Renormalization-group%20investigation%20of%20the%202D%20Hubbard%20model

1
Partnergroup
Renormalization-group investigationof the 2D
Hubbard model
A. A. Katanina,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)
2008
2
Content

1. The model
2. The field theoretical and functional RG
   approaches
3. Phase diagrams
4. Fulfillment of Ward Identities
5. The two-loop corrections
6. Conclusions and future perspectives

3
The 2D Hubbard model
t
t'
U

• Provides a prototype model of interacting
  fermionic systems leading to nontrivial physics

• The weak-coupling regime U lt W/2

• Why it is interesting
• Non-trivial
• Gives a possibility of rigorous numerical and
  semi- analytical RG treatment.

4
The case of general Fermi surface
k1,s
k3,s
kF
k1 k2k3 k4
k4,s'
k2 ,s'
There is no interference between different
channels (channel separation)
k1 - k2 k3 - k4 BCS channel k1 k3 k2 k4
ZS channel k1 k4 k2 k3 ZS' channel
k
The Fermi liquid

kq
k

q-k

• Possible types of instabilities
• Superconducting (only for Ult0)
• Ferro- and antiferromagnetic instabilities are
  not in the weak-coupling regime

5
The parameter space

• 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. ?

The line of van Hove singularities
???
t'/t
???
???
n
Nesting
6
Theoretical approaches

• Parquet approach (V.V. Sudakov, 1957 I.E.
  Dzyaloshinskii, 1966 I.E. Dzyaloshinskii and
  V.M. Yakovenko, 1988)

• 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)

• 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)

7
The field-theory (two-patch) approach
B
2?
A
Similar to the left and right moving
particles in 1D
But the geometry of the Fermi surface and the
dispersion are different !
8
The two patch equations at T ??
9
The vertices scale dependence
U2t, t'/t0.1 (nVH0.92)
g3 (umklapp)
g2 (inter-patch direct)
g1
(????)
?
g4
U2t, t'/t0.45 (nVH0.47)
g1(inter-patch exchange)
g4 (intra-patch)
g2
(????)
g3
?
10
Phase diagram vH band fillings
T0, ?0
32 - patch fRG approach
11
Functional renormalization group

• Projecting momenta to the Fermi surface
• Projecting frequencies to zero
• 32-48 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)

• 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
T0, ?0
32 - patch fRG approach
15
Ward identities
Ward identity
is fulfilled up to the order V2 only
Replacement
in the equation for the vertex
(A. Katanin, Phys. Rev. B 70, 115109 (2005))
improves fulfillment of Ward identities

• 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,
  2005-2008)

(23) and the mean-field-type result for the
self-energy,


16
Half filling, non-nested Fermi surface
MF
n1
d-wave superconducting
antiferromagnetic
48-patch fRG approach
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)
17
Angular dependence of the order parameter
Hole-doped sc, A. A. Katanin and A. P. Kampf,
Phys. Rev. 2005
Electron-doped sc, A. A. Katanin, Phys. Rev. 2006
Max
Hot spot
J. Mesot et al., Phys. Rev. Lett. 83, 840 (1999).
Pr0.89LaCe0.11CuO4
H. Matsui et al., Phys. Rev.Lett. 95, 017003
(2005).
18
Taking 6-point vertex into account
U?? 2.5t
????? 0.1t
??? 0.1t
A. Katanin, arXiv2008
19
Scattering rates
??? 0.1t
NFL
Landau
Landau
FL
FL
NFL
??? 0.1t
????? 0.1t
From spin-fermion theory
(R. Haslinger, Ar. Abanov, andA. Chubukov, 2001)
20
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 T2 dependence at low T and T1-a
  dependence at higher T, a 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

21
Future perspectives

• 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.

similarity to CSB in QCD ?
T
QC
vs.
RC
QD
m
QPT
O(N) or O(N)/O(N-2) NLs-model
La2CuO4
field-theor. RG 1/N expansion, V. Yu. Irkhin,
A. Katanin et al., PRB 1997
Frustration and quantum criticality
Thank you for the attention!
We are open for collaboration