Q-Dependent Susceptibilities in Ferromagnetic Quasiperiodic Z-Invariant Ising Models
We study the q-dependent susceptibility χ(q) of a series of quasiperiodic Ising models on the square lattice. Several different kinds of aperiodic sequences of couplings are studied, including the Fibonacci and silver-mean sequences. Some identities and theorems are generalized and simpler derivations are presented. We find that the q-dependent susceptibilities are periodic, with the commensurate peaks of χ(q) located at the same positions as for the regular Ising models. Hence, incommensurate everywhere-dense peaks can only occur in cases with mixed ferromagnetic–antiferromagnetic interactions or if the underlying lattice is aperiodic. For mixed-interaction models the positions of the peaks depend strongly on the aperiodic sequence chosen.
Key WordsIsing model Z-invariance quasiperiodicity golden ratio silver mean correlation functions wavevector-dependent susceptibility
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- 1.H. Au-Yang and J. H. H. Perk, Wavevector-dependent susceptibility in Z-invariant pentagrid Ising model. J. Stat. Phys. DOI: 10.1007/s10955-006-9212-x (2006).Google Scholar
- 3.H. Au-Yang and J. H. H. Perk, Wavevector-dependent susceptibility in aperiodic planar Ising models, in MathPhys Odyssey 2001: Integrable Models and Beyond, M. Kashiwara and T. Miwa, eds. (Birkhäuser, Boston, 2002), pp. 1–21.Google Scholar
- 10.G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th edition (Oxford University Press, London, 1960), Ch. XXIII Kronecker’s Theorem.Google Scholar
- 11.H. Au-Yang and J. H. H. Perk, Correlation functions and susceptibility in the Z-invariant Ising model, in MathPhys Odyssey 2001: Integrable Models and Beyond, M. Kashiwara and T. Miwa, eds. (Birkhäuser, Boston, 2002), pp. 23–48.Google Scholar
- 14.U. Grimm, M. Baake and H. Simon, Ising spins on the labyrinth, in Proc. of the 5th International Conference on Quasicrystals, C. Janot and R. Mosseri, eds. (World Scientific, Singapore, 1995), pp. 80–83.Google Scholar
- 15.U. Grimm and M. Baake, Aperiodic Ising models, in The Mathematics of Long-Range Aperiodic Order, R. V. Moody, ed. (Kluwer, Dordrecht, 1997), pp. 199–237.Google Scholar