Fixed Point Action Research Articles

Bound states for overlap and fixed point actions close to the chiral limit

We study the overlap and the fixed point Dirac operators for massive fermions in the two-flavor lattice Schwinger model. The masses of the triplet (pion) and singlet (eta) bound states are determined down to small fermion masses and the mass dependence is compared with various continuum model approximations. Near the chiral limit, at very small fermion masses the fixed point operator has stability problems, which in this study are dominated by finite size effects,

Local field theory for disordered itinerant quantum ferromagnets

An effective field theory is derived that describes the quantum critical behavior of itinerant ferromagnets in the presence of quenched disorder. In contrast to previous approaches, all soft modes are kept explicitly. The resulting effective theory is local and allows for an explicit perturbative treatment. It is shown that previous suggestions for the critical fixed point and the critical behavior are recovered under certain assumptions. The validity of these assumptions is discussed in the light of the existence of two different time scales. It is shown that, in contrast to previous suggestions, the correct fixed point action is not Gaussian, and that the previously proposed critical behavior was correct only up to logarithmic corrections. The connection with other theories of disordered interacting electrons, and in particular with the resolution of the runaway flow problem encountered in these theories, is also discussed.

Fixed point SU(3) gauge actions: scaling properties and glueballs

We present a new parametrization of a SU(3) fixed point (FP) gauge action using smeared (“fat”) gauge links. We report on the scaling behaviour of the FP action on coarse lattices by means of the static quark-antiquark potential, the hadronic scale r 0, the string tension σ and the critical temperature T c of the deconfining phase transition. In addition, we investigate the low lying glueball masses where we observe no scaling violations within the statistical errors.

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.

CP(N-1) model and topological term

We study CP(N-1) model with topological term for N = 3 and 4. The topological charge distribution P(Q) is calculated. The distribution P(Q) is then transformed to the partition function Z(θ) as a function of θ parameter. P(Q) is Gaussian in the strong coupling region, leading to a first order phase transition at θ = π. In the weak coupling region, P(Q) deviates from Gaussian. Intermediate and weak coupling regions are characterized by the leading power of the fits to 1n P(Q). Free energy as a function of θ shows “flattening” behavior at θ = θ f < π, when we calculate the free energy directly from topological charge distribution. We also show some preliminary results for the fixed point action.

Fixed-point pure gauge action using b = √3 RGT

We present a status report on the construction of the classical perfect action using the b = √3 renormalization group transformation (RGT) [1]. We investigate finite volume corrections and map the locality of the fixed-point action by tuning the RGT parameter, κ. We compare results with the previous calculation for b = 2 RGT [2].

Fixed-point pure gauge action using b = √3 RGT

We present a status report on the construction of the classical perfect action using the $ b=\sqrt{3} $ renormalization group transformation (RGT). We investigate finite volume corrections and map the locality of the fixed-point action by tuning the RGT parameter, $\kappa$. We compare results with the previous calculation for b=2 RGT.

Comparison of the O(3) bootstrap σ model with lattice regularization at low energies

The renormalized coupling $\gr$ defined through the connected 4-point function at zero external momentum in the non-linear O(3) sigma-model in two dimensions, is computed in the continuum form factor bootstrap approach with estimated error $\sim 0.3%$. New high precision data are presented for $\gr$ in the lattice regularized theory with standard action for nearly thermodynamic lattices $L/\xi\sim 7$ and correlation lengths $\xi$ up to $\sim 122$ and with the fixed point action for correlation lengths up to $\sim 12$. The agreement between the form factor and lattice results is within $\sim 1%$. We also recompute the phase shifts at low energy by measuring the two-particle energies at finite volume, a task which was previously performed by L\"uscher and Wolff using the standard action, but this time using the fixed point action. Excellent agreement with the Zamolodchikov S-matrix is found.

Simple observables from fat link fermion actions

A comparison is made of the (quenched) light hadron spectrum and of simple matrix elements for a hypercubic fermion action (based on a fixed point action) and the clover action, both using fat links, at a lattice spacing a= 0.18 fm. Renormalization constants for the naive and improved vector current and the naive axial current are computed using Ward identities. The renormalization factors are very close to unity, and the spectroscopy of light hadrons and the pseudoscalar and vector decay constants agree well with simulations at smaller lattice spacings (and with experiment).

Eigenvalue spectrum of massless Dirac operators on the lattice

We present a detailed study of the interplay between chiral symmetry and spectral properties of the Dirac operator in lattice gauge theories. We consider, in the framework of the Schwinger model, the fixed point action and a fermion action recently proposed by Neuberger. Both actions show the remnant of chiral symmetry on the lattice as formulated in the Ginsparg-Wilson relation. We check this issue for practical implementations, also evaluating the fermion condensate in a finite volume by a subtraction procedure. Moreover, we investigate the distribution of the eigenvalues of a properly defined anti-hermitian lattice Dirac operator, studying the statistical properties at the low lying edge of the spectrum. The comparison with the predictions of chiral Random Matrix Theory enables us to obtain an estimate of the infinite volume fermion condensate.

