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New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain

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Abstract

In this paper we show how an infinite system of coupled Toda-type nonlinear differential equations derived by one of us can be used efficiently to calculate the time-dependent pair-correlations in the Ising chain in a transverse field. The results are seen to match extremely well long large-time asymptotic expansions newly derived here. For our initial conditions we use new long asymptotic expansions for the equal-time pair correlation functions of the transverse Ising chain, extending an old result of T.T. Wu for the 2d Ising model. Using this one can also study the equal-time wavevector-dependent correlation function of the quantum chain, a.k.a. the q-dependent diagonal susceptibility in the 2d Ising model, in great detail with very little computational effort.

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References

  1. Pfeuty, P.: The one-dimensional Ising model with a transverse field. Ann. Phys. 57, 79–90 (1970)

    Article  ADS  Google Scholar 

  2. Nambu, Y.: A note on the eigenvalue problem in crystal statistics. Progr. Theor. Phys. 5, 1–13 (1950)

    Article  ADS  MathSciNet  Google Scholar 

  3. Lieb, E., Schultz, T., Mattis, D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Katsura, S.: Statistical mechanics of the anisotropic linear Heisenberg model. Phys. Rev. 127, 1508–1518, 2835 (1962)

    Article  MATH  ADS  Google Scholar 

  5. Niemeijer, Th.: Some exact calculations on a chain of spins \(\frac{1}{2}\) . Physica 36, 377–419 (1967)

    Article  ADS  Google Scholar 

  6. Katsura, S., Horiguchi, T., Suzuki, M.: Dynamical properties of the isotropic XY model. Physica 46, 67–86 (1970)

    Article  ADS  Google Scholar 

  7. Tommet, T.N., Huber, D.L.: Dynamical correlation functions of the transverse spin and energy density for the one-dimensional spin-1/2 Ising model with a transverse field. Phys. Rev. B 11, 450–457 (1975)

    Article  ADS  Google Scholar 

  8. Huber, D.L., Tommet, T.: Spin and energy coupling in the Ising model with a transverse field: One dimension, T=∞. Solid State Commun. 13, 1973–1976 (1973)

    Article  ADS  Google Scholar 

  9. Perk, J.H.H., Capel, H.W., Siskens, Th.J.: Time-correlation functions and ergodic properties in the alternating XY-chain. Physica A 89, 304–325 (1977)

    Article  ADS  Google Scholar 

  10. Pesch, W., Mikeska, H.J.: Dynamical correlation functions in the x-y model. Z. Phys. B 30, 177–182 (1978)

    Article  ADS  Google Scholar 

  11. McCoy, B.M., Barouch, E., Abraham, D.B.: Statistical mechanics of the XY model. IV. Time-dependent spin-correlation functions. Phys. Rev. A 4, 2331–2341 (1971)

    Article  ADS  Google Scholar 

  12. Perk, J.H.H.: Equations of motion for the transverse correlations of the one-dimensional XY-model at finite temperature. Phys. Lett. A 79, 1–2 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  13. Perk, J.H.H., Capel, H.W., Quispel, G.R.W., Nijhoff, F.W.: Finite-temperature correlations for the Ising chain in a transverse field. Physica A 123, 1–49 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  14. McCoy, B.M., Wu, T.T.: Nonlinear partial difference equations for the two-dimensional Ising model. Phys. Rev. Lett. 45, 675–678 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  15. Perk, J.H.H.: Quadratic identities for Ising model correlations. Phys. Lett. A 79, 3–5 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  16. Sur, A., Jasnow, D., Lowe, I.J.: Spin dynamics for the one-dimensional XY model at infinite temperature. Phys. Rev. B 12, 3845–3848 (1975)

    Article  ADS  Google Scholar 

  17. Brandt, U., Jacoby, K.: Exact results for the dynamics of one-dimensional spin-systems. Z. Phys. B 25, 181–187 (1976)

