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Erhard Seiler

Publications and source records attributed to Erhard Seiler.

At least 55 records · Page 3Linked to original sources
arXiv2011

Complex Langevin dynamics: criteria for correctness

The complex Langevin method is a leading candidate for solving the sign problem occurring in various physical situations, notably QCD at finite chemical potential. Its most vexing problem is `convergence to the wrong limit', where the simulation gives a well defined, but incorrect, result. Here, we first outline a formal justification of the method and identify points at which it might fail. From these we derive a condition that must be satisfied in order for correct results to be obtained. We then apply these ideas to the three-imensional SU(3) spin model at finite chemical potential and show strong indications that complex Langevin dynamics yields correct results in this theory.

hep-lat↗
arXiv2011

Complex Langevin: Etiology and Diagnostics of its Main Problem

The complex Langevin method is a leading candidate for solving the so-called sign problem occurring in various physical situations. Its most vexing problem is that in some cases it produces `convergence to the wrong limit'. In the first part of the paper we go through the formal justification of the method, identify points at which it may fail and identify a necessary and sufficient criterion for correctness. This criterion would, however, require checking infinitely many identities, and therefore is somewhat academic. We propose instead a truncation to the check of a few identities; this still gives a necessary criterion, but a priori it is not clear whether it remains sufficient. In the second part we carry out a detailed study of two toy models: first we identify the reasons why in some cases the method fails, second we test the efficiency of the truncated criterion and find that it works perfectly at least in the toy models studied.

hep-lat↗
arXiv2010

The Strange World of Non-amenable Symmetries

Nonlinear sigma models with non-compact target space and non-amen-able symmetry group were introduced long ago in the study of disordered electron systems. They also occur in dimensionally reduced quantum gravity; recently they have been considered in the context of the AdS/CFT correspondence. These models show spontaneous symmetry breaking in {\it any} dimension, even one and two (superficially in contradiction with the Mermin-Wagner theorem) as a consequence of the non-amenability of their symmetry group. The low-dimensional models show other peculiarities: invariant observables remain dependent on boundary conditions in the thermodynamic limit and the Osterwalder-Schrader reconstruction yields a non-separable Hilbert space. The ground state space, however, under quite general conditions, carries a unique unitary and continuous representation. The existence of a continuum limit in 2D is an open question: while the perturbative Renormalization Group suggests triviality, other arguments hint at the existence of a conformally invariant continuum limit at least for suitable observables. This talk gives an overview of the work done during the last several years in collaboration first of all with Max Niedermaier, some of it also with Peter Weisz and Tony Duncan.

math-ph↗
arXiv2010

Adaptive stepsize and instabilities in complex Langevin dynamics

Stochastic quantization offers the opportunity to simulate field theories with a complex action. In some theories unstable trajectories are prevalent when a constant stepsize is employed. We construct algorithms for generating an adaptive stepsize in complex Langevin simulations and find that unstable trajectories are completely eliminated. To illustrate the generality of the approach, we apply it to the three-dimensional XY model at nonzero chemical potential and the heavy dense limit of QCD.

hep-lat↗
arXiv2010

The Complex Langevin method: When can it be trusted?

We analyze to what extent the complex Langevin method, which is in principle capable of solving the so-called sign problems, can be considered as reliable. We give a formal derivation of the correctness and then point out various mathematical loopholes. The detailed study of some simple examples leads to practical suggestions about the application of the method.

hep-lat↗
arXiv2008

On the large N expansion in hyperbolic sigma-models

Invariant correlation functions for ${\rm SO}(1,N)$ hyperbolic sigma-models are investigated. The existence of a large $N$ asymptotic expansion is proven on finite lattices of dimension $d \geq 2$. The unique saddle point configuration is characterized by a negative gap vanishing at least like 1/V with the volume. Technical difficulties compared to the compact case are bypassed using horospherical coordinates and the matrix-tree theorem.

math-ph↗
arXiv2007

Perturbative and nonperturbative correspondences between compact and non-compact sigma-models

Compact (ferro- and antiferromagnetic) sigma-models and noncompact (hyperbolic) sigma-models are compared in a lattice formulation in dimensions $d \geq 2$. While the ferro- and antiferromagnetic models are essentially equivalent, the qualitative difference to the noncompact models is highlighted. The perturbative and the large $N$ expansions are studied in both types of models and are argued to be asymptotic expansions on a finite lattice. An exact correspondence between the expansion coefficients of the compact and the noncompact models is established, for both expansions, valid to all orders on a finite lattice. The perturbative one involves flipping the sign of the coupling and remains valid in the termwise infinite volume limit. The large $N$ correspondence concerns the functional dependence on the free propagator and holds directly only in finite volume.

hep-th↗
arXiv2006

Quantum Field Theory: Where We Are

We comment on the present status, the concepts and their limitations, and the successes and open problems of the various approaches to a relativistic quantum theory of elementary particles, with a hindsight to questions concerning quantum gravity and string theory.

