Advertisement

Transfer-matrix approach to classical systems

  • T. Nishino
  • K. Okunishi
  • Y. Hieida
  • T. Hikihara
  • H. Takasaki
Introductory Lectures
Part of the Lecture Notes in Physics book series (LNP, volume 528)

Keywords

Partition Function Ising Model Transfer Matrix Classical System Open Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S.R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10345 (1993)CrossRefADSGoogle Scholar
  2. 2.
    H.F. Trotter, Proc. Am. Math. Soc. 10, 545 (1959)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. Suzuki, Prog. Theor. Phys. 56, 1454 (1976)zbMATHCrossRefADSGoogle Scholar
  4. 4.
    R.P. Feynmann and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill (1965)Google Scholar
  5. 5.
    T. Nishino, J. Phys. Soc. Jpn. 64, 3598 (1995)zbMATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    E. Carlon and A. Drzewiński, Phys. Rev. Lett. 79, (1997) 1591; Phys. Rev. E 57, 2626 (1998)CrossRefADSGoogle Scholar
  7. 7.
    E. Carlon and F. Igloi, Phys. Rev. B 57, 7877 (1998); F. Igloi and E. Carlon, cond-mat/9805083CrossRefADSGoogle Scholar
  8. 8.
    E. Carlon, A. Drzewiński and J. Rogiers, Phys. Rev. B 58, 5070 (1998)CrossRefADSGoogle Scholar
  9. 9.
    R.J. Bursill, T. Xiang and G.A. Gehring, J. Phys. Condensed Matter L583–L590 (1996)Google Scholar
  10. 10.
    X. Wang and T. Xiang, Phys. Rev. B 56, 5061 (1997)CrossRefADSGoogle Scholar
  11. 11.
    F. Naef, X. Wang, X. Zotos and W. van der Linden, cond-mat/9812117, to appear in Phys. Rev. B (1999)Google Scholar
  12. 12.
    N. Shibata, J. Phys. Soc. Jpn 66, 2221 (1997)CrossRefADSGoogle Scholar
  13. 13.
    B. Ammon, M. Troyer, T.M. Rice and N. Shibata, cond-mat/9812144Google Scholar
  14. 14.
    Y. Honda and T. Horiguchi, Phys. Rev. E 56, 3920 (1997)CrossRefADSGoogle Scholar
  15. 15.
    S. Östlund and S. Rommer, Phys. Rev. Lett 75, 3537 (1995)CrossRefADSGoogle Scholar
  16. 15a.
    S. Rommer and S. Östlund, Phys. Rev. B 55, 2164 (1997); M. Andersson, M. Boman and S. Östlund, cond-mat/9810093CrossRefADSGoogle Scholar
  17. 16.
    It is possible to fix the boundary spins to consider more general boundary conditions.Google Scholar
  18. 17.
    There is a bibliography of Ising in cond-mat/9605174.Google Scholar
  19. 18.
    R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, (1982)zbMATHGoogle Scholar
  20. 19.
    R.B. Potts, Proc. Camb. Phil. Soc. 48, 106Google Scholar
  21. 20.
    F.Y. Wu, Rev. Mod. Phys. 54, 235 (1982) and references therein.CrossRefADSGoogle Scholar
  22. 21.
    Baxter used another definition of the density submatrix in his variational method [18], where his density submatrix is block diagonal, see (42)-(44).Google Scholar
  23. 22.
    T. Nishino and K. Okunishi, in Strongly Correlated Magnetic and Superconducting Systems, Eds. G. Sierra and M.A. Martín-Delgado, Springer (1997)Google Scholar
  24. 23.
    Eigenvalues of the DSM can be negative when the system contains randomness.Google Scholar
  25. 24.
    I. Peschel, M. Kaulke and Ö. Legeza, Ann. Physik (Leipzig) 8, 153 (1999); cond-mat/9810174zbMATHCrossRefADSMathSciNetGoogle Scholar
  26. 25.
    Strictly speaking, the density matrix eigenvalues do not decay exponentially; See K. Okunishi, Y. Hieida and Y. Akutsu, cond-mat/9810239Google Scholar
  27. 26.
    C. Lanczos: J. Res. Nat. Bur. Std. 45, 255 (1950)MathSciNetGoogle Scholar
  28. 27.
    The Numerical recipes home page (http://cfata2.harvard.edu/numerical-recipes/) is useful to know about numerical linear algebra. Also it is worth reading J. Wilkinson, The Algebraic Eigenvalue Problem, Oxford (1965)Google Scholar
  29. 28.
    L. Onsager, Phys. Rev. 65, 117 (1944)zbMATHCrossRefADSMathSciNetGoogle Scholar
  30. 29.
    H.A. Kramers and G.H. Wannier, Phys. Rev. 60, 263 (1941)zbMATHCrossRefADSMathSciNetGoogle Scholar
  31. 30.
    R. Kikuchi, Phys. Rev. 81, 988 (1951)zbMATHCrossRefADSMathSciNetGoogle Scholar
  32. 31.
    H.A. Bethe, Proc. Roy. Soc. A 150, 552 (1935)zbMATHADSCrossRefGoogle Scholar
  33. 32.
    M.C. Gutzwiller, Phys. Rev. 137, A1726 (1965)Google Scholar
  34. 33.
    J. Kanamori, J. Phys. Soc. Jpn. 30, 275 (1963)zbMATHGoogle Scholar
  35. 34.
    J. Hubbard, Proc. Roy. Soc. A 276, 238 (1963); A 281, 401 (1964)ADSCrossRefGoogle Scholar
  36. 35.
    R.J. Baxter, J. Math. Phys. 9, 650 (1968)CrossRefADSGoogle Scholar
  37. 35a.
    R.J. Baxter, J. Stat. Phys. 19, 461 (1978)CrossRefMathSciNetADSGoogle Scholar
