A.A. Katanin a,b and A.P. Kampf a

1. Renormalization-group studiesof the 2D Hubbard model A.A. Katanina,b and A.P. Kampfa aTheoretische Physik III, Institut für Physik, Universität Augsburg, Germany b Institute of Metal Physics, Ekaterinburg, Russia 2003

2. Content • The model • The weak-coupling regime: • motivation and methods • III. Results • Standard Hubbard model: • a) the phase diagram • b) the vicinity of half-filling • c) low-density flat-band ferromagnetism • Extended Hubbard model • IV. Conclusions

3. The 2D Hubbard model Experimental relevance: cuprates AB Cuprates (Bi2212) B Bi2212 La2-x SrxCuO4 D.L. Feng et al., Phys. Rev. B 65, 220501 (2002) A. Ino et al., Journ. Phys. Soc. Jpn, 68, 1496 (1999). Ruthenate Sr2RuO4 • A. Damascelli et al,J. Electron Spectr. Relat. Phenom. 114, 641 (2001). b a g

4. The weak coupling regime • U < W/2 • Why it is interesting: • Non-trivial • Gives the possibility of rigorous numerical and semi analytical RG treatment. • Questions that we want to answer: • What are the possible instabilities ? • How do they depend on the form of the Fermi surface, model parameters e.t.c. ? Interaction alone is not enough to produce magnetic or superconducting instabilities in the weak-coupling regime • However, instabilities are possible due to the peculiarities of the electron spectrum: • nesting (ek=- ek+Q) n=1; t'=0; • van Hove singularities (k=0) n=nVH; any t'

5. The parameter space The line of van Hove singularities m>0 t'/t m=0 m<0 n Nesting The simplest mean-field (RPA) approach becomes inapplicable close to the line m=0due to “the interference” of different channels of electron scattering: ph-scattering pp-scattering

6. Theoretical approaches • Parquet approach (V.V. Sudakov, 1957; I.E. Dzyaloshinskii, 1966; I.E. Dzyaloshinskii and V.M. Yakovenko, 1988) • Many-patch renormalization group approaches: • Polchinskii RG equations (D. Zanchi and H.J. Schulz, 1996) • Wick-ordered RG equations (M. Salmhofer, 1998; C.J. Halboth and W. Metzner, 2000) • RG equations for 1PI Green functions (M. Salmhofer, T.M. Rice, N. Furukawa, and C. Honerkamp, 2001) • RG equations for 1PI Green functions with temperature cutoff (M. Salmhofer and C. Honerkamp, 2001) • Two-patch renormalization group approach (P. Lederer et al., 1987; T.M. Rice, N. Furukawa, and M. Salmhofer, 1999; A.A. Katanin, V.Yu. Irkhin and M.I. Katsnelson, 2001; B. Binz, D. Baeriswyl, and B. Doucot, 2001) • Continuous unitary transformations (C.P. Heidbrink and G. Uhrig, 2001; I. Grote, E. Körding and F. Wegner, 2001)

7. The two-patch approach B 2 A Similar to the “left” and “right” moving particles in 1D But the topology of the Fermi surface is different ! Possible types of vertices There is no separation of the channels: each vertex is renormalized by all the channels

8. The two patch equations at T » |m|

9. The vertices: scale dependence U=2t, t'/t=0.1; nVH=0.92 g3(umklapp) g2 (inter-patch direct) g1 (0++-) l g4 U=2t, t'/t=0.45; nVH=0.47 g1(inter-patch exchange) g4 (intra-patch) g2 (++0+) g3 l

10. Many-patch renormalization group

11. The phase diagram: vH band fillings T=0, m=0 32 - patch RG approach

12. The vicinity of half filling QMC: H.Q. Lin and J.E. Hirsch, Phys. Rev. B 35, 3359 (1987). PIRG: T. Kashima and M. Imada Journ. Phys. Soc. Jpn 70, 3052 (2001). MF: W. Hofstetter and D. Vollhardt, Ann. Phys. 7, 48 (1998) n=1 d-wave superconducting antiferromagnetic 48-patch RG approach: t'=0; n<1

13. The flat-band ferromagnetism U>0 Mielke and Tasaki (1993. 1994) t’/t=1/2 ky kx r(e) ~1/e1/2 • The system is ferromagnetic at t/t~1/2, cf. Refs. R. Hlubina, Phys. Rev. B 59, 9600 (1999) (T - matrix approach) R.Hlubina, S.Sorella and F.Guinea, Phys. Rev. Lett. 78, 1343 (1997) (projected QMC)

14. Ferromagnetism and RG L FS Momentum cutoff: no Temperature cutoff: yes

15. The flat-band ferromagnetism  T-matrix result for FM instability by Hlubina et al.

16. Ferromagnetism due to vHS t’/t=0.45 • Similar peaks occur due to “merging” of vHS in 3D • FCC Ca, Sr, …. (M.I.Katsnelson and A.Peschanskih)

17. Possible order parameters Charge-density wave Spin-density wave ph, q=Q Charge-flux Spin-flux Ferromagnetism Bond-charge order (PI) ph, q=0 Bond-spin order (A) Phase separation s - wave supercond. pp, q=0 d - wave supercond. h - Pairing pp, q=Q p - Pairing

18. The phase diagram at U=2t (nVH=1) SDW spin-density wave; CDW charge-density wave dSC d - wave superconductivityCF charge flux; SF – spin flux; PS phase separation

19. The phase diagram at U=2t (nVH=0.92)

20. The phase diagram at U=2t (nVH=0.73)

21. Conclusions • The two-patch approach gives qualitatively correct predictions for competition of phases with different symmetry • Many-patch generalization is necessary a) To resolve between the phases with the same symmetry b) To go away from the van Hove band fillingc) To consider nearly flat bands • The phase diagrams of the t-t' Hubbard model and the extended Hubbard model are obtained • The extended U-V-J model at J>0 allows for a variety of ordering tendencies. There is a close competition between charge-flux, spin-density wave and d-wave superconducting instabilities in certain region of the parameter space (J>0)

22. The patching scheme

23. From: J.V. Alvarez et al., J. Phys. Soc. Jpn., 67, 1868 (1998)