Perfect Lattice Action Research Articles

Optimised Dirac operators on the lattice: construction, properties and applications

We review a number of topics related to block variable renormalisation group transformations of quantum fields on the lattice, and to the emerging perfect lattice actions. We first illustrate this procedure by considering scalar fields. Then we proceed to lattice fermions, where we discuss perfect actions for free fields, for the Gross-Neveu model and for a supersymmetric spin model. We also consider the extension to perfect lattice perturbation theory, in particular regarding the axial anomaly and the quark gluon vertex function. Next we deal with properties and applications of truncated perfect fermions, and their chiral correction by means of the overlap formula. This yields a formulation of lattice fermions, which combines exact chiral symmetry with an optimisation of further essential properties. We summarise simulation results for these so-called overlap-hypercube fermions in the two-flavour Schwinger model and in quenched QCD. In the latter framework we establish a link to Chiral Perturbation Theory, both, in the p-regime and in the ϵ-regime. In particular we present an evaluation of the leading Low Energy Constants of the chiral Lagrangian – the chiral condensate and the pion decay constant – from QCD simulations with extremely light quarks.

Almost perfect quantum lattice action for low-energy SU(2) gluodynamics

We study various representations of infrared effective theory of SU(2) Gluodynamics as a (quantum) perfect lattice action. In particular we derive a monopole action and a string model of hadrons from SU(2) Gluodynamics. These are lattice actions which give almost cut-off independent physical quantities even on coarse lattices. The monopole action is determined by numerical simulations in the infrared region of SU(2) Gluodynamics. The string model of hadrons is derived from the monopole action by using BKT transformation. We illustrate the method and evaluate physical quantities such as the string tension and the mass of the lowest state of the glueball analytically using the string model of hadrons. It turns out that the classical results in the string model is near to the one in quantum SU(2) Gluodynamics.

PERFECT SCALARS ON THE LATTICE

We perform renormalization group transformations to construct optimally local perfect lattice actions for free scalar fields of any mass. Their couplings decay exponentially. The spectrum is identical to the continuum spectrum, while thermodynamic quantities have tiny lattice artifacts. To make such actions applicable in simulations, we truncate the couplings to a unit hypercube and observe that spectrum and thermodynamics are still drastically improved compared to the standard lattice action. We show how preconditioning techniques can be applied successfully to this type of action. We also consider a number of variants of the perfect lattice action, such as the use of an anisotropic or triangular lattice, and modifications of the renormalization group transformations motivated by wavelets. Along the way we illuminate the consistent treatment of gauge fields, and we find a new fermionic fixed point action with attractive properties.

Recent topics of infrared effective lattice QCD

Three topics concerning infrared effective lattice QCD are discussed. (1) Perfect lattice action of infrared SU(3) QCD and perfect operators for the static potential are analytically given when we assume two-point monopole interactions alone. The assumption seems to be justified from numerical analyses of pure SU(3) QCD in maximally abelian gauge. (2) Gauge invariance of monopole dominance can be proved theoretically if the gauge invariance of abelian dominance is proved. The gauge invariance of monopole condensation leads us to confinement of abelian neutral but color octet states after abelian projection. (3) A stochastic gauge fixing method is developed to study the gauge dependence of the Abelian projection, which interpolates between the maximally abelian (MA) gauge and no gauge fixing. Abelian dominance for the heavy quark potential holds even in the gauge which is far from Maximally Abelian one.

A perfect monopole action for SU(2) QCD

We found a quantum perfect lattice action in the 4-dimensional monopole current theory which is known as an effective theory in the infrared region of QCD. The perfect monopole action is transformed exactly into a lattice action of a string model. When the monopole interactions are weak as in the case of infrared SU(2) QCD, the string interactions are strong. The static potential and the string tension in this region can be estimated analytically by the use of the strong coupling expansion.

