arXiv subjects

W. Bietenholz

Publications and source records attributed to W. Bietenholz.

At least 73 records · Page 4Linked to original sources

arXiv2001

Ginsparg-Wilson Fermions in Odd Dimensions

The Ginsparg-Wilson relation, if written in a suitable form, can be used as a condition for lattice Dirac operators of massless fermions also in odd dimensions. The fermion action with such a Dirac operator is invariant under a generalized parity transformation, which reduces to the ordinary parity transformation in the (naive) continuum limit. The fermion measure, however, transforms non-trivially under the generalized parity transformation, and hence the parity anomaly arises solely from the fermion measure. The analogy to the lattice construction of chiral gauge theories in even dimensions is clarified by considering a dimensional reduction. We also propose a natural definition of a lattice Chern-Simons term, which is consistent with odd dimensional Ginsparg-Wilson fermions.

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arXiv2001

On the Spontaneous Breakdown of Lorentz Symmetry in Matrix Models of Superstrings

In string or M theories, the spontaneous breaking of 10D or 11D Lorentz symmetry is required to describe our space-time. A direct approach to this issue is provided by the IIB matrix model. We study its 4D version, which corresponds to the zero volume limit of 4D super SU(N) Yang-Mills theory. Based on the moment of inertia as a criterion, spontaneous symmetry breaking (SSB) seems to occur, so that only one extended direction remains, as first observed by Bialas, Burda et al. However, using Wilson loops as probes of space-time we do not observe any sign of SSB in Monte Carlo simulations where N is as large as 48. This agrees with an earlier observation that the phase of the fermionic integral, which is absent in the 4D model, should play a crucial role if SSB of Lorentz symmetry really occurs in the 10D IIB matrix model.

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arXiv2001

Monte Carlo Studies of the Dimensionally Reduced 4d SU(N) Super Yang-Mills Theory

We simulate a supersymmetric matrix model obtained from dimensional reduction of 4d SU(N) super Yang-Mills theory. The model is well defined for finite N and it is found that the large N limit obtained by keeping g^2 N fixed gives rise to well defined operators which represent string amplitudes. The space-time structure which arises dynamically from the eigenvalues of the bosonic matrices is discussed, as well as the effect of supersymmetry on the dynamical properties of the model. Eguchi-Kawai equivalence of this model to ordinary gauge theory does hold within a finite range of scale. We report on new simulations of the bosonic model for N up to 768 that confirm this property, which comes as a surprise since no quenching or twist is introduced.

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arXiv2000

Constructing Improved Overlap Fermions in QCD

We describe an explicit construction of approximate Ginsparg-Wilson fermions for QCD. We use ingredients of perfect action origin, and further elements. The spectrum of the lattice Dirac operator reveals the quality of the approximation. We focus on beta =6 for optimisation. Such fermions are intended to be inserted into the overlap formula. Hence we also test the speed of convergence under polynomial evaluation of the overlap formula.

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arXiv2000

Simulating Simplified Versions of the IKKT Matrix Model

We simulate a supersymmetric matrix model obtained from dimensional reduction of 4d SU(N) super Yang-Mills theory (a 4d counter part of the IKKT model or IIB matrix model). The eigenvalue distribution determines the space structure. The measurement of Wilson loop correlators reveals a universal large N scaling. Eguchi-Kawai equivalence may hold in a finite range of scale, which is also true for the bosonic case. We finally report on simulations of a low energy approximation of the 10d IKKT model, where we omit the phase of the Pfaffian and look for evidence for a spontaneous Lorentz symmetry breaking.

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arXiv2000

Approximate Ginsparg-Wilson Fermions for QCD

Lattice fermions obeying the Ginsparg-Wilson relation do correctly represent the physical properties related to chirality. This can be achieved by local fermions, which involve an infinite number of couplings, however. For practical purposes, it is useful to first construct approximate Ginsparg-Wilson fermions within a short range. We report on a successful construction in QCD at β=6. The good quality of the approximation is observed from the spectrum, which is situated close to a Ginsparg-Wilson circle. These fermions also provide an excellent approximation to rotational symmetry and they are promising for a good scaling, since they arise from the perfect action framework. Their insertion into the overlap formula renders the Ginsparg-Wilson relation exact. It leads to an improved overlap fermion with a high level of locality. This insertion is statistically on safe grounds at about β> 5.6.

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arXiv2000

Monte Carlo Studies of the IIB Matrix Model at Large N

The low-energy effective theory of the IIB matrix model developed by H. Aoki et al. is written down explicitly in terms of bosonic variables only. The effective theory is then studied by Monte Carlo simulations in order to investigate the possibility of a spontaneous breakdown of Lorentz invariance. The imaginary part of the effective action, which causes the so-called sign problem in the simulation, is dropped by hand. The extent of the eigenvalue distribution of the bosonic matrices shows a power-law large N behavior, consistent with a simple branched-polymer prediction. We observe, however, that the eigenvalue distribution becomes more and more isotropic in the ten-dimensional space-time as we increase N. This suggests that if the spontaneous breakdown of Lorentz invariance really occurs in the IIB matrix model, a crucial role must be played by the imaginary part of the effective action.

