Renormalization-group
studies
of 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
1
Content
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
2
The 2D Hubbard model
H
  k ck ck  U  ni ni
k ,
i
 k  2t (cos k x  cos k y )  4t ' (cos k x cos k y  1)  
t, t '  0
Experimental relevance: cuprates
AB
Cuprates (Bi2212)
B
Bi2212
La2-x SrxCuO4
A. Ino et al., Journ. Phys. Soc.
Jpn, 68, 1496 (1999).
D.L. Feng et al.,
Phys. Rev. B 65, 220501 (2002)
Ruthenate Sr2RuO4
A. Damascelli et al,
J. Electron Spectr. Relat. Phenom.
114, 641 (2001).
b
g
a
3
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
 qph
 qph ,0

,
ph , 0
1  U q
 qpp
 qpp ,0

,
1  U qpp ,0
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=nVH; any t'
4
The parameter space
0.5
The line of
van Hove
singularities
t'/t
0
<0
0.0
1.0
n
0.0
Nesting
The simplest mean-field (RPA) approach
becomes inapplicable close to the line 0 due to
“the interference” of different channels of
electron scattering:
pp-scattering
ph-scattering
5
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)
6
The two-patch approach
B
 kA  2t (sin 2  k x2  cos2  k y2 )  
 kB  2t (cos2  k x2  sin 2  k y2 )  
2
A
Similar to the “left” and “right”
moving particles in 1D
  (1 / 2) arccos(2t ' / t )
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
7
The two patch equations at T » ||

T

 ln
T
 00, ph 
 ln
 Q0, ph 
 00, pp 
 ln 2
 Q0, pp
 ln


T

T
dg1 / d  2d1 (  )g1 (g 2  g1 )  2d 2 g1 g 4  2d 3 g1 g 2
dg 2 / d  d1 (  )(g 2  g 3 )  2d 2 (g1  g 2 )g 4  d 3 (g1  g 2 )
2
2
2
2
dg 3 / d  2d 0 (  )g 3 g 4  2d1 (  )g 3 ( 2 g 2  g1 )
dg 4 / d   d 0 (  )(g 3  g 4 )  d 2 (g1  2 g1 g 2  2 g 2  g 4 )
2
2
2
d 0 ( )   'pp ,0 ( )  2/ 1  R 2 ;
d1 ( )   'ph ,Q ( )  2 min(  , ln[(1  1  R 2 )/R ]);
2
2
R  2t ' / t
  ln(  / T )
d 2   'ph ,0 ( )  2/ 1  R 2 ;
d 3   'pp ,Q ( )  2 tan 1 ( R/ 1  R 2 )/R
8
The vertices: scale dependence
U=2t, t'/t=0.1; nVH=0.92
g3
(umklapp)
g2
(inter-patch direct)
g1

(0)
g4
U=2t, t'/t=0.45; nVH=0.47
g1
(inter-patch exchange)
g4
(intra-patch)
g2
g3
(0)

9
Many-patch renormalization group
dV ( p1 , p2 , p3 )
   dpVT ( p1 , p2 , p ) Lpp ( p, p  p1  p2 )VT ( p, p  p1  p2 , p3 )
dT
  dp [  2VT ( p1 , p, p3 )VT ( p  p1  p3 , p2 , p )  VT (p1,p,p3 )VT (p 2 ,p  p1  p3,p)
 VT ( p1 , p, p  p1  p3 )VT ( p  p1  p3 , p2 , p )]Lph ( p, p  p1  p3 )
  dpVT ( p1 , p  p2 - p3 , p )VT ( p, p2 , p3 ) Lph ( p, p  p2  p3 )
10
The phase diagram: vH band fillings
T=0, =0
32 - patch
RG approach
11
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).
n=1
48-patch RG approach:
antiferromagnetic
MF: W. Hofstetter and
D. Vollhardt, Ann. Phys. 7, 48
(1998)
d-wave superconducting
t'=0; n<1
12
The flat-band ferromagnetism
U>0
Mielke and Tasaki (1993. 1994)
t’/t=1/2
ky
kx
r() ~1/1/2
 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)
13
(projected QMC)
Ferromagnetism and RG
 q 0  
f ( k )  f ( k q )
 k q   k
k

 
k
f ( k )
 k
FS
Momentum cutoff: no
Temperature cutoff: yes
14
The flat-band ferromagnetism
 T-matrix result
for FM instability
by Hlubina et al.
15
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)
16
Possible order parameters
Charge-density wave
OCDW   ck ck Q,
Spin-density wave
OSDW   ck ck Q,
Charge-flux
OCF   f k ck ck Q,
Spin-flux
OSF   f k ck ck Q,
Ferromagnetism
OF   ck ck,
k,
k,
ph,
q=Q
k,
k,
k,
Bond-charge order (PI) OBC   f k ck ck,
k,

 f k ck ck,
k,
Bond-spin order (A)
OBS 
Phase separation
OPS   ck ck q,
s - wave supercond.
OsSC   ck c-k,-
k,
k,
d - wave supercond.
OdSC   f k ck c-k,-
 - Pairing
O   ck c-k Q,-
pp,
q=0
k,
k,
 - Pairing
ph,
q=0
O   f k ck c-k Q,-
pp,
q=Q
k,
f k  cos k x  cos k y
17
The phase diagram at U=2t
(nVH=1)
SDW spin-density wave;
CDW charge-density wave
dSC d - wave superconductivity
CF charge flux; SF – spin flux;
PS phase separation
18
The phase diagram at U=2t
(nVH=0.92)
19
The phase diagram at U=2t
(nVH=0.73)
20
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)
21
The patching scheme
22
From: J.V. Alvarez et al.,
J. Phys. Soc. Jpn., 67, 1868 (1998)
23