Skip to main content
SpringerLink
Log in
Book cover

MathPhys Odyssey 2001 pp 23–48Cite as

Birkhäuser

Correlation Functions and Susceptibility in the Z-Invariant Ising Model

  • Chapter

Part of the book series: Progress in Mathematical Physics ((PMP,volume 23))

Abstract

We discuss historical and recent developments in the calculation of two-point correlation functions and susceptibilities in planar Ising models. We note some remarkable simplifications of approach within Baxter’s Z-invariant generalization of the usual uniform model. We highlight several of the many important contributions of McCoy to the field.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. IsingBeitrag zur Theorie des FerromagnetismusZ. Physik 31 (1925) 253–258

    Article  Google Scholar 

  2. S. Kobe Ernst Ising 1900–1998 Brazil. J. Phys. 30 (2000) 649–653.

    Article  Google Scholar 

  3. H. A. Kramers and G. H. WannierStatistics of the Two-Dimensional Ferromagnet. Part 1 & 11Phys. Rev. 60 (1941) 252–262, 263–276.

    Article  MathSciNet  Google Scholar 

  4. L. OnsagerCrystal Statistics. I. A Two-Dimensional Model with an Order-Disorder TransitionPhys. Rev. 65 (1944) 117–149.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. J. BaxterSolvable Eight Vertex Model on an Arbitrary Planar LatticePhil. Trans. R. Soc. Lond. A 289 (1978) 315–346.

    Article  MathSciNet  Google Scholar 

  6. L. OnsagerDiscussion, Nuovo Cimento (ser. 9) 6(Sup.) (1949) 261.

    MathSciNet  Google Scholar 

  7. C. N. YangThe Spontaneous Magnetization of a Two-Dimensional Ising ModelPhys. Rev. 85 (1952) 808–816.

    Article  MATH  Google Scholar 

  8. I. Syozi and S. Naya, Spontaneous Magnetization, Generalized Square Lattice, Progr. Theor. Phys.24(1960) 374–376.

    MathSciNet  Google Scholar 

  9. E. W. Montroll, R. B. Potts, and J. C. WardCorrelations and Spontaneous Magnetization of the Two-Dimensional Ising ModelJ. Math. Phys. 4 (1963) 308–322.

    Article  MathSciNet  Google Scholar 

  10. H. S. Green and C. A. HurstOrder-Disorder Phenomena(Interscience Publ., London, 1964).

    MATH  Google Scholar 

  11. T. D. Schultz, D. C. Mattis, and E. H. LiebTwo-Dimensional Ising Model as a Soluble Problem of Many FermionsRev. Mod. Phys. 36 (1964) 856–871.

    Article  MathSciNet  Google Scholar 

  12. L. OnsagerThe Ising Model in Two Dimensionsin Critical Phenomena in Alloys Magnets and SuperconductorsR.E. Mills, E. Ascher, R.I. Jaffee, eds., (McGraw-Hill, New York, 1971), pp. xix—xxiv, 3–12.

    Google Scholar 

  13. B. M. McCoy and T. T. WuThe Two-Dimensional Ising Model(Harvard Univ. Press, Cambridge, Mass., 1973).

    Google Scholar 

  14. B. KaufmanCrystal Statistics. II. Partition Function Evaluated by Spinor AnalysisPhys. Rev. 76 (1949) 1232–1243.

    Article  MATH  Google Scholar 

  15. B. Kaufman and L. OnsagerCrystal Statistics. Ill. Short-Range Order in a Binary Ising LatticePhys. Rev. 76 (1949) 1244–1252.

    Article  MATH  Google Scholar 

  16. R. J. BaxterExactly Solved Models in Statistical Mechanics(Academic Press, London, 1982).

    MATH  Google Scholar 

  17. M. E. FisherThe Susceptibility of the Plane Ising ModelPhysica 25 (1959) 521–524.

    Article  Google Scholar 

  18. M. E. FisherTransformations of Ising ModelsPhys. Rev. 113 (1959) 969–981.

    Article  Google Scholar 

  19. M. E. Fisher and R. J. BurfordTheory of Critical-Point Scattering and Correlations. I. The Ising ModelPhys. Rev. 156 (1967) 583–622.