    Article  ADS  Google Scholar 

  18. Capel, H.W., Perk, J.H.H.: Autocorrelation function of the x-component of the magnetization in the one-dimensional XY-model. Physica A 87, 211–242 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  19. Brandt, U., Jacoby, K.: The transverse correlation function of anisotropic X–Y-chains: exact results at T=∞. Z. Phys. B 26, 245–252 (1977)

    Article  ADS  Google Scholar 

  20. Perk, J.H.H., Capel, H.W.: Time-dependent xx-correlations in the one-dimensional XY-model. Physica A 89, 265–303 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  21. Perk, J.H.H., Capel, H.W.: Transverse correlations in the inhomogeneous XY-model at infinite temperature. Physica A 92, 163–184 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  22. Perk, J.H.H., Capel, H.W.: Time- and frequency-dependent correlation functions for the homogeneous and alternating XY-models. Physica A 100, 1–23 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  23. Stolze, J., Viswanath, V.S., Müller, G.: Dynamics of semi-infinite quantum spin chains at T=∞. Z. Phys. B 89, 45–55 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  24. Johnson, J.D., McCoy, B.M.: Off-diagonal time-dependent spin-correlation functions of the XY model. Phys. Rev. A 4, 2314–2324 (1971)

    Article  ADS  Google Scholar 

  25. Vaidya, H.G., Tracy, C.A.: Transverse time-dependent spin-correlation functions for the one-dimensional XY model at zero temperature. Physica A 92, 1–41 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  26. McCoy, B.M., Perk, J.H.H., Shrock, R.E.: Time-dependent correlation functions of the transverse Ising chain at the critical magnetic field. Nucl. Phys. B 220(FS8), 35–47 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  27. McCoy, B.M., Perk, J.H.H., Shrock, R.E.: Correlation functions of the transverse Ising chain at the critical field for large temporal and spatial separations. Nucl. Phys. B 220(FS8), 269–282 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  28. Müller, G., Shrock, R.E.: Dynamic correlation functions for quantum spin chains. Phys. Rev. Lett. 51, 219–222 (1983)

    Article  ADS  Google Scholar 

  29. Müller, G., Shrock, R.E.: Dynamic correlation functions for one-dimensional quantum-spin systems: New results based on a rigorous approach. Phys. Rev. B 29, 288–301 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  30. Müller, G., Shrock, R.E.: Susceptibilities of one-dimensional quantum spin models at zero temperature. Phys. Rev. B 30, 5254–5264 (1984)

    Article  ADS  Google Scholar 

  31. Müller, G., Shrock, R.E.: Wave-number-dependent susceptibilities of one-dimensional quantum spin models at zero temperature. Phys. Rev. B 31, 637–640 (1985)

    Article  ADS  Google Scholar 

  32. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. Int. J. Mod. Phys. B 4, 1003–1037 (1990)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. Its, A.R., Izergin, A.G., Korepin, V.E., Novokshenov, V.Ju.: Temperature autocorrelations of the transverse Ising chain at the critical magnetic field. Nucl. Phys. B 340, 752–758 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  34. Colomo, F., Izergin, A.G., Korepin, V.E., Tognetti, V.: Temperature correlation functions in the XX0 Heisenberg chain. I. Teor. Mat. Fiz. 94, 19–51 (1993) [Theor. Math. Phys. 94, 11–38 (1993)]

    MathSciNet  Google Scholar 

  35. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Temperature correlations of quantum spins. Phys. Rev. Lett. 70, 1704–1706, 2357 (1993)

    Article  ADS  Google Scholar 

  36. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Integrable differential equations for temperature correlation functions of the XXO Heisenberg chain. Zap. Nauch. Sem. POMI 205, 6–20 (1993) [J. Math. Sciences 80, 1747–1759 (1996)]

