# High-temperature expansion for Ising models on quasiperiodic tilings

@article{Repetowicz1999HightemperatureEF, title={High-temperature expansion for Ising models on quasiperiodic tilings}, author={Przemysław Repetowicz and Uwe Grimm and Michael Schreiber}, journal={Journal of Physics A}, year={1999}, volume={32}, pages={4397-4418} }

We consider high-temperature expansions for the free energy of zero-field Ising models on planar quasiperiodic graphs. For the Penrose and the octagonal Ammann-Beenker tiling, we compute the expansion coefficients up to 18th order. As a by-product, we obtain exact vertex-averaged numbers of self-avoiding polygons on these quasiperiodic graphs. In addition, we analyse periodic approximants by computing the partition function via the Kac-Ward determinant. It turns out that the series expansions…

## 13 Citations

Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

- Physics
- 2000

We demonstrate that the invaded cluster algorithm, introduced by Machta et al (1995 Phys. Rev. Lett. 75 2792-5), is a fast and reliable tool for determining the critical temperature and the magnetic…

Wavevector-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

- Mathematics, Physics
- 2007

We study the q-dependent susceptibility χ(q) of a Z-invariant ferromagnetic Ising model on a Penrose tiling, as first introduced by Korepin using de Bruijn's pentagrid for the rapidity lines. The…

Wavevector-Dependent Susceptibility in Aperiodic Planar Ising Models

- Physics
- 2002

We study the q-dependent susceptibility of Fibonacci triangular and honeycomb Ising lattices, with edge interactions that take on both positive and negative values in an aperiodical way, according to…

Partition function zeros of aperiodic Ising models

- Mathematics, Physics
- 2001

We consider Ising models defined on periodic approximants of aperiodic graphs. The model contains only a single coupling constant and no magnetic field, so the aperiodicity is entirely given by the…

Wavevector-Dependent Susceptibility in Quasiperiodic Ising Models

- Mathematics
- 2001

Using the various functional relations for correlation functions in planar Ising models, new results are obtained for the correlation functions and the q-dependent susceptibility for Ising models on…

Finite-lattice expansion for the Ising model on the Penrose tiling

- Physics
- 2002

Low-temperature series are calculated for the free energy, magnetization, susceptibility and field derivatives of the susceptibility in the Ising model on the quasiperiodic Penrose lattice. The…

Self-avoiding walks and polygons on quasiperiodic tilings

- Mathematics, Physics
- 2003

We enumerate self-avoiding walks and polygons, counted by perimeter, on the quasiperiodic rhombic Penrose and Ammann-Beenker tilings, thereby considerably extending previous results. In contrast to…

Energy spectra and eigenstates of quasiperiodic tight-binding Hamiltonians

- Mathematics, Physics
- 2003

Analytic and numerical results for quasiperiodic tight-binding models are reviewed, with emphasis on two and three-dimensional models which so far are beyond a mathematically rigorous treatment. In…

Random walks on carbon nanotubes and quasicrystals

- Mathematics
- 2000

An analytic formula for the total number of k -step walks between given sites on a carbon nanotube is obtained by using a new mathematical model based on a three-axes description of the honeycomb…

## References

SHOWING 1-10 OF 89 REFERENCES

Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

- Physics
- 2000

We demonstrate that the invaded cluster algorithm, introduced by Machta et al (1995 Phys. Rev. Lett. 75 2792-5), is a fast and reliable tool for determining the critical temperature and the magnetic…

High Temperature Expansion for the Ising Model on the Dual Penrose Lattice

- Physics
- 1990

High temperature expansion for ln Z ( Z : the partition function in the absence of magnetic field) of the Ising model on the Penrose lattice is discussed. The terms up to the order of w 8 are…

Aperiodic Ising Models

- Mathematics, Physics
- 1996

We consider several aspects of non-periodic Ising models in one and two dimensions. Here we are not interested in random systems, but rather in models with intrinsic long-range aperiodic order. The…

Lee - Yang zeros in the scaling region of a two-dimensional quasiperiodic Ising model

- Mathematics
- 1997

Quasiperiodic, planar Ising models with ferromagnetic nearest-neighbour interactions should show the same universal critical behaviour as the classical Ising model on the square lattice. We use the…

A critical Ising model on the Labyrinth

- Physics, Mathematics
- 1994

A zero-field Ising model with ferromagnetic coupling constants on the so-called Labyrinth tiling is investigated. Alternatively, this can be regarded as an Ising model on a square lattice with a…

Static critical behavior of the ferromagnetic Ising model on the quasiperiodic octagonal tiling.

- Physics, MedicinePhysical review. B, Condensed matter
- 1995

The static critical behavior of the nonfrustrated ferromagnetic Ising model on the two-dimensional quasiperiodic octagonal tiling with free boundary conditions is studied by means of Monte Carlo simulations and finite-size scaling analysis and shows that tendency toferromagnetic ordering is higher in the octagonal quasilattice than in the square lattice.

Duality in the Ising Model on the Quasicrystals

- Physics
- 1988

The ferromagnetic Ising model on the dual Penrose lattice is simulated by the Monte Carlo method. Using a phenomenological Monte Carlo renormalization group based on finite-size scaling, the critical…

Exact enumeration study of free energies of interacting polygons and walks in two dimensions

- Mathematics, Physics
- 1998

We present analyses of substantially extended series for both interacting self-avoiding walks (ISAW) and polygons (ISAP) on the square lattice. We argue that these provide good evidence that the free…

Statistics of the Two-Dimensional Ferromagnet. Part II

- Physics
- 1941

In an effort to make statistical methods available for the treatment of cooperational phenomena, the Ising model of ferromagnetism is treated by rigorous Boltzmann statistics. A method is developed…