    Article  Google Scholar 

  20. M. E. FisherThe Perpendicular Susceptibility of an AnisotropicAntiferromagnetPhysica 26 (1960) 618–622, 1028.

    Google Scholar 

  21. L. P. KadanoffSpin-Spin Correlations in the Two-Dimensional Ising ModelNuovo Cimento (ser. 10) B44 (1966) 276–305.

    Google Scholar 

  22. T. T. WuTheory of Toeplitz Determinants and Spin Correlations of the Two-Dimensional Ising Model. I,Phys. Rev.149 (1966) 380–401.

    Article  Google Scholar 

  23. H. Cheng and T. T. WuTheory of Toeplitz Determinants and Spin Correlations of the Two-Dimensional Ising Model. IIIPhys. Rev.164 (1967) 719–735.

    Article  Google Scholar 

  24. R. E. Hartwig and J. StephensonIsing-model Spin CorrelationsJ. Math. Phys. 9 (1968) 836–848.

    Article  Google Scholar 

  25. D. S. FisherCritical Behavior of Random Transverse-Field Ising Spin ChainsPhys. Rev. B 51 (1995) 6411–6461.

    Article  Google Scholar 

  26. E. Barouch, B. M. McCoy, and T. T. WuZero-Field Susceptibility of the Two-Dimensional Ising Model near T c Phys. Rev. Lett.31 (1973) 1409–1411.

    Article  Google Scholar 

  27. C. A. Tracy and B. M. McCoyNeutron Scattering and the Correlation Functions of the Two-Dimensional Ising Model near T c Phys. Rev. Lett.31 (1973) 1500–1504.

    Article  Google Scholar 

  28. T. T. Wu, B. M. McCoy, C. A. Tracy, and E. BarouchSpin-Spin Correlation Functions for the Two-Dimensional Ising Model: Exact Theory in the Scaling RegionPhys. Rev. B 13 (1976) 316–374.

    Article  Google Scholar 

  29. B. M. McCoy, C. A. Tracy, and T. T. WuPainlevé Functions of the Third KindJ. Math. Phys. 18 (1977) 1058–1092.

    Article  MathSciNet  MATH  Google Scholar 

  30. H. G. VaidyaThe Spin-Spin Correlation Functions and Susceptibility Amplitudes for the Two-Dimensional Ising Model: Triangular LatticePhys. Lett. A 57 (1976) 1–4.

    Article  Google Scholar 

  31. C. A. TracyAsymptotics of a r-Function Arising in the Two-Dimensional Ising ModelCommun. Math. Phys. 142 (1991) 297–311.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Sato, T. Miwa, and M. Jimbo, Studies on Holonomic Quantum Fields. 1—V, Proc. Japan Acad. A 53 (1977) 6–10,147–152,153–158,183–185,219–224.

    Google Scholar 

  33. B. M. McCoyThe Connection between Statistical Mechanics and Quantum Field Theoryin Statistical Mechanics and Field TheoryV.V. Bazhanov and C.J. Burden, eds., (World Scientific, Singapore, 1995), pp. 26–128.

    Google Scholar 

  34. D. B. AbrahamSusceptibility of the Rectangular Ising FerromagnetJ. Stat. Phys. 19 (1978) 349–358.

    Article  Google Scholar 

  35. J. Palmer and C. A. TracyTwo-Dimensional Ising Correlations: Convergence of the Scaling LimitAdv. Appl. Math. 2 (1981) 329–388.

    MathSciNet  MATH  Google Scholar 

  36. B. NickelOn the Singularity Structure of the 2D Ising Model SusceptibilityJ. Phys. A 32 (1999) 3889–3906.

    Article  MathSciNet  MATH  Google Scholar 

  37. B. NickelAddendum to `On the Singularity Structure of the 2D Ising Model Susceptibility’J. Phys. A 33 (2000) 1693–1711.

    Article  MathSciNet  MATH  Google Scholar 

  38. M. Jimbo and T. MiwaStudies on Holonomic Quantum Fields. XVIIProc. Japan Acad. A56 (1980) 405–410, 57 (1981) 347.

    Google Scholar 

  39. M. Jimbo, T. Miwa, Y. Mori, and M. SatoDensity Matrix of an Impenetrable Bose Gas and the Fifth Painlevé TranscendentPhysica D 1 (1980) 80–158.

