15,613 research outputs found
Algebraic reduction of the Ising model
We consider the Ising model on a cylindrical lattice of L columns, with
fixed-spin boundary conditions on the top and bottom rows. The spontaneous
magnetization can be written in terms of partition functions on this lattice.
We show how we can use the Clifford algebra of Kaufman to write these partition
functions in terms of L by L determinants, and then further reduce them to m by
m determinants, where m is approximately L/2. In this form the results can be
compared with those of the Ising case of the superintegrable chiral Potts
model. They point to a way of calculating the spontaneous magnetization of that
more general model algebraically.Comment: 25 pages, one figure, last reference completed. Various typos fixed.
Changes on 12 July 2008: Fig 1, 0 to +1; before (2.1), if to is; after (4.6),
from to form; before (4.46), first three to middle two; before (4.46), last
to others; Conclusions, 2nd para, insert how ; renewcommand \i to be \rm
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
Oneness, Aspects, and the Neo-Confucians
Confucius gave counsel that is notoriously hard to follow: "What you do not wish for
yourself, do not impose on others" (Huang 1997: 15.24). People tend to be concerned
with themselves and to be indifferent to most others. We are distinct from others so our self-concern does not include them, or so it seems. Were we to realize this distinctness is merely apparent--that our true self includes others--Confucius's counsel would be easier to follow. Concern for our true self would extend concern beyond the narrow selves we appear to be. The neo-Confucians held just such a view. They espoused an identity with the universe and everything in it, arguing that this identity explains a natural concern for everyone and everything, not just for our narrow selves. However, many things in the universe differ from each other, that is, some have qualities others lack. If they are all one and the same thing then that one thing differs from itself. I will suggest that the objection can be answered with some metaphysical
innovation. I will address the objection by sketching a theory--call it the theory of aspects--that explains how numerically identical things can differ qualitatively
Hume on Substance: A Critique of Locke
The ancient theory of substance and accident is supposed to make sense of complex unities in a way that respects both their unity and their complexity. On Hume’s view such complex unities are only fictitiously unities. This result follows from his thoroughgoing critique of the theory of substance.
I will characterize the theory Hume is critiquing as it is presented in Locke, presupposing what Bennett calls the “Leibnizian interpretation.” Locke uses the word ‘substance’ in two senses. Call substance in the first sense “individual substance” and in the second sense “pure substance.”
I will discuss the seven main parts of Hume’s view: (I) that we have no idea of pure substance, (II) that there is no complex individual substance, except in a loose sense, (III) that the fiction of complex individual substance arises in a way parallel to that of the fiction of identity through time, and (IV) results in the fiction of pure substance, (V) that simple qualities and perceptions satisfy the definition of individual substance, (VI) that there is no such thing as inherence, and (VII) that there is no such thing as pure substance.
Hume’s views on substance are often mentioned without being discussed in detail. Kemp Smith, Stroud, and Garrett, for example, mostly summarize various claims of Hume in the course of expounding on his theory of the idea of personal identity. In contrast, I will attempt to present a systematic treatment of Hume on substance as a refutation of Locke
Corner transfer matrices in statistical mechanics
Corner transfer matrices are a useful tool in the statistical mechanics of
simple two-dimensinal models. They can be very effective way of obtaining
series expansions of unsolved models, and of calculating the order parameters
of solved ones. Here we review these features and discuss the reason why the
method fails to give the order parameter of the chiral Potts model.Comment: 18 pages, 4 figures, for Proceedings of Conference on Symmetries and
Integrability of Difference Equations. (SIDE VII), Melbourne, July 200
Gaudin Hypothesis for the XYZ Spin Chain
The XYZ spin chain is considered in the framework of the generalized
algebraic Bethe ansatz developed by Takhtajan and Faddeev. The sum of norms of
the Bethe vectors is computed and expressed in the form of a Jacobian. This
result corresponds to the Gaudin hypothesis for the XYZ spin chain.Comment: 12 pages, LaTeX2e (+ amssymb, amsthm); to appear in J. Phys.
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
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
A Bethe Ansatz Study of Free Energy and Excitation Spectrum for Even Spin Fateev Zamolodchikov Model
A Bethe Ansatz study of a self dual Z_N spin model is undertaken for even
spin system. One has to solve a coupled system of Bethe Ansatz Equations (BAE)
involving zeroes of two families of transfer matrices. A numerical study on
finite size lattices is done for identification of elementary excitations over
the Ferromagnetic and Antiferromagnetic ground states. The free energies for
both Ferromagnetic and Antiferromagnetic ground states and dispersion relation
for elementary excitations are found.Comment: 25 pages, 4 figure
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
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