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Modern finite-size criticality: Dirichlet and Neumann boundary conditions

  • Messias V. S. Santos
  • José B. da Silva Jr.
  • Marcelo M. LeiteEmail author
Regular Article
  • 29 Downloads

Abstract.

Finite-size critical systems defined on a parallel-plate geometry of finite extent along one single (z) direction with Dirichlet and Neumann boundary conditions at z = 0, L are analyzed in momentum space. We introduce a modified representation for the discrete eigenfunctions in a renormalized one-particle-irreducible (1PI) vertex part scalar field-theoretic framework using either massless or massive fields. The appearance of multiplicities in the Feynman rules to construct diagrams due to this choice of representation of the basis functions is discussed along with the modified normalization conditions. For nonvanishing external quasi-momenta, Dirichlet and Neumann boundary conditions are shown to be unified within a single formalism. We examine the dimensional crossover regimes for these and show a correspondence with those from antiperiodic and periodic boundary conditions. It is demonstrated that finite-size effects for Dirichlet and Neumann boundary conditions do not require surface fields necessarily but are implemented nontrivially from the Feynman rules involving only bulk terms in the Lagrangian. As an application, the critical exponents \(\eta\) and \(\nu\) are evaluated at least up to two-loop level through diagrammatic means. We show that the critical indices are the same as those from the bulk (infinite) system irrespective of the boundary conditions.

