Field theories in replica formalism and fixed points

W. Ishizuka; Y. Kikuchi

doi:10.1103/PhysRevD.32.422

Access by Xinjiang University

Field theories in replica formalism and fixed points

W. Ishizuka and Y. Kikuchi

Department of Physics, Tohoku University, Sendai, 980 Japan

PDF

Phys. Rev. D 32, 422 – Published 15 July, 1985

DOI: https://doi.org/10.1103/PhysRevD.32.422

Abstract

Scalar field theories, four-dimensional $φ^{4}$ and three-dimensional $φ^{6}$ theories, under the influences of random external fields are investigated in the replica formalism. First this formalism is reviewed by taking the effective potential as an example. Next the loop expansion is developed based on the effective replica Lagrangian. The β functions and their fixed points are studied in detail. The ( $φ^{4}$ $)_{4}$ theory is found to have a nontrivial infrared fixed point due to the interaction with the random external field in this formalism. The behavior of the coupling constant and the associated exponents are also calculated. In the replica formalism of the ( $φ^{6}$ $)_{3}$ theory, the modification of the ultraviolet fixed point in the original theory is studied.

References (26)

G. Parisi and Y.-S. Wu, Sci. Sin. 24, 484 (1981).
V. N. Gribov, Nucl. Phys. 139, 7 (1978).
Omitted end note.
Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 (1975).
G. Grinstein, Phys. Rev. Lett. 37, 944 (1976).
G. Parisi and N. Sourlas, Phys. Rev. Lett. 46, 371 (1981).
G. Parisi and N. Sourlas, Phys. Rev. Lett. 43, 774 (1979).
B. McClain, A. Niemi and C. Taylor, Ann. Phys. (N.Y.) 140, 232 (1982).
B. McClain, A. Niemi, C. Taylor, and L. C. R. Wijewardhana, Nucl. Phys. 217, 430 (1983).
A. Niemi and L. C. R. Wijewardhana, Nucl. Phys. 220, 213 (1983).
H. B. Nielsen and M. Ninomiya, Nucl. Phys. 156, 1 (1979).
For a review see, for example, L. P. Kadanoff et al., Rev. Mod. Phys. 39, 395 (1967).
S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975).
S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).
We have corrected an error in Ref. 10.
J. C. Collins, Phys. Rev. D 10, 1213 (1974).
G. 't Hooft, Nucl. Phys. 61, 455 (1973).
S. Weinberg, Phys. Rev. D 8, 3497 (1973).
K. G. Wilson, Phys. Rev. 179, 1499 (1969).
P. Ramond, Field Theory (Benjamin/Cummings, Reading, Mass., 1974).
K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972).
S. Weinberg, in Understanding the Fundamental Constituents of Matter, edited by A. Zichichi (Plenum, New York, 1976).
T. Appelquist and U. Heinz, Phys. Rev. D 24, 2169 (1981); ibid. 25, 2620 (1982).
K. H. Fischer, Solid State Commun. 18, 1515 (1976).
M. A. Shifman, A. I. Vainstein, and V. I. Zakharov, Nucl. Phys. 147, 385 (1979).
P. Olesen, Nucl. Phys. 200, 381 (1982).

[c1] G. Parisi and Y.-S. Wu, Sci. Sin. 24, 484 (1981).

[c2] V. N. Gribov, Nucl. Phys. 139, 7 (1978).

[c3] Omitted end note.

[c4] Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 (1975).

[c5] G. Grinstein, Phys. Rev. Lett. 37, 944 (1976).

[c6] G. Parisi and N. Sourlas, Phys. Rev. Lett. 46, 371 (1981).

[c7] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43, 774 (1979).

[c8] B. McClain, A. Niemi and C. Taylor, Ann. Phys. (N.Y.) 140, 232 (1982).

[c9] B. McClain, A. Niemi, C. Taylor, and L. C. R. Wijewardhana, Nucl. Phys. 217, 430 (1983).

[c10] A. Niemi and L. C. R. Wijewardhana, Nucl. Phys. 220, 213 (1983).

[c11] H. B. Nielsen and M. Ninomiya, Nucl. Phys. 156, 1 (1979).

[c12] For a review see, for example, L. P. Kadanoff et al., Rev. Mod. Phys. 39, 395 (1967).

[c13] S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975).

[c14] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).

[c15] We have corrected an error in Ref. 10.

[c16] J. C. Collins, Phys. Rev. D 10, 1213 (1974).

[c17] G. 't Hooft, Nucl. Phys. 61, 455 (1973).

[c18] S. Weinberg, Phys. Rev. D 8, 3497 (1973).

[c19] K. G. Wilson, Phys. Rev. 179, 1499 (1969).

[c20] P. Ramond, Field Theory (Benjamin/Cummings, Reading, Mass., 1974).

[c21] K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972).

[c22] S. Weinberg, in Understanding the Fundamental Constituents of Matter, edited by A. Zichichi (Plenum, New York, 1976).

[c23] T. Appelquist and U. Heinz, Phys. Rev. D 24, 2169 (1981); ibid. 25, 2620 (1982).

[c24] K. H. Fischer, Solid State Commun. 18, 1515 (1976).

[c25] M. A. Shifman, A. I. Vainstein, and V. I. Zakharov, Nucl. Phys. 147, 385 (1979).

[c26] P. Olesen, Nucl. Phys. 200, 381 (1982).

Physical Review D

Export citation

Export citation

Field theories in replica formalism and fixed points

Abstract

References (26)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation