Wavevector-dependent susceptibility in Z-invariant pentagrid ising model

Journal of Statistical Physics

Volumn 127, Issue 2, 2007, Pages 221-264

  Au Yang, Helen ^a   Perk, Jacques H H ^a  

^a OKLAHOMA STATE UNIVERSITY   (United States)

Author keywords

Correlation functions; Fibonacci sequence; Ising model; Penrose tiles; Pentagrid; Q dependent susceptibility; Quasiperiodicity; Z invariance

Indexed keywords

EID: 34247342560     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-006-9212-x     Document Type: Article

References (87)

1. D. Shechtman, I. Blech, D. R. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53:1951-1953 (1984).
2. R. Penrose, Tilings and quasi-crystals; a non-local growth problem?, in Introduction to The Mathematics of Quasicrystals, Aperiodicity and Order, Vol. 2, M. V Jarić, ed. (Academic Press, Boston, 1989), pp. 53-79.
3. R. Penrose, The rôle of aesthetics in pure and applied mathematical research, Bull. Inst. Math. Appl. 10:266-271 (1974).
4. R. Penrose, Pentaplexity: A class of non-periodic tilings of the plane, The Mathematical Intelligencer 2:32-37 (1979), [reprinted from Eureka No. 39].
5. N. G. de Bruijn, Algebraic theory of Penrose's non-periodic tilings of the plane. I, Indagationes Mathematicae 84:38-52 (1981);
6. -. II, ibid. 84:53-66 (1981).
7. A. L. Mackay, De nive quinquangula: On the pentagonal snowflake, Kristallografiya 26:910-919 (1981)
8. [Sov. Phys. Crystallogr. 26:517-522 (1981)].
9. A. L. Mackay, Crystallography on the Penrose pattern, Physica A 114:609-613 (1982).
10. D. Levine and P. J. Steinhardt, Quasicrystals: A new class of ordered structures, Phys. Rev. Lett. 53:2477-2480 (1984).
11. M. Baake, U. Grimm, and R. V. Moody, Die verborgene Ordnung der Quasikristalle, Spektrum der Wissenschaft, Heft 02 (Februar 2002) 64-74,
12. [in German, English translation: What is Aperiodic Order?, Preprint math.HO/0203252].
13. C. Janot, Quasicrystals: A Primer, 2nd Ed. (Clarendon Press, Oxford, 1994).
14. C. L. Henley, Quasicrystal order, its origins and its consequences: A survey of current models, Comments Cond. Mat. Phys. 13:59-117 (1987).
15. M. Baake, A guide to mathematical quasicrystals, in Quasicrystals - An Introduction to the Structure, Physical Properties, and Applications, Material Science, Vol. 55, J.-B. Suck, M. Schreiber, and P. Häußler, eds., (Springer, Berlin, 2002), pp. 17-48.
16. V. E. Korepin, Eight-vertex model of the quasicrystal, Phys. Lett. A 118:285-287 (1986).
17. R. J. Baxter, Solvable eight vertex model on an arbitrary planar lattice, Phil. Trans. R. Soc. Lond. A 289:315-346 (1978).
18. V.E. Korepin, Completely integrable models in quasicrystals, Commun. Math. Phys. 110:157-171 (1987).
19. N. V. Antonov and V. E. Korepin, Critical properties and correlation functions of the eight-vertex model on a quasicrystal, Zap. Nauch. Semin. LOMI 161:13-23 (1987)
20. [J. Sov. Math. 46:2058-2065 (1989)].
21. N. V. Antonov and V. E. Korepin, Critical properties of completely integrable spin models in quasicrystals, Teor. Mat. Fiz. 77:402-411 (1988)
22. [Theor. Math. Phys. 77:1282-1288 (1988)].
23. T. C. Choy, Ising models on two-dimensional quasi-crystals: Some exact results, Intern. J. Mod Phys. B 2:49-63 (1988).
24. M. Baake, U. Grimm, and R. J. Baxter, A critical Ising model on the labyrinth, Intern. J. Mod. Phys. B 8:3579-3600 (1994).
25. 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.
26. 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.
27. H. Aoyama and T. Odagaki, Eight-parameter renormalization group for Penrose lattices, J. Stat. Phys. 48:503-511 (1987).
28. G. Amarendra, G. Ananthakrishna, and G. Athithan, Critical behavior of the Ising model on a two-dimensional Penrose lattice, Europhys. Lett. 5:181-184 (1988).