References

  1. 1.
    K.G. Wilson, Phys. Rev. B 4, 3174 (1971)CrossRefGoogle Scholar
  2. 2.
    K.G. Wilson, Phys. Rev. B 4, 3184 (1971)CrossRefGoogle Scholar
  3. 3.
    K.G. Wilson, M.E. Fisher, Phys. Rev. Lett. 28, 240 (1972)CrossRefGoogle Scholar
  4. 4.
    K.G. Wilson, Phys. Rev. Lett. 28, 548 (1972)CrossRefGoogle Scholar
  5. 5.
    D.J. Amit, V. Martin-Mayor, Field Theory, the Renormalization Group and Critical Phenomena, 3rd edition (World Scientific, Singapore, 2005)Google Scholar
  6. 6.
    L.M. Falicov, D.T. Pierce, S.D. Bader, R. Gronsky, K.B. Hathaway, H.J. Hopster, D.N. Lambeth, S.S.P. Parkin, G. Prinz, M. Salamon, I.K. Schuller, R.H. Victora, J. Mater. Res. 5, 1299 (1990)CrossRefGoogle Scholar
  7. 7.
    A. Monsen, J.E. Boschker, F. Macià, J.J. Wells, P. Nordblad, A. Kent, R. Mathieu, T. Tybell, E. Walström, J. Magn. & Magn. Mater. 369, 197 (2014)CrossRefGoogle Scholar
  8. 8.
    H.W. Diehl, in Phase Transitions and Critical Phenomena, edited by C. Domb, J.L. Lebowitz, Vol. 10 (Academic, London, 1986) p. 76Google Scholar
  9. 9.
    A.M. Nemirovsky, K.F. Freed, Phys. Rev. B 31, 3161 (1985)CrossRefGoogle Scholar
  10. 10.
    A.M. Nemirovsky, K.F. Freed, J. Phys. A 18, 3275 (1985)CrossRefGoogle Scholar
  11. 11.
    A.M. Nemirovsky, K.F. Freed, J. Phys. A 19, 591 (1986)CrossRefGoogle Scholar
  12. 12.
    A.M Nemirovsky, Z.-G. Wang, K.F. Freed, Phys. Rev. B 34, 7886 (1996)CrossRefGoogle Scholar
  13. 13.
    A.M. Nemirovsky, Z.-G. Wang, K.F. Freed, Phys. Rev. B 36, 3755 (1987)CrossRefGoogle Scholar
  14. 14.
    A.M. Nemirovsky, in Field Theory, Quantum Gravity and Strings II, edited by H.J. de Vega, N. Sánchez, Vol. 280 (Springer-Verlag, Berlin, 1986) p. 229Google Scholar
  15. 15.
    J.G. Brankov, D.M. Danchev, N.S. Tonchev, Theory of Critical Phenomena in Finite-Size Systems: Scaling and Quantum Effects (World Scientific, Singapore, 2000) chapt. 7Google Scholar
  16. 16.
    H.B. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948)Google Scholar
  17. 17.
    Z. Borjan, P.J. Upton, Phys. Rev. Lett. 101, 125702 (2008)CrossRefGoogle Scholar
  18. 18.
    A. Gambassi, J. Phys.: Conf. Ser. 161, 012037 (2009)Google Scholar
  19. 19.
    C. Farina, Braz. J. Phys. 36, 1137 (2006)CrossRefGoogle Scholar
  20. 20.
    R. Garcia, M.H.W. Chan, Phys. Rev. Lett. 83, 1187 (1999)CrossRefGoogle Scholar
  21. 21.
    A. Ganshin, S. Scheidemantel, R. Garcia, M.H.W. Chan, Phys. Rev. Lett. 97, 075301 (2006)CrossRefGoogle Scholar
  22. 22.
    G. Bimonte, E. Calloni, G. Esposito, L. Milano, L. Rosa, Phys. Rev. Lett. 94, 180402 (2005)CrossRefGoogle Scholar
  23. 23.
    J. Goyon, A. Colin, G. Ovariez, A. Ajdari, L. Bocquet, Nature 454, 84 (2008)CrossRefGoogle Scholar
  24. 24.
    J. Goyon, A. Colin, L. Bocquet, Soft Matter 6, 2668 (2010)CrossRefGoogle Scholar
  25. 25.
    T.P. Chen, F.M. Gasparini, Phys. Rev. Lett. 40, 331 (1978)CrossRefGoogle Scholar
  26. 26.
    F.M. Gasparini, G. Agnolet, J.D. Reppy, Phys. Rev. B 29, 138 (1984)CrossRefGoogle Scholar
  27. 27.
    F.M. Gasparini, M.O. Kimball, K.P. Mooney, M. Diaz-Avila, Rev. Mod. Phys. 80, 1009 (2008)CrossRefGoogle Scholar
  28. 28.
    B.A. Scheibner, M.R. Meadows, R.C. Mockler, W.J. O'Sullivan, Phys. Rev. Lett. 43, 590 (1979)CrossRefGoogle Scholar
  29. 29.
    M.R. Meadows, B.A. Scheibner, R.C. Mockler, W.J. O'Sullivan, Phys. Rev. Lett. 43, 592 (1979)CrossRefGoogle Scholar
  30. 30.
    R. Höhmann, U. Kuhl, H.J. Stöckmann, J.D. Urbina, M.R. Dennis, Phys. Rev. E 79, 016203 (2009)MathSciNetCrossRefGoogle Scholar
  31. 31.
    S. Sandfeld, Z. Budrikis, S. Zapperi, D.F. Castellanos, J. Stat. Mech. 2, 02011 (2015)Google Scholar
  32. 32.
    X. Zhou, Z. Zhang, Int. J. Mol. Sci. 14, 24135 (2013)CrossRefGoogle Scholar
  33. 33.
