Classification and stability of phases of the multicomponent one-dimensional electron gas

Physical Review B - Condensed Matter and Materials Physics

Volumn 59, Issue 24, 1999, Pages 15641-15653

  Zachar, O ^c  

^a BROOKHAVEN NATIONAL LABORATORY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITÉ PARIS SUD   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000567614     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.59.15641     Document Type: Article

References (51)

1. For a review, see E. Dagotto and T. M. Rice, Science 271, 618 (1996).
2. M. Fabrizio, Phys. Rev. B 48, 15 838 (1993);
3. Phys. Rev. BH. J. Schulz, 53, 2959 (1996);
4. Phys. Rev. BD.V. Khveshchenko and T. M. Rice, 50, 252 (1994).
5. T. M. Rice, et al., Phys. Rev. B 56, 14 655 (1997);
6. Phys. Rev. BS. R. White and D. J. Scalapino, 57, 3031 (1998).
7. S. R. White and D. J. Scalapino, Phys. Rev. Lett. 81, 3227 (1998).
8. V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. B 56, 6120 (1997).
9. O. Zachar, S. A. Kivelson, and V. J. Emery, Phys. Rev. Lett. 77, 1342 (1996).
10. A. E. Sikkema, I. Affleck, and S. R. White, Phys. Rev. Lett. 79, 929 (1997).
11. P. W. Anderson, Adv. Phys. 46, 3 (1997).
12. V. J. Emery in Proceedings of the Kyoto Summer Institute, 1979, edited by Y. Nagaoka and S. Hikami (Publication Office of Theoretical Physics, Kyoto, 1979), p. 1.
13. V. J. Emery in Highly Conducting One-Dimensional Solids, edited by J. T. Devreese, R. P. Evrard, and V. E. van Doren (Plenum, New York, 1979), p. 327;
14. J. Solyom, Adv. Phys. 28, 201 (1979).
15. L. Balents and M. P. A. Fisher, Phys. Rev. B 53, 12 133 (1996);
16. Phys. Rev. BH. H. Lin, L. Balents, and M. P. A. Fisher, 56, 6569 (1997).
17. K. Wilson, Rev. Mod. Phys. 47, 773 (1975).
18. V. J. Emery and S. A. Kivelson in Fundamental Problems in Statistical Mechanics VIII, Proceedings of the 1993 Altenberg Summer School, edited by H. van Beijeren and M. H. Ernst (Elsevier, Amsterdam, 1994), p. 1.
19. See, e.g., V. J. Emery and C. Noguera, Phys. Rev. Lett. 60, 631 (1988), and references therein.
20. M. Yamanaka, M. Oshikawa, and I. Affleck, Phys. Rev. Lett. 79, 1110 (1997).
21. See also P. Gagliardini, S. Haas, and T. M. Rice, Phys. Rev. B 58, 9603 (1998).
22. C. N. Yang, Phys. Rev. Lett. 63, 2144 (1989).
23. O. Zachar (unpublished).
24. Note that if there exist gapless spin-(Formula presented) modes with wave vectors (Formula presented) and any integer (Formula presented), it follows that one can construct gapless states out of multiple excitations of these gapless modes with any integer spin. Thus, the case we have considered here, in which the spin mode has spin 1, is in fact completely general. There are, however, some exotic possibilities that we have not considered here explicitly, as they do not seem to occur in any model problems of which we are aware. For example, it is conceivable that the fundamental gapless mode is spin 1, but charge (Formula presented).
25. If there exists a gapless, spinless charge-(Formula presented) mode, one can clearly construct gapless, spinless excitation with any charge (Formula presented). One can imagine cases in which there are no gapless charge-(Formula presented) excitations, but there are gapless spinless excitations with charge (Formula presented) and multiples thereof. We will adopt the convention that the fundamental charged, spinless excitation is assumed to have charge (Formula presented), unless otherwise specified.
26. H. Frahm and V. Korepin, Phys. Rev. B 42, 10 553 (1990). The findings of this paper, which is based on the Bethe ansatz, are consistent with bosonization results for the oscillatory (Formula presented) part of the density-density correlation function, but inconsistent with the conclusions about the long-wavelength density fluctuations. Frahm and Korepin found a nonoscillatory part of the density-density correlation function which falls with distance like (Formula presented), whereas, from bosonization, one would deduce that this result is valid only in the absence of a charge gap, which would eliminate any power-law behavior. At present, the source of this discrepancy is not clear, although it is possible that the amplitude of this term (which Frahm and Korepin did not evaluate explicitly) actually vanishes.
