1. 1 A.A. Katanin a,b and A.P. Kampf a 2003 a Theoretische Physik III, Institut für Physik, Universität Augsburg, Germany b Institute of Metal Physics, Ekaterinburg, Russia Renormalization-group studies of the 2D Hubbard model

2. 2 I.The model II.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 Content

3. 3 The 2D Hubbard model Cuprates (Bi2212) AB B La 2-x Sr x CuO 4 Bi2212 Experimental relevance: cuprates Ruthenate Sr 2 RuO 4 A. Ino et al., Journ. Phys. Soc. Jpn, 68, 1496 (1999). D.L. Feng et al., Phys. Rev. B 65, 220501 (2002) A. Damascelli et al, J. Electron Spectr. Relat. Phenom. 114, 641 (2001).   

4. 4 The weak coupling regime 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. ? Why it is interesting: Non-trivial Gives the possibility of rigorous numerical and semi analytical RG treatment. U < W /2 However, instabilities are possible due to the peculiarities of the electron spectrum: nesting (  k  k+Q ) n=1; t'=0; van Hove singularities (  k =0 ) n=n VH ; any t' Interaction alone is not enough to produce magnetic or superconducting instabilities in the weak-coupling regime

5. 5 The parameter space  t'/t n The line of van Hove singularities   Nesting The simplest mean-field (RPA) approach becomes inapplicable close to the line  due to “the interference” of different channels of electron scattering: pp-scattering ph-scattering

6. 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. 7 The two-patch approach Similar to the “left” and “right” moving particles in 1D But the topology of the Fermi surface is different ! 22 B A Possible types of vertices There is no separation of the channels: each vertex is renormalized by all the channels

8. 8 The two patch equations at T » | 

9. 9 The vertices: scale dependence g1g1 g 2 (inter-patch direct) g 3 (umklapp) g4g4 g 1 (inter-patch exchange) g2g2 g3g3 g 4 (intra-patch) (  ) (  ) U=2t, t'/t=0.45; n VH =0.47 U=2t, t'/t=0.1; n VH =0.92

10. 10 Many-patch renormalization group

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

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

13. 13 The flat-band ferromagnetism  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) kxkx kyky  ~1/  1/2 t’/t=1/2 U>0 Mielke and Tasaki (1993. 1994)

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

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

16. 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. 17 Possible order parameters Charge-density wave Spin-density wave Charge-flux Spin-flux Phase separation Ferromagnetism Bond-charge order (PI) Bond-spin order (A) s - wave supercond. d - wave supercond.  - Pairing  - Pairing ph, q=Q ph, q=0 pp, q=0 pp, q=Q

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

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

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

21. 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 filling c) 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. 22 The patching scheme

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