    Google Scholar 

  37. Deift, P., Zhou, X.: Long-time asymptotics for the autocorrelation function of the transverse Ising chain at the critical magnetic field. In: Singular Limits of Dispersive Waves, Lyon, 1991. NATO Adv. Sci. Inst. Ser. B Phys., vol. 320, pp. 183–201. Plenum, New York (1994)

    Google Scholar 

  38. Stolze, J., Nöppert, A., Müller, G.: Gaussian, exponential, and power-law decay of time-dependent correlation functions in quantum spin chains. Phys. Rev. B 52, 4319–4326 (1995). arXiv:cond-mat/9501079

    Article  ADS  Google Scholar 

  39. Sachdev, S.: Universal, finite temperature, crossover functions of the quantum transition in the Ising chain in a transverse field. Nucl. Phys. B 464, 576–595 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. Sachdev, S.: Finite temperature correlations in the one-dimensional quantum Ising model. Nucl. Phys. B 482, 579–612 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Doyon, B., Gamsa, A.: Integral equations and long-time asymptotics for finite-temperature Ising chain correlation functions. J. Stat. Mech. P03012, 40 pp. (2008). arXiv:0711.4619

  42. Jimbo, M., Miwa, T.: Studies on holonomic quantum fields. XVII. Proc. Japan Acad. A 56, 405–410 (1980)

    Article  MathSciNet  Google Scholar 

  43. Jimbo, M., Miwa, T.: Errata to Studies on holonomic quantum fields. XVII. Proc. Japan Acad. A 57, 347 (1987)

    Google Scholar 

  44. Witte, N.S.: Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model. J. Phys. A 40, F491–F501 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  45. Wu, T.T.: Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. I. Phys. Rev. 149, 380–401 (1966)

    Article  ADS  Google Scholar 

  46. McCoy, B.M., Wu, T.T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973)

    MATH  Google Scholar 

  47. Au-Yang, H., Perk, J.H.H.: Correlation functions and susceptibility in the Z-invariant Ising model. In: Kashiwara, M., Miwa, T. (eds.) MathPhys Odyssey 2001: Integrable Models and Beyond, pp. 23–48. Birkhäuser, Boston (2002)

    Google Scholar 

  48. Li, N.Y., Mansour, T.: An identity involving Narayana numbers. European J. Combin. 29, 672–675 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  49. Ghosh, R.K.: On the low-temperature series expansion for the diagonal correlation functions in the two-dimensional Ising model. arXiv:cond-mat/0505166 (7 pp.)

  50. Orrick, W.P., Nickel, B., Guttmann, A.J., Perk, J.H.H.: The susceptibility of the square lattice Ising model: New developments. J. Stat. Phys. 102, 795–841 (2001). arXiv:cond-mat/0103074. See http://www.ms.unimelb.edu.au/~tonyg for the complete set of series coefficients

    Article  MATH  MathSciNet  Google Scholar 

  51. Orrick, W.P., Nickel, B.G., Guttmann, A.J., Perk, J.H.H.: Critical behavior of the two-dimensional Ising susceptibility. Phys. Rev. Lett. 86, 4120–4123 (2001). arXiv:cond-mat/0009059

    Article  ADS  Google Scholar 

  52. Fisher, M.E., Burford, R.J.: Theory of critical-point scattering and correlations. I. The Ising model. Phys. Rev. 156, 583–622 (1967). See footnote 25 on p. 591

    ADS  Google Scholar 

  53. Wu, T.T., McCoy, B.M., Tracy, C.A., Barouch, E.: Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region. Phys. Rev. B 13, 316–374 (1976)

    Article  ADS  Google Scholar 

  54. Kong, X.-P., Au-Yang, H., Perk, J.H.H.: New results for the susceptibility of the two-dimensional Ising model at criticality. Phys. Lett. A 116, 54–56 (1986)

    Article  ADS  Google Scholar 

  55. Kong, X.-P., Au-Yang, H., Perk, J.H.H.: Logarithmic singularities of Q-dependent susceptibility of 2-d Ising model. Phys. Lett. A 118, 336–340 (1986)

