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Transfer-matrix approach to classical systems

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Density-Matrix Renormalization

Part of the book series: Lecture Notes in Physics ((LNP,volume 528))

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References

  1. S.R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10345 (1993)

    Article  ADS  Google Scholar 

  2. H.F. Trotter, Proc. Am. Math. Soc. 10, 545 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Suzuki, Prog. Theor. Phys. 56, 1454 (1976)

    Article  MATH  ADS  Google Scholar 

  4. R.P. Feynmann and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill (1965)

    Google Scholar 

  5. T. Nishino, J. Phys. Soc. Jpn. 64, 3598 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. E. Carlon and A. Drzewiński, Phys. Rev. Lett. 79, (1997) 1591; Phys. Rev. E 57, 2626 (1998)

    Article  ADS  Google Scholar 

  7. E. Carlon and F. Igloi, Phys. Rev. B 57, 7877 (1998); F. Igloi and E. Carlon, cond-mat/9805083

    Article  ADS  Google Scholar 

  8. E. Carlon, A. Drzewiński and J. Rogiers, Phys. Rev. B 58, 5070 (1998)

    Article  ADS  Google Scholar 

  9. R.J. Bursill, T. Xiang and G.A. Gehring, J. Phys. Condensed Matter L583–L590 (1996)

    Google Scholar 

  10. X. Wang and T. Xiang, Phys. Rev. B 56, 5061 (1997)

    Article  ADS  Google Scholar 

  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. N. Shibata, J. Phys. Soc. Jpn 66, 2221 (1997)

    Article  ADS  Google Scholar 

  13. B. Ammon, M. Troyer, T.M. Rice and N. Shibata, cond-mat/9812144

    Google Scholar 

  14. Y. Honda and T. Horiguchi, Phys. Rev. E 56, 3920 (1997)

    Article  ADS  Google Scholar 

  15. S. Östlund and S. Rommer, Phys. Rev. Lett 75, 3537 (1995)

    Article  ADS  Google Scholar 

  16. S. Rommer and S. Östlund, Phys. Rev. B 55, 2164 (1997); M. Andersson, M. Boman and S. Östlund, cond-mat/9810093

    Article  ADS  Google Scholar 

  17. It is possible to fix the boundary spins to consider more general boundary conditions.

    Google Scholar 

  18. There is a bibliography of Ising in cond-mat/9605174.

    Google Scholar 

  19. R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, (1982)

    MATH  Google Scholar 

  20. R.B. Potts, Proc. Camb. Phil. Soc. 48, 106

    Google Scholar 

  21. F.Y. Wu, Rev. Mod. Phys. 54, 235 (1982) and references therein.

    Article  ADS  Google Scholar 

  22. 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. 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. Eigenvalues of the DSM can be negative when the system contains randomness.

    Google Scholar 

  25. I. Peschel, M. Kaulke and Ö. Legeza, Ann. Physik (Leipzig) 8, 153 (1999); cond-mat/9810174

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Strictly speaking, the density matrix eigenvalues do not decay exponentially; See K. Okunishi, Y. Hieida and Y. Akutsu, cond-mat/9810239

    Google Scholar 

  27. C. Lanczos: J. Res. Nat. Bur. Std. 45, 255 (1950)

    MathSciNet  Google Scholar 

  28. 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. L. Onsager, Phys. Rev. 65, 117 (1944)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. H.A. Kramers and G.H. Wannier, Phys. Rev. 60, 263 (1941)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. R. Kikuchi, Phys. Rev. 81, 988 (1951)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. H.A. Bethe, Proc. Roy. Soc. A 150, 552 (1935)

    Article  MATH  ADS  Google Scholar 

  33. M.C. Gutzwiller, Phys. Rev. 137, A1726 (1965)

    Google Scholar 

  34. J. Kanamori, J. Phys. Soc. Jpn. 30, 275 (1963)

    MATH  Google Scholar 

  35. J. Hubbard, Proc. Roy. Soc. A 276, 238 (1963); A 281, 401 (1964)

    Article  ADS  Google Scholar 

  36. R.J. Baxter, J. Math. Phys. 9, 650 (1968)

    Article  ADS  Google Scholar 

  37. R.J. Baxter, J. Stat. Phys. 19, 461 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  38. N.P. Nightingale and H.W. Blöte: Phys. Rev. B 33, 659 (1986)