29. S. M. Bhattacharjee, J.-S. Ho, and J. A. Y. Johnson, Translational invariance in critical phenomena: Ising model on a quasi-lattice, J. Phys. A 20:4439-4448 (1987).
30. Y. Okabe and K. Niizeki, Monte Carlo simulation of the Ising model on the Penrose lattice, J. Phys. Soc. Jpn. 57:16-19 (1988).
31. Y. Okabe and K. Niizeki, Duality in the Ising model on the quasicrystals, J. Phys. Soc. Jpn. 57:1536-1539 (1988).
32. E. S. Sørensen, M. V. Jarić, and M. Ronchetti, Ising model on the Penrose lattice: Boundary conditions, Phys. Rev. B 44:9271-9282 (1991).
33. L.-H. Tang and M. V. Jarić, Equilibrium quasicrystal phase of a Penrose tiling model, Phys. Rev. B 41:4524-4546 (1990), see p. 4537.
34. O. Redner and M. Baake, Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models, J. Phys. A 33:3097-3109 (2000).
35. R. Abe and T. Dotera, High temperature expansion for the Ising model on the Penrose lattice, J. Phys. Soc. Jpn. 58:3219-3226 (1989).
36. T. Dotera and R. Abe, High temperature expansion for the Ising model on the dual Penrose lattice, J. Phys. Soc. Jpn. 59:2064-2077 (1990).
37. P. Repetowicz, Finite-lattice expansion for the Ising model on the Penrose tiling, J. Phys. A 35:7753-7772 (2002).
38. H. Simon, M. Baake, and U. Grimm, Lee-Yang zeros for substitutional systems, in Proc. of the 5th International Conference on Quasicrystals, C. Janot and R. Mosseri, eds., (World Scientific, Singapore, 1995), pp. 100-103.
39. H. Simon and M. Baake, Lee-Yang zeros in the scaling region of a two-dimensional quasiperiodic Ising model, J. Phys. A 30:5319-5327 (1997).
40. P. Repetowicz, U. Grimm, and M. Schreiber, Planar quasiperiodic Ising models, Mat. Science Eng. A 294-296:638-641 (2000).
41. H. Au-Yang and J. H. H. Perk, Susceptibility calculations in periodic and quasiperiodic planar Ising models, Physica A 321:81-89 (2003).
42. D. Levine and P. J. Steinhardt, Quasicrystals. I. Definition and structure, Phys. Rev. B 34:596-616 (1986).
43. J. E. S. Socolar and P. J. Steinhardt, Quasicrystals. II. Unit-cell configurations, Phys. Rev. B 34:617-647 (1986).
44. T. C. Lubensky, J. E. S. Socolar, P. J. Steinhardt, P. A. Bancel, and P. A. Heiney, Distortion and peak broadening in quasicrystal diffraction patterns, Phys. Rev. Lett. 57:1440-1443 (1986).
45. A. Hof, Diffraction of aperiodic structures at high temperatures, J. Phys. A 28:57-62 (1995).
46. B. Kaufman and L. Onsager, Crystal statistics. III. Short-range order in a binary Ising lattice, Phys. Rev. 76:1244-1252 (1949).
47. M. E. Fisher, The susceptibility of the plane Ising model, Physica 25:521-524 (1959).
48. E. W. Montroll, R. B. Potts, and J. C. Ward, Correlations and spontaneous magnetization of the two-dimensional Ising model, J. Math. Phys. 4:308-322 (1963).
49. B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (Harvard Univ. Press, Cambridge, Mass., 1973).
50. T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region, Phys. Rev. B 13:316-374 (1976).
51. D. B. Abraham, Pair function for the rectangular Ising ferromagnet, Commun. Math. Phys. 60:181-191 (1978).
52. .T. H. H. Perk, Quadratic identities for Ising correlations, Phys. Lett. A 79:3-5 (1980).
53. H. Au-Yang and I H. H. Perk, Critical correlations in a Z-invariant inhomogeneous Ising model, Physica A 144:44-104 (1987).
54. C. A. Tracy, Universality class of a Fibonacci Ising model, J. Stat. Phys. 51:481-490 (1988).
55. C. A. Tracy, Universality classes of some aperiodic Ising models, J. Phys. A 21:L603-L605 (1988).
56. H. Au-Yang and B. M. McCoy, Theory of layered Ising models. II. Spin correlation functions parallel to the layering, Phys. Rev. B 10:3885-3905 (1974).