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arXiv2000

Large N Dynamics of Dimensionally Reduced 4D SU(N) Super Yang-Mills Theory

We perform Monte Carlo simulations of a supersymmetric matrix model, which is obtained by dimensional reduction of 4D SU(N) super Yang-Mills theory. The model can be considered as a four-dimensional counterpart of the IIB matrix model. We extract the space-time structure represented by the eigenvalues of bosonic matrices. In particular we compare the large N behavior of the space-time extent with the result obtained from a low energy effective theory. We measure various Wilson loop correlators which represent string amplitudes and we observe a nontrivial universal scaling in N. We also observe that the Eguchi-Kawai equivalence to ordinary gauge theory does hold at least within a finite range of scale. Comparison with the results for the bosonic case clarifies the role of supersymmetry in the large N dynamics. It does affect the multi-point correlators qualitatively, but the Eguchi-Kawai equivalence is observed even in the bosonic case.

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arXiv2000

Optimizing Chirality and Scaling of Lattice Fermions

If we construct a lattice fermion formulation, there are a number of goals to be considered: doubling should be avoided; even at finite lattice spacing, we want to represent chiral symmetry in a sound way; and we are seeking a good scaling behavior. Conceptually we have to require locality, and for practical purposes we even desire a high level of locality. A further issue is a good approximation to rotation invariance. Last but not least the formulation should be simple enough to allow for efficient simulations. Here we report on a construction, which is designed to do justice to all of these goals.

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arXiv2000

The Absence of Ultralocal Ginsparg-Wilson Fermions

It was shown recently by I. Horvath that lattice fermions obeying the standard form of the Ginsparg-Wilson relation cannot be ultralocal. However, there are more general forms of the Ginsparg-Wilson relation, which also guarantee the physical properties related to chirality, but which are not covered by Horvath's consideration. Here we present a proof which applies to all Ginsparg-Wilson fermions, demonstrating that they can only be local in the sense of an exponential decay of their couplings, but not ultralocal.

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arXiv1999

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.

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arXiv1999

The Scaling of Exact and Approximate Ginsparg-Wilson Fermions

We construct a number of lattice fermions, which fulfill the Ginsparg-Wilson relation either exactly or approximately, and test them in the framework of the 2-flavor Schwinger model. We start from explicit approximations within a short range, and study this formulation, as well as its correction to an exact Ginsparg-Wilson fermion by the ``overlap formula''. Then we suggest a new method to realize this correction perturbatively, without using the tedious square root operator. In this way we combine many favorable properties: good chiral behavior, small mass renormalization, excellent scaling and rotational invariance, as well as a relatively modest computational effort, which makes such formulations most attractive for QCD.

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arXiv1999

Improved Overlap Fermions

We test exact and approximate Ginsparg-Wilson fermions with respect to their chiral and scaling behavior in the 2-flavor Schwinger model. We first consider explicit approximate GW fermions in a short range, then we proceed to their chiral correction by means of the ``overlap formula'', and finally we discuss a numerically efficient perturbative chiral correction. In this way we combine very good chiral and scaling properties with a relatively modest computational effort.

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arXiv1998

The Schwinger Model with Perfect Staggered Fermions

We construct and test a quasi-perfect lattice action for staggered fermions. The construction starts from free fermions, where we suggest a new blocking scheme, which leads to excellent locality of the perfect action. An adequate truncation preserves a high quality of the free action. An Abelian gauge field is inserted in d=2 by effectively tuning the couplings to a few short-ranged lattice paths, based on the behavior of topological zero modes. We simulate the Schwinger model with this action, applying a new variant of Hybrid Monte Carlo, which damps the computational overhead due to the non-standard couplings. We obtain a tiny ``pion'' mass down to very small β, while the ``η'' mass follows very closely the prediction of asymptotic scaling. The observation that even short-ranged quasi-perfect actions can yield strong improvement is most relevant in view of QCD.

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arXiv1998

The Schwinger Model with Perfect Staggered Fermions

We construct a new perfect action for free staggered fermions, which is more local than the one obtained from the standard block average scheme. This pays off in superior properties after a short ranged truncation. This action is ``gauged by hand'' and tested in Schwinger model simulations by means of a new variant of hybrid MC. Using ``fat links'' for the gauge field, we obtain a tiny ``pion'' mass down to β\approx 1.5, and the ``eta'' mass follows very closely the prediction of asymptotic scaling.

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arXiv1998

A Preconditioner for Improved Fermion Actions

SSOR preconditioning of fermion matrix inversions which is parallelized using a locally-lexicographic lattice sub-division, has been shown to be very efficient for standard Wilson fermions. We demonstrate here the power of this method for the Sheikholeslami-Wohlert improved fermion action and for a renormalization group improved action incorporating couplings of the lattice fermion fields up to the diagonal in the unit hypercube.

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arXiv1998

Solutions of the Ginsparg-Wilson relation and improved domain wall fermions

We discuss a number of lattice fermion actions solving the Ginsparg-Wilson relation. We also consider short ranged approximate solutions. In particular, we are interested in reducing the lattice artifacts, while avoiding (or suppressing) additive mass renormalization. In this context, we also arrive at a formulation of improved domain wall fermions.

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arXiv1998

Preconditioning of Improved and ``Perfect'' Fermion Actions

We construct a locally-lexicographic SSOR preconditioner to accelerate the parallel iterative solution of linear systems of equations for two improved discretizations of lattice fermions: the Sheikholeslami-Wohlert scheme where a non-constant block-diagonal term is added to the Wilson fermion matrix and renormalization group improved actions which incorporate couplings beyond nearest neighbors of the lattice fermion fields. In case (i) we find the block llssor-scheme to be more effective by a factor about 2 than odd-even preconditioned solvers in terms of convergence rates, at beta=6.0. For type (ii) actions, we show that our preconditioner accelerates the iterative solution of a linear system of hypercube fermions by a factor of 3 to 4.

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