for the CG algorithm, we are able to use a considerably superior spectral pre-conditioner. The superior performance of the spectral pre-conditioner, and its need for less accurate eigenvalues, almost erodes SUMR’s advantage over CG as an inversion algorithm.</P> <P>We see factor of three gains for the inversion algorithm from the deflation on our small test lattices; we expect larger gains over the undeflated algorithms in realistic simulations on larger lattices and with smaller masses. There is, however, a significant cost in the eigenvalue calculation because we cannot relax the accuracy of the matrix sign function as aggressively when calculating the eigenvalues as we do while performing the inversions. This set-up cost is, however, more than compensated for the gain in the deflation if enough right hand sides are required.</P>

The static quark potential from the gauge invariant Abelian decomposition

Cundy, N.,Cho, Y.M.,Lee, W.,Leem, J. North-Holland Pub. Co 2014 Physics letters. Section B Vol.729 No.-

We investigate the relationship between colour confinement and topological structures derived from the gauge invariant Abelian (Cho-Duan-Ge) decomposition. This Abelian decomposition is made imposing an isometry on a colour field n which selects the Abelian direction; the principle novelty of our study is that we have defined this field in terms of the eigenvectors of the Wilson loop. This allows us to establish an equivalence between the path ordered integral of the non-Abelian gauge fields with an integral over an Abelian restricted gauge field which is tractable both theoretically and numerically in lattice QCD. By using Stokes' theorem, we can relate the Wilson loop in terms of a surface integral over a restricted field strength, and show that the restricted field strength may be dominated by topological structures, which occur when one of the parameters parametrising the colour field n winds itself around a non-analyticity in the colour field. If they exist, these objects will lead to an area law scaling for the Wilson loop and provide a mechanism for quark confinement. We search for these structures in quenched lattice QCD. We perform the Abelian decomposition, and find that the restricted field strength is dominated by peaks on the lattice. Wilson loops containing these peaks show a stronger area-Law and thus provide the dominant contribution to the string tension.

The static quark potential from the gauge independent Abelian decomposition

Cundy, Nigel,Cho, Y.M.,Lee, Weonjong,Leem, Jaehoon Elsevier 2015 Nuclear Physics, Section B Vol.895 No.-

<P><B>Abstract</B></P> <P>We investigate the relationship between colour confinement and the gauge independent Cho–Duan–Ge Abelian decomposition. The decomposition is defined in terms of a colour field <I>n</I>; the principle novelty of our study is that we have used a unique definition of this field in terms of the eigenvectors of the Wilson Loop. This allows us to establish an equivalence between the path-ordered integral of the non-Abelian gauge fields and an integral over an Abelian restricted gauge field which is tractable both theoretically and numerically in lattice QCD. We circumvent path ordering without requiring an additional path integral. By using Stokes' theorem, we can compute the Wilson Loop in terms of a surface integral over a restricted field strength, and show that the restricted field strength may be dominated by certain structures, which occur when one of the quantities parametrising the colour field <I>n</I> winds itself around a non-analyticity in the colour field. If they exist, these structures will lead to an area law scaling for the Wilson Loop and provide a mechanism for quark confinement. Unlike most studies of confinement using the Abelian decomposition, we do not rely on a dual-Meissner effect to create the inter-quark potential.</P> <P>We search for these structures in quenched lattice QCD. We perform the Abelian decomposition, and compare the electric and magnetic fields with the patterns expected theoretically. We find that the restricted field strength is dominated by objects which may be peaks of a single lattice spacing in size or extended string-like lines of electromagnetic flux. The objects are not isolated monopoles, as they generate electric fields in addition to magnetic fields, and the fields are not spherically symmetric, but may be either caused by a monopole/anti-monopole condensate, some other types of topological objects, or a combination of these. Removing these peaks removes the area law scaling of the string tension, suggesting that they are responsible for confinement.</P>

A lattice Dirac operator for QCD with light dynamical quarks

Cundy, N.,Kennedy, A.D.,Schafer, A. North Holland 2011 Nuclear physics, B Vol.845 No.1

In QCD chiral symmetry is explicitly broken by quark masses, the effect of which can be described reliably by chiral perturbation theory. Effects of explicit chiral symmetry breaking by the lattice regularisation of the Dirac operator, typically parametrised by the residual mass, should be negligible for almost all observables if the residual mass of the Dirac operator is much smaller than the quark mass. However, maintaining a small residual mass becomes increasingly expensive as the quark mass decreases towards the physical value and the continuum limit is approached. We investigate the feasibility of using a new approximately chiral Dirac operator with a small residual mass as an alternative to overlap and domain wall fermions for lattice simulations. Our Dirac operator is constructed from a Zolotarev rational approximation for the matrix sign function that is optimal for bulk modes of the hermitian kernel Dirac operator but not for the low-lying parts of its spectrum. We test our operator on various 32<SUP>3</SUP>x64 lattices, comparing the residual mass and the performance of the Hybrid Monte Carlo algorithm at a similar lattice spacing and pion mass with a hyperbolic tangent operator as used by domain wall fermions. We find that our approximations have a significantly smaller residual mass than domain wall fermions at a similar computational cost, and still admit topological charge change.

Deconfinement transition in two-flavor lattice QCD with dynamical overlap fermions in an external magnetic field

Bornyakov, V. G.,Buividovich, P. V.,Cundy, N.,Kochetkov, O. A.,Schä,fer, A. American Physical Society 2014 PHYSICAL REVIEW D - Vol.90 No.3