Exact Supersymmetry on the Lattice
We discuss the possibility of representing supersymmetry exactly in a lattice discretized system. In particular, we construct a perfect supersymmetric action for the Wess-Zumino model.
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arXiv subjects
W. Bietenholz
Publications and source records attributed to W. Bietenholz.
Improved Lattice Actions with Chemical Potential
We give a prescription how to include a chemical potential μinto a general lattice action. This inclusion does not cause any lattice artifacts. Hence its application to an improved - or even perfect - action at μ=0 yields an improved resp. perfect action at arbitrary μ. For short-ranged improved actions, a good scaling behavior holds over a wide region, and the upper bound for the baryon density - which is known for the standard lattice actions - can be exceeded.
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Truncated Perfect Actions for Staggered Fermions
We discuss the behavior of free perfect staggered fermions and truncated versions thereof. The study includes flavor non-degenerate masses. We suggest a new blocking scheme, which provides excellent locality of the perfect lattice action. A truncation procedure adequate for the structure of staggered fermions is applied. We consider spectral and thermodynamic properties and compare truncated perfect actions, Symanzik improved and standard staggered fermions in two and four dimensions.
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Perfect and Quasi-Perfect Lattice Actions
Perfect lattice actions are exiting with several respects: they provide new insight into conceptual questions of the lattice regularization, and quasi-perfect actions could enable a great leap forward in the non-perturbative solution of QCD. We try to transmit a flavor of them, also beyond the lattice community.
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Perfect Actions with Chemical Potential
We show how to include a chemical potential μin perfect lattice actions. It turns out that the standard procedure of multiplying the quark fields Ψ, \barΨat Euclidean time t by \exp(\pm μt), 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.
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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 \propto 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 ΔE_1, ΔE_2 and for the scaling quantity ΔE_2 / ΔE1 in different regimes of the interaction parameter, and of the correlation length.
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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).
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SSOR Preconditioning of Improved Actions
We generalize local lexicographic SSOR preconditioning for the Sheikholeslami-Wohlert improved Wilson fermion action and the truncated perfect free fermion action. In our test implementation we achieve performance gains as known from SSOR preconditioning of the standard Wilson fermion action.
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The Perfect Quark-Gluon Vertex Function
We evaluate a perfect quark-gluon vertex function for QCD in coordinate space and truncate it to a short range. We present preliminary results for the charmonium spectrum using this quasi-perfect action.
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Perfect Lattice Topology: The Quantum Rotor as a Test Case
Lattice actions and topological charges that are classically and quantum mechanically perfect (i.e. free of lattice artifacts) are constructed analytically for the quantum rotor. It is demonstrated that the Manton action is classically perfect while the Villain action is quantum perfect. The geometric construction for the topological charge is only perfect at the classical level. The quantum perfect lattice topology associates a topological charge distribution, not just a single charge, with each lattice field configuration. For the quantum rotor with the classically perfect action and topological charge, the remaining cut-off effects are exponentially suppressed.
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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 fermions.
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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.
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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 1-d 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 2-d 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 3-gluon perfect vertex functions are determined to leading order in the gauge coupling. We also construct a new block factor $n$ renormalization group transformation for QCD that allows to extend our results beyond perturbation theory. For weak fields it leads to the same perfect action as blocking from the continuum. For arbitrarily strong 2-d Abelian gauge fields the Manton plaquette action is classically perfect for this transformation.
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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.
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Testing Haldane's Conjecture in the O(3) Model by a Meron Cluster Simulation
Twelve years ago, Haldane formulated his famous conjecture for 1-d antiferromagnetic quantum spin chains. In the context of the 2-d O(3) model with a θterm, it predicts a phase transition at θ= π, which has not yet been verified reliably. To simulate this we use the Wolff cluster algorithm together with an improved estimator for the topological charge distribution. Each cluster carries integer or half integer charge. Clusters with charge 1/2 are identified with merons. At θ= πthey are inactive, such that the mass gap vanishes. We obtain critical exponents which are consistent with predictions from the k=1 WZNW model, therefore confirming a second order phase transition.
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Meron-Cluster Simulation of the $θ$-Vacuum in the 2-d $O(3)$-Model
The 2-d $O(3)$-model with a $θ$-vacuum term is formulated in terms of Wolff clusters. Each cluster carries a half-integer topological charge. The clusters with charge $\pm 1/2$ are identified as merons. At $θ= π$ the merons are bound in pairs inducing a second order phase transition at which the mass-gap vanishes. The construction of an improved estimator for the topological charge distribution makes numerical simulations of the phase transition feasible. The measured critical exponents agree with those of the $k = 1$ Wess-Zumino-Novikov-Witten model. Our results are consistent with Haldane's conjecture for 1-d antiferromagnetic quantum spin chains.
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A Perturbative Construction of Lattice Chiral Fermions
We perform a renormalization group transformation to construct a lattice theory of chiral fermions. The field variables of the continuum theory are averaged over hypercubes to define lattice fields. Integrating out the continuum variables in perturbation theory we derive a chirally invariant effective action for the lattice fields. This is consistent with the Nielsen-Niniomiya theorem because the effective action is nonlocal. We also construct the axial current on the lattice and we show that the axial anomaly of the continuum theory is reproduced in the Schwinger model. This shows that chiral fermions can be regularized on the lattice.
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Perfect Lattice Actions for the Gross-Neveu Model
We apply the method of Hasenfratz and Niedermayer to analytically construct perfect lattice actions for the Gross--Neveu model. In the large $N$ limit these actions display an exactly perfect scaling, i.e. cut-off artifacts are completely eliminated even at arbitrarily short correlation length. Also the energy spectrum coincides with the spectrum in the continuum and continuous translation and rotation symmetries are restored in physical observables. This is the first (analytic) construction of an exactly perfect lattice action at finite correlation length.
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