27. Where there is a gapless spin mode, a CDW fluctuation can be viewed as a total spin-zero composite of three spin modes. Where there is a spin gap but no charge gap, the generalized Luttinger’s theorem implies that there must be a gapless, zero-momentum charge (Formula presented) “BCS mode,” which can be thought of as a composite of an (Formula presented)-pairing mode and a CDW mode; conversely, the CDW mode can be thought of as a composite of oppositely charged (Formula presented)-pairing and BCS modes. This ambiguity is a reflection of the number-phase duality of the charge modes.
28. The Abelian boson representation used here is not explicitly SU(2) invariant, and indeed is valid even if there is Ising-Heisenberg symmetry which breaks the spin symmetry down to (Formula presented). There is a subtlety in the SU(2) invariant case when the spin modes are gapless, in that the cosine term is only marginally irrelevant, so that to compute the asymptotic behavior of correlation functions completely correctly, one cannot simply set (Formula presented), but must rather let it renormalize to zero. Even when there is a spin gap, SU(2) symmetry constrains the values of the renormalized Luttinger exponents in the fixed-point Hamiltonian, since there are special degeneracies of the gapped excitations implied by this symmetry. For instance, if (Formula presented) (i.e., if (Formula presented), the lowest breather (or soliton anti-soliton bound state) has the same energy as the single-soliton state, so together they form the triplet (massive-magnon) state; for slightly different values of (Formula presented) this degeneracy, which is required by SU(2), is lifted.
29. Here we have considered only the case in which a gap results from a relevant interaction that pins the value of the density-wave phase, (Formula presented). While it is typically forbidden by symmetry in most cases, it is certainly possible to imagine cases in which there is a relevant interaction of the form (Formula presented) which would pin the dual phase; we do not explore this possibility in the present paper.
30. K. A. Muttalib and V. J. Emery, Phys. Rev. Lett. 57, 1370 (1986).
31. For an explicit discussion, see D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988).
32. For a review, see H. J. Schulz in Strongly Correlated Electronic Materials: The Los Alamos Symposium 1993, edited by K. S. Bedell et al. (Addison-Wesley, Redwood City, CA, 1994) p. 187.
33. C. M. Varma and A. Zawadowski, Phys. Rev. B 32, 7399 (1985).
34. H. J. Schulz, Phys. Rev. B 53, R2959 (1996).
35. See, Refs. 1, 3, and 4, and R. M. Noack, et al., Phys. Rev. B 56, 7162 (1997);
36. Phys. Rev. BS. R. White and D. J. Scalapino, 55, R14 701 (1997).
37. S. Hellberg and E. Manousakis, Phys. Rev. Lett. 78, 4609 (1997).
38. S. R. White and I. Affleck, Phys. Rev. B 54, 9862 (1996);
39. A. E. Sikkema, I. Affleck, and S. R. White, Phys. Rev. Lett. 79, 929 (1997).
40. O. Zachar and I. Affleck (unpublished).
41. P. Coleman, A. George, and A. Tsvelik, Phys. Rev. B 49, 8955 (1994).
42. Oron Zachar (unpublished).
43. M. Sigrist, H. Tsunetsugu, K. Ueda, and T. M. Rice, Phys. Rev. B 46, 13 838 (1992);
44. Phys. Rev. BH. Tsunetsugu, Y. Hatsugai, K. Ueda, and M. Sigrist, 46, 3175 (1992).
45. V. L. Berezinskii, Pis’ma Zh. Éksp. Teor. Fiz. 20, 628 (1974) [JETP Lett. 20, 287 (1974)];
46. For recent interest in odd-frequency pairing, see E. Abrahams and A. V. Balatsky, Phys. Rev. B 45, 13 125 (1992);
47. F. Mila and E. Abrahams, Phys. Rev. Lett. 67, 2379 (1991);
48. E. Abrahams, A. Balatsky, J. R. Schrieffer, and P. B. Allen, Phys. Rev. B 47, 513 (1993).
49. P. Coleman, E. Miranda, and A. M. Tsvelik, Phys. Rev. Lett. 70, 2960 (1993);
50. Phys. Rev. Lett.P. ColemanE. MirandaA. M. Tsvelik49, 8955 (1994);
51. Phys. Rev. Lett.P. ColemanE. MirandaA. M. Tsvelik74, 1653 (1995).