2008 Renormalization-group investigation of the 2D Hubbard model A. A. Katanin a,b a Institute of Metal Physics, Ekaterinburg, Russia b Max-Planck Institut für Festkörperforschung, Stuttgart Many thanks for collaboration to: A. P. Kampf (Institut für Physik, Universität Augsburg) W. Metzner (Max-Planck Institut für Festkörperforschung, Stuttgart) Partnergroup

2 I.The model II.The field theoretical and functional RG approaches III.Phase diagrams IV.Fulfillment of Ward Identities V.The two-loop corrections VI.Conclusions and future perspectives Content

3 The 2D Hubbard model Why it is interesting: Non-trivial Gives a possibility of rigorous numerical and semi- analytical RG treatment. The weak-coupling regime U < W /2 Cuprates (Bi2212) La 2-x Sr x CuO 4 Bi2212 Experimental relevance: high-T c cuprates A. Ino et al., Journ. Phys. Soc. Jpn, 68, 1496 (1999). D.L. Feng et al., Phys. Rev. B 65, (2002) U t t' Provides a prototype model of interacting fermionic systems leading to nontrivial physics

The case of general Fermi surface  Possible types of instabilities:  Superconducting (only for U<0)  Ferro- and antiferromagnetic instabilities are not in the weak-coupling regime k1,k1, k 2,  ' k3,k3, k4,'k4,' k 1 =  k 2 ; k 3 =  k 4 : BCS channel k 1 = k 3 ; k 2 = k 4 : ZS channel k 1 = k 4 ; k 2 = k 3 : ZS' channel There is no ‘interference’ between different channels (channel separation) kFkF The Fermi liquid k 1 +  k 2 =k 3 + k 4 = = k k+qk+q k q-k

5 The parameter space  t'/t n The line of van Hove singularities   Nesting Questions to answer: What are the possible instabilities for t-t' dispersion? How do they depend on the form of the Fermi surface, model parameters e.t.c. ? 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'

Theoretical approaches  Parquet approach (V.V. Sudakov, 1957; I.E. Dzyaloshinskii, 1966; I.E. Dzyaloshinskii and V.M. Yakovenko, 1988)  Functional renormalization group approach  Polchinskii equations (D. Zanchi and H.J. Schulz, 1996; 2000)  Wick-ordered equations (M. Salmhofer, 1998; C.J. Halboth and W. Metzner, 2000; D. Rohe and W. Metzner, 2005)  Equations for 1PI functional (M. Salmhofer, T.M. Rice, N. Furukawa, and C. Honerkamp, 2001)  Equations for 1PI functional with temperature cutoff (M. Salmhofer and C. Honerkamp, 2001; A. Katanin and A. P. Kampf, 2003, 2004)  Continuous unitary transformations (C.P. Heidbrink and G. Uhrig, 2001; I. Grote, E. Körding and F. Wegner, 2001)  Field-theory 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)

7 The field-theory (two-patch) approach Similar to the “left” and “right” moving particles in 1D But the geometry of the Fermi surface and the dispersion are different ! 22 B A

8 The two patch equations at T » |  Possible types of vertices There is no separation of the channels: each vertex is renormalized by all the channels

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 Phase diagram: vH band fillings 32 - patch fRG approach T=0,  =0

Functional renormalization group o Projecting momenta to the Fermi surface o Projecting frequencies to zero o patches on the Fermi surface (after M. Salmhofer and C. Honerkamp, 2001)

12 1PI functional RG Considers the evolution of the 1PI generating functional (T. Morris, 1994; M. Salmhofer and C. Honerkamp, 2001) Expands in fields Obtains the equations for the coefficients of the expansion where … Truncates the hierarchy of equations, e.g.

13 1PI scheme  Temperature cutoff

14 Phase diagram: vH band fillings 32 - patch fRG approach T=0,  =0

15 Ward identities (23) and the mean-field-type result for the self-energy, is fulfilled up to the order V 2 only Ward identity: Replacement:in the equation for the vertex Applications: Zero-dimensional impurity problems (C. Schönhammer, V. Meden, and T. Pruschke, 2005, 2008) Flow into symmetry-broken phases (W. Metzner, M. Salmhofer, C. Honerkamp, and R. Gersch, ) (A. Katanin, Phys. Rev. B 70, (2005)) improves fulfillment of Ward identities

16 MF: W. Hofstetter and D. Vollhardt, Ann. Phys. 7, 48 (1998) Half filling, non-nested Fermi surface 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 fRG approach: MF

17 Angular dependence of the order parameter Hole-doped sc, A. A. Katanin and A. P. Kampf, Phys. Rev Electron-doped sc, A. A. Katanin, Phys. Rev J. Mesot et al., Phys. Rev. Lett. 83, 840 (1999). H. Matsui et al., Phys. Rev. Lett. 95, (2005). Pr 0.89 LaCe 0.11 CuO 4 Hot spot Max

18 Taking 6-point vertex into account  0.1t  0.1t U  2.5t A. Katanin, arXiv:2008

19 Scattering rates From spin-fermion theory: (R. Haslinger, Ar. Abanov, and A. Chubukov, 2001) FL NFL FL NFL  0.1t  0.1t Landau

Summary of the results of fRG approach  fRG allows for treating competing instabilities in fermionic systems and obtain information about  susceptibilities  phase diagrams  symmetries of the order parameter  quasiparticle characteristics  The ferro-, antiferromagnetic, and superconducting instabilities occur in different regions of phase diagram; the order parameter symmetry deviates from the standard s- and d-wave forms  The quasiparticle residues remain finite in the paramagnetic state; the quasiparticle damping shows a T 2 dependence at low T and T 1-   dependence  at higher T,  0  The truncation at 4-point vertex yields results compatible with more complicated truncations; the divergence of vertices and susceptibilities is however suppressed including the 6-point vertex

Detail description of quantum critical points Application to the localized Heisenberg (e.g. frustrated) magnets – bosonic (magnons) vs. fermionic (spinons) excitations Combination with other nonperturbative approaches Including long-range interactions, gauge fields etc. Future perspectives La 2 CuO 4 field-theor. RG + 1/N expansion, V. Yu. Irkhin, A. Katanin et al., PRB 1997  QPT QD QC RC  vs. similarity to CSB in QCD ? O(N) or O(N)/O(N-2) NL  -model Frustration and quantum criticality