    Article  MathSciNet  MATH  Google Scholar 

  40. B. M. McCoy and J. H. H. PerkRelation of conformal field theory and deformation theory for the Ising modelNucl. Phys. B 285[FS19] (1987) 279–294.

    MathSciNet  Google Scholar 

  41. J.H. H. Perk and H. W. Capel, Time-Dependent xx-Correlations in the One-Dimensional XY-Model, Physica A 89(1977) 265–303.

    Article  Google Scholar 

  42. B. M. McCoy and T. T. WuNonlinear Partial Difference Equations for the Two-Dimensional Ising ModelPhys. Rev. Lett. 45 (1980) 675–678.

    Article  MathSciNet  Google Scholar 

  43. J. H. H. PerkEquations of Motion for the Transverse Correlations of the One-Dimensional XY-Model at Finite TemperaturePhys. Lett. A 79 (1980) 1–2.

    MathSciNet  Google Scholar 

  44. J. H. H. PerkQuadratic Identities for Ising Model CorrelationsPhys. Lett. A 79 (1980) 3–5.

    MathSciNet  Google Scholar 

  45. B. M. McCoy and T. T. WuSome Properties of the Nonlinear Partial Difference Equation for the Ising-Model Correlation Function, Phys. Lett. A 80(1980) 159–162.

    Article  MathSciNet  Google Scholar 

  46. B. M. McCoy and T. T. WuNonlinear Partial Difference Equations for the Two-Spin Correlation Function of the Two-Dimensional Ising ModelNucl. Phys. B 180[FS2] (1980) 89–115.

    MathSciNet  Google Scholar 

  47. B. M. McCoy, J. H. H. Perk, and T. T. Wu, Ising Field Theory: Quadratic Difference Equations for the n-Point Green’s Functions on the Square LatticePhys.

    Google Scholar 

  48. J. H. H. Perk, H. W. Capel, G. R. W. Quispel, and F. W. NijhoffFinite-Temperature Correlations for the Ising Chain in a Transverse FieldPhysica A 123 (1984) 149.

    Article  MathSciNet  Google Scholar 

  49. C. Itzykson and J.-M. DrouffeStatistical Field TheoryVol. 1, (Cambridge Univ. Press, Cambridge, 1989), Ch. 2.

    Book  Google Scholar 

  50. L. R Kadanoff and H. CevaDetermination of an Operator Algebra for the Two-Dimensional Ising ModelPhys. Rev. B 3 (1971) 3918–3939.

    Article  MathSciNet  Google Scholar 

  51. H. Au-Yang, B.-Q. Jin, and J. H. H. PerkWavevector-Dependent Susceptibility in Quasiperiodic Ising ModelsJ. Stat. Phys. 102 (2001) 501–543.

    Article  MathSciNet  MATH  Google Scholar 

  52. H. Au-Yang and J. H. H. PerkCritical Correlations in a Z-Invariant Inhomogeneous Ising ModelPhysica A 144 (1987) 44–104.

    Article  MathSciNet  Google Scholar 

  53. H. Au-Yang and J. H. H. PerkSolution of Hirota’s Discrete-Time Toda Lattice Equation and the Critical Correlations in the Z-Invariant Ising Modelin Proc. 1987 Summer Research Institute on Theta FunctionsProc. Symp. Pure Math., Vol. 49, part 1, L. Ehrenpreis and R. C. Gunning, eds., (Am. Math. Soc., Providence, R.I., 1989), pp. 287–293.

    Google Scholar 

  54. W. P. Orrick, B. Nickel, A. J. Guttmann, and J. H. H. PerkThe Susceptibility of the Square Lattice Ising Model: New DevelopmentsJ. Stat. Phys. 102 (2001) 795–841. For the complete set of series coefficients, see http://www.ms.unimelb.edu.aurtonyg

    Article  MathSciNet  MATH  Google Scholar 

  55. W. P. Orrick, B. G. Nickel, A. J. Guttmann, and J. H. H. PerkCritical Behavior of the Two-Dimensional !sing SusceptibilityPhys. Rev. Lett.86 (2001) 4120–4123.

