Ferromagnetic phase transitions of inhomogeneous systems modelled by square Ising models with diamond-type bond-decorations. (English) Zbl 1011.82506
Summary: The two-dimensional Ising model defined on square lattices with diamond-type bond-decorations is employed to study the nature of the ferromagnetic phase transitions of inhomogeneous systems. The model is studied analytically under the bond-renormalization scheme. For an \(n\)-level decorated lattice, the long-range ordering occurs at the critical temperature given by the fitting function \((k_BT_c/J)_n=1.6410+(0.6281) \exp[-(0.5857)n]\), and the local ordering inside \(n\)-level decorated bonds occurs at the temperature given by the fitting function \((k_BT_m/J)_n=1.6410-(0.8063) \exp[-(0.7144)n]\). The critical amplitude \(A_{\sin g}^{(n)}\) of the logarithmic singularity in specific heat characterizes the width of the critical region, and it varies with the decoration-level \(n\) as \(A_{\sin g}^{(n)}=(0.2473) \exp[-(0.3018)n]\), obtained by fitting the numerical results. The cross over from a finite-decorated system to an infinite-decorated system is not a smooth continuation. For the case of infinite decorations, the critical specific heat becomes a cusp with the height \(c^{(n)}=0.639852\). The results are compared with those obtained in the cell-decorated Ising model.
MSC:
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
| 82B27 | Critical phenomena in equilibrium statistical mechanics |
References:
| [1] | Griffiths, R. B.; Kaufman, M., Phys. Rev. B, 26, 5022 (1982) |
| [2] | Svrakic, N. M.; Kertesz, J.; Selke, W., J. Phys. A, 15, L427 (1982) |
| [3] | Derrida, B.; Eckmann, J. P.; Erzan, A., J. Phys. A, 16, 893 (1983) |
| [4] | Kaufman, M.; Griffiths, R. B., Phys. Rev. B, 24, 496 (1981) |
| [5] | Erzan, A., Phys. Lett. A, 93, 237 (1983) |
| [6] | Derrida, B.; De. Seze, L.; Itzykson, C., J. Stat. Phys., 33, 559 (1983) |
| [7] | Derrida, B.; Itzykson, C.; Luck, J. M., Commun. Math. Phys., 94, 115 (1985) |
| [8] | Lee, F. T.; Huang, M. C., J. Stat. Phys., 75, 1119 (1994) |
| [9] | Lee, F. T.; Huang, M. C., Chin. J. Phys., 37, 398 (1999) · Zbl 07844758 |
| [10] | Migdal, A. A., Sov. Phys. JETP, 42, 743 (1976) |
| [11] | Kadanoff, L. P., Ann. Phys. (N.Y.), 100, 359 (1976) |
| [12] | Berker, A. N.; Qstlund, S., J. Phys. C, 12, 4961 (1979) |
| [13] | Plechko, V. N., Physica A, 152, 51 (1988) |
| [14] | Plechko, V. N.; Sobolev, I. K., Physica A, 197, 323 (1993) |
| [15] | Gefen, Y.; Mandelbrot, B. B.; Aharony, A., Phys. Rev. Lett., 45, 855 (1980) |
| [16] | Onsager, L., Phys. Rev., 65, 117 (1944) · Zbl 0060.46001 |
| [17] | Kac, M.; Ward, J. C., Phys. Rev., 88, 1332 (1952) · Zbl 0048.45804 |
| [18] | Green, H. S.; Hurst, C. A., Order-Disorder Phenomena (1964), Interscience: Interscience New York · Zbl 0138.22301 |
| [19] | Plechko, V. N., Theor. Math . Phys., 64, 748 (1985) |
| [20] | Plechko, V. N., Phys. Lett. A, 157, 335 (1991) |
| [21] | Liaw, T. M.; Huang, M. C.; Lin, S. C.; Wu, M. C., Phys. Rev. B, 60, 12994 (1999) |
| [22] | McCoy, B. M.; Wu, T. S., The Two-Dimensional Ising Model (1973), Harvard University Press: Harvard University Press Cambridge, MA · Zbl 1094.82500 |
| [23] | Hu, B., Phys. Rev. Lett., 55, 2316 (1985) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
