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2
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0003242030
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C. Domb and J. L. Lebowitz (Academic, London
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W. Selke, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London, 1992), Vol. 15, pp. 1–72.
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(1992)
Phase Transitions and Critical Phenomena
, vol.15
, pp. 1-72
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Selke, W.1
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3
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0036025979
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cond-mat/0205284
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H.W. Diehl, Acta Phys. Slov. 52, 271 (2002); cond-mat/0205284.
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(2002)
Acta Phys. Slov.
, vol.52
, pp. 271
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Diehl, H.W.1
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5
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4243374619
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C.C. Becerra, Y. Shapira, N.F. Oliveira, Jr., and T.S. Chang, Phys. Rev. Lett. 44, 1692 (1980).
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(1980)
Phys. Rev. Lett.
, vol.44
, pp. 1692
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Becerra, C.C.1
Shapira, Y.2
Oliveira, N.F.3
Chang, T.S.4
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8
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0034289666
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C.C. Becerra, V. Bindilatti, and N.F. Oliveira, Jr., Phys. Rev. B 62, 8965 (2000).
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(2000)
Phys. Rev. B
, vol.62
, pp. 8965
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Becerra, C.C.1
Bindilatti, V.2
Oliveira, N.F.3
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22
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85038995832
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i ≠ 1 are conceivable
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i ≠ 1 are conceivable.
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24
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85039022424
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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
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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.
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26
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85039010186
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*(7) − 1 = 13/2. Thus the largest permissible integer value of m is m = 6
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*(7) − 1 = 13/2. Thus the largest permissible integer value of m is m = 6.
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28
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0003130040
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C. Domb and J. L. Lebowitz (Academic, London
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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.
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(1986)
Phase Transitions and Critical Phenomena
, vol.10
, pp. 75-267
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Diehl, H.W.1
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31
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85038984508
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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
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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.
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33
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85039024068
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DD(u/2) = Z(u)
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DD(u/2) = Z(u).
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34
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85038993334
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m, d(υ) can be found in this reference as well
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m, d(υ) can be found in this reference as well.
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35
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85038988033
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* = 2 of the O(n) vector model
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* = 2 of the O(n) vector model.
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42
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0004081609
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F. J. W. Hahne, Vol. 186 of Lecture Notes in Physics (Springer-Verlag, Berlin
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M. E. Fisher, in Critical Phenomena, edited by F. J. W. Hahne, Vol. 186 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1983), pp. 1–139.
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(1983)
Critical Phenomena
, pp. 1-139
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Fisher, M.E.1
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43
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85038994503
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For background on dangerous irrelevant variables, see, e.g., Appendix D of Ref. 41
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For background on dangerous irrelevant variables, see, e.g., Appendix D of Ref. 41.
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44
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85038992941
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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
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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.
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45
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85039008482
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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
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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.
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46
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85038975095
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τ and σ
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τ and σ.
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51
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85039006139
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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
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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.
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53
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85038999499
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l4⊥
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⊥.
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54
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0001856561
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C. Domb and J. L. Lebowitz (Academic, London
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K. Binder, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London, 1983), Vol. 8, pp. 1–144.
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(1983)
Phase Transitions and Critical Phenomena
, vol.8
, pp. 1-144
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Binder, K.1
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60
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85039021279
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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
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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.
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62
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85038992680
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MATHEMATICA, version 3.0, a product of Wolfram Research
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MATHEMATICA, version 3.0, a product of Wolfram Research.
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