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Volumn 68, Issue 22, 2003, Pages

Boundary critical behavior at m-axial Lifshitz points for a boundary plane parallel to the modulation axes

Author keywords

[No Author keywords available]

Indexed keywords

ANISOTROPY; ARTICLE; ENERGY; FOURIER TRANSFORMATION; MATHEMATICAL ANALYSIS; MONTE CARLO METHOD; THEORY;

EID: 0742305042     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.68.224428     Document Type: Article
Times cited : (9)

References (62)
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    • (1992) Phase Transitions and Critical Phenomena , vol.15 , pp. 1-72
    • Selke, W.1
  • 3
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    • cond-mat/0205284
    • H.W. Diehl, Acta Phys. Slov. 52, 271 (2002); cond-mat/0205284.
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    • Diehl, H.W.1
  • 22
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    • i ≠ 1 are conceivable
    • i ≠ 1 are conceivable.
  • 24
    • 85039022424 scopus 로고    scopus 로고
    • A brief account of parts of this work has been given in Ref. 25. Note that Eq. (12) of this letter contains a sign error, which is corrected by Eq. (58) of the present paper
    • A brief account of parts of this work has been given in Ref. 25. Note that Eq. (12) of this letter contains a sign error, which is corrected by Eq. (58) of the present paper.
  • 26
    • 85039010186 scopus 로고    scopus 로고
    • *(7) − 1 = 13/2. Thus the largest permissible integer value of m is m = 6
    • *(7) − 1 = 13/2. Thus the largest permissible integer value of m is m = 6.
  • 28
    • 0003130040 scopus 로고
    • C. Domb and J. L. Lebowitz (Academic, London
    • H. W. Diehl, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London, 1986), Vol. 10, pp. 75–267.
    • (1986) Phase Transitions and Critical Phenomena , vol.10 , pp. 75-267
    • Diehl, H.W.1
  • 31
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    • As usual, these hold inside of averages (i.e., in an “operator sense”) and follow from the contributions localized on the surface of the equations of motion; see, e.g., Refs. 28, 29, and 30
    • As usual, these hold inside of averages (i.e., in an “operator sense”) and follow from the contributions localized on the surface of the equations of motion; see, e.g., Refs. 28, 29, and 30.
  • 33
    • 85039024068 scopus 로고    scopus 로고
    • DD(u/2) = Z(u)
    • DD(u/2) = Z(u).
  • 34
    • 85038993334 scopus 로고    scopus 로고
    • m, d(υ) can be found in this reference as well
    • m, d(υ) can be found in this reference as well.
  • 35
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    • * = 2 of the O(n) vector model
    • * = 2 of the O(n) vector model.
  • 42
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    • F. J. W. Hahne, Vol. 186 of Lecture Notes in Physics (Springer-Verlag, Berlin
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    • (1983) Critical Phenomena , pp. 1-139
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  • 43
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    • For background on dangerous irrelevant variables, see, e.g., Appendix D of Ref. 41
    • For background on dangerous irrelevant variables, see, e.g., Appendix D of Ref. 41.
  • 44
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    • * and (Formula presented) utilized in Eqs. (3.105b) and (3.105d) of Ref. 28. In the present m > 0 case, they do not only depend on u, the initial value of the running coupling constant ū, but also on λ
    • * and (Formula presented) utilized in Eqs. (3.105b) and (3.105d) of Ref. 28. In the present m > 0 case, they do not only depend on u, the initial value of the running coupling constant ū, but also on λ.
  • 45
    • 85039008482 scopus 로고    scopus 로고
    • Strictly speaking, this is guaranteed at this stage only for sufficiently small ε > 0 because the ε dependent terms of the exponent 1 + O(ε) appearing in (Formula presented) can then be trusted to be positive. However, from the well-studied m = 0 case we know that this exponent is proportional to the difference (Formula presented) of the surface critical exponents (Formula presented) of the ordinary and special CP transitions, respectively. This difference is positive and expected to be so even in the m > 0 case
    • Strictly speaking, this is guaranteed at this stage only for sufficiently small ε > 0 because the ε dependent terms of the exponent 1 + O(ε) appearing in (Formula presented) can then be trusted to be positive. However, from the well-studied m = 0 case we know that this exponent is proportional to the difference (Formula presented) of the surface critical exponents (Formula presented) of the ordinary and special CP transitions, respectively. This difference is positive and expected to be so even in the m > 0 case.
  • 46
    • 85038975095 scopus 로고    scopus 로고
    • τ and σ
    • τ and σ.
  • 51
    • 85039006139 scopus 로고    scopus 로고
    • In other words, the analogous bulk model with homogeneous values of (Formula presented) and (Formula presented) and the Hamiltonian (Formula presented) should have these properties
    • In other words, the analogous bulk model with homogeneous values of (Formula presented) and (Formula presented) and the Hamiltonian (Formula presented) should have these properties.
  • 53
    • 85038999499 scopus 로고    scopus 로고
    • l4⊥
    • ⊥.
  • 54
    • 0001856561 scopus 로고
    • C. Domb and J. L. Lebowitz (Academic, London
    • K. Binder, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London, 1983), Vol. 8, pp. 1–144.
    • (1983) Phase Transitions and Critical Phenomena , vol.8 , pp. 1-144
    • Binder, K.1
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    • 85039021279 scopus 로고    scopus 로고
    • 2 of their external points off the surface. Amputated legs whose external points are located on the surface, such as oz − − − − − − | − ⊗, are exponentials in z and hence good test functions of z. With regard to the coordinates r parallel to the surface situation is even simpler and completely analogous to the bulk case. Owing to the translational invariance we have parallel to the surface, the required integrals over r variables can be written as convolutions. Such convolutions of well-defined generalized functions of r variables (i.e., of the amputated graph with the two amputated legs) are well-defined, becoming simple products in p space
    • 2 of their external points off the surface. Amputated legs whose external points are located on the surface, such as oz − − − − − − | − ⊗, are exponentials in z and hence good test functions of z. With regard to the coordinates r parallel to the surface situation is even simpler and completely analogous to the bulk case. Owing to the translational invariance we have parallel to the surface, the required integrals over r variables can be written as convolutions. Such convolutions of well-defined generalized functions of r variables (i.e., of the amputated graph with the two amputated legs) are well-defined, becoming simple products in p space.
  • 62
    • 85038992680 scopus 로고    scopus 로고
    • MATHEMATICA, version 3.0, a product of Wolfram Research
    • MATHEMATICA, version 3.0, a product of Wolfram Research.


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.