494 research outputs found
Free energy of the three-state model as a product of elliptic functions
{We show that the free energy of the three-state
model can be expressed as products of Jacobi elliptic
functions, the arguments being those of an hyperelliptic parametrization of the
associated chiral Potts model. This is the first application of such a
parametrization to the -state chiral
Potts free energy problem for .Comment: 20 pages, 3 figure
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
Planar lattice gases with nearest-neighbour exclusion
We discuss the hard-hexagon and hard-square problems, as well as the
corresponding problem on the honeycomb lattice. The case when the activity is
unity is of interest to combinatorialists, being the problem of counting binary
matrices with no two adjacent 1's. For this case we use the powerful corner
transfer matrix method to numerically evaluate the partition function per site,
density and some near-neighbour correlations to high accuracy. In particular
for the square lattice we obtain the partition function per site to 43 decimal
places.Comment: 16 pages, 2 built-in Latex figures, 4 table
New Q matrices and their functional equations for the eight vertex model at elliptic roots of unity
The Q matrix invented by Baxter in 1972 to solve the eight vertex model at
roots of unity exists for all values of N, the number of sites in the chain,
but only for a subset of roots of unity. We show in this paper that a new Q
matrix, which has recently been introduced and is non zero only for N even,
exists for all roots of unity. In addition we consider the relations between
all of the known Q matrices of the eight vertex model and conjecture functional
equations for them.Comment: 20 pages, 2 Postscript figure
A conjecture for the superintegrable chiral Potts model
We adapt our previous results for the ``partition function'' of the
superintegrable chiral Potts model with open boundaries to obtain the
corresponding matrix elements of e^{-\alpha H}, where H is the associated
hamiltonian. The spontaneous magnetization M_r can be expressed in terms of
particular matrix elements of e^{-\alpha H} S^r_1 \e^{-\beta H}, where S_1 is a
diagonal matrix.We present a conjecture for these matrix elements as an m by m
determinant, where m is proportional to the width of the lattice. The author
has previously derived the spontaneous magnetization of the chiral Potts model
by analytic means, but hopes that this work will facilitate a more algebraic
derivation, similar to that of Yang for the Ising model.Comment: 19 pages, one figure; Corrections made between 28 March 2008 and 28
April 2008: (1) 2.10: q to p; (2) 3.1: epsilon to 0 (not infinity); (3) 5.29:
p to q; (4) p14: sub-head: p, q to q,p; (5) p15: sub-head: p, q to q,p; (6)
7.5 second theta to -theta ; (7) before 7.6: make more explicit definition of
lambda_j. Several other typos fixed late
Spin operator matrix elements in the superintegrable chiral Potts quantum chain
We derive spin operator matrix elements between general eigenstates of the
superintegrable Z_N-symmetric chiral Potts quantum chain of finite length. Our
starting point is the extended Onsager algebra recently proposed by R.Baxter.
For each pair of spaces (Onsager sectors) of the irreducible representations of
the Onsager algebra, we calculate the spin matrix elements between the
eigenstates of the Hamiltonian of the quantum chain in factorized form, up to
an overall scalar factor. This factor is known for the ground state Onsager
sectors. For the matrix elements between the ground states of these sectors we
perform the thermodynamic limit and obtain the formula for the order
parameters. For the Ising quantum chain in a transverse field (N=2 case) the
factorized form for the matrix elements coincides with the corresponding
expressions obtained recently by the Separation of Variables Method.Comment: 24 pages, 1 figur
Some remarks on a generalization of the superintegrable chiral Potts model
The spontaneous magnetization of a two-dimensional lattice model can be
expressed in terms of the partition function of a system with fixed
boundary spins and an extra weight dependent on the value of a particular
central spin. For the superintegrable case of the chiral Potts model with
cylindrical boundary conditions, W can be expressed in terms of reduced
hamiltonians H and a central spin operator S. We conjectured in a previous
paper that W can be written as a determinant, similar to that of the Ising
model. Here we generalize this conjecture to any Hamiltonians that satisfy a
more general Onsager algebra, and give a conjecture for the elements of S.Comment: 18 pages, one figur
Competing density-wave orders in a one-dimensional hard-boson model
We describe the zero-temperature phase diagram of a model of bosons,
occupying sites of a linear chain, which obey a hard-exclusion constraint: any
two nearest-neighbor sites may have at most one boson. A special case of our
model was recently proposed as a description of a ``tilted'' Mott insulator of
atoms trapped in an optical lattice. Our quantum Hamiltonian is shown to
generate the transfer matrix of Baxter's hard-square model. Aided by exact
solutions of a number of special cases, and by numerical studies, we obtain a
phase diagram containing states with long-range density-wave order with period
2 and period 3, and also a floating incommensurate phase. Critical theories for
the various quantum phase transitions are presented. As a byproduct, we show
how to compute the Luttinger parameter in integrable theories with
hard-exclusion constraints.Comment: 16 page
Order Parameters of the Dilute A Models
The free energy and local height probabilities of the dilute A models with
broken \Integer_2 symmetry are calculated analytically using inversion and
corner transfer matrix methods. These models possess four critical branches.
The first two branches provide new realisations of the unitary minimal series
and the other two branches give a direct product of this series with an Ising
model. We identify the integrable perturbations which move the dilute A models
away from the critical limit. Generalised order parameters are defined and
their critical exponents extracted. The associated conformal weights are found
to occur on the diagonal of the relevant Kac table. In an appropriate regime
the dilute A model lies in the universality class of the Ising model in a
magnetic field. In this case we obtain the magnetic exponent
directly, without the use of scaling relations.Comment: 53 pages, LaTex, ITFA 93-1
The "inversion relation" method for obtaining the free energy of the chiral Potts model
We derive the free energy of the chiral Potts model by the infinite lattice
``inversion relation'' method. This method is non-rigorous in that it always
needs appropriate analyticity assumptions. Guided by previous calculations
based on exact finite-lattice functional relations, we find that in addition to
the usual assumption that the free energy be analytic and bounded in some
principal domain of the rapidity parameter space that includes the physical
regime, we also need a much less obvious symmetry. We can then obtain the free
energy by Wiener-Hopf factorization in the complex planes of appropriate
variables. Together with the inversion relation, this symmetry relates the
values of the free energy in all neighbouring domains to those in the principal
domain.Comment: 27 pages, 4 figure
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