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Prog. Theor. Phys. Vol. 49 No. 1 (1973) pp. 113-128

[ Full Text PDF : FREE ACCESS (1075K) ]

Expansion of a Critical Exponent in Inverse Powers of Spin Dimensionality

Ryuzo Abe

Department of Pure and Applied Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo

(Received July 10, 1972)

Abstract:

A systematic expansion method is developed for studying a critical exponent. The expansion parameter in the present theory is 1/n where n is the spin dimensionality. Since the limit n →∞ corresponds to the spherical model, the present method means that the deviation of critical exponent from its spherical model value is expanded in powers of 1/n. The susceptibility exponent γ is discussed and the terms up to 1/n are exactly derived for the d-dimensional simple hypercubical system with nearest-neighbor interaction (d is assumed to take a continuous value). The result is proved to be consistent with recent Wilson's ε expansion.


URL : http://ptp.ipap.jp/link?PTP/49/113/
DOI : 10.1143/PTP.49.113

[ Full Text PDF : FREE ACCESS (1075K) ] Citation:

References:

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Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 49 No. 2 (1973) pp. 424-441 :
    Critical Exponents for Long-Range Interactions. I
    Masuo Suzuki
  2. Progress of Theoretical Physics Vol. 49 No. 2 (1973) pp. 442-452 :
    Critical Exponents and Scaling Relations in 1/n Expansion
    Ryuzo Abe and Shinobu Hikami
  3. Progress of Theoretical Physics Vol. 49 No. 3 (1973) pp. 1074-1075 :
    Critical Exponent η up to 1/n2 for the Three-Dimensional System with Short-Range Interaction
    Ryuzo Abe
  4. Progress of Theoretical Physics Vol. 49 No. 4 (1973) pp. 1096-1105 :
    Critical Exponents and Scaling Relations in 1/n Expansion. II
    Shinobu Hikami
  5. Progress of Theoretical Physics Vol. 49 No. 4 (1973) pp. 1106-1120 :
    Critical Exponents for Long-Range Interactions. II
    Masuo Suzuki
  6. Progress of Theoretical Physics Vol. 49 No. 5 (1973) pp. 1440-1450 :
    Critical Exponents for Long-Range Interactions. III
    Masuo Suzuki
  7. Progress of Theoretical Physics Vol. 49 No. 5 (1973) pp. 1451-1458 :
    Critical Behavior of the Anisotropic Classical n-Vector Model
    Masuo Suzuki
  8. Progress of Theoretical Physics Vol. 49 No. 6 (1973) pp. 1851-1857 :
    Breakdown of Some Scaling Law Relations in 1/n Expansion
    Ryuzo Abe and Shinobu Hikami
  9. Progress of Theoretical Physics Vol. 49 No. 6 (1973) pp. 1877-1888 :
    Critical Exponent η up to 1/n2 for the Three-Dimensional System with Short-Range Interaction
    Ryuzo Abe
  10. Progress of Theoretical Physics Vol. 50 No. 2 (1973) pp. 393-408 :
    On the Expansion Theory of Critical Exponents
    Masuo Suzuki
  11. Progress of Theoretical Physics Vol. 51 No. 4 (1974) pp. 1041-1052 :
    Discontinuities of Critical Amplitude for Specific Heat
    Ryuzo Abe and Shinobu Hikami
  12. Progress of Theoretical Physics Vol. 52 No. 2 (1974) pp. 369-378 :
    Crossover Exponent of the Spin Anisotropic n-Vector Model with Short-Range Interaction in 1/n Expansion
    Shinobu Hikami and Ryuzo Abe
  13. Progress of Theoretical Physics Vol. 52 No. 4 (1974) pp. 1135-1145 :
    Quantum Effect, Critical Dynamics and One-Particle Excitation in 1/n Expansion. I
    Ryuzo Abe
  14. Progress of Theoretical Physics Vol. 52 No. 5 (1974) pp. 1431-1437 :
    Critical Behavior of an Extremely Anisotropic n-Vector Model
    Shinobu Hikami
  15. Progress of Theoretical Physics Vol. 54 No. 2 (1975) pp. 325-332 :
    Scaling Function for Critical Scattering in 1/n Expansion
    Ryuzo Abe and Shinobu Hikami
  16. Progress of Theoretical Physics Vol. 54 No. 6 (1975) pp. 1693-1706 :
    Study of Specific Heat below Tc in 1/n Expansion
    Ryuzo Abe and Shinobu Hikami
  17. Progress of Theoretical Physics Vol. 55 No. 4 (1976) pp. 1108-1115 :
    Critical Behavior of Classical n-Vector Model near Zero Temperature
    Shinobu Hikami
  18. Progress of Theoretical Physics Vol. 55 No. 5 (1976) pp. 1412-1418 :
    A Few Layered n-Vector Model in the Limit n → ∞
    Shinobu Hikami and Ryuzo Abe
  19. Progress of Theoretical Physics Vol. 56 No. 1 (1976) pp. 258-269 :
    The Form of Effective Potential
    Reijiro Fukuda
  20. Progress of Theoretical Physics Vol. 56 No. 3 (1976) pp. 990-991 :
    Critical Temperature of n-Vector Model near Two-Dimensions
    Shinobu Hikami
  21. Progress of Theoretical Physics Vol. 57 No. 3 (1977) pp. 785-796 :
    Fixed Points and Anomalous Dimensions in a Thirring-Type Model in 2+ε Dimensions
    Shinobu Hikami and Taizo Muta
