3,370 research outputs found
Logarithmic Corrections to Scaling in the --Model
We study the distribution of partition function zeroes for the --model in
two dimensions. In particular we find the scaling behaviour of the end of the
distribution of zeroes in the complex external magnetic field plane in the
thermodynamic limit (the Yang--Lee edge) and the form for the density of these
zeroes. Assuming that finite--size scaling holds, we show that there have to
exist logarithmic corrections to the leading scaling behaviour of thermodynamic
quantities in this model. These logarithmic corrections are also manifest in
the finite--size scaling formulae and we identify them numerically. The method
presented here can be used to check the compatibility of scaling behaviour of
odd and even thermodynamic functions in other models too.Comment: 3 pages, latex, 2 figure
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
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
Phase Transition Strength through Densities of General Distributions of Zeroes
A recently developed technique for the determination of the density of
partition function zeroes using data coming from finite-size systems is
extended to deal with cases where the zeroes are not restricted to a curve in
the complex plane and/or come in degenerate sets. The efficacy of the approach
is demonstrated by application to a number of models for which these features
are manifest and the zeroes are readily calculable.Comment: 16 pages, 12 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
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
Logarithmic Corrections to Scaling in the Two Dimensional --Model
By expressing thermodynamic functions in terms of the edge and density of
Lee--Yang zeroes, we relate the scaling behaviour of the specific heat to that
of the zero field magnetic susceptibility in the thermodynamic limit of the
--model in two dimensions. Assuming that finite--size scaling holds, we
show that the conventional Kosterlitz--Thouless scaling predictions for these
thermodynamic functions are not mutually compatable unless they are modified by
multiplicative logarithmic corrections. We identify these logarithmic
corrections analytically in the case of the specific heat and numerically in
the case of the susceptibility. The techniques presented here are general and
can be used to check the compatibility of scaling behaviour of odd and even
thermodynamic functions in other models too.Comment: 11 pages, latex, 4 figure
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