The matrix sign function in lattice QCD

QCD (Quantum Chromodynamics) is the fundamental physical theory of the quarks as the constituents of matter. Lattice QCD is a discretization of QCD, making QCD tractable computationally where it cannot be handled analytically.

Recent advances in lattice QCD have put forward a discretization scheme which for the first time respects the fundamental chiral symmetry of QCD. This overlap fermion model results in operators using the sign function of the (symmetrized) Wilson fermion matrix $Q$ , a 4d cartesian nearest neighbour coupling matrix with stochastic coefficients.

In this talk, we focus on the main computational problem associated with overlap fermions: How efficiently solve the basic equation $$(I+\rho {\mathrm{\Gamma }}_5\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(Q))x=b$$ Here, ${\mathrm{\Gamma }}_5$ is a simple permutation matrix and $0<\rho \le 1$ is a regularization parameter. Our approach is to consider inner-outer iteration methods, where each inner iteration computes the action of $\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(Q)$ on a vector $y$. We propose several alternatives to compute this action, all based on projections on Krylov subspaces, and we prove some results on the accuracy of these approaches which can be verified computationally. We also discuss accuracy issues associated with the inner-outer paradigm with the purpose to minimize the overall computational effort.