A quantum perfect lattice action for monopoles and strings

A quantum perfect lattice action in four dimensions can be derived analytically as a renormalized trajectory when we perform a block spin transformation of monopole currents in a simple but non-trivial case of quadratic monopole interactions. The spectrum of the lattice theory is identical to that of the continuum theory. The perfect monopole action is transformed exactly into a lattice action of a string model. A perfect operator evaluating a static potential between electric charges is also derived explicitly. If the monopole interactions are weak as in the case of infrared SU(2) QCD, the string interactions become strong. The static potential and the string tension is estimated analytically by the use of the strong coupling expansion and the continuum rotational invariance is restored completely.

PERFECT LATTICE PERTURBATION THEORY: A STUDY OF THE ANHARMONIC OSCILLATOR

As an application of perfect lattice perturbation theory, we construct an O(λ) perfect lattice action for the anharmonic oscillator analytically in momentum space. In coordinate space, we obtain a set of 2-spin and 4-spin couplings ∝λ, which we evaluate for various masses. These couplings never involve variables separated by more than two lattice spacings. The O(λ) perfect action is simulated and compared to the standard action. We discuss the improvement for the first two energy gaps ΔE1, ΔE2 and for the scaling quantity ΔE2/ΔE1 in different regimes of the interaction parameter, and of the correlation length. For the quartic oscillator — which corresponds to an asymptotically free theory — we also discuss a classically perfect action. The single gaps perform very well, which corresponds to a clearly improved asymptotic scaling. On the other hand, it turns out to be difficult to demonstrate an improvement for the scaling ratio.

On the perfect lattice actions of abelian-projected SU(2) QCD

We study the perfect lattice actions of abelian-projected SU(2) gluodynamics. Using the BKT and duality transformations on the lattice, an effective string model is derived from the direction-dependent quadratic monopole action, obtained numerically from SU(2) gluodynamics in maximally abelian gauge. The string tension and the restoration of continuum rotational invariance are investigated using strong coupling expansion of lattice string model analytically. We also found that the block spin transformation can be performed analytically for the quadratic monopole action.

Perfect monopole action for infrared SU(2) QCD

Performing block-spin transformations on the dual lattice after abelian projection, we obtain an almost perfect lattice action in terms of monopole currents in the infrared region of pure SU(2) QCD. The action depends only on a physical scale b and is free from a cut-off a( β). It reproduces analytically the string tension which is near the physical one for a wide physical-scale region (b ⩾ 1.0(κ −1 2 )) .

Perfect actions with chemical potential

We show how to include a chemical potential \mu in perfect lattice actions. It turns out that the standard procedure of multiplying the quark fields \Psi, \bar\Psi at Euclidean time t by \exp(\pm \mu t), respectively, is perfect. As an example, the case of free fermions with chemical potential is worked out explicitly. Even after truncation, cut-off effects in the pressure and the baryon density are small. Using a (quasi-)perfect action, numerical QCD simulations for non-zero chemical potential become more powerful, because coarse lattices are sufficient for extracting continuum physics.

Perfect actions for scalar theories

We construct an optimally local perfect lattice action for free scalars of arbitrary mass, and truncate its couplings to a unit hypercube. Spectral and thermodynamic properties of this “hypercube scalar” are drastically improved compared to the standard action. We also discuss new variants of perfect actions, using anisotropic or triangular lattices, or applying new types of RGTs. Finally we add a λΦ 4 term and address perfect lattice perturbation theory. We report on a lattice action for the anharmonic oscillator, which is perfect to O( λ).

Perfect lattice actions

The improvement program based on Renormalization Group approach is discussed for various models.

Perfect lattice actions for staggered fermions

We construct a perfect lattice action for staggered fermions by blocking from the continuum. The locality, spectrum and pressure of such perfect staggered fermions are discussed. We also derive a consistent fixed point action for free gauge fields and discuss its locality as well as the resulting static quark-antiquark potential. This provides a basis for the construction of (classically) perfect lattice actions for QCD using staggered fennions.

Progress on perfect lattice actions for QCD

We describe a number of aspects in our attempt to construct an approximately perfect lattice action for QCD. Free quarks are made optimally local on the whole renormalized trajectory and their couplings are then truncated by imposing 3-periodicity. The spectra of these short ranged fermions are excellent approximations to continuum spectra. The same is true for free gluons. We evaluate the corresponding perfect quark-gluon vertex function, identifying in particular the “perfect clover term”. First simulations for heavy quarks show that the mass is strongly renormalized, but again the renormalized theory agrees very well with continuum physics. Furthermore we describe the multigrid formulation for the non-perturbative perfect action and we present the concept of an exactly (quantum) perfect topological charge on the lattice.

