175 research outputs found

### Series studies of the Potts model. I: The simple cubic Ising model

The finite lattice method of series expansion is generalised to the $q$-state
Potts model on the simple cubic lattice.
It is found that the computational effort grows exponentially with the square
of the number of series terms obtained, unlike two-dimensional lattices where
the computational requirements grow exponentially with the number of terms. For
the Ising ($q=2$) case we have extended low-temperature series for the
partition functions, magnetisation and zero-field susceptibility to $u^{26}$
from $u^{20}$. The high-temperature series for the zero-field partition
function is extended from $v^{18}$ to $v^{22}$. Subsequent analysis gives
critical exponents in agreement with those from field theory.Comment: submitted to J. Phys. A: Math. Gen. Uses preprint.sty: included. 24
page

### Osculating and neighbour-avoiding polygons on the square lattice

We study two simple modifications of self-avoiding polygons. Osculating
polygons are a super-set in which we allow the perimeter of the polygon to
touch at a vertex. Neighbour-avoiding polygons are only allowed to have nearest
neighbour vertices provided these are joined by the associated edge and thus
form a sub-set of self-avoiding polygons. We use the finite lattice method to
count the number of osculating polygons and neighbour-avoiding polygons on the
square lattice. We also calculate their radius of gyration and the first
area-weighted moment. Analysis of the series confirms exact predictions for the
critical exponents and the universality of various amplitude combinations. For
both cases we have found exact solutions for the number of convex and
almost-convex polygons.Comment: 14 pages, 5 figure

### Low temperature series expansions for the square lattice Ising model with spin S > 1

We derive low-temperature series (in the variable $u = \exp[-\beta J/S^2]$)
for the spontaneous magnetisation, susceptibility and specific heat of the
spin-$S$ Ising model on the square lattice for $S=\frac32$, 2, $\frac52$, and
3. We determine the location of the physical critical point and non-physical
singularities. The number of non-physical singularities closer to the origin
than the physical critical point grows quite rapidly with $S$. The critical
exponents at the singularities which are closest to the origin and for which we
have reasonably accurate estimates are independent of $S$. Due to the many
non-physical singularities, the estimates for the physical critical point and
exponents are poor for higher values of $S$, though consistent with
universality.Comment: 14 pages, LaTeX with IOP style files (ioplppt.sty), epic.sty and
eepic.sty. To appear in J. Phys.

### High-temperature expansion of the magnetic susceptibility and higher moments of the correlation function for the two-dimensional XY model

We calculate the high-temperature series of the magnetic susceptibility and
the second and fourth moments of the correlation function for the XY model on
the square lattice to order $\beta^{33}$ by applying the improved algorithm of
the finite lattice method. The long series allow us to estimate the inverse
critical temperature as $\beta_c=1.1200(1)$, which is consistent with the most
precise value given previously by the Monte Carlo simulation. The critical
exponent for the multiplicative logarithmic correction is evaluated to be
$\theta=0.054(10)$, which is consistent with the renormalization group
prediction of $\theta={1/16}$.Comment: 13 pages, 8 Postscript figure

### Size and area of square lattice polygons

We use the finite lattice method to calculate the radius of gyration, the
first and second area-weighted moments of self-avoiding polygons on the square
lattice. The series have been calculated for polygons up to perimeter 82.
Analysis of the series yields high accuracy estimates confirming theoretical
predictions for the value of the size exponent, $\nu=3/4$, and certain
universal amplitude combinations. Furthermore, a detailed analysis of the
asymptotic form of the series coefficients provide the firmest evidence to date
for the existence of a correction-to-scaling exponent, $\Delta = 3/2$.Comment: 12 pages 3 figure

### A parallel algorithm for the enumeration of benzenoid hydrocarbons

We present an improved parallel algorithm for the enumeration of fixed
benzenoids B_h containing h hexagonal cells. We can thus extend the enumeration
of B_h from the previous best h=35 up to h=50. Analysis of the associated
generating function confirms to a very high degree of certainty that $B_h \sim
A \kappa^h /h$ and we estimate that the growth constant $\kappa =
5.161930154(8)$ and the amplitude $A=0.2808499(1)$.Comment: 14 pages, 6 figure

### Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices

We present an efficient algorithm for computing the partition function of the
q-colouring problem (chromatic polynomial) on regular two-dimensional lattice
strips. Our construction involves writing the transfer matrix as a product of
sparse matrices, each of dimension ~ 3^m, where m is the number of lattice
spacings across the strip. As a specific application, we obtain the large-q
series of the bulk, surface and corner free energies of the chromatic
polynomial. This extends the existing series for the square lattice by 32
terms, to order q^{-79}. On the triangular lattice, we verify Baxter's
analytical expression for the bulk free energy (to order q^{-40}), and we are
able to conjecture exact product formulae for the surface and corner free
energies.Comment: 17 pages. Version 2: added 4 further term to the serie

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