    Article  ADS  Google Scholar 

  39. I. Affleck, T. Kennedy, E.H. Lieb and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987)

    Article  ADS  Google Scholar 

  40. M. Fannes, B. Nachtergale and R.F. Werner, Europhys. Lett. 10, 633 (1989)

    Article  ADS  Google Scholar 

  41. M. Fannes, B. Nachtergale and R.F. Werner, Commun. Math. Phys. 144, 443 (1992)

    Article  MATH  ADS  Google Scholar 

  42. M. Fannes, B. Nachtergale and R.F. Werner, Commun. Math. Phys. 174, 477 (1995)

    Article  ADS  Google Scholar 

  43. A. Klümper, A. Schadschneider and J. Zittarz, Z. Phys. B 87, 281 (1992)

    Article  ADS  Google Scholar 

  44. H. Niggemann, A. Klümper and J. Zittartz, Z. Phys. B 104, 103 (1997)

    Article  ADS  Google Scholar 

  45. B. Derrida and M.R. Evans, J. Phys. A: Math. Gen. 26, 1493 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  46. N. Rajewsky, L. Santen, A. Schadschneider and M. Schreckenberg, cond-mat/9710316

    Google Scholar 

  47. A. Honecker and I. Peschel, J. Stat. Phys. 88, 319 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  48. Y. Hieida, J. Phys. Soc. Jpn. 67, 369 (1998)

    Article  MATH  ADS  Google Scholar 

  49. S.R. White, Phys. Rev. Lett. 77, 3633 (1996)

    Article  ADS  Google Scholar 

  50. T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 64, 4084 (1995)

    Article  ADS  Google Scholar 

  51. U. Schollwöck, Phys. Rev. B 58, 8194 (1998)

    Article  ADS  Google Scholar 

  52. 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. 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)

    Article  ADS  Google Scholar 

  54. M.E. Fisher, in Proc. Int. School of Physics ‘Enrico Fermi’, Ed. M.S. Green, Academic Press, 51, p 1 (1971)

    Google Scholar 

  55. 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. 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. T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 65, 891 (1996)

    Article  MATH  ADS  Google Scholar 

  58. T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 66, 3040 (1997)

    Article  MATH  ADS  Google Scholar 

  59. T. Nishino, K. Okunishi, and M. Kikuchi, Physics Letters A 213, 69 (1996)

    Article  ADS  Google Scholar 

  60. T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 67, 1492 (1998)

    Article  ADS  Google Scholar 

  61. T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 67, 3066 (1998)

    Article  ADS  Google Scholar 

  62. G. Sierra and M.A. Martín-Delgado, cond-mat/9811170.

    Google Scholar 

  63. Y. Hieida, K. Okunishi and Y. Akutsu, cond-mat/9901155

    Google Scholar 

  64. 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. 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. Position dependence in the quantum Hamiltonian can be treated by quantum DMRG; K. Hida, J. Phys. Soc. Jpn. 65, 895 (1996)

    Article  ADS  Google Scholar 

  67. 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. The authors will list up the latest information about DMRG in this URL: http://quattro.phys.sci.kobe-u.ac.jp/dmrg.html

    Google Scholar 

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Ingo Peschel Matthias Kaulke Xiaoqun Wang Karen Hallberg

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Nishino, T., Okunishi, K., Hieida, Y., Hikihara, T., Takasaki, H. (1999). Transfer-matrix approach to classical systems. In: Peschel, I., Kaulke, M., Wang, X., Hallberg, K. (eds) Density-Matrix Renormalization. Lecture Notes in Physics, vol 528. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0106067

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  • DOI: https://doi.org/10.1007/BFb0106067

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