[Paper Review] On Gauged Renormalisation Group Transformations of Lattice Fermions

Artan Boriçi|ArXiv.org|Apr 18, 2007

advanced mathematical theories4 citations

TL;DR

This paper proposes a novel gauged renormalisation group transformation for lattice fermions using the Schur complement of a block UL decomposition of the fine lattice Dirac operator, enabling exact chiral symmetry preservation via the Ginsparg-Wilson relation. The method allows for the first time a full multigrid algorithm in lattice QCD, with numerical tests showing convergence rates reduced from 0.89 to 0.48 at β=5.9 using a 3-level V-cycle preconditioner with type II inverse permutation.

ABSTRACT

We construct a hierarchy of lattice fermions, where the coarser lattice Dirac operator is the Schur complement of the block UL decomposition of the finer lattice operator. We show that the construction is an exact gauged renormalisation group transformation of the lattice action. In particular, using such a transformation and the QCDLAB tool, it is shown how to implement the Ginsparg-Wilson strategy for chiral fermions in the presence of a dynamical gauge field. The scheme allows, for the first time, a full multigrid algorithm for lattice quarks.

Motivation & Objective

• To develop an exact gauged renormalisation group transformation for lattice fermions that preserves chiral symmetry.
• To implement the Ginsparg-Wilson strategy for chiral fermions in the presence of dynamical gauge fields.
• To enable a full multigrid algorithm for lattice quarks by constructing a stable, sparsity-preserving Schur complement approximation.
• To test the effectiveness of Schur complement approximations in accelerating linear solvers and simulation algorithms in lattice QCD.

Proposed method

• The coarse lattice Dirac operator is defined as the Schur complement of the block UL decomposition of the fine lattice Dirac operator.
• The method uses gauge-covariant blocking kernels derived from the inverse of the $D_{bb}$ block to ensure consistency with the fermion measure.
• A permutation operator $P(p)$ is applied to reorder lattice sites so that coarse sites are labeled first, enabling efficient computation of the Schur complement.
• Four types of site permutations (I, II, and their inverses) are used to explore different Schur complement constructions and their spectral properties.
• A second-order Schur complement approximation is derived using diagonal and row-sum matrices of $D_{rr}$, preserving sparsity and stability.
• A 3-level V-cycle preconditioner is constructed using exact inversion of the Schur complement approximation and BiCGstab for $D_{rr}$, with convergence tested at multiple β values.

Experimental results

Research questions

• RQ1 Can the Schur complement of a block UL decomposition of the lattice Dirac operator serve as an exact gauged renormalisation group transformation preserving chiral symmetry?
• RQ2 Does the use of a Schur complement approximation maintain sufficient spectral stability to allow iterative multigrid application in lattice QCD?
• RQ3 Can the proposed method achieve significant acceleration of linear solvers and simulation algorithms in dynamical fermion simulations?
• RQ4 How do different site permutations (type I vs. II) affect the spectral properties and convergence of the Schur complement approximation?

Key findings

• The Schur complement of the block UL decomposition yields a coarse Dirac operator that satisfies the Ginsparg-Wilson relation, ensuring exact chiral symmetry in the effective action.
• The spectrum of the Schur complement exhibits an 'owl eyes' pattern, indicating improved chiral properties and potential for use as a kernel in domain wall or overlap fermions.
• The second-order Schur complement approximation preserves spectral clustering and maintains computational cost comparable to the original operator.
• The 3-level V-cycle preconditioner reduced the average convergence rate from 0.89 (unpreconditioned) to 0.48 at β=5.9, demonstrating robust performance near the chiral limit.
• The type II inverse permutation provided superior convergence stability, especially at β=5.9, where the type I inverse permutation failed to converge.
• The method enables the first full multigrid algorithm for lattice quarks by combining exact Schur complement structure with stable, sparsity-preserving approximations.