57. B. M. McCoy and T. T. Wu, Theory of Toeplitz determinants and spin correlations of the two-dimensional Ising model. II, Phys. Rev. 155:438-452 (1967).
58. H. Au-Yang, B.-Q. Jin, and J. H. H. Perk, Wavevector-dependent susceptibility in quasiperiodic Ising models, J. Stat. Phys. 102:501-543 (2001).
59. 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.
60. X.-P. Kong, Wave-Vector Dependent Susceptibility of the Two-Dimensional Ising Model, (Ph. D. Thesis, State University of New York at Stony Brook, September 1987).
61. P. W. Stephens, The icosahedral glass model, in Extended Icosahedral Structures, Aperiodicity and Order, Vol. 3, M. V. Jarić and D. Gratias, eds. (Academic Press, Boston, 1989), pp. 37-104.
62. M. Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles, Scientific American 236#1:110-121 (January 1977).
63. F. Y. Wu, Ising model with four-spin interactions, Phys. Rev. B 4:2312-2314 (1971).
64. L. P. Kadanoff and F. J. Wegner, Some critical properties of the eight-vertex model, Phys. Rev. B 4:3989-3993 (1971).
65. 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.
66. N. G. de Bruijn, Sequences of zeros and ones generated by special production rules, Indagationes Mathematicae 84:27-37 (1981).
67. 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.
68. H. Au-Yang and J. H. H. Perk, New results for susceptibilities in planar Ising models, Int. J. Mod. Phys. B 16:2089-2095 (2002).
69. H. Au-Yang and J. H. H. Perk, Susceptibility calculations in periodic and quasiperiodic planar Ising models, Physica A 321:81-89 (2003).
70. N. G. de Bruijn, Quasicrystals and their Fourier transform, Indagationes Mathematicae 89:123-152 (1986).
71. m obtained by projection, Acta Crystallogr. A 40:580-587 (1984).
72. F. Gähler and J. Rhyner, Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings, J. Phys. A 19:267-277 (1986).
73. V.E. Korepin, F. Gähler. and J. Rhyner, Quasiperiodic tilings: A generalized grid-projection method, Acta Crystallogr. A 44:667-672 (1988).
74. M. Duneau and A. Katz, Quasiperiodic patterns, Phys. Rev. Lett. 54:2688-2691 (1985).
75. 0.14; A six-dimensional crystal, P'isma Zh. Eksp. Teor. Fiz. 41:119-121 (1985)
76. [JETP Lett. 41:145-149 (1985)].
77. M. V. Jarić, Diffraction from quasicrystals: Geometric structure factor, Phys. Rev. B 34:4685-4698 (1986).
78. V. Elser, The diffraction pattern of projected structures, Acta Crystallogr. A 42:36-43 (1986).
79. P. Repetowicz, U. Grimm, and M. Schreiber, High-temperature expansion for Ising models on quasiperiodic tilings, J. Phys. A 32:4397-4418 (1999).
80. A. N. Rogers, C. Richard, and A. J. Guttmann, Self-avoiding walks and polygons on quasiperiodic tilings, J. Phys. A 36:6661-6673 (2003).
81. M. Baake, P. Kramer, M. Schlottmann, and D. Zeidler, Planar patterns with fivefold symmetry as sections of periodic structures in 4-space, Intern. J. Mod. Phys. A 15-16:2217-2268 (1990).
82. M. Baake, D. Joseph, P. Kramer, and M. Schlottmann, Root lattices and quasicrystals, J. Phys. A 23:L1037-L1041 (1990).
83. M. Baake and U. Grimm, Combinatorial problems of (quasi-)crystallography, in Quasicrystals -Structure and Physical Properties, H.-R. Trebin, ed. (Wiley-VCH, New York, 2003), pp. 160-171.
84. H. Tsunetsugu, T. Fujiwara, K. Ueda, and T. Tokihiro, Eigenstates in 2-dimensional Penrose tiling, J. Phys. Soc. Jpn. 55:1420-1423 (1986).
85. H. Tsunetsugu, T. Fujiwara, K. Ueda, and T. Tokihiro, Electronic properties of the Penrose lattice. I. Energy spectrum and wave functions, Phys. Rev. B 43:8879-8891 (1991).
86. H. Tsunetsugu and K. Ueda, Electronic properties of the Penrose lattice. II. Conductance at zero temperature, Phys. Rev. B 43:8892-8902 (1991).
87. B. Grünbaum and G.C. Shephard, Tilings and Patterns (W. H. Freeman and Co., New York, 1987), Ch. 11, Wang Tiles.