Loop Lattice Research Articles

Dual formulations of Polyakov loop lattice models

Dual representations are constructed for non-abelian lattice spin models with U(N) and SU(N) symmetry groups, for all N and in any dimension. These models are usually related to the effective models describing the interaction between Polyakov loops in the strong coupled QCD. The original spin degrees of freedom are explicitly integrated out and a dual theory appears to be a local theory for the dual integer-valued variables. The construction is performed for the partition function and for the most general correlation function. The latter include the two-point function corresponding to quark-anti-quark free energy and the N-point function related to the free energy of a baryon. We consider both pure gauge models and models with static fermion determinant for both the staggered and Wilson fermions with an arbitrary number of flavours. While the Boltzmann weights of such models are complex in the presence of non-zero chemical potential the dual Boltzmann weights appear to be strictly positive on admissible configurations. An essential part of this work with respect to previous studies is an extension of the dual representation to the case of 1) an arbitrary value of the temporal coupling constant in the Wilson action and 2) an arbitrary number of flavours of static quark determinants. The applications and extensions of the results are discussed in detail. In particular, we outline a possible approach to Monte-Carlo simulations of the dual theory, to the large N expansion and to the development of a tensor renormalization group.

Euclidean Gibbs measures on loop lattices: Existence and a priori estimates

We present a new method to prove existence and uniform a priori estimates for Euclidean Gibbs measures corresponding to quantum anharmonic crystals. It is based first on the alternative characterization of Gibbs measures in terms of their logarithmic derivatives through integration by parts formulas, and second on the choice of appropriate Lyapunov functionals.

Moments of singlet parton densities on the lattice in the Schrödinger functional scheme

A non-perturbative computation of the evolution of singlet parton densities without gauge-fixing requires a gauge invariant gluon source operator. Within the Schrödinger Functional scheme (SF), such a source can be defined in terms of path ordered products of gauge links, connected to the time boundaries. In this paper we adopt this definition and perform a one loop lattice computation of the renormalization constants of the twist-2 operators that correspond to the second moment of singlet parton densities. This calculation fixes the connection between the lattice SF scheme where a non-perturbative evaluation of the absolute normalization of singlet parton densities can be made at low energy and the MS scheme where one can extract the experimental values.

Gibbs states on loop lattices: existence and a priori estimates

We prove existence and uniform a priori estimates for Euclidean Gibbs states of quantum lattice systems with unbounded spins. These results substantially extend all previous existence results. Detailed proofs are contained in [3].

Two loop lattice expansion of the Schrödinger functional coupling in improved QCD

The contributions of the improved fermion action of Sheikholeslami and Wohlert to the two loop coefficient of the expansion of the Schrödinger functional coupling in terms of the lattice coupling are calculated for the gauge group SU(3). These coefficients are required for the second order relation of lattice data to the MS -coupling. By taking into account all improvement coefficients we are able to improve the Schrödinger functional to two loop order.

Two loop lattice expansion of the Schrödinger functional coupling in improved QCD

Two loop lattice expansion of the Schrödinger functional coupling in improved QCD

Clustering theory of atomic defects

Clustering of atomic defects leads to changes in the microstructure of materials, and hence induces drastic variations in their properties. In many technical fields, the role of defect clustering is very significant, and is sometimes limiting to further progress. We present here a comprehensive review of the theory of atomic defect clustering under non-equilibrium conditions, particularly encountered during irradiation of materials with energetic particles, as well as during material processing by energetic sources. These conditions are met in a wide range of technical applications, ranging from nuclear and fusion energy to microelectronics and surface engineering. We first present a general stochastic framework for the evolution of atomic clusters, and show how this can be described within the context of death-and-birth processes. This leads to the well-known master equation for microscopic atomic clusters. In the limiting case of a Poissonian process for the transition probabilities between cluster sizes, the master equation tends, in the macroscopic limit, to the mean-field approximation embodied by the theory of rate processes. When atomic clusters grow or shrink by the absorption of single atomic defects, a continuum Fokker-Planck approximation can be derived. Within this approximation, the evolution of interstitial loops, voids, bubbles, and general clusters of complex phases is presented, and in some cases, good agreement with experiments is obtained. It is shown that because of coalescence reactions, the evolution of surface atomic clusters during atom deposition processes is best described by kinetic moment equations, directly derived from rate equations. It is shown that breaking the symmetry of space or time leads to drastic variations in the size and space distributions of defect clusters. Examples are given for pulsed irradiation conditions, where it is shown that non-linear rate processes enhance cluster formation during on-time, and could lead to their dissolution during the off-time at high temperature. On the other hand, fluctuations are shown to result in instabilities and spatial self-organization of defect clusters. Description of pattern formation during irradiation, such as void and interstitial loop lattices, is very well described by a Ginzburg-Landau type equation, reminiscent of phase transitions under thermodynamic equilibrium conditions.