  38. 36.
    N.P. Nightingale and H.W. Blöte: Phys. Rev. B 33, 659 (1986)CrossRefADSGoogle Scholar
  39. 37.
    I. Affleck, T. Kennedy, E.H. Lieb and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987)CrossRefADSGoogle Scholar
  40. 38.
    M. Fannes, B. Nachtergale and R.F. Werner, Europhys. Lett. 10, 633 (1989)ADSCrossRefGoogle Scholar
  41. 38a.
    M. Fannes, B. Nachtergale and R.F. Werner, Commun. Math. Phys. 144, 443 (1992)zbMATHCrossRefADSGoogle Scholar
  42. 38b.
    M. Fannes, B. Nachtergale and R.F. Werner, Commun. Math. Phys. 174, 477 (1995)CrossRefADSGoogle Scholar
  43. 39.
    A. Klümper, A. Schadschneider and J. Zittarz, Z. Phys. B 87, 281 (1992)CrossRefADSGoogle Scholar
  44. 39a.
    H. Niggemann, A. Klümper and J. Zittartz, Z. Phys. B 104, 103 (1997)CrossRefADSGoogle Scholar
  45. 40.
    B. Derrida and M.R. Evans, J. Phys. A: Math. Gen. 26, 1493 (1993)zbMATHCrossRefADSMathSciNetGoogle Scholar
  46. 41.
    N. Rajewsky, L. Santen, A. Schadschneider and M. Schreckenberg, cond-mat/9710316Google Scholar
  47. 42.
    A. Honecker and I. Peschel, J. Stat. Phys. 88, 319 (1997)zbMATHCrossRefMathSciNetADSGoogle Scholar
  48. 43.
    Y. Hieida, J. Phys. Soc. Jpn. 67, 369 (1998)zbMATHCrossRefADSGoogle Scholar
  49. 44.
    S.R. White, Phys. Rev. Lett. 77, 3633 (1996)CrossRefADSGoogle Scholar
  50. 45.
    T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 64, 4084 (1995)CrossRefADSGoogle Scholar
  51. 46.
    U. Schollwöck, Phys. Rev. B 58, 8194 (1998)CrossRefADSGoogle Scholar
  52. 47.
    A modification of the whole variational state is more time consuming. The situation is similar to the ‘Order N’ problem in the density functional formalism.Google Scholar
  53. 48.
    S.R. White, D.J. Scalapino, R.L. Sugar, E.Y. Loh, J.E. Gubernatis and R.T. Scalettar, Phys. Rev. B 40, 506 (1989)CrossRefADSGoogle Scholar
  54. 49.
    M.E. Fisher, in Proc. Int. School of Physics ‘Enrico Fermi’, Ed. M.S. Green, Academic Press, 51, p 1 (1971)Google Scholar
  55. 50.
    M.N. Barber, in Phase Transitions and Critical Phenomena, Ed. C. Domb and J.L. Lebowitz, Academic Press, 8, p. 146 (1983) and references therein.Google Scholar
  56. 51.
    K. Okunishi, Thesis, Osaka University 1996 (in Japanese); Thesis, Osaka University 1999 (in English). (Contact to okunishi@godzilla.phys.sci.osaka-u.ac.jp)Google Scholar
  57. 52.
    T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 65, 891 (1996)zbMATHCrossRefADSGoogle Scholar
  58. 52a.
    T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 66, 3040 (1997)zbMATHCrossRefADSGoogle Scholar
  59. 53.
    T. Nishino, K. Okunishi, and M. Kikuchi, Physics Letters A 213, 69 (1996)CrossRefADSGoogle Scholar
  60. 54.
    T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 67, 1492 (1998)CrossRefADSGoogle Scholar
  61. 55.
    T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 67, 3066 (1998)CrossRefADSGoogle Scholar
  62. 56.
    G. Sierra and M.A. Martín-Delgado, cond-mat/9811170.Google Scholar
  63. 57.
    Y. Hieida, K. Okunishi and Y. Akutsu, cond-mat/9901155Google Scholar
  64. 58.
    A way to decrease the computational effort in higher dimension is to perform RG transformations before creating the new RG transformations; it is possible, because the (infinite-system) DMRG is a self consistent method. [1,45]Google Scholar
  65. 59.
    The numerical precision in DMRG for a system with periodic boundary conditions is lower than that for a system with open boundary conditions. The reason can be understood by looking at the variational state written as a matrix product.Google Scholar
  66. 60.
    Position dependence in the quantum Hamiltonian can be treated by quantum DMRG; K. Hida, J. Phys. Soc. Jpn. 65, 895 (1996)CrossRefADSGoogle Scholar
  67. 61.
    S.G. Chung, J. Phys.: Cond. Matt. 9, L619 (1997); Current Topics in Physics, Ed. Y.M. Cho, J.B. Hong, and C.N. Yang, World Scientific, 1, p 295 (1998)Google Scholar
  68. 62.
    The authors will list up the latest information about DMRG in this URL: http://quattro.phys.sci.kobe-u.ac.jp/dmrg.htmlGoogle Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • T. Nishino
    • 1
  • K. Okunishi
    • 2
  • Y. Hieida
    • 2
  • T. Hikihara
    • 1
  • H. Takasaki
    • 1
  1. 1.Department of PhysicsGraduate School of Science and Technology Kobe UniversityKobeJapan
  2. 2.Department of Physics, Graduate School of ScienceOsaka University ToyonakaOsakaJapan

Personalised recommendations