D. J. Wallace

Monte Carlo renormalisation group studies of SU(3) lattice gauge theory

Results are reported of Monte Carlo renormalisation group studies of the approach to asymptotic scaling in SU(3) lattice gauge theory. By comparing measurements on 8 4 and 16 4 lattices, estimates are obtained for the shift, [Delta][beta], in the fundamental plaquette coupling, [beta9, corresponding to a change of length scale by a factor of 2. The definitions of block link variables contain a free parameter whose value can be optimised to minimise the transient flow to a renormalised trajectory. Our results, at [beta] = 6.0, 6.3 and 6.6, are consistent with those obtained previously with the improved ratio method, which is also briefly discussed. In both methods simulation is performed only with the standard Wilson action. An important feature of the results is the appearance of a pronounced dip in [Delta][beta] which implies that in the presently accesible range of [beta] the asymptotic value is approached from below, and its onset is delayed.

Hadron Mass Calculations With Susskind and Wilson Fermions in the Fundamental Adjoint Plane

Abstract We report the results of hadron mass calculations in the valence (quenched) approximation, on an 8 3 × 16 lattice. For Wilson fermions with the standard Wilson action we find good agreement with results form the hopping parameter expansion on a 16 4 lattice at β = 5.7, with only a small finite size effect in the anticipated direction. The proton-to-rho mass ration is however too high by 60% and the delta-proton mass difference is too small. We have repeated these calculations at two points of the same string tension in the plane of the fundamental and adjoint couplings, in an attempt to avoid the unphysical critical point there; within statistical errors the meson masses remain the same, there is an improvement in the delta-proton splitting and the proton mass decreases slightly, but not enough to produce agreement with experiment. The estimates of lines of constant string tension obtained as a preliminary to the mass calculation are in good agreement with weak coupling expansions at the larger β values and cross over towards strong coupling predictions around β = 5.7. Also, crude estimates reveal the disappearance of the specific heat peak as one moves away from the unphysical singularity. For Susskind fermions we find some measure of agreement with other results form a 10 3 × 16 lattice, but large, apparently finite size, effects in the rho mass at low quark mass. Meson masses in lattice units disagree with the Wilson fermion results by as much as a factor of 2. This disagreement persists in the fundamental-adjoint plane, suggesting the importance of studying improved fermion actions. At β = 6.0, Wilson fermion results show clear finite size effects on the 8 3 × 16 lattice.

Implementing neural network models on parallel computers

The remarkable processing capabilities of the nervous system must derive from the large numbers of neurons participating (roughly 10 10 ), since the time-scales involved are of the order of a millisecond, rather than the nanoseconds of modern computers. The neural network models which attempt to capture this behaviour are inherently parallel. We review the implementation of a range of neural network models on SIMD and MIMD computers. On the ICL Distributed Array Processor (DAP), a 4096-processor SIMD machine, we have studied training algorithms in the context of the Hopfield net, with specific applications including the storage of words and continuous text in contentaddressable memory. The Hopfield and Tank analogue neural net has been used for image restoration with the Geman and Geman algorithm. We compare the performance of this scheme on the DAP and on a Meiko Computing Surface, a reconfigurable MIMD array of transputers. We describe also the strategies which we have used to implement the Durbin and Willshaw elastic net model on the Computing Surface. Received June 1987

Dynamics and statistical mechanics of the Hopfield model

The authors present a study of the Hopfield model of the memory characteristics of a network of interconnected two-state neuron variables. The fraction of nominated configurations which the model stores without error is calculated analytically as a function of the number, N, of neurons and the number, n, of the nominated configurations. The calculation is tested by computer simulation. The noise-free (zero-temperature) phase diagram of the model is determined within a replica-symmetric solution of the mean-field equations. The model exhibits a phase transition at alpha ( identical to n/N)= alpha c approximately=0.069; at this point the thermodynamic states having macroscopic overlap with the nominated configurations disappear, implying a discontinuous change in the fraction of bits (of any nominated configuration) recalled correctly. Large scale Monte Carlo simulations using a distributed array processor provide some support for the existence of a phase transition close to the predicted value.

Critical behaviour of the continuous n-component Potts model

The continuous n-component Potts model is studied in the framework of Wilsons multiplicative renormalization group. Apart from the isotropic phi 4 interaction, there is another one inherent in the model for n>2. For n>or=4, it is distinct from any phi 4 interaction that has been investigated in detail. Critical exponents are calculated. The n dependence of the fixed points shows new behaviour; in particular for n in the neighbourhood of five, the epsilon expansion must be reformulated as a power series in epsilon 12/.

Gradient properties of the renormalisation group equations in multicomponent systems

Abstract The existence of a metric, which enables the renormalisation group β functions of a multicomponent field theory to be written as a gradient, has very important implications for the asymptotic behavior of the renormalisation group equations. It is shown that a very simple metric exists in a field theory with n-component Bose fields and arbitrary φ4 interaction, when the β functions are calculated perturbatively up to and including the 2-loop diagrams. This same metric is shown to work for all irreducible diagrams, but it must and can be modified to accommodate reducible 3-loop contributions. The prospects and outlook of this aspect of the renormalisation group are discussed.

Goldstone modes in vacuum decay and first-order phase transitions

The authors introduce effective Hamiltonians for Goldstone modes of the Euclidean group, representing fluctuations in the surface of a critical droplet or in the interface between two phases. The Euclidean invariance is non-linearly realised on the Goldstone fields. The Hamiltonians are non-renormalisable in more than one dimension, showing that the disappearance of a phase transition in one dimension for systems with a discrete symmetry may be interpreted in terms of the infrared instabilities induced by these modes. The existence and form of these Hamiltonians indicates the universality of the essential singularity at a first-order phase transition in models with Euclidean invariance.

Phase diagrams of U(1) lattice Higgs models

Abstract Phase diagrams for charge-one and charge-two scalar particles in compact QED are constructed using Monte Carlo methods on the ICL DAP computer. The results are in agreement with the features predicted by Fradkin and Shenker and with known and new limiting cases.

The renormalisation group approach to scaling in physics

The auth ors review the application of the renormalisation group to many-body problems, with particular emphasis on the scaling behaviour which may arise. An introductory section discusses the difficulties of calculating the many-body effects near a magnetic phase transition, and shows by a simple example how the renormalisation group may be used to predict scaling behaviour. A general discussion of scaling and universality follows, with a review of techniques for calculation in lattice models, and an outline of the approaches in field theory. The range of applications of these techniques is discussed.

Large scale applications of transputers in HEP: The Edinburgh concurrent supercomputer project

Abstract The Edinburgh Concurrent Supercomputer Project is built around a Meiko Computing Surface, with presently some 400 floating-point transputers and 1.6 Gbytes of memory. The first part of this paper gives a brief overview of the projects origins and status. In the second part we review the results of applications in high energy physics, including lattice gauge theory and Monte Carlo event simulation.