Uniqueness of Gibbs states on loop lattices

We prove the uniqueness of Gibbs states for certain quantum lattice systems with unbounded spins, based on Dobrushin's fundamental uniqueness criterion. The necessary estimates for the Vasershtein distance between one-point conditional distributions, with boundary conditions differing only at one site, are obtained from the Log-Sobolev inequality which holds on the infinite dimensional single spin (= loop) spaces. Detailed proofs are contained in [2] and [3]. The latter deals with the more general case of nonharmonic pair potentials with possibly infinite radius of interaction.

Computer two dimensional maps of loop soliton lattice systems using the new approach to the no integrability Aesthetic field equations

We show that there are varieties of somewhat different loop soliton lattices when we specify an integration path in No Integrability Aesthetic Field Theory. These are illustrated using two dimensional computer maps. We have previously studied several such systems using the new approach to non‐integrable systems developed in previous papers. [1‐3]. The results of these earlier papers indicated that the solitons were rearranged by the new integration scheme in an erratic looking manner. However, we were restricted to regions close to the origin in these studies. With additional computer time made available and the use of tapes to store large amounts of information we have studied the above loop soliton systems and have been able to map considerable larger regions of the x, y plane. Symmetries for locations of the planar maxima and minima have been uncovered within a particular quadrant, although the symmetry found is not as great as the lattice. The type symmetry found is not maintained when more than one quadrant is involved. We also study a system that can be looked at as a perturbation of a loop lattice system. A brief discussion of the background material appears in the appendix.

Rearrangement of lattice particles

We have studied origin point data which lead to soliton loop lattice systems when we specify an integration path in no integrability Aesthetic Field Theory. When we applied the integration scheme developed in previous paper we found that the solitons get rearranged. Close to the origin we saw a system more disorderly than the lattice. However, farther from the origin in two dimensional maps the location of planar maxima (minima) for fixedybecame regular. In this paper, we investigate various approaches with the aim of enlarging the nonsymmetric regions. Integrating inzdid not lead to an enlarged nonsymmetric region. We were able to enlarge the region by altering the magnitudes appearing in the origin point data. It is not clear if we can continually enlarge the nonsymmetric region by this method. We studied what we call an “imperfect” lattice which in a coarse sense can be thought of as being comprised of soliton loops when we specify an integration path. Here the integration scheme did not lead to an exact symmetry, but there was a repeat of “type” structures (as indicated by observations of contour lines in the maps). We then extended the system to higher dimensions. In particular, we studied a complex six dimensional space which is a natural extension of Minkowski space as an example. The system studied gave rise to a loop lattice, but with magnitudes of maxima (minima) of the different loops varying in an oscillatory way. When we applied the integration scheme to this system found no sign of the previously discussed symmetry in the domain studied although the system is not free from other regularities (this is also the case when magnitudes are altered).

Summation-over-paths degree of freedom and the loop lattice

Previously we found different kinds of lattice solutions in the no-integrability aesthetic field theory when we specify an integration path. When we combine the lattice origin-point data and aesthetic field equations with the summation-over-paths degree of freedom, the lattice solution is altered in a dramatic way. In this paper we study particular how a loop lattice system is affected by the summation over paths. We find likely evidence for two loop particles. The other potential particle systems could be closed strings (loops), but increased computer capability would be necessary to confirm this.

Use of commutation relations in no integrability aesthetic field theory

We work with a simple set of no integrability field equations obtained from the aesthetic field theory. We show that with the use of commutation relations we can calculate the field at any point by integrating only along a single path, together with a correction term which is different at each point. This approach is superior, from a numerical point of view, to the summation over paths technique which we developed in a previous paper, because we can better avoid the path proliferation problem. Furthermore, we can make use of considerably smaller grid sizes, which allows for greater accuracy. The resulting map obtained for the simple system of equations considered was similar to the summation over paths results. We found no sign of any two dimensional maxima or minima in the region studied. Combining this result with our previous studies, we conclude that the type of result that we get, when we consider the summation over paths degree of freedom, depends on the kind of lattice solution we are working with. In particular, the summation over paths degree of freedom appears consistent with multiparticles for a “loop” lattice. A loop lattice is illustrated in the Appendix. It appears when we specify an integration path.

