arXiv subjects

A J Guttmann

Publications and source records attributed to A J Guttmann.

8 recordsLinked to original sources

arXiv1996

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

We derive low-temperature series (in the variable $u = \exp[-βJ/S^2]$) 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.

cond-mat↗

arXiv1996

Extrapolation procedure for low-temperature series for the square lattice spin-1 Ising model

The finite-lattice method of series expansions has been combined with a new extrapolation procedure to extend the low-temperature series for the specific heat, spontaneous magnetisation and susceptibility of the spin-1 Ising model on the square lattice. The extended series were derived by directly calculating the series to order 99 (in the variable $u=\exp[-J/k_B T])$ and using the new extrapolation procedure to calculate an additional 13--14 terms.

cond-mat↗

arXiv1996

Directed percolation near a wall

Series expansion methods are used to study directed bond percolation clusters on the square lattice whose lateral growth is restricted by a wall parallel to the growth direction. The percolation threshold $p_c$ is found to be the same as that for the bulk. However the values of the critical exponents for the percolation probability and mean cluster size are quite different from those for the bulk and are estimated by $β_1 = 0.7338 \pm 0.0001$ and $γ_1 = 1.8207 \pm 0.0004$ respectively. On the other hand the exponent $Δ_1=β_1 +γ_1$ characterising the scale of the cluster size distribution is found to be unchanged by the presence of the wall. The parallel connectedness length, which is the scale for the cluster length distribution, has an exponent which we estimate to be $ν_{1\parallel} = 1.7337 \pm 0.0004$ and is also unchanged. The exponent $τ_1$ of the mean cluster length is related to $β_1$ and $ν_{1\parallel}$ by the scaling relation $ν_{1\parallel} = β_1 + τ_1$ and using the above estimates yields $τ_1 = 1$ to within the accuracy of our results. We conjecture that this value of $τ_1$ is exact and further support for the conjecture is provided by the direct series expansion estimate $τ_1= 1.0002 \pm 0.0003$.

cond-mat↗

arXiv1993

Series studies of the Potts model. III: The 3-state model on the simple cubic lattice

The finite lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the $3$-state Potts model on the simple cubic lattice to order $z^{43}$ and the high-temperature expansion of the partition function to order $v^{21}$. We use the numerical data to show that the transition is first-order, and estimate the latent heat, the discontinuity in the magnetisation, and a number of other critical parameters.

hep-lat↗

arXiv1993

Series studies of the Potts model. II: Bulk series for the square lattice

The finite lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the $q$-state Potts model to order $z^{56}$ (i.e. $u^{28}$), $z^{47}$, $z^{43}$, $z^{39}$, $z^{39}$, $z^{39}$, $z^{35}$, $z^{31}$ and $z^{31}$ for $q = 2$, 3, 4, \dots 9 and 10 respectively. These series are used to test techniques designed to distinguish first-order transitions from continuous transitions. New numerical values are also obtained for the $q$-state Potts model with $q>4$.

hep-lat↗

arXiv1992

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.

hep-lat↗

arXiv1992

Algebraic Techniques for Enumerating Self-Avoiding Walks on the Square Lattice

We describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of $N$ steps is of order $3^{N/4}$ times a polynomial in $N$, and so the approach is greatly superior to direct counting techniques. We have enumerated walks of up to 39 steps. As a consequence, we are able to accurately estimate the critical point, critical exponent, and critical amplitude.

hep-lat↗

arXiv1992

Enumeration of self avoiding trails on a square lattice using a transfer matrix technique

We describe a new algebraic technique, utilising transfer matrices, for enumerating self-avoiding lattice trails on the square lattice. We have enumerated trails to 31 steps, and find increased evidence that trails are in the self-avoiding walk universality class. Assuming that trails behave like $A λ^n n^{11 \over 32}$, we find $λ= 2.72062 \pm 0.000006$ and $A = 1.272 \pm 0.002$.

hep-lat↗