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Erhard Seiler

Publications and source records attributed to Erhard Seiler.

68 records · Page 4Linked to original sources
arXiv2000

Lattice artefacts and the running of the coupling constant

We study the running of the Lüscher-Weisz-Wolff (LWW) coupling constant in the two dimensional O(3) nonlinear $σ$ model. To investigate the continuum limit we refine the lattice spacing from the $1\over 16$ value used by LWW up to $1\over 160$. We find that the lattice artefacts are much larger than estimated by LWW and that most likely the coupling constant runs slower than predicted by perturbation theory. A precise determination of the running in the continuum limit would require a controlled ansatz of extrapolation, which, we argue, is not presently available.

hep-lat↗
arXiv2000

Nontrivial fixed point in nonabelian models

We investigate the percolation properties of equatorial strips in the two-dimensional O(3) nonlinear $σ$ model. We find convincing evidence that such strips do not percolate at low temperatures, provided they are sufficiently narrow. Rigorous arguments show that this implies the vanishing of the mass gap at low temperature and the absence of asymptotic freedom in the massive continuum limit. We also give an intuitive explanation of the transition to a massless phase and, based on it, an estimate of the transition temperature.

hep-lat↗
arXiv1999

Comparison of the O(3) Bootstrap $σ$-Model with the 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 $L/ξ\sim 7$ and correlation lengths $ξ$ 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 Lüscher and Wolff using the standard action, but this time using the fixed point action. Excellent agreement with the Zamolodchikov S-matrix is found.

hep-lat↗
arXiv1999

The Difference between Abelian and Non-Abelian Models: Fact and Fancy

The commonly accepted belief that non-Abelian and Abelian models are different because of the presence/absence of instantons and/or perturbative asymptotic freedom is analyzed from a historical perspective. The presentation covers the major developments which brought about this dogma, as well as all the supportive evidence produced since. For a model possessing both asymptotic freedom and instantons it is shown rigorously that a disorder variable varies nonanalytically with the temperature.

math-ph↗
arXiv1998

Critical behavior of classical spin models and local cohomology

Using reflection positivity as the main tool, we establish a connection between the existence of a critical point in classical spin models and the triviality of a certain local cohomology class related to the Noether current of the model in the continuum limit. Furthermore we find a relation between the location of the critical point and the momentum space autocorrelation function of the Noether current.

hep-th↗
arXiv1998

Nonlinear $σ$-model, form factors and universality

We report the results of a very high statistics Monte Carlo study of the continuum limit of the two dimensional O(3) non-linear $σ$ model. We find a significant discrepancy between the continuum extrapolation of our data and the form factor prediction of Balog and Niedermaier, inspired by the Zamolodchikovs' S-matrix ansatz. On the other hand our results for the O(3) and the dodecahedron model are consistent with our earlier finding that the two models possess the same continuum limit.

hep-th↗
arXiv1997

Is the 2D O(3) Nonlinear $σ$ Model Asymptotically Free?

We report the results of a Monte Carlo study of the continuum limit of the two dimensional O(3) non-linear $σ$ model. The notable finding is that it agrees very well with both the prediction inspired by Zamolodchikovs' S-matrix ansatz and with the continuum limit of the dodecahedron spin model. The latter finding renders the existence of asymptotic freedom in the O(3) model rather unlikely.

hep-lat↗
arXiv1997

Does Conformal Quantum Field Theory Describe the Continuum Limits of 2D Spin Models with Continuous Symmetry?

It is generally taken for granted that two-dimensional critical phenomena can be fully classified by the well known two-dimensional (rational) conformal quantum field theories (CQFTs). In particular it is believed that in models with a continuous symmetry characterized by a Lie group $G$ the continuum theory enjoys an enhanced symmetry $G\times G$ due to the decoupling of right and left movers. In this letter we review the conventional arguments leading to this conclusion, point out two gaps and provide a conterexample. Nevertheless we justify in the end the conventional conclusions by additional arguments.

hep-lat↗
arXiv1997

Continuum Limit of $2D$ Spin Models with Continuous Symmetry and Conformal Quantum Field Theory

According to the standard classification of Conformal Quantum Field Theory (CQFT) in two dimensions, the massless continuum limit of the $O(2)$ model at the Kosterlitz-Thouless (KT) transition point should be given by the massless free scalar field; in particular the Noether current of the model should be proportional to (the dual of) the gradient of the massless free scalar field, reflecting a symmetry enhanced from $O(2)$ to $O(2)\times O(2)$. More generally, the massless continuum limit of a spin model with a symmetry given by a Lie group $G$ should have an enhanced symmetry $G\times G$. We point out that the arguments leading to this conclusion contain two serious gaps: i) the possibility of `nontrivial local cohomology' and ii) the possibility that the current is an ultralocal field. For the $2D$ $O(2)$ model we give analytic arguments which rule out the first possibility and use numerical methods to dispose of the second one. We conclude that the standard CQFT predictions appear to be borne out in the $O(2)$ model, but give an example where they would fail. We also point out that all our arguments apply equally well to any $G$ symmetric spin model, provided it has a critical point at a finite temperature.

hep-lat↗
arXiv1997

Questionable Arguments for the Correctness of Perturbation Theory in Non-Abelian Models

We analyze the arguments put forward recently by Niedermayer et al in favor of the correctness of conventional perturbation theory in non-Abelian models and supposedly showing that our super-instanton counterexample was sick. We point out that within their own set of assumptions, the proof of Niedermayer et al regarding the correctness of perturbation theory is incorrect and provide a correct proof under more restrictive assumptions. We reply also to their claim that the S-matrix bootstrap approach of Balog et al supports the existence of asymptotic freedom in the O(3) model.

hep-lat↗
arXiv1996

New Universality Classes in One--Dimensional $O(N)$--Invariant Spin--Models with an $n$--Parametric Action

An action with $n$ parameters, which generalizes the $O(N) - R P^{N-1}$ -model, is considered in one dimension for general $N$. We use asymptotic expansion techniques to determine where the model becomes critical and show that for the actions considered there exists a family of hypersurfaces whose asymptotic behaviour determines a one-parameter family of new universality classes. They interpolate between the $O(N)$-vector-model-class and the $R P^{N-1}$-model-class. Furthermore continuum limits are discussed, including the exceptional case $N=2$.

hep-lat↗
arXiv1995

Universality Class of $O(N)$ Models

We point out that existing numerical data on the correlation length and magnetic susceptibility suggest that the two dimensional $O(3)$ model with standard action has critical exponent $η=1/4$, which is inconsistent with asymptotic freedom. This value of $η$ is also different from the one of the Wess-Zumino-Novikov-Witten model that is supposed to correspond to the $O(3)$ model at $θ=π$.

hep-lat↗
arXiv1993

Functional Integral Approach to the N-Flavor Schwinger Model

We study massless QED_2 with N flavors using path integrals. We identify the sector that is generated by the N^2 classically conserved vector currents. One of them (the U(1) current) creates a massive particle, while the others create massless ones. We show that the mass spectrum obeys a Witten-Veneziano type formula. Two theorems on n-point functions clarify the structure of the Hilbert space. Evaluation of the Fredenhagen-Marcu order parameter indicates that a confining force exists only between charges that are integer multiples of +/- Ne, whereas charges that are nonzero mod(N) screen their confining forces and lead to non-vacuum sectors. Finally we identify operators that violate clustering, and decompose the theory into clustering theta vacua.

hep-th↗