  22. Progress of Theoretical Physics Vol. 57 No. 4 (1977) pp. 1197-1208 :
    Equation of State in 1/n Expansion
    Ryuzo Abe and Shinobu Hikami
  23. Progress of Theoretical Physics Vol. 57 No. 5 (1977) pp. 1538-1546 :
    Consistency for a Critical Amplitude Ratio Rχ in 1/n and ε Expansions
    Ryuzo Abe and Shinobu Hikami
  24. Progress of Theoretical Physics Vol. 57 No. 5 (1977) pp. 1705-1719 :
    Anomalous Dimensions of Composite Operators in the Gross-Neveu Model in 2+ε Dimensions
    Taizo Muta and Dragan S. Popović
  25. Progress of Theoretical Physics Vol. 58 No. 1 (1977) pp. 300-316 :
    Anomalous Dimensions in the (\overlineψφ)ψφ Model Field Theory in 1/N Expansion
    Dragan S. Popović
  26. Progress of Theoretical Physics Vol. 58 No. 2 (1977) pp. 506-515 :
    Fixed Points of a Coupled Order System in 1/n Expansion
    Shinobu Hikami
  27. Progress of Theoretical Physics Vol. 58 No. 4 (1977) pp. 1166-1176 :
    Phase Transitions in Gaussian Random Magnetic Fields
    Fumihiko Tanaka
  28. Progress of Theoretical Physics Vol. 58 No. 6 (1977) pp. 1722-1727 :
    Critical Behavior of Annealed Random Spin System at n = -2
    Ryuzo Abe
  29. Progress of Theoretical Physics Vol. 59 No. 2 (1978) pp. 386-392 :
    Logarithmic Corrections to a Simple Power Law at d = 4 in 1/n Expansion
    Yutaka Okabe
  30. Progress of Theoretical Physics Vol. 59 No. 3 (1978) pp. 742-750 :
    1/n Expansion for Weakly Random System
    Ryuzo Abe
  31. Progress of Theoretical Physics Vol. 59 No. 5 (1978) pp. 1478-1482 :
    Scaling Function for Equation of State
    Ryuzo Abe
  32. Progress of Theoretical Physics Vol. 59 No. 6 (1978) pp. 1825-1833 :
    1/n Expansion up to Order 1/n2. I
    Yutaka Okabe, Michiyoshi Oku and Ryuzo Abe
  33. Progress of Theoretical Physics Vol. 60 No. 5 (1978) pp. 1277-1286 :
    1/n Expansion Up to Order 1/n2. II
    Yutaka Okabe and Michiyoshi Oku
  34. Progress of Theoretical Physics Vol. 60 No. 5 (1978) pp. 1287-1297 :
    1/n Expansion Up to Order 1/n2. III
    Yutaka Okabe and Michiyoshi Oku
  35. Progress of Theoretical Physics Vol. 61 No. 2 (1979) pp. 443-451 :
    1/n Expansion Up to Order1/n2. IV
    Michiyoshi Oku and Yutaka Okabe
  36. Progress of Theoretical Physics Vol. 62 No. 1 (1979) pp. 98-106 :
    Some Remarks on the n=0 Problem in Critical Phenomena
    Ryuzo Abe
  37. Progress of Theoretical Physics Vol. 63 No. 5 (1980) pp. 1528-1540 :
    Spherical Limit of Coupled Time-Dependent n-Vector Models. I
    Michiyoshi Oku
  38. Progress of Theoretical Physics Vol. 64 No. 3 (1980) pp. 813-819 :
    Modification of 1/n Expansion in Critical Phenomena
    Ryuzo Abe
  39. Progress of Theoretical Physics Vol. 65 No. 4 (1981) pp. 1179-1190 :
    Higher-Order Calculations for Weakly Random System in 1/n Expansion
    Kazuo Ideura, Yutaka Okabe and Ryuzo Abe
  40. Progress of Theoretical Physics Vol. 65 No. 4 (1981) pp. 1237-1245 :
    d'-Dimensional Defect in d-Dimensional Lattice. I
    Ryozo Abe
  41. Progress of Theoretical Physics Vol. 65 No. 6 (1981) pp. 1835-1843 :
    d'-Dimensional Defect in d-Dimensional Lattice. II
    Ryuzo Abe
  42. Progress of Theoretical Physics Vol. 66 No. 1 (1981) pp. 77-89 :
    Quenched System in the Spherical Limit of the Random n-Vector Model
    Michiyoshi Oku
  43. Progress of Theoretical Physics Vol. 66 No. 5 (1981) pp. 1604-1611 :
    1/ n Expansion for Weakly Random System with Impurity Correlation
    Kazuhisa Yamamoto and Ryuzo Abe
  44. Progress of Theoretical Physics Vol. 66 No. 6 (1981) pp. 1959-1969 :
    Critical Amplitude Ratio ξ0+/ξ0T in 1/n and ε Expansions
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  45. Progress of Theoretical Physics Vol. 67 No. 1 (1982) pp. 118-130 :
    Gaussian Random Bond Problem for Annealed System in 1/n Expansion
    Michiyoshi Oku and Ryuzo Abe
  46. Progress of Theoretical Physics Vol. 70 No. 2 (1983) pp. 366-371 :
    d'-Dimensional Defect in d-Dimensional Lattice. III
    Ryuzo Abe
  47. Progress of Theoretical Physics Vol. 70 No. 5 (1983) pp. 1226-1239 :
    The 1/n Expansion for the n-Vector Model in the Semi-Infinite Space
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  48. Progress of Theoretical Physics Vol. 71 No. 4 (1984) pp. 707-713 :
    Equation for Pair Correlation Function of n-Vector Model
    Ryuzo Abe
  49. Progress of Theoretical Physics Vol. 78 No. 1 (1987) pp. 97-104 :
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    Ryuzo Abe
  50. Progress of Theoretical Physics Vol. 119 No. 1 (2008) pp. 9-27 :
    1/n Expansion of the Random Easy Axis Model
    Kazuo Ideura

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