Universal scaling relations for logarithmic-correction exponents

By the early 1960's advances in statistical physics had established the existence of universality classes for systems with second-order phase transitions and characterized these by critical exponents which are different to the classical ones. There followed the discovery of (now famous) scaling relations between the power-law critical exponents describing second-order criticality. These scaling relations are of fundamental importance and now form a cornerstone of statistical mechanics. In certain circumstances, such scaling behaviour is modified by multiplicative logarithmic corrections. These are also characterized by critical exponents, analogous to the standard ones. Recently scaling relations between these logarithmic exponents have been established. Here, the theories associated with these advances are presented and expanded and the status of investigations into logarithmic corrections in a variety of models is reviewed.Comment: Review prepared for the book "Order, Disorder, and Criticality. Vol. III", ed. by Yu. Holovatch and based on the Ising Lectures in Lviv. 48 pages, 1 figur

Two cultures: "them and us" in the academic world

Impact of academic research onto the non-academic world is of increasing importance as authorities seek return on public investment. Impact opens new opportunities for what are known as "professional services": as scientometrical tools bestow some with confidence they can quantify quality, the impact agenda brings lay measurements to evaluation of research. This paper is partly inspired by the famous "two cultures" discussion instigated by C.P. Snow over 60 years ago. He saw a chasm between different academic disciplines and I see a chasm between academics and professional services, bound into contact through competing targets. This paper draws on my personal experience and experiences recounted to me by colleagues in different universities in the UK. It is aimed at igniting discussions amongst people interested in improving the academic world and it is intended in a spirit of collaboration and constructiveness. As a professional services colleague said, what I have to say "needs to be said". It is my pleasure to submit this paper to the Festschrift devoted to the 60th birthday of a renowned physicist, my good friend and colleague Ihor Mryglod. Ihor's role as leader of the Institute for Condensed Matter Physics in Lviv has been essential to generating some of the impact described in this paper and forms a key element of the story I wish to tell.Comment: 20 pages, 9 figure

Homotopy in statistical physics

In condensed matter physics and related areas, topological defects play important roles in phase transitions and critical phenomena. Homotopy theory facilitates the classification of such topological defects. After a pedagogic introduction to the mathematical methods involved in topology and homotopy theory, the role of the latter in a number of mainly low-dimensional statistical-mechanical systems is outlined. Some recent activities in this area are reviewed and some possible future directions are discussed.Comment: Significant extensions and updates: 29 pages, 11 figures. Lecture given at the Mochima Spring School, Mochima, Venezuela, June 2006. To appear in Cond. Matt. Phy

The Strength of First and Second Order Phase Transitions from Partition Function Zeroes

We present a numerical technique employing the density of partition function zeroes (i) to distinguish between phase transitions of first and higher order, (ii) to examine the crossover between such phase transitions and (iii) to measure the strength of first and second order phase transitions in the form of latent heat and critical exponents. These techniques are demonstrated in applications to a number of models for which zeroes are available.Comment: 18 pages, LaTeX, 6 postscript figures, accepted for publication in J. Stat. Phy

Phase transition strengths from the density of partition function zeroes

We report on a new method to extract thermodynamic properties from the density of partition function zeroes on finite lattices. This allows direct determination of the order and strength of phase transitions numerically. Furthermore, it enables efficient distinguishing between first- and second-order transitions, elucidates crossover between them and illuminates the origins of finite-size scaling. The power of the method is illustrated in typical applications for both Fisher and Lee-Yang zeroes.Comment: 3 pages, LaTeX, 4 postscript figures, Lattice2001(spin

New methods to measure phase transition strength

A recently developed technique to determine the order and strength of phase transitions by extracting the density of partition function zeroes (a continuous function) from finite-size systems (a discrete data set) is generalized to systems for which (i) some or all of the zeroes occur in degenerate sets and/or (ii) they are not confined to a singular line in the complex plane. The technique is demonstrated by application to the case of free Wilson fermions.Comment: 3 pages, 2 figures, Lattice2002(spin

Normalization of peer-evaluation measures of group research quality across academic disciplines

Peer-evaluation based measures of group research quality such as the UK's Research Assessment Exercise (RAE), which do not employ bibliometric analyses, cannot directly avail of such methods to normalize research impact across disciplines. This is seen as a conspicuous flaw of such exercises and calls have been made to find a remedy. Here a simple, systematic solution is proposed based upon a mathematical model for the relationship between research quality and group quantity. This model manifests both the Matthew effect and a phenomenon akin to the Ringelmann effect and reveals the existence of two critical masses for each academic discipline: a lower value, below which groups are vulnerable, and an upper value beyond which the dependency of quality on quantity reduces and plateaus appear when the critical masses are large. A possible normalization procedure is then to pitch these plateaus at similar levels. We examine the consequences of this procedure at RAE for a multitude of academic disciplines, corresponding to a range of critical masses.Comment: 5 figures, each with 2 panels. To appear in the journal Research Evaluatio

Finite-size scaling and corrections in the Ising model with Brascamp-Kunz boundary conditions

The Ising model in two dimensions with the special boundary conditions of Brascamp and Kunz is analysed. Leading and sub-dominant scaling behaviour of the Fisher zeroes are determined exactly. The finite-size scaling, with corrections, of the specific heat is determined both at the critical and pseudocritical points. The shift exponents associated with scaling of the pseudocritical points are not the same as the inverse correlation length critical exponent. All corrections to scaling are analytic.Comment: 15 page

Critical phenomena for systems under constraint

It is well known that the imposition of a constraint can transform the properties of critical systems. Early work on this phemomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the leading critical exponents describing phase transitions. Recent work extended these considerations to critical amplitudes and to exponents governing logarithmic corrections in certain marginal scenarios. Here these old and new results are gathered and summarised. The involutory nature of transformations between the critical parameters describing ideal and constrained systems are also discussed, paying particular attention to matters relating to universality