    Article  ADS  Google Scholar 

  56. Kong, X.-P., Au-Yang, H., Perk, J.H.H.: Comment on a paper by Yamada and Suzuki. Progr. Theor. Phys. 77, 514–516 (1987)

    Article  ADS  Google Scholar 

  57. Kong, X.-P.: Wave-vector dependent susceptibility of the two-dimensional Ising model. Ph.D. Thesis, State University of New York at Stony Brook (September, 1987)

  58. Au-Yang, H., Jin, B.-Q., Perk, J.H.H.: Wavevector-dependent susceptibility in quasiperiodic Ising models. J. Stat. Phys. 102, 501–543 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  59. Au-Yang, H., Perk, J.H.H.: Wavevector-dependent susceptibility in aperiodic planar Ising models. In: Kashiwara, M., Miwa, T. (eds.) MathPhys Odyssey 2001: Integrable Models and Beyond, pp. 1–21. Birkhäuser, Boston (2002)

    Google Scholar 

  60. Au-Yang, H., Perk, J.H.H.: Q-dependent susceptibilities in Z-invariant pentagrid Ising models. J. Stat. Phys. 127, 221–264 (2007). arXiv:cond-mat/0409557

    Article  MATH  ADS  MathSciNet  Google Scholar 

  61. Au-Yang, H., Perk, J.H.H.: Q-dependent susceptibilities in ferromagnetic quasiperiodic Z-invariant Ising models. J. Stat. Phys. 127, 265–286 (2007). arXiv:cond-mat/0606301

    Article  MATH  ADS  MathSciNet  Google Scholar 

  62. Baxter, R.J.: Solvable eight-vertex model on an arbitrary planar lattice. Philos. Trans. R. Soc. Lond. Ser. A 289, 315–346 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  63. Au-Yang, H., Perk, J.H.H.: Critical correlations in a Z-invariant inhomogeneous Ising model. Physica A 144, 44–104 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  64. Au-Yang, H., Perk, J.H.H.: New results for susceptibilities in planar Ising models. Int. J. Mod. Phys. B 16, 2089–2095 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  65. Au-Yang, H., Perk, J.H.H.: Susceptibility calculations in periodic and quasiperiodic planar Ising models. Physica A 321, 81–89 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  66. McCoy, B.M., Tang, S.: Connection formulae for Painlevé V functions. Physica D 19, 42–72 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  67. Barouch, E., McCoy, B.M.: Statistical mechanics of the XY model. II. Spin-correlation functions. Phys. Rev. A 3, 786–804 (1971)

    Article  ADS  Google Scholar 

  68. Lajzerowicz, J., Pfeuty, P.: Space-time-dependent spin correlation of the one-dimensional Ising model with a transverse field. Application to higher dimension. Phys. Rev. B 11, 4560–4562 (1975)

    Article  ADS  Google Scholar 

  69. Hamer, C.J., Oitmaa, J., Zheng, W.: One-particle dispersion and spectral weights in the transverse Ising model. Phys. Rev. B 74, 174428 (2006), 10 pp.

    Article  ADS  Google Scholar 

  70. Hamer, C.J., Oitmaa, J., Zheng, W., McKenzie, R.H.: Critical behavior of one-particle spectral weights in the transverse Ising model. Phys. Rev. B 74, 060402(R) (2006), 4 pp.

    ADS  Google Scholar 

  71. Boukraa, S., Hassani, S., Maillard, J.-M., McCoy, B.M., Zenine, N.: The diagonal Ising susceptibility. J. Phys. A: Math. Theor. 40, 8219–8236 (2007). arXiv:math-ph/0703009

    Article  MATH  ADS  MathSciNet  Google Scholar 

  72. Bostan, A., Boukraa, S., Hassani, S., Maillard, J.-M., Weil, J.-A., Zenine, N.: Globally nilpotent differential operators and the square Ising model. J. Phys. A: Math. Theor. 42, 125206 (2009), 50 pp. arXiv:0812.4931

    Article  ADS  Google Scholar 

  73. Campbell, I.A., Butera, P.: Extended scaling for the high-dimension and square-lattice Ising ferromagnets. Phys. Rev. B 78, 024435 (2008), 7 pp.