    Article  Google Scholar 

  56. B. M. McCoyDo Hard Spheres have Natural Boundaries?e-print condmat/0103556.

    Google Scholar 

  57. M. Caselle, M. Hasenbusch, A. Pelissetto, and E. VicariIrrelevant Operators in the Two-Dimensional Ising Modele-print cond-mat/0106372.

    Google Scholar 

  58. R. J. BaxterFree-Fermion Checkerboard and Z-Invariant Lattice Models in Statistical MechanicsProc. R. Soc. Lond. A 404 (1986) 1–33.

    Article  MathSciNet  Google Scholar 

  59. J. R. Reyes MartínezCorrelation Functions for the Z-Invariant Ising ModelPhys. Lett. A227 (1997) 203–208.

    Article  MATH  Google Scholar 

  60. J. R. Reyes MartínezMulti-Spin Correlation Functions for the Z-Invariant ¡sing ModelPhysica A256 (1998) 463–484.

    Article  Google Scholar 

  61. H. Au-Yang and J.H.H. PerkIsing Correlations at the Critical TemperaturePhys. Lett. A104 (1984) 131–134.

    MathSciNet  Google Scholar 

  62. B.M. McCoy, J.H.H. Perk, and R.E. ShrockTime-Dependent Correlation Functions of the Transverse !sing Chain at the Critical Magnetic FieldNucl. Phys. B 220[FS8] (1983) 35–47.

    Article  MathSciNet  Google Scholar 

  63. B.M. McCoy, J.H.H. Perk, and R.E. ShrockCorrelation Functions of the Transverse Ising Chain at the Critical Field for Large Temporal and Spatial SeparationsNucl. Phys. B 220[FS8] (1983) 269–282.

    MathSciNet  Google Scholar 

  64. M. F. Sykes, D. G. Gaunt, P. D. Roberts, and J. A. WylesHigh Temperature Series for the Susceptibility of the Ising Model. I. Two Dimensional Lattices,J.Phys A5 (1972) 624–639.

    Article  Google Scholar 

  65. C.A. Tracy and B.M. McCoyExamination of the Phenomenological Scaling Functions for Critical ScatteringPhys. Rev. B12 (1975) 358–387.

    Article  Google Scholar 

  66. C.A. TracyCritical Scattering Scaling Functions and the Measurement of rlin Magnetism and Magnetic Materials-1975, AIP Conference Proceedings, Vol. No. 29, J.J. Becker, G.H. Lander, J.J. Rhyne, eds, (American Institute of Physics, New York, 1975), pp. 483–487.

    Google Scholar 

  67. K. YamadaPair Correlation Function in the Ising Square Lattice—Generalized Wronskian FormProgr. Theor. Phys. 76 (1986) 602–612.

    Google Scholar 

  68. H. Au-Yang and J.H.H. PerkWavevector-Dependent Susceptibility in Aperiodic Planar Ising ModelsChapter 1, pp. 1–21, this volume.

    Google Scholar 

  69. I. Syozi and S. NayaSymmetry Properties of Two-Dimensional Ising LatticesProgr. Theor. Phys. 24 (1960) 829–839.

    MathSciNet  Google Scholar 

  70. I. Syozi, Transformation of Ising Models, in Phase Transitions and Critical Phenomena, Vol. 1, C. Domb and M.S. Green, eds., (Academic Press, London, 1972), pp. 269–329.

    Google Scholar 

  71. M. Jimbo and T. MiwaAlgebraic Analysis of Solvable Lattice ModelsRegional Conference Series in Mathematics, Nr. 85, (American Mathematical Society, Providence, R.I., 1995).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics 145 Physical Sciences, Oklahoma State University, Stillwater, OK, 74078-3072, USA

    Helen Au-Yang & Jacques H. H. Perk

Authors
  1. Helen Au-Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacques H. H. Perk
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RIMS, Kyoto University, Kyoto, 606-8502, Japan

    Masaki Kashiwara

  2. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Tetsuji Miwa

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer Science+Business Media New York

About this chapter

Cite this chapter

Au-Yang, H., Perk, J.H.H. (2002). Correlation Functions and Susceptibility in the Z-Invariant Ising Model. In: Kashiwara, M., Miwa, T. (eds) MathPhys Odyssey 2001. Progress in Mathematical Physics, vol 23. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0087-1_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0087-1_2

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6605-1

  • Online ISBN: 978-1-4612-0087-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation