Coexistence of ferromagnetism and superconductivity

Physical Review B - Condensed Matter and Materials Physics

Volumn 67, Issue 2, 2003, Pages

  Kirkpatrick, T R ^a   Belitz, D ^b  

^a Bldg 142   (United States)

^b UNIVERSITY OF OREGON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

FERROMAGNETIC MATERIAL;

ARTICLE; MAGNETISM; MATHEMATICAL ANALYSIS; PHASE TRANSITION; PHYSICS; SEMICONDUCTOR; TEMPERATURE DEPENDENCE; THEORY;

EID: 0037219802     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.67.024515     Document Type: Article

References (47)

1. Early work on this subject was done in the context of superfluidity in (Formula presented) K. A. Brueckner, T. Soda, P. W. Anderson, and P. Morel, Phys. Rev. 118, 1442 (1960);
2. P. W. Anderson and W. F. Brinkman, Phys. Rev. Lett. 30, 1108 (1973).
3. D. Belitz, T. R. Kirkpatrick, and Thomas Vojta, Phys. Rev. Lett. 82, 4707 (1999).
4. T. R. Kirkpatrick and D. Belitz (unpublished).
5. First order ferromagnetic transitions at low temperatures have been observed in MnSi, C. Pfleiderer, G. J. McMullan, S. R. Julian, and G. G. Lonzarich, Phys. Rev. B 55, 8330 (1997), and in (Formula presented) Ref. 5.
6. S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley, L. Sheikin, D. Braithwaite, and J. Floquet, Nature (London) 406, 587 (2000);
7. A. Huxley, I. Sheikin, E. Ressouche, N. Kernavanois, D. Braithwaite, R. Calemczuk, and J. Floquet, Phys. Rev. B 63, 144519 (2001), and references therein.
8. D. Fay and J. Appel, Phys. Rev. B 22, 3173 (1980).See also W. F. Brinkman and S. Engelsberg, Ref. 35.
9. In (Formula presented) a weak anomaly has been found in the transport properties, A. Huxley, I. Sheikin, E. Ressouche, N. Kernavanois, D. Braithwaite, R. Calemczuk, and J. Flouquet, Phys. Rev. B 63, 144519 (2001).
10. and in the temperature dependence of the magnetization, N. Tateiwa, K. Hanazono, T. C. Kobayashi, K. Amaya, T. Inoue, K. Kindo, Y. Koike, N. Metoki, Y. Haga, R. Settai, and Y. Onuki, J. Phys. Soc. Jpn. 70, 2876 (2001), within the ferromagnetic phase. The position of the superconducting phase with respect to this line is more symmetric. However, the nature of this anomaly, and whether it has anything to do with the observed superconductivity, is currently unclear.
11. C. Pfleiderer, M. Uhlarz, S. M. Hayden, R. Vollmer, H. von Löhneysen, N. R. Bernhoeft, and G. G. Lonzarich, Nature (London) 412, 58 (2001).
12. D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Floquet, J.-P. Brison, E. Lhotel, and C. Paulsen, Nature (London) 413, 613 (2001).
13. A. B. Shick and W. E. Pickett, Phys. Rev. Lett. 86, 300 (2001).
14. H. Shimahara and M. Kohmoto, Europhys. Lett. 57, 247 (2002).
15. G. Santi, S. B. Dugdale, and T. Jarlborg, Phys. Rev. Lett. 87,247004 (2001).
16. S. Watanabe and K. Miyake, cond-mat/0110492 (unpublished).
17. K. B. Blagoev, J. R. Engelbrecht, and K. S. Bedell, Phys. Rev. Lett. 82, 133 (1999).
18. K. Machida and T. Ohmi, Phys. Rev. Lett. 86, 850 (2001).
19. K. V. Samokhin and M. B. Walker, cond-mat/0203309 (unpublished).
20. J. Spalek and P. Wrobel, cond-mat/0202043 (unpublished).
21. Z. Wang, W. Mao, and K. Bedell, Phys. Rev. Lett. 87, 257001 (2001).
22. R. Roussev and A. J. Millis, Phys. Rev. B 63, 140504 (2001).
23. Contrary to the earlier work in Ref. 6, these authors find a nonzero value of (Formula presented) at the FM transition. Simple approximations for (Formula presented) such as the one we will employ, do not capture this feature. Since this effect is small compared to the enhancement of (Formula presented) in the ferromagnetic phase that is our main focus, we will neglect it for our purposes. We also note that the point is moot if the magnetic transition at low temperatures is of first order, as is the case in (Formula presented)
24. T. R. Kirkpatrick, D. Belitz, Thomas Vojta, and R. Narayanan, Phys. Rev. Lett. 87, 127003 (2001).