    Article  ADS  Google Scholar 

  74. Au-Yang, H., Perk, J.H.H.: Ising correlations at the critical temperature. Phys. Lett. A 104, 131–134 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  75. Perk, J.H.H., Au-Yang, H.: Some recent results on pair correlation functions and susceptibilities in exactly solvable models. J. Phys.: Conf. Ser. 42, 231–238 (2006). arXiv:math-ph/0606046

    Article  ADS  Google Scholar 

  76. Lukyanov, S., Terras, V.: Long-distance asymptotics of spin-spin correlation functions for the XXZ spin chain. Nucl. Phys. B 654(FS), 323–356 (2003). arXiv:hep-th/0206093

    Article  MATH  ADS  MathSciNet  Google Scholar 

  77. Sato, J., Shiroishi, M., Takahashi, M.: Evaluation of dynamic spin structure factor for the spin-1/2 XXZ chain in a magnetic field. J. Phys. Soc. Japan 73, 3008–3014 (2004). arXiv:cond-mat/0410102

    Article  MATH  ADS  Google Scholar 

  78. Kitanine, N., Maillet, J.M., Slavnov, N.A., Terras, V.: Dynamical correlation functions of the XXZ spin-1/2 chain. Nucl. Phys. B 729(FS), 558–580 (2005). arXiv:hep-th/0407108

    Article  MATH  ADS  MathSciNet  Google Scholar 

  79. Caux, J.-S., Maillet, J.-M.: Computation of dynamical correlation functions of Heisenberg chains in a magnetic field. Phys. Rev. Lett. 95, 077201 (2005), 3 pp. arXiv:cond-mat/0502365

    Article  ADS  Google Scholar 

  80. Caux, J.-S., Hagemans, R., Maillet, J.-M.: Computation of dynamical correlation functions of Heisenberg chains: the gapless anisotropic regime. J. Stat. Mech. 95, P09003 (2005), 20 pp. arXiv:cond-mat/0506698

    Article  Google Scholar 

  81. Pereira, R.G., Sirker, J., Caux, J.-S., Hagemans, R., Maillet, J.M., White, S.R., Affleck, I.: The dynamical spin structure factor for the anisotropic spin-1/2 Heisenberg chain. Phys. Rev. Lett. 96, 257202 (2006), 4 pp. arXiv:cond-mat/0603681

    Article  ADS  Google Scholar 

  82. Hagemans, R., Caux, J.-S., Maillet, J.M.: How to calculate correlation functions of Heisenberg chains. AIP Conf. Proc. 846, 245–254 (2006). arXiv:cond-mat/0611467

    Article  ADS  Google Scholar 

  83. Pereira, R.G., Sirker, J., Caux, J.-S., Hagemans, R., Maillet, J.M., White, S.R., Affleck, I.: Dynamical structure factor at small q for the XXZ spin-1/2 chain. J. Stat. Mech. P08022 (2007), 64 pp. arXiv:0706.4327

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  1. Jacques H. H. Perk & Helen Au-Yang

    Present address: Department of Theoretical Physics, (RSPE), and Centre for Mathematics and its Applications (CMA), Australian National University, Canberra, ACT, 2600, Australia

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

    Jacques H. H. Perk & Helen Au-Yang

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Supported in part by the National Science Foundation under grant PHY 07-58139 and by the Australian Research Council under Project ID: LX0989627.

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Perk, J.H.H., Au-Yang, H. New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain. J Stat Phys 135, 599–619 (2009). https://doi.org/10.1007/s10955-009-9758-5

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