hep-th↗
arXiv2005

The clash of positivities in topological density correlators

We discuss the apparent conflict between reflection positivity and positivity of the topological susceptibility in two-dimensional nonlinear sigma models and in four-dimensional gauge theories. We pay special attention to the fact that this apparent conflict is already present on the lattice; its resolution puts some nontrivial restrictions on the short-distance behavior of the lattice correlator. It is found that these restrictions can be satisfied both in the case of asymptotic freedom and the dissident scenario of a critical point at finite coupling.

hep-lat↗
arXiv2005

The high density region of QCD in a large mass and chemical potential model

We study the high density region of QCD within an effective model obtained in the frame of the hopping parameter expansion. The model still acknowledges the sign problem peculiar to non-zero chemical potential, but it permits the development of refined algorithms which ensure a good overlap of the Monte Carlo ensemble with the true one. We review the main features of the model, including the most explicit form of the resumed expansion, and present calculations concerning the dependence of various observables on the chemical potential and on the temperature, in particular of the charge density and the diquark susceptibility, which may be used to characterize the various phases expected at high baryonic density.

hep-lat↗
arXiv2005

Perturbation theory for O(3) topological charge correlators

To check the consistency of positivity requirements for the two-point correlation function of the topological charge density, which were identified in a previous paper, we are computing perturbatively this two-point correlation function in the two-dimensional O(3) model. We find that at the one-loop level these requirements are fulfilled.

hep-lat↗
arXiv2005

Non-amenability and spontaneous symmetry breaking -- The hyperbolic spin-chain

The hyperbolic spin chain is used to elucidate the notion of spontaneous symmetry breaking for a non-amenable internal symmetry group, here SO(1,2). The noncompact symmetry is shown to be spontaneously broken -- something which would be forbidden for a compact group by the Mermin-Wagner theorem. Expectation functionals are defined through the L \to \infty limit of a chain of length L; the functional measure is found to have its weight mostly on configurations boosted by an amount increasing at least powerlike with L. This entails that despite the non-amenability a certain subclass of noninvariant functions is averaged to an SO(1,2) invariant result. Outside this class symmetry breaking is generic. Performing an Osterwalder-Schrader reconstruction based on the infinite volume averages one finds that the reconstructed quantum theory is different from the original one. The reconstructed Hilbert space is nonseparable and contains a separable subspace of ground states of the reconstructed transfer operator on which SO(1,2) acts in a continuous, unitary and irreducible way.

hep-th↗
arXiv2003

The case against asymptotic freedom

In this talk I give an overview of the work done during the last 15 years in collaboration with the late Adrian Patrascioiu. In this work we accumulated evidence against the commonly accepted view that theories with nonabelian symmetry -- either two dimensional nonlinear $σ$ models or four dimensional Yang-Mills theories -- have the property of asymptotic freedom (AF) usually ascribed to them.

hep-th↗
arXiv2002

Critical behavior in a quasi D dimensional spin model

We study a classical spin model (more precisely a class of models) with O(N) symmetry that can be viewed as a simplified $D$ dimensional lattice model. It is equivalent to a non-translationinvariant one dimensional model and contains the dimensionality $D$ as a parameter that need not be an integer. The critical dimension turns out to be 2, just as in the usual translation invariant models. We study the phase structure, critical phenomena and spontaneous symmetry breaking. Furthermore we compute the perturbation expansion to low order with various boundary conditions. In our simplified models a number of questions can be answered that remain controversial in the translation invariant models, such as the asymptoticity of the perturbation expansion and the role of super-instantons. We find that perturbation theory produces the right asymptotic expansion in dimension $D\leq 2$ only with special boundary conditions. Finally the model allows a test of the percolation ideas of Patrascioiu and Seiler.

hep-lat↗
arXiv2001

Testing Asymptotic Scaling and Nonabelian Symmetry Enhancement

We determine some points on the finite size scaling curve for the correlation length in the two dimensional O(3) and icosahedron spin models. The Monte Carlo data are consistent with the two models possessing the same continuum limit. The data also suggest that the continuum scaling curve lies above the estimate of Kim and of Caracciolo et al and thus leads to larger thermodynamic values of of the correlation length than previously reported.

hep-lat↗
arXiv2000

Percolation and the existence of a soft phase in the classical Heisenberg model

We present the results of a numerical investigation of percolation properties in a version of the classical Heisenberg model. In particular we study the percolation properties of the subsets of the lattice corresponding to equatorial strips of the target manifold ${\cal S}^2$. As shown by us several years ago, this is relevant for the existence of a massless phase of the model. Our investigation yields strong evidence that such a massless phase does indeed exist. It is further shown that this result implies lack of asymptotic freedom in the massive continuum limit. A heuristic estimate of the transition temperature is given which is consistent with the numerical data.

hep-th↗