    H. Chamati, J. Phys. A 41, 375002 (2008)MathSciNetCrossRefGoogle Scholar
  34. 34.
    M.E. Fisher, in Critical Phenomena, Proceedings of the 1970 Enrico Fermi International School of Physics, Course LI, edited by M.S. Green (Academic, New York, 1971) p. 1Google Scholar
  35. 35.
    M.E. Fisher, Rev. Mod. Phys. 46, 597 (1974)CrossRefGoogle Scholar
  36. 36.
    M.E. Fisher, M.N. Barber, Phys. Rev. Lett. 28, 1516 (1972)CrossRefGoogle Scholar
  37. 37.
    M.N. Barber, in Phase Transitions and Critical Phenomena, edited by C. Domb, J.L. Lebowitz, Vol. 8 (Academic, New York, 1983) p. 145Google Scholar
  38. 38.
    V. Privman, in Finite Size Scaling and Numerical Simulations in Statistical Mechanics, edited by V. Privman (World Scientific, Singapore, 1990) p. 1Google Scholar
  39. 39.
    V. Privman, M.E. Fisher, Phys. Rev. B 30, 322 (1984)MathSciNetCrossRefGoogle Scholar
  40. 40.
    A.M. Nemirovsky, K.F. Freed, J. Phys. A 18, L319 (1985)CrossRefGoogle Scholar
  41. 41.
    A.M. Nemirovsky, K.F. Freed, Nucl. Phys. B 270, 423 (1986)CrossRefGoogle Scholar
  42. 42.
    J.B. da Silva Jr., M.M. Leite, J. Math. Phys. 53, 043303 (2012)MathSciNetCrossRefGoogle Scholar
  43. 43.
    N.F. Svaiter, J. Math. Phys. 45, 4524 (2004)MathSciNetCrossRefGoogle Scholar
  44. 44.
    E. Brezin, J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. D 8, 434 (1973)CrossRefGoogle Scholar
  45. 45.
    E. Brezin, J.C. Le Guillou, J. Zinn-Justin, in Phase Transitions and Critical Phenomena, edited by C. Domb, M.S. Green, Vol. 6 (Academic Press, London, 1976) p. 127Google Scholar
  46. 46.
    P.R.S. Carvalho, M.M. Leite, J. Math. Phys. 54, 093301 (2013)MathSciNetCrossRefGoogle Scholar
  47. 47.
    A.A. Vladimirov, D.I. Kazakov, O.V. Tarasov, Sov. Phys. JETP 50, 521 (1979)Google Scholar
  48. 48.
    J. Naud, I. Nemenmann, M. Van Raamsdonk, V. Periwal, Nucl. Phys. B 540, 533 (1999)CrossRefGoogle Scholar
  49. 49.
    H. Boschi-Filho, C. Farina, Phys. Lett. A 205, 255 (1995)MathSciNetCrossRefGoogle Scholar
  50. 50.
    I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals Series and Products (Academic Press, New York, 2000)Google Scholar
  51. 51.
    M. Asorey, D. Garcia-Álvarez, J.M. Muñoz-Castañeda, J. Phys. A 40, 6667 (2007)CrossRefGoogle Scholar
  52. 52.
    M.M. Leite, A.M. Nemirovsky, M.D. Coutinho-Filho, J. Magn. & Magn. Mater. 104-107, 181 (1992)CrossRefGoogle Scholar
  53. 53.
    M.M. Leite, M. Sardelich, M.D. Coutinho-Filho, Phys. Rev. E 59, 2683 (1999)CrossRefGoogle Scholar
  54. 54.
    C.D. Fosco, N.F. Svaiter, J. Math. Phys. 42, 5185 (2001)MathSciNetCrossRefGoogle Scholar
  55. 55.
    M.I. Caicedo, N.F. Svaiter, J. Math. Phys. 45, 179 (2004)MathSciNetCrossRefGoogle Scholar
  56. 56.
    M.M. Leite, Phys. Rev. B 67, 104415 (2003)CrossRefGoogle Scholar
  57. 57.
    P.R.S. Carvalho, M.M. Leite, Ann. Phys. 324, 178 (2009)CrossRefGoogle Scholar
  58. 58.
    M.M. Leite, Phys. Rev. B 61, 14691 (2000)CrossRefGoogle Scholar
  59. 59.
    M.M. Leite, Phys. Rev. B 68, 052408 (2003)CrossRefGoogle Scholar
  60. 60.
    M.M. Leite, Phys. Lett. A 326, 281 (2004)CrossRefGoogle Scholar
  61. 61.
    M.M. Leite, Phys. Rev. B 72, 224432 (2005)CrossRefGoogle Scholar
  62. 62.
    P.R.S. Carvalho, M.M. Leite, Ann. Phys. 325, 151 (2010)CrossRefGoogle Scholar
  63. 63.
    C.F. Farias, M.M. Leite, J. Stat. Phys. 148, 972 (2012)MathSciNetCrossRefGoogle Scholar
  64. 64.
    M.I. Sena Jr., M.M. Leite, J. Phys. Conf. Ser. 574, 012170 (2015)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratório de Física Teórica e Computacional, Departamento de FísicaUniversidade Federal de PernambucoRecife, PEBrazil
  2. 2.C N Yang Institute for Theoretical PhysicsState University of New YorkStony BrookUSA

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