25. A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1975).
26. The such defined matrices (Formula presented) are not Dirac matrices, but they serve a purpose that is very similar to that of Dirac matrices in relativistic electron theories.
27. P. G. De Gennes, Superconductivity of Metals and Alloys (Addison-Wesley, Redwood City, CA, 1989), Chap. 4.2.
28. This is most obvious in a paramagnetic phase, where the positive definite tensor (Formula presented) is diagonal: Parallel vectors (Formula presented) and (Formula presented) lower the energy, while antiparallel vectors increase it.
29. We recall that the action does not yet contain an explicit particle-particle interaction that one could decouple by means of a Hubbard-Stratonovich transformation. Rather, this interaction will be generated by integrating out the magnetic fluctuations, see also the remark after Eq. (2.2b). This is the reason why we treat the magnetic and superconducting order parameters differently.
30. Consistent with general requirements for the coexistence of superconductivity and ferromagnetism, this order parameter is a special case of nonunitary triplet pairing, see Ref. 15.
31. This feature of our theory is in exact analogy to the Bogoliubov theory of superconductivity or superfluidity, see, e.g., Ref. 24, Chap. 5.1.
32. This is true for magnetic susceptibilities that are smooth functions of the wave number, as is the case in the models that we will consider. If (Formula presented) had pronounced features in wave number space, then some channel with (Formula presented) could be favored.
33. T. R. Kirkpatrick and D. Belitz, Phys. Rev. B 62, 952 (2000).
34. The Gaussian approximation to the current field theory normalizes the magnetic susceptibilities by (Formula presented) while Ref. 21, used susceptibilites normalized by (Formula presented) (Recall that at magnetic criticality, (Formula presented) in our mean-field theory.) For consistency, we keep this factor in all susceptibilities, although doing so has less of an effect on (Formula presented) than some of our approximations. As a result, our numerical results for (Formula presented) for given parameter values are slightly different from those in Ref. 21.
35. See, e.g., S.-K. Ma, Modern Theory of Critical Phenomena (Perseus Publishing, New York, 2000), Chap. IX.7.
36. E. Brézin and D. J. Wallace, Phys. Rev. B 7, 1967 (1973).
37. In a magnetic context, this effect was first obtained by E. Brézin, D. J. Wallace, and K. G. Wilson, Phys. Rev. Lett. 29, 591 (1972);
38. E. Brézin and D. J. Wallace, Phys. Rev. B 7, 1967 (1973).
39. in a (Formula presented) expansion. [An analogous phenomenon in particle physics had been discussed earlier by L. F. Li and H. Pagels, Phys. Rev. Lett. 27, 1089 (1972);
40. H. Pagels, Phys. Rep. 16, 219 (1975), Sec. 6, and references therein.]
41. D. Belitz and T. R. Kirkpatrick, Phys. Rev. B 56, 6513 (1997) used renormalization group techniques to show that this is actually the exact long-wavelength behavior of the longitudinal susceptibility, independent of perturbation theory. This was achieved by demonstrating that the (Formula presented) is a manifestation of the leading correction to scaling near the stable fixed point that describes the ferromagnetic phase.
42. W. F. Brinkman and S. Engelsberg, Phys. Rev. 169, 417 (1968).
43. See, e.g., P. B. Allen and R. C. Dynes, Phys. Rev. B 12, 905 (1975), and references therein.
44. See K. Levin and O. Valls, Phys. Rev. B 17, 191 (1978) for a discussion of paramagnon-induced pairing in (Formula presented)
45. A. V. Chubukov, A. M. Finkel’stein, R. Haslinger, and D. K. Morr, cond-mat/0210653 (unpublished).
46. See, e.g., V. N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics (Reidel, Dordrecht, 1983);N. Nagaosa, Quantum Field Theory in Condensed Matter Physics (Springer, Berlin, 1999).
47. P. Morel and P. W. Anderson, Phys. Rev. 125, 1263 (1962).