Scaling topological charge in the CP 3 model using a fixed point action

We define a fixed point action in two-dimensional lattice CP N−1 models. The fixed point action is a classical perfect lattice action, which is expected to show strongly reduced cut-off effects in numerical simulations. Furthermore, the action has scale invariant instanton solutions, which enables us to define a topological charge without topological defects. We present results for the scaling of the topological suceptibility from a Monte Carlo simulation in the CP 3 model.

Fixed point action and topology in the CP3 model.

We define a fixed point action in two-dimensional lattice ${\rm CP}^{N-1}$ models. The fixed point action is a classical perfect lattice action, which is expected to show strongly reduced cutoff effects in numerical simulations. Furthermore, the action has scale-invariant instanton solutions, which enables us to define a correct topological charge without topological defects. Using a parametrization of the fixed point action for the ${\rm CP}^{3}$ model in a Monte Carlo simulation, we study the topological susceptibility.

Perfect 3-dimensional lattice actions for 4-dimensional quantum field theories at finite temperature

We propose a two-step procedure to study the order of phase transitions at finite temperature in electroweak theory and in simplified models thereof. In a first step a coarse grained free energy is computed by perturbative methods. It is obtained in the form of a 3-dimensional perfect lattice action by a block spin transformation. It has finite temperature-dependent coefficients. In this way the UV-problem and the infrared problem are separated in a clean way. In the second step the effective 3-dimensional lattice theory is treated in a nonperturbative way, either by the Feynman-Bogoliubov method (solution of a gap equation), by real space renormalization group methods, or by computer simulations. In this paper we outline the principles for ϕ 4-theory and scalar electrodynamics. The Bałaban-Jaffe block spin transformation for the gauge field is used. It is known how to extend this transformation to the nonabelian case, but this will not be discussed here.

Perfect lattice actions for quarks and gluons

We use perturbation theory to construct perfect lattice actions for quarks and gluons. The renormalized trajectory for free massive quarks is identified by blocking directly from the continuum. We tune a parameter in the renormalization group transformation such that for 1d configurations the perfect action reduces to the nearest-neighbor Wilson fermion action. The fixed point action for free gluons is also obtained by blocking from the continuum. For 2d configurations it reduces to the standard plaquette action. Classically perfect quark and gluon fields, quark-gluon composite operators and vector and axial vector currents are constructed as well. Also the quark-antiquark potential is derived from the classically perfect Polyakov loop. The quark-gluon and three-gluon perfect vertex functions are determined to leading order in the gauge coupling. This work provides a basis for the numerical construction of a lattice action for QCD, which is (approximately) perfect even beyond perturbation theory.

Perfect lattice actions with and without chiral symmetry

We use perturbation theory to construct perfect lattice actions for fermions and gauge fields by blocking directly from the continuum. When one uses a renormalization group transformation that preserves chiral symmetry the resulting lattice action for massless fermions is chirally symmetric but nonlocal. When the renormalization group transformation breaks chiral symmetry, the lattice action becomes local but chiral symmetry is explicitly broken. In particular, starting with a chiral gauge theory in the continuum one either obtains a lattice theory which is gauge invariant but nonlocal, or a local theory with explicitly broken gauge invariance. In both cases the spectrum of the lattice theory is identical with the one of the continuum and the anomaly is correctly reproduced. We also apply our techniques to vector-like theories. In particular we propose a new renormalization group transformation for QCD and we optimize its parameters for locality of the perfect action.

Block spin transformation on the dual lattice and monopole action

To find a perfect lattice action in terms of monopole action on the dual lattice, we performed simulations of a monopole effective action obtained numerically from vacuum configurations in SU(2) QCD. Although the Polyakov loop behavior near $T_c$ is well reproduced by the action, a small but repulsive term is needed in addition to get the string tension correctly. It is reported also a monopole effective action in $SU(3)$ QCD which is expressed by one kind of monopole currents.