Spectrum of the fixed point Dirac operator in the Schwinger model

Recently, properties of the fixed point action for fermion theories have been pointed out indicating realization of chiral symmetry on the lattice. We check these properties by numerical analysis of the spectrum of a parametrized fixed point Dirac operator investigating also microscopic fluctuations and fermion condensation.

Fixed point actions and on-shell tree-level Symanzik improvement

In this paper it is argued that the properties of the fixed point action of a renormalization group transformation can be used to implement the on-shell tree-level Symanzik improvement of lattice actions to any given order in the expansion in the lattice spacing, in a way which does not involve any perturbative calculations. In particular, a well-known technique for the lowest order improvement of SU(N) lattice gauge theories is revisited from the point of view of fixed point actions, which allows to shed light on some subtle points.

Lattice QCD without tuning, mixing and current renormalization

The classically perfect action of QCD requires no tuning to get the pion massless in the broken phase: the critical bare mass m q c is zero. Neither the vector nor the flavour non-singlet axial vector currents need renormalization. Further, there is no mixing between 4-fermion operators in different chiral representations. The order parameter of chiral symmetry requires, however, a subtraction which is given here explicitly. These results are based on the fact that the fixed point action satisfies the Ginsparg-Wilson remnant chiral symmetry condition. On chiral symmetry related questions any other local solution of this condition will produce similar results.

Chiral properties of the fixed point action of the Schwinger model

We study the spectrum properties for a recently constructed fixed point lattice Dirac operator. We also consider the problem of the extraction of the fermion condensate, both by direct computation, and through the Banks-Casher formula by analyzing the density of eigenvalues of a redefined antihermitean lattice Dirac operator.

Instanton classical solutions of SU(3) fixed point actions on open lattices

We construct instanton-like classical solutions of the fixed point action of a suitable renormalization group transformation for the SU(3) lattice gauge theory. The problem of the non-existence of one-instantons on a lattice with periodic boundary conditions is circumvented by working on open lattices. We consider instanton solutions for values of the size (0.6-1.9 in lattice units) which are relevant when studying the SU(3) topology on coarse lattices using fixed point actions. We show how these instanton configurations on open lattices can be taken into account when determining a few-couplings parametrization of the fixed point action.

The fixed point action for the Schwinger model: a perturbative approach

We compute the fixed point action of a properly defined renormalization group transformation for the Schwinger model through an expansion in the gauge field. It is local, with couplings exponentially suppressed with the distance. We check its perfection by computing the 1-loop mass gap at finite spatial volume, finding only exponentially vanishing cut-off effects, in contrast with the standard action, which is affected by large power-like cut-off effects. We point out that the 1-loop mass gap calculation provides a check of the classical perfection of the fixed point action, and not of the 1-loop perfection, as could be naively expected.

The index theorem in QCD with a finite cut-off

The fixed point Dirac operator on the lattice has exact chiral zero modes on topologically non-trivial gauge field configurations independently whether these configurations are smooth, or coarse. The relation $n_L-n_R = Q^{FP}$, where $n_L$ $(n_R)$ is the number of left (right)-handed zero modes and $Q^{FP}$ is the fixed point topological charge holds not only in the continuum limit, but also at finite cut-off values. The fixed point action, which is determined by classical equations, is local, has no doublers and complies with the no-go theorems by being chirally non-symmetric. The index theorem is reproduced exactly, nevertheless. In addition, the fixed point Dirac operator has no small real eigenvalues except those at zero, i.e. there are no 'exceptional configurations'.

Prospects for perfect actions

The fixed-point (FP) action in QCD, although it is local and determined by classical equations, is difficult to parametrize well and is expensive to simulate. But the stake is high: the FP action has scale invariant instanton solutions, has no topological artifacts, satisfies the index theorem on the lattice, does not allow exceptional configurations, requires no tuning to get the pion massless and is expected to reduce the cut-off effects significantly. An overview is given including a discussion on tests in Yang-Mills theory, QCD and d = 2 spin and gauge models.

Fixed point actions in SU(3) gauge theory: surface tension and topology

This work is organized in two independent parts. In the first part are presented some results concerning the surface tension in SU(3) obtained with a parametrized fixed point action. In the second part, a new, approximately scale-invariant, parametrized fixed point action is proposed which is suitable to study the topology in SU(3).

A fixed-point action for the lattice Schwinger model

We determine non-perturbatively a fixed-point (FP) action for fermions in the two-dimensional U(1) gauge (Schwinger) model. Our parameterization for the fermionic action has terms within a 7 × 7 square on the lattice, using compact link variables. With the Wilson fermion action as starting point we determine the FP-action by iterating a block spin transformation (BST) with a blocking factor of 2 in the background of non-compact gauge field configurations sampled according to the (perfect) Gaussian measure. We simulate the model at various values of β and find excellent improvement for the studied observables.