11,719 research outputs found
The order parameter of the chiral Potts model
An outstanding problem in statistical mechanics is the order parameter of the
chiral Potts model. An elegant conjecture for this was made in 1983. It has
since been successfully tested against series expansions, but as far as the
author is aware there is as yet no proof of the conjecture. Here we show that
if one makes a certain analyticity assumption similar to that used to derive
the free energy, then one can indeed verify the conjecture. The method is based
on the ``broken rapidity line'' approach pioneered by Jimbo, Miwa and
Nakayashiki.Comment: 29 pages, 7 figures. Citations made more explicit and some typos
correcte
Ex-nihilo: Obstacles Surrounding Teaching the Standard Model
The model of the Big Bang is an integral part of the national curriculum for
England. Previous work (e.g. Baxter 1989) has shown that pupils often come into
education with many and varied prior misconceptions emanating from both
internal and external sources. Whilst virtually all of these misconceptions can
be remedied, there will remain (by its very nature) the obstacle of ex-nihilo,
as characterised by the question `how do you get something from nothing?' There
are two origins of this obstacle: conceptual (i.e. knowledge-based) and
cultural (e.g. deeply held religious viewpoints). The article shows how the
citizenship section of the national curriculum, coming `online' in England from
September 2002, presents a new opportunity for exploiting these.Comment: 6 pages. Accepted for publication in Physics E
Some comments on developments in exact solutions in statistical mechanics since 1944
Lars Onsager and Bruria Kaufman calculated the partition function of the
Ising model exactly in 1944 and 1949. Since then there have been many
developments in the exact solution of similar, but usually more complicated,
models. Here I shall mention a few, and show how some of the latest work seems
to be returning once again to the properties observed by Onsager and Kaufman.Comment: 28 pages, 5 figures, section on six-vertex model revise
Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz
We connect two alternative concepts of solving integrable models, Baxter's
method of auxiliary matrices (or Q-operators) and the algebraic Bethe ansatz.
The main steps of the calculation are performed in a general setting and a
formula for the Bethe eigenvalues of the Q-operator is derived. A proof is
given for states which contain up to three Bethe roots. Further evidence is
provided by relating the findings to the six-vertex fusion hierarchy. For the
XXZ spin-chain we analyze the cases when the deformation parameter of the
underlying quantum group is evaluated both at and away from a root of unity.Comment: 32 page
Conformal invariance studies of the Baxter-Wu model and a related site-colouring problem
The partition function of the Baxter-Wu model is exactly related to the
generating function of a site-colouring problem on a hexagonal lattice. We
extend the original Bethe ansatz solution of these models in order to obtain
the eigenspectra of their transfer matrices in finite geometries and general
toroidal boundary conditions. The operator content of these models are studied
by solving numerically the Bethe-ansatz equations and by exploring conformal
invariance. Since the eigenspectra are calculated for large lattices, the
corrections to finite-size scaling are also calculated.Comment: 12 pages, latex, to appear in J. Phys. A: Gen. Mat
Auxiliary matrices on both sides of the equator
The spectra of previously constructed auxiliary matrices for the six-vertex
model at roots of unity are investigated for spin-chains of even and odd
length. The two cases show remarkable differences. In particular, it is shown
that for even roots of unity and an odd number of sites the eigenvalues contain
two linear independent solutions to Baxter's TQ-equation corresponding to the
Bethe ansatz equations above and below the equator. In contrast, one finds for
even spin-chains only one linear independent solution and complete strings. The
other main result is the proof of a previous conjecture on the degeneracies of
the six-vertex model at roots of unity. The proof rests on the derivation of a
functional equation for the auxiliary matrices which is closely related to a
functional equation for the eight-vertex model conjectured by Fabricius and
McCoy.Comment: 22 pages; 2nd version: one paragraph added in the conclusion and some
typos correcte
Theory of the Quantum Critical Fluctuations in Cuprates
The statistical mechanics of the time-reversal and inversion symmetry
breaking order parameter, possibly observed in the pseudogap region of the
phase diagram of the Cuprates, can be represented by the Ashkin-Teller model.
We add kinetic energy and dissipation to the model for a quantum generalization
and show that the correlations are determined by two sets of charges, one
interacting locally in time and logarithmically in space and the other locally
in space and logarithmically in time. The quantum critical fluctuations are
derived and shown to be of the form postulated in 1989 to give the marginal
fermi-liquid properties. The model solved and the methods devised are likely to
be of interest also to other quantum phase transitions
Analyticity and Integrabiity in the Chiral Potts Model
We study the perturbation theory for the general non-integrable chiral Potts
model depending on two chiral angles and a strength parameter and show how the
analyticity of the ground state energy and correlation functions dramatically
increases when the angles and the strength parameter satisfy the integrability
condition. We further specialize to the superintegrable case and verify that a
sum rule is obeyed.Comment: 31 pages in harvmac including 9 tables, several misprints eliminate
Star-Triangle Relation for a Three Dimensional Model
The solvable -chiral Potts model can be interpreted as a
three-dimensional lattice model with local interactions. To within a minor
modification of the boundary conditions it is an Ising type model on the body
centered cubic lattice with two- and three-spin interactions. The corresponding
local Boltzmann weights obey a number of simple relations, including a
restricted star-triangle relation, which is a modified version of the
well-known star-triangle relation appearing in two-dimensional models. We show
that these relations lead to remarkable symmetry properties of the Boltzmann
weight function of an elementary cube of the lattice, related to spatial
symmetry group of the cubic lattice. These symmetry properties allow one to
prove the commutativity of the row-to-row transfer matrices, bypassing the
tetrahedron relation. The partition function per site for the infinite lattice
is calculated exactly.Comment: 20 pages, plain TeX, 3 figures, SMS-079-92/MRR-020-92. (corrupted
figures replaced
On kernel engineering via PaleyâWiener
A radial basis function approximation takes the form
where the coefficients a 1,âŠ,a n are real numbers, the centres b 1,âŠ,b n are distinct points in â d , and the function Ï:â d ââ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which Ï is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure ÎŒ for which the convolution Ï=ÎŒ Ï is a function of compact support, and when Ï is polyharmonic. The novelty of this construction is its use of the PaleyâWiener theorem to identify compact support via analysis of the Fourier transform of the new kernel Ï, so providing a new form of kernel engineering
- âŠ