Breakdown of the perturbative renormalization group at certain quantum critical points

Physical Review Letters

Volumn 93, Issue 15, 2004, Pages

  Belitz, D ^a   Kirkpatrick, T R ^b   Rollbühler, Jörg ^a  

^a UNIVERSITY OF OREGON   (United States)

^b Bldg 142   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; CORRELATION METHODS; FERROMAGNETIC MATERIALS; PERTURBATION TECHNIQUES; RELAXATION PROCESSES; THERMODYNAMICS; TIME SERIES ANALYSIS;

COUPLING CONSTANTS; FERROMAGNETIC TRANSITION; MOMENTUM-SHELL METHOD; RENORMALIZATION GROUPS (RG);

QUANTUM THEORY;

EID: 19644400911     PISSN: 00319007     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevLett.93.155701     Document Type: Article

References (22)

1. See, e.g., P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
2. J. Hertz, Phys. Rev. B 14, 1165 (1976).
3. A. J. Millis, Phys. Rev. B 48, 7183 (1993).
4. T. R. Kirkpatrick and D. Belitz, Phys. Rev. B 53, 14364 (1996).
5. For a recent review, see D. Belitz, T. R. Kirkpatrick, and T. Vojta, cond-mat/0403182.
6. K.G. Wilson and J. Kogut, Phys. Rep. 12, 75 (1974).
7. S.-K. Ma, Modern Theory of Critical Phenomena (Benjamin, Reading, MA, 1976).
8. See, e.g., J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford University Press, Oxford, 1996).
9. S. Weinberg, The Quantum Theory of Fields (Cambridge University, Cambridge, 1996), Vol. II.
10. M. E. Fisher, in Advanced Course on Critical Phenomena, edited by F. W. Hahne (Springer, Berlin, 1983), p. 1.
11. D. Belitz, T.R. Kirkpatrick, M.T. Mercaldo, and S.L. Sessions, Phys. Rev. B 63, 174427 (2001).
12. D. Belitz, T.R. Kirkpatrick, M.T. Mercaldo, and S.L. Sessions, Phys. Rev. B 63, 174428 (2001).
13. An analogous irrelevant variable appears at the liquid-gas critical point, but it is not dangerous [22].
14. For a review, see J. R. Dorfman, T.R. Kirkpatrick, and J.V. Sengers, Annu Rev. Phys. Chem. 45, 213 (1994).
15. + = 0 [2], This is fallacious, however, as can be seen from the fact that y becomes relevant with respect to the mean-field fixed point for d < 4. This is the first manifestation of the dangerous irrelevancy of w, as we see below.
16. 4 vertex [11,12]. Here we suppress these technicalities, which only distract from our main message, consider Eqs. (4) and (5) the definition of our model, and use the schematic action to explain only the structure of the RG analysis.
17. T.R. Kirkpatrick and D. Belitz, Phys. Rev. B 45, 3187 (1992).
18. We choose to introduce only one dynamical critical exponent z, despite the fact that there are two time scales in the problem. The second one is implicit in the scale dimension of H. This is same procedure as in Ref. [22].
19. We use the term, "dangerous irrelevant variable" in a very general sense: The loop expansion parameter itself depends singularly on the irrelevant variable h, not on just some scaling function.
20. To one-loop order one can use either a momentum-shell RG, or a field-theoretic one, to derive these results. At two-loop order, only the field-theoretic method is practical, as the shell integrals become too complicated.
21. T. R. Kirkpatrick and D. Belitz, Phys. Rev. B 67, 024419 (2003).
22. C. DeDominicis and L. Peliti, Phys. Rev. B 18, 353 (1978).