Computational Strategies in Lattice QCD
Lectures given at the Summer School on "Modern perspectives in lattice QCD", Les Houches, August 3-28, 2009
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arXiv subjects
Martin Lüscher
Publications and source records attributed to Martin Lüscher.
Trivializing maps, the Wilson flow and the HMC algorithm
In lattice gauge theory, there exist field transformations that map the theory to the trivial one, where the basic field variables are completely decoupled from one another. Such maps can be constructed systematically by integrating certain flow equations in field space. The construction is worked out in some detail and it is proposed to combine the Wilson flow (which generates approximately trivializing maps for the Wilson gauge action) with the HMC simulation algorithm in order to improve the efficiency of lattice QCD simulations.
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Chiral symmetry breaking and the Banks--Casher relation in lattice QCD with Wilson quarks
The Banks--Casher relation links the spontaneous breaking of chiral symmetry in QCD to the presence of a non-zero density of quark modes at the low end of the spectrum of the Dirac operator. Spectral observables like the number of modes in a given energy interval are renormalizable and can therefore be computed using the Wilson formulation of lattice QCD even though the latter violates chiral symmetry at energies on the order of the inverse lattice spacing. Using numerical simulations, we find (in two-flavour QCD) that the low quark modes do condense in the expected way. In particular, the chiral condensate can be accurately calculated simply by counting the low modes on large lattices. Other spectral observables can be considered as well and have a potentially wide range of uses.
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Fluctuations and reweighting of the quark determinant on large lattices
We propose to stabilise HMC simulations of lattice QCD with very light Wilson quarks by splitting the quark determinant into two factors and by treating the factor that includes the contribution of the low modes of the Dirac operator as a reweighting factor. In general, determinant reweighting becomes inefficient on large lattices, because the statistical fluctuations of quark determinants increase exponentially with the lattice volume. Random matrix theory and some numerical studies now suggest that the low-mode contribution to the determinant behaves differently, which allows factorisations to be devised that preserve the efficiency of the simulation on large lattices.
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Deflation acceleration of lattice QCD simulations
Close to the chiral limit, many calculations in numerical lattice QCD can potentially be accelerated using low-mode deflation techniques. In this paper it is shown that the recently introduced domain-decomposed deflation subspaces can be propagated along the field trajectories generated by the Hybrid Monte Carlo (HMC) algorithm with a modest effort. The quark forces that drive the simulation may then be computed using a deflation-accelerated solver for the lattice Dirac equation. As a consequence, the computer time required for the simulations is significantly reduced and an improved scaling behaviour of the simulation algorithm with respect to the quark mass is achieved.
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Local coherence and deflation of the low quark modes in lattice QCD
The spontaneous breaking of chiral symmetry in QCD is known to be linked to a non-zero density of eigenvalues of the massless Dirac operator near the origin. Numerical studies of two-flavour QCD now suggest that the low quark modes are locally coherent to a certain extent. As a consequence, the modes can be simultaneously deflated, using local projectors, with a total computational effort proportional to the lattice volume (rather than its square). Deflation has potentially many uses in lattice QCD. The technique is here worked out for the case of quark propagator calculations, where large speed-up factors and a flat scaling behaviour with respect to the quark mass are achieved.
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The Schrödinger functional in lattice QCD with exact chiral symmetry
Similarly to the interaction lagrangian, the possible boundary conditions in quantum field theories on space-time manifolds with boundaries are strongly constrained by the symmetry and scaling properties of the theory. Based on this general insight, a lattice formulation of the QCD Schrödinger functional is proposed for the case where the lattice Dirac operator in the bulk of the lattice coincides with the Neuberger--Dirac operator. The construction satisfies all basic requirements (locality, symmetries, hermiticity) and is suitable for numerical simulations.
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Lattice QCD with light Wilson quarks
Wilson's formulation of lattice QCD is attractive for many reasons, but perhaps mainly because of its simplicity and conceptual clarity. Numerical simulations of the Wilson theory (and of its improved versions) tend to be extremely demanding, however, to the extent that they rapidly become impractical at small quark masses. Recent advances in simulation algorithms now allow such simulations to be pushed to significantly smaller masses without having to compromise in other ways. Contact with chiral perturbation theory can thus be made and many physics questions can be addressed that previously appeared to be inaccessible.
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Schwarz-preconditioned HMC algorithm for two-flavour lattice QCD
The combination of a non-overlapping Schwarz preconditioner and the Hybrid Monte Carlo (HMC) algorithm is shown to yield an efficient simulation algorithm for two-flavour lattice QCD with Wilson quarks. Extensive tests are performed, on lattices of size up to 32x24x24x24, with lattice spacings a~0.08 fm and at bare current-quark masses as low as 21 MeV.
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String excitation energies in SU(N) gauge theories beyond the free-string approximation
In the presence of a static quark--antiquark pair, the spectrum of the low-lying states in SU($N$) gauge theories is discrete and likely to be described, at large quark separations $r$, by an effective string theory. The expansion of the excitation energies in powers of $1/r$, which derives from the latter, involves an increasing number of unknown couplings that characterize the string self-interactions. Using open--closed string duality, we show that the possible values of the couplings are constrained by a set of algebraic relations. In particular, the corrections of order $1/r^2$ must vanish, while the $1/r^3$ terms (which we work out for the few lowest levels) depend on a single adjustable coupling only.
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Topological effects in QCD and the problem of short-distance singularities
The topological susceptibility and the higher moments of the topological charge distribution in QCD are expressed through certain n-point functions of the scalar and pseudo-scalar quark densities at vanishing momenta, which are free of short-distance singularities. Since the normalization of the correlation functions is determined by the non-singlet chiral Ward identities, these formulae provide an unambiguous regularization-independent definition of the moments and thus of the charge distribution.
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Solution of the Dirac equation in lattice QCD using a domain decomposition method
Efficient algorithms for the solution of partial differential equations on parallel computers are often based on domain decomposition methods. Schwarz preconditioners combined with standard Krylov space solvers are widely used in this context, and such a combination is shown here to perform very well in the case of the Wilson--Dirac equation in lattice QCD. In particular, with respect to even-odd preconditioned solvers, the communication overhead is significantly reduced, which allows the computational work to be distributed over a large number of processors with only small parallelization losses.
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Lattice QCD in the epsilon-regime and random matrix theory
In the epsilon-regime of QCD the main features of the spectrum of the low-lying eigenvalues of the (euclidean) Dirac operator are expected to be described by a certain universality class of random matrix models. In particular, the latter predict the joint statistical distribution of the individual eigenvalues in any topological sector of the theory. We compare some of these predictions with high-precision numerical data obtained from lattice QCD for a range of lattice spacings and volumes. While no complete matching is observed, the results agree with theoretical expectations at volumes larger than about 5 fm^4.
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Lattice QCD and the Schwarz alternating procedure
A numerical simulation algorithm for lattice QCD is described, in which the short- and long-distance effects of the sea quarks are treated separately. The algorithm can be regarded, to some extent, as an implementation at the quantum level of the classical Schwarz alternating procedure for the solution of elliptic partial differential equations. No numerical tests are reported here, but theoretical arguments suggest that the algorithm should work well also at small quark masses.
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Lattice QCD -- from quark confinement to asymptotic freedom
According to the present understanding, the observed diversity of the strong interaction phenomena is described by Quantum Chromodynamics, a gauge field theory with only very few parameters. One of the fundamental questions in this context is how precisely the world of mesons and nucleons is related to the properties of the theory at high energies, where quarks and gluons are the important degrees of freedom. The lattice formulation of QCD combined with numerical simulations and standard perturbation theory are the tools that allow one to address this issue at a quantitative level.
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Quark confinement and the bosonic string
Using a new type of simulation algorithm for the standard SU(3) lattice gauge theory that yields results with unprecedented precision, we confirm the presence of a $γ/r$ correction to the static quark potential at large distances $r$, with a coefficient $γ$ as predicted by the bosonic string theory. In both three and four dimensions, the transition from perturbative to string behaviour is evident from the data and takes place at surprisingly small distances.
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Lattice QCD on PCs?
Current PC processors are equipped with vector processing units and have other advanced features that can be used to accelerate lattice QCD programs. Clusters of PCs with a high-bandwidth network thus become powerful and cost-effective machines for numerical simulations.
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Locality and exponential error reduction in numerical lattice gauge theory
In non-abelian gauge theories without matter fields, expectation values of large Wilson loops and loop correlation functions are difficult to compute through numerical simulation, because the signal-to-noise ratio is very rapidly decaying for increasing loop sizes. Using a multilevel scheme that exploits the locality of the theory, we show that the statistical errors in such calculations can be exponentially reduced. We explicitly demonstrate this in the SU(3) theory, for the case of the Polyakov loop correlation function, where the efficiency of the simulation is improved by many orders of magnitude when the area bounded by the loops exceeds 1 fm^2.
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