Ferenc Niedermayer

Perfect lattice action for asymptotically free theories

Abstract There exist lattice actions which give cut-off independent physical predictions even on coarse grained lattices. Rotation symmetry is restored, the spectrum becomes exact and, in addition, the classical equations have scale invariant instanton solutions. This perfect action can be made short ranged. It can be determined by combining analytical calculations with numerical simulations on small lattices. We illustrate the method and the benefits on the d = 2 non-linear σ-model.

The exact mass gap of the O(3) and O(4) non-linear σ-models in d = 2☆

Abstract By comparing the Bethe Ansatz solution with perturbation theory we obtain the exact results m = 8 e Λ MS and m = ( 32 π e ) 1 2 Λ MS for the O(3) and O(4) non-linear σ-models, respectively. Here Λ MS is the renormalization group invariant scale in the MS scheme.

The exact mass gap of the O(N) σ-model for arbitrary N ⩾ 3 in d = 2

Abstract Starting from the exact S -matrix of the O( N ) model we derive an integral equation, which determines the free energy as the function of the chemical potential coupled to a Noether charge. For N = 3 and 4 the equation agrees with that obtained earlier by Polyakov and Wiegmann with Bethe Ansatz for the related fermionic models. Using a method discussed in a recent paper by M. Maggiore and the authors, this integral equation predicts for the mass gap m = ( 8 e ) 1 (N−2) × [ 1 Γ ( 1 + 1 (N − 2) ) ] Λ MS .

Finite size and temperature effects in the AF Heisenberg model

The low temperature and large volume effects in thed=2+1 antiferomagnetic quantum Heisenberg model are dominated by magnon excitations. The leading and next-to-leading corrections are fully controlled by three physical constants, the spin stiffness, the spin wave velocity and the staggered magnetization. Among others, the free energy, the ground state energy, the low lying excitations, staggered magnetization, staggered and uniform susceptibilities are studied here. The special limits of very low temperature and infinite volume are considered also.

Quenched spectroscopy with fixed-point and chirally improved fermions

Abstract We present results from quenched spectroscopy calculations with the parametrized fixed-point and the chirally improved Dirac operators. Both these operators are approximate solutions of the Ginsparg–Wilson equation and have good chiral properties. This allows us to work at small quark masses and we explore pseudoscalar-mass to vector-mass ratios down to 0.28. We discuss meson and baryon masses, their scaling properties, finite volume effects and compare our results with recent large scale simulations. We find that the size of quenching artifacts of the masses is strongly correlated with their experimentally observed widths and that the gauge and hadronic scales are consistent.

THE CONSTRUCTION OF GENERALIZED DIRAC OPERATORS ON THE LATTICE

We discuss the steps to construct Dirac operators, which have arbitrary fermion offsets, gauge paths, a general structure in Dirac space and satisfy the basic symmetries (gauge symmetry, hermiticity condition, charge conjugation, hypercubic rotations and reflections) on the lattice. We give an extensive set of examples and offer help to add further structures.

Finite-size effects, goldstone bosons and critical exponents in the d = 3 Heisenberg model☆

The d = 3 classical O(3) Heisenberg model is studied numerically in the broken phase close to the critical point. The finite-size behaviour of the magnetisation and the correlation functions are shown to be in excellent agreement with the theoretical predictions obtained by chiral perturbation theory. The finite-size effects are governed by two constants, which are defined at infinite volume and zero magnetic field: the Goldstone boson decay constant F (or helicity modulus T = F2) and the magnetisation Σ. The data determine the scaling behaviour of Σ and F leading to the prediction 0.6930(2) for the critical coupling on simple cubic lattices and v′ = 0.73(4) and β = 0.36(2) for the correlation length and magnetisation critical indices, respectively.

Panel discussion on the cost of dynamical quark simulations

Abstract This is a transcript of the recorded panel discussion on the cost of dynamical quark simulations at Lattice2001.

Instantons and the fixed point topological charge in the two-dimensional O(3) sigma model.

We define a fixed point topological charge for the two-dimensional O(3) lattice sigma-model which is free of topological defects. We use this operator in combination with the fixed point action to measure the topological susceptibility for a wide range of correlation lengths. The results strongly suggest that it is not a physical quantity in this model. The procedure, however, can be applied to other asymptotically free theories as well.

Mass gap, scaling and universality in the d = 2 O(3) σ-model

Using direct mass measurements and different MCRG methods we investigated the onset of asymptotic scaling and the universality of the mass gap. The standard nearest-neighbour action and a modified action with next-to-nearest neighbour coupling were studied up to correlation length ξ ∼ 1090 and ξ ∼ 330, respectively. The different methods give consistently that asymptotic scaling sets in only above ξ ∼ 500 for the standard action, while much earlier, at ξ ∼ 40, for the modified one. In full agreement with universality, the results m = (3.3 ± 0.1)ΛMS and m = (3.4 ± 0.1)ΛMS were obtained for these two cases, respectively.