Abstract
The critical behaviour of a semi-infinite system withO(n) spin symmetry is studied in 4-ɛ dimensions near the ordinary transition using renormalization-group methods of field theory and ɛ-expansion techniques. It is found that, to all orders in ɛ, all surface exponents can be expressed in terms of two bulk exponents and a single surface exponent which follows from the anomalous dimension of the derivative ∂⊥ φ(x ∥,0) of The order parameter ϕ(x‖,x ⊥) at the surface (x ⊥=0). As a byproduct, Barber's scaling law 2γ1 − γ11 = γ + ν is obtained. The surface exponents are calculated to second order in ɛ. Our results show that the scaling relationη ∥ = ν−1 proposed by Bray and Moore is incorrect. The behaviour of various scaling functions close to the surface (i.e. forx ⊥ ≪correlation length) is determined with the help of short-distance expansions. We also treat corrections to scaling and logarithmic corrections in four dimensions. Our results for the logarithmic corrections of the layer and local susceptibilities disagree with those obtained by Guttmann and Reeve.
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Diehl, H.W., Dietrich, S.: Phys. Lett.80A, 408 (1980)
Reeve, J.S., Guttmann, A.J.: Phys. Rev. Lett.45, 1581 (1980)
For a more complete list of earlier references see [5] and [8]
Binder, K., Hohenberg, P.C.: Phys. Rev. B6, 3461 (1972)
Binder, K., Hohenberg, P.C.: Phys. Rev. B9, 2194 (1974)
Lubensky, T.C., Rubin, M.H.: Phys. Rev. B11, 4533 (1975)
Lubensky, T.C., Rubin, M.H.: Phys. Rev. B12, 3885 (1975): Phys. Rev. Lett.31, 1469 (1973)
Bray, A.J., Moore, M.A.: J. Phys. A10, 1927 (1977); Phys. Rev. Lett.38, 1046 (1977)
Bray, A.J., Moore, M.A.: Phys. Rev. Lett.38, 735 (1977)
Švrakić, N.M., Wortis, M.: Phys. Rev. B15, 396 (1977)
Švrakić, N.M., Pandit, R., Wortis, M.: Phys. Rev. B22, 1286 (1980)
Burkhardt, T.W., Eisenriegler, E.: Phys. Rev. B16, 3213 (1977); Phys. Rev. B17, 318 (1978)
McCoy, B.M., Wu, T.T.: Phys. Rev.162, 436 (1967)
Fisher, M.E., Ferdinand, A.E.: Phys. Rev. Lett.19, 169 (1967)
Bariev, R.Z.: Teor. Mat. Fiz.40, 95 (1979); Teor. Mat. Fiz.42, 262 (1980); Sov. Phys. JETP50, 613 (1979)
Enting, I.G., Guttmann, A.J.: J. Phys. A13, 1043 (1980)
Whittington, S.G., Torrie, G.M., Guttmann, A.J.: J. Phys. A12, 2449 (1979)
Barber, M.N., Guttmann, A.J., Middlemiss, K.M., Torrie, G.M., Whittington, S.G.: J. Phys. A11, 1833 (1978)
De'Bell, K., Essam, J.W.: Preprint (1980)
Guttmann, A.J., Reeve, J.S.: J. Phys. A13, 3495 (1980)
Wilson, C.A.: J. Phys. C13, 925 (1980)
Grempel, D.R., Houghton, A., Ying, S.C.: Phys. Lett.78A, 295 (1980)
Barber, M.N.: Phys. Rev. B8, 407 (1973)
Fisher, M.E., Barber, M.N.: Phys. Rev. Lett.28, 1516 (1972)
Fisher, M.E.: In: Proceedings of the 1970 Enrico Fermi Summer School, Course No. 51. New York: Academic Press 1972
Brézin, E., Le Guillou, J.C., Zinn-Justin, J.: In: Phase Transitions and Critical Phenomena. Domb, C., Green, M.S. (ed.), Vol. 6. New York: Academic Press 1976
Amit, D.J.: Field theory, the renormalization group, and critical phenomena. New York: McGraw-Hill 1978
Kawasaki, K., Ohta, T., Onuki, A.: Prog. Theor. Phys.63, 821 (1980)
t'Hooft, G., Veltman, M.: Nucl. Phys. B44, 189 (1972)
Itzykson, C., Zuber, J.-B.: Quantum Field Theory. New York: McGraw-Hill 1980
Zimmermann, W.: In: Lectures on Elementary Particles and Quantum Field Theory. Deser, S., Grisaru, M., Pendleton, H. (ed.). Cambridge: MIT Press 1971
Jasnow, D., Rudnick, J.: Phys. Rev. B17, 1351 (1978); Phys. Rev. Lett.41, 698 (1978)
Ohta, T., Kawasaki, K.: Progr. Theor. Phys.58, 467 (1977)
Collins, J.C., Macfarlane, A.J.: Phys. Rev. D10, 1201 (1974)
Barber, M.N.: J. Stat. Phys.10, 59 (1974)
Costache, G.: Private communication
Nakanishi, N.: Graph Theory and Feynman Integrals, New York: Gordon and Breach, Science Publishers, Inc. 1971
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A brief account of some of the results presented here was given in [1]. The surface exponents were independently calculated to order ε2 by Reeve and Guttmann [2] using an alternative method
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Diehl, H.W., Dietrich, S. Field-theoretical approach to static critical phenomena in semi-infinite systems. Z. Physik B - Condensed Matter 42, 65–86 (1981). https://doi.org/10.1007/BF01298293
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DOI: https://doi.org/10.1007/BF01298293