On Gauged Renormalisation Group Transformations of Lattice Fermions
aa r X i v : . [ h e p - l a t ] A p r On Gauged Renormalisation GroupTransformations of Lattice Fermions
Artan Boric¸i
University of TiranaDepartment of Physics, Faculty of Natural SciencesKing Zog I Boulevard, Tirana, [email protected]
Abstract
We construct a hierarchy of lattice fermions, where the coarser lattice Dirac operatoris the Schur complement of the block UL decomposition of the finer lattice operator. Weshow that the construction is an exact gauged renormalisation group transformation of thelattice action. In particular, using such a transformation and the QCDLAB tool, it is shownhow to implement the Ginsparg-Wilson strategy for chiral fermions in the presence of adynamical gauge field. The scheme allows, for the first time, a full multigrid algorithm forlattice quarks. . A renormalisation group transformation for quadratic actions, as it is the case of latticefermions, is a simple Gaussian integration, det ˜
D e − ¯ ψ b S bb ψ b = Z ¯ φφ e − ( ¯ ψ b − ¯ φ ¯ B ) D bb ( ψ b − Bφ ) − ¯ φDφ , where D and S bb are Dirac operators on the fine and coarse lattices, D bb is a naive Dirac operatoron the coarse lattice, B, ¯ B are blocking operators, and, by evaluating the right hand side, onecan show that ˜ D = D + ¯ BD bb B, S bb = D bb − D bb B ˜ D − ¯ BD bb . This general transformation can be made concrete by the following prescription:
1. Partition the Dirac operator as a 2x2 block operator in the form , D = D bb D br D rb D rr
2. Define the gauge covariant blocking kernels , B = (cid:18) D − bb D br (cid:19) , ¯ B = D rb D − bb . In this case, one can show that: det ˜ D = det D bb det D rr , S bb = D bb − D br D − rr D rb . On the other hand, a block UL decomposition of D gives D bb D br D rb D rr = I bb D br D − rr I rr S bb D rb D rr ,S bb being the Schur complement . Therefore, we may conclude that:i) the coarse lattice Dirac operator is the Schur complement S bb of the block UL decompo-sition of the fine lattice Dirac operator; 2i) the fermion measure of the original theory is det D = det S bb det D rr ;iii) if D is the continuum Dirac operator, the Ginsparg-Wilson relation reads [1]: { γ , S − bb } = { γ , D − bb } . In their paper, Ginsparg and Wilson note that “any h , in particular a fixed point h approached after many iterations of the block spin transformation is said to be chirally invariantif it satisfies” the relation. They conclude the paper by noting: “Finding a way to go aheadand actually gauge a symmetry present only in remnant form stands as a further challenge” .Since then, a lot of efforts have been devoted to solve the Ginsparg-Wilson relation: theoverlap fermion is an exact solution to this relation [2, 3], whereas domain wall fermions [4, 5]and classical fixed point actions [6] lead to approximate solutions.Since a fixed point approach in the presence of gauge fields will be a very complicatedfunction of the original fine lattice operator, we propose a hybrid approach: trading the Dirackernel of the one step gauged renormalisation group, or possibly its approximation, for thekernel of the overlap or domain wall fermions . With one step only, we retain simplicity andhope to go away from the Aoki phase. In that case, we would get domain wall fermions withsmaller residual mass and overlap fermions with better localisation properties.We test the idea using the QCDLAB tool [7] and the new functions,
Permutation block ,and
Schur complement . Permutation block returns a permutation operator P ( p ) ,which exchanges the rows of D , whereas Schur complement is used to compute S bb . Thepermutation p is such that the coarser lattice sites are labelled first, as is shown in Figure 1.We consider two types of permutation: In their notations the relation is { γ , h − } = { γ , α − } .
13 14 15 161 2 3 45 6 7 89 10 11 1213 14 15 16 Type I permutation
67 8 9 10 Figure 1:
Example of a site permutation on a 4x4 lattice. Coarser lattice sites are labelled by boldface font. • Type I permutation: block the lattice in all directions as shown in the upper panel ofFigure 1. • Type II permutation: block the lattice in all directions, one direction at a time, as shownin the lower panel of Figure 1.
Since to every permutation there is an inverse permutation, we have in all four types of permu-tations, p I , p II , p − I , p − II , the corresponding permutation operators being P I , P II , P TI , P TII .Using these permutations, we have computed four different Schur complements in a SU(3)background field using the plaquette action at β = 5 . on a lattice. At this coupling, thespectrum of the Wilson operator should have no spectral gap in the real axes [8]. In Figure2 we show the effect of one step renormalisation group transformation on the spectrum of theWilson-Dirac operator. 4 Figure 2:
The spectrum of the blocked Wilson-Dirac operator using various permutations of lattice sites asdescribed in the titles of the plots.