Alternative approach to no integrability field theory

This is another paper in a series which illustrates that foregoing the rules of traditional calculus opens up a realm of possibilities. We study the “aesthetic field theory”, although the techniques are more general. We introduce a new integration procedure which is consistent with aesthetic principles. There is a single change function associated with each point rather than many change quantities at a point (there is a change quantity associated with each path going through any point in our previous work). The integration procedure is consistent with the rules of traditional calculus when the integrability equations are satisfied. We find that the new procedure, when applied to the data associated with the loop lattice, leads to multiple maxima and minima on a plane. Another lattice solution (point lattice) does not show evidence of being bounded in certain domains of the region studied. In the summation-over-paths method all the paths from the origin point are needed to calculate the field. Knowledge of the field on one hypersurface is not sufficient to obtain the field on the succeeding hypersurface. In the present approach to no integrability, we can calculate the field on one hypersurface from knowledge of the field on the proceeding hypersurface (without requiring knowledge of the past history). Thus, the new approach enables us to obtain this desirable feature of a hyperbolic theory without requiring the field to be arbitrary on a hypersurface.

Some implications of anisotropic self-interstitial diffusion on void swelling in metals

Earlier work has suggested the probable role of two-dimensional diffusion of self-interstitial atoms (SIA) in the phenomena of void, bubble and vacancy loop lattice formation in metals. In the present paper other consequences of 2-D SIA diffusion are examined. These include enhanced void swelling rates at low displacement doses and the later saturation of swelling, boundary effects and correlated recombination of SIA and vacancies.

A computer simulation of the two-dimensional SIA diffusion model for void lattice formation

In previous work it was suggested that the two-dimensional diffusion of self-interstitial atoms (SIA) could be the cause of void, bubble and vacancy loop lattice formation in metals. The model was based on the property that 2-d SIA diffusion can promote the planar ordering of vacancy sinks, such as voids, in planes parallel to the plane of SIA diffusion. Most of the discussion centred on molybdenum, where a 2-d SIA jump-step on {011} planes had recently been proposed. Although a number of experimental observations, such as the evolution of the lattice from long range planar ordering through to a perfect lattice, fitted the model, it was not possible to demonstrate formally that independent planar alignment on all six sets of {011} planes must lead to a perfect bcc lattice. The purpose of the present paper is to describe a computer approach in which the evolution of a random distribution of voids, subjected to 2-d SIA diffusion on the different sets of planes, was simulated. The results showed that the independent planar ordering can readily combine to form a regular bcc void lattice characteristic of those found in molybdenum and other bcc metals. This demonstration clearly strengthens the proposed model; when supporting evidence is taken into account the case for its validity appears very strong. The paper includes a relevant digression on the small effect (relative to pre-exponential parameters) that low temperature activation energies need have on high temperature defect behaviour. In addition the paper discusses the extension of the model to fcc and hcp metals, where 2-d SIA diffusion on the close packed planes is required.

The depth dependent damage profile in nickel irradiated with nickel or copper ions

High purity nickel samples have been irradiated with high energy (> 14 MeV) heavy ions and the resultant microstructure examined in cross section (i.e., in a plane parallel to the incident ions). The microstructure observed after irradiation with copper ions was not significantly different from that observed after nickel ion irradiation. Swelling near the ion end-of-range was less than that in the mid-range in both the self-ion and copper ion irradiated samples. A 100 nm void denuded zone was observed at the front surface in samples irradiated at 525 °C, and voids were observed at depths ≈ 15% greater than the calculated damage curves. An irradiation at 200 °C produced a loop lattice structure aligned along {001} with a spacing of ≈40 nm. The steady-state rate equations were formulated and solved to give the depth dependent defect concentrations. The calculated void growths rate was found to qualitatively fit the experimental results. A depth dependent radiation enhanced diffusion model predicted the result that for ion fluences used in this study, the injected copper impurities were confined to the end-of-range region.

Ferromagnetic Spin Excitations in a One-Band Metallic Loop, the Case of Long-Range Interactions

We extend the previous treatment of one-band spin excitations to include an arbitrary band term and a long-range two-body interaction. The system still consists of $N$ electrons in a periodic loop lattice with $N$ sites. It is found that the number of bound states depends on the range of the effective two-body interaction, and that the distribution in energy scale of these bound states depends on the spatial variation of the two-body interaction strength.