Field Theory, Disorder and Simulations

The following sections are included:

• CONTENTS

The following sections are included:

• Field Theory

• Disordered Systems

• Simulations

We review the physical principles which are at the basis of recent field-theoretic computations of the critical exponents in two- and three-dimensional systems. We concentrate on those points that do not show up explicitly in the more standard ε-expansion: they must be discussed with care if one uses a perturbative approach at fixed space dimensions (the loop expansion). We present in detail simple computations of the critical exponents, while we summarize the results of longer and more accurate computations.

The following sections are included:

• Introduction

• The Relation with Critical Phenomena

• Two Examples

• The Large N Expansion

• The Structure around g = 0

• The Solution of the Callan Symanzik Equation

• The Counterterms

• The Callan Symanzik Equation as a Substitute of the Equations of Motion

• Conclusions

• References

The theory of scaling violations in deep inelastic scattering is presented using the parton model language; intuitive physical arguments are used as far as possible. In the comparison between theory and experiments particular attention is payed to the consequences of the opening of the threshold for charm production.

The following sections are included:

• INTRODUCTION

• THE UNSTABLE VACUUM

• THE WRONG VACUUM

• THE LANDAU GHOST

• GAUGE THEORIES

• REFERENCES

The principles of the recent calculations of critical exponents from three- and two-dimensional field theory are reviewed. They rely on the Callan–Symanzik equations, diagram calculations, and on the characterization of the asymptotic behavior of perturbation series at large order. We then present new results concerning the normalization of the large-order behavior.

We study the consequences of the renormalization group on the large momenta behaviour of the Borel transform of the Green's functions. We discuss the implications on the structure of the ultraviolet singularities of the Borel transform.

The singularities of the Borel transform in gauge theories are studied using a few modes approximation. The asymptotic freedom of the theory implies the presence of infrared singularities on the positive real axis; these singularities cannot be computed using semi-classical methods.

We study the structure of infrared divergences in superrenormalizable interactions. We conjecture that there is an extension of the Bogoliubov-Parasiuk-Hepp theorem which copes also with infrared divergences. We discuss the consequences of this conjecture on the singularities of the Borel transform in a massless asymptotic free field theory. The application of these ideas to gauge theories is briefly discussed.

On the example of the semi-classical expansion for the levels of the quartic oscillator −(d²/dq²) + q⁴, we show how the complex WKB method provides information about the singularities of the Borel transform of the semi-classical series. In this problem there occurs a tunneling effect between complex turning points, by which those singularities generate exponentially small, yet detectable, corrections to the energy levels.

Trace identities are derived for the operator p² + x^q, q positive even. For q = 4 these identities can be used to estimate the ground state energy within an accuracy of about 10⁻⁴.

The following sections are included:

• Introduction

• Random magnetic field (mean field approximation)

• An intermezzo on supersymmetry

• Random magnetic field (the replica method)

• Spin glasses (the naive approach)

• Spin glasses (breaking the replica symmetry)

• Spin glasses (the physical interpretation)

• Random potential model (the density of states)

• Random potential model (localisation)

• Acknowledgements

• References

We show that some supersymmetric models can be written as classical stochastic equations. We give examples in one and two dimensions, but in four dimensions the only rotationally invariant supersymmetric model we succeed in writing as a classical stochastic equation is the free field theory. We also discuss spontaneous breaking of supersymmetry; we show it is related to the number of solutions of the stochastic equations.

The following sections are included:

• Introduction

• The naive approach to spin glasses

• The structure of pure equilibrium states

• The solution of the model

• An intermezzo on polymers

• The matching

• Conclusions

• Acknowledgment

• References

A spin glass is a disordered magnet: in a typical example 50% of the bonds (randomly chosen) between two spins are ferromagnetic while the others are antiferromagnetic. From the experimental point of view⁽¹⁾ spin glasses are characterized by the presence of a low temperature phase in which the relaxation time toward equilibrium is very large (as in real glasses). The magnetic susceptibility in the presence of a time dependent magnetic field depends on the frequency also in the region of a few Hertz…

The following sections are included:

• INTRODUCTION

• COMPLEX SYSTEMS

• SPIN GLASSES

• OTHER FRUSTRATED MODELS

• THE TAXONOMY OF NEARLY OPTIMAL CONFIGURATIONS

• MORE INFORMATION OF THE TREE

• DYNAMICS WITH AN APPLICATION TO PROTEINS

• A COMPARISON WITH REAL TAXONOMY

• ON EVOLUTION

• CONCLUSIONS AND GENERAL OUTLOOK

• Acknowledgements

• REFERENCES

It is generally argued that the energy dissipation of three-dimensional turbulent flow is concentrated on a set with non-integer Hausdorff dimension. Recently, in order to explain experimental data, it has been proposed that this set does not possess a global dilatation invariance: it can be considered to be a multifractal set. In this paper we review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the β-model.

The following sections are included:

• INTRODUCTION

• EUCLIDEAN FIELD THEORY

• CONFINEMENT

• LATTICE GAUGE THEORIES

• COMPUTER SIMULATION

• THE HIGH TEMPERATURE EXPANSION

• THE ROUGHENING TRANSITION

• THE LARGE N EXPANSION

• DISCUSSION

• FOOTNOTES AND REFERENCES

The following sections are included:

• Introduction

• Statistical errors

• Finite lattice spacing

• Finite time

• Finite volume

• Extrapolation in the quark masses

• References

The following sections are included:

• INTRODUCTION

• BASIC FORMULAE

• FINITE VOLUME EFFECTS

• THE LATTICE SPACING

• FIGHTING AGAINST STATISTICAL ERRORS

• FIGHTING AGAINST THE CRITICAL SLOWING DOWN

• CONCLUSIONS

• ACKNOWLEDGMENTS

• REFERENCES

• FOOTNOTES

The following sections are included:

• Introduction

• Numerical simulations

• The Langevin equation

• Fermions

• The Monte Carlo Method

• Some general considerations

• Gauge theories

• Lattice gauge theories

• Abelian gauge theories

• SU(3) gauge theories

• Statistical errors

• Appendix 1. Monte Carlo simulation; program listing

• References

The following sections are included:

• Introduction

• The floating numbers

• Random numbers

• Solving differential equations

• Simulating statistical mechanics

• Preliminaries on field theories

• Pure gauge QCD: the spectrum

• The deconfinement transition in pure gauge QCD

• Quenched fermions

• Open problems

• References

The APE computer is a high performance processor designed to provide massive computational power for intrinsically parallel and homogeneous applications. APE is a linear array of processing elements and memory boards that execute in parallel in SIMD mode under the control of a CERN/SLAC 3081/E. Processing elements and memory boards are connected by a ‘circular’ switchnet. The hardware and software architecture of APE, as well as its implementation are discussed in this paper. Some physics results obtained in the simulation of lattice gauge theories are also presented.

We describe APE-100, a SIMD, modular and fine-grained parallel processor architecture for large scale scientific computations. The largest configuration that will be implemented in the present design delivers a peak speed of 100 Gflops. This performance is, for instance, required for high precision computations in Quantum Chromo Dynamics, for which APE-100 is very well suited.