5n the case of type II inverse permutation, one can see the emergence of an “owl eyes” plot.If such a picture is generic for this particular permutation, we could define the shifted Schurcomplement to be the domain wall or overlap kernel: if c is the centre of the right “eye”, thekernel of the chiral operator will be cI − S bb . In this case, the benefits are threefold:1. better chiral properties/localisation of the domain wall/overlap operator;2. faster inversions of the kernel;3. better scaling properties of the lattice Dirac operator.However, this construction could be computationally expensive: in order to compute theSchur complement one has to invert the D rr matrix. In fact, as it will be clear below, a Schurcomplement approximation to the exact one will do. In principle, one can stay with the exact Schur complement if there is no need to iteratethe renormalisation group transformation. One exmaple is the domain decomposition approachof L¨uscher [9]. He treats exactly the Schur complement and gets excellent results for the two-level algorithm. However, if one wants to iterate the scheme to full multigrid one has to rely onsome approximation of the Schur complement. The reason is twofold: one would like to retainthe sparsity of the operator and keep the computational overhead under control.Earlier papers on Schur complement approximation have ignored the D rr matrix altogether[10]. Ignoring D rr has the advantage of conserving the sparsity pattern of the coarser operator.This is similar to Migdal-Kadanoff approximation, which according to Creutz, “its primarydrawback lies in the difficulty of assessing the severity of the approximations involved” [11].On the other hand, an exact Schur complement may not be practical. The compromise is toallow the appearance of one power of D rr . 6 Figure 3:
The spectrum of the Schur complement (circles) and its approximation (crosses).
The Schur complement approximation would be in the spirit of the renormalisation grouptransformation if the coarse lattice operator inherits basic properties of the fine lattice opera-tor. In case of the Wilson operator one would like to conserve the positive definiteness, thecovariance of the hopping matrix, and perhaps γ -Hermiticity. Reusken shows that employ-ing point-Gaussian elimination on a weakly diagonally dominant M-matrix, which is similar towhat we have, one gets a stable Schur complement approximation [12]. To the second order,his result reads: ˜ S bb = D bb − D br (cid:16) d − rr + ˜ d − rr − d − rr D rr ˜ d − rr (cid:17) D rb , where d rr is the diagonal of D rr and ˜ d rr is a diagonal matrix, its entries being the sum of D rr rows. Using this approximation, one can compare the properties of S bb and ˜ S bb : the spectrumof ˜ S bb is again an “owl eyes” plot, as in Figure 3, whereas the cost of multiplication by ˜ S bb isapproximately the same as the cost of multiplication by D . In fact, if there is a place to test the Schur complement approximation, this is the fullmultigrid algorithm. Inheritance or stability is of great importance here, as well as regularity,7 .8 1 1.2−0.2−0.100.10.2 Type I 0.8 1 1.2−0.2−0.100.10.2 Type II0.8 1 1.2−0.2−0.100.10.2 Inverse Type I 0.8 1 1.2−0.2−0.100.10.2 Inverse Type II
Figure 4:
The spectrum of ˜ S − bb S bb using variuos permutations of lattice sites as described in the titles of theplots. i.e. how close an approximation is to the exact Schur complement. Without regularity, it wouldbe impossible to iterate the two-grid algorithm to a full multigrid. From Figure 4 we see that,indeed, such an approximation gives a spectrum of ˜ S − bb S bb clustered around one. Moreover, weobserve that the type II and type II inverse permutations give excellent results.Using such approximations of the Schur complement, one can construct a cyclic reductionpreconditioner as in [12]. In our tests we used a 3-level V-cycle preconditioner: at the highestlevel, the Schur complement approximation is inverted exactly, whereas for the D rr inversionwe used vanilla BiCGstab at one percent accuracy. We solved three linear systems to − accuracy at κ = 1 / on three lattices at β = 5 . , . , . . The performance of the precondi-8 R (BiCGstab) 0.76 0.83 0.89 R (Preconditioned BiCGstab) 0.05 0.16 0.48Table 1: Performance of the 3-level V-cycle preconditioner on lattices for various couplings. tioner is measured using the average convergence rate, R , i.e. the average ratio of the next tothe current residual error. In the second row of Table 1 we give the estimated condition num-bers for each case. In the third and fourth row we show the results for the unpreconditionedand preconditioned BiCGstab algorithm. For the first two lattices we used the type I inversepermutation, whereas for the third one the type II inverse permutation. It should be noted thatthe preconditioned BiCGstab with the type I inverse permutation did not converge in the thirdcase. According to [9], the lattice at β = 5 . and bare quark mass κ = 0 . correspondsto m π = 320 MeV, which indicates that our pion mass is close to its physical value. Thisshows that, even that close to criticality, a type II inverse permutation will yield a sufficientapproximation to the Schur complement.Applications of this preconditioner in lattice QCD with Wilson fermions are obvious: • Acceleration of linear solvers: we didn’t make any effort to construct a cheap precon-ditioner or to optimise its parameters; our prime interest was to test its potential. Aproduction code should take this into account. • Acceleration of simulation algorithms: as it is already shown for DD-HMC algorithm[13], the fermion determinant factorisation using the exact Schur complement can effec-tively be employed to accelerate the simulation algorithm. A full multigrid preconditioner9ffers the flexibility to employ more than two factors.
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