9,414 research outputs found
The challenge of the chiral Potts model
The chiral Potts model continues to pose particular challenges in statistical
mechanics: it is ``exactly solvable'' in the sense that it satisfies the
Yang-Baxter relation, but actually obtaining the solution is not easy. Its free
energy was calculated in 1988 and the order parameter was conjectured in full
generality a year later.
However, a derivation of that conjecture had to wait until 2005. Here we
discuss that derivation.Comment: 22 pages, 3 figures, 29 reference
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
Exact results for the one-dimensional many-body problem with contact interaction: Including a tunable impurity
The one-dimensional problem of particles with contact interaction in the
presence of a tunable transmitting and reflecting impurity is investigated
along the lines of the coordinate Bethe ansatz. As a result, the system is
shown to be exactly solvable by determining the eigenfunctions and the energy
spectrum. The latter is given by the solutions of the Bethe ansatz equations
which we establish for different boundary conditions in the presence of the
impurity. These impurity Bethe equations contain as special cases well-known
Bethe equations for systems on the half-line. We briefly study them on their
own through the toy-examples of one and two particles. It turns out that the
impurity can be tuned to lift degeneracies in the energies and can create bound
states when it is sufficiently attractive. The example of an impurity sitting
at the center of a box and breaking parity invariance shows that such an
impurity can be used to confine asymmetrically a stationary state. This could
have interesting applications in condensed matter physics.Comment: 20 pages, 5 figures, version accepted for publication: some typos
corrected, references and comments adde
Tetromino tilings and the Tutte polynomial
We consider tiling rectangles of size 4m x 4n by T-shaped tetrominoes. Each
tile is assigned a weight that depends on its orientation and position on the
lattice. For a particular choice of the weights, the generating function of
tilings is shown to be the evaluation of the multivariate Tutte polynomial
Z\_G(Q,v) (known also to physicists as the partition function of the Q-state
Potts model) on an (m-1) x (n-1) rectangle G, where the parameter Q and the
edge weights v can take arbitrary values depending on the tile weights.Comment: 8 pages, 6 figure
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
Eigenvectors of Baxter-Bazhanov-Stroganov \tau^{(2)}(t_q) model with fixed-spin boundary conditions
The aim of this contribution is to give the explicit formulas for the
eigenvectors of the transfer-matrix of Baxter-Bazhanov-Stroganov (BBS) model
(N-state spin model) with fixed-spin boundary conditions. These formulas are
obtained by a limiting procedure from the formulas for the eigenvectors of
periodic BBS model. The latter formulas were derived in the framework of the
Sklyanin's method of separation of variables. In the case of fixed-spin
boundaries the corresponding T-Q Baxter equations for the functions of
separated variables are solved explicitly. As a particular case we obtain the
eigenvectors of the Hamiltonian of Ising-like Z_N quantum chain model.Comment: 14 pages, paper submitted to Proceedings of the International
Workshop "Classical and Quantum Integrable Systems" (Dubna, January, 2007
General scalar products in the arbitrary six-vertex model
In this work we use the algebraic Bethe ansatz to derive the general scalar
product in the six-vertex model for generic Boltzmann weights. We performed
this calculation using only the unitarity property, the Yang-Baxter algebra and
the Yang-Baxter equation. We have derived a recurrence relation for the scalar
product. The solution of this relation was written in terms of the domain wall
partition functions. By its turn, these partition functions were also obtained
for generic Boltzmann weights, which provided us with an explicit expression
for the general scalar product.Comment: 24 page
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
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