Partnergroup
Renormalization-group
investigation
of 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
Content
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
2
The 2D Hubbard model


H

c
c

U
n
n


k
k

k

i

i



k
,
i
t
t'



2
t
(
cos
k

cos
k
)

4
t
'
(
cos
k
cos
k

1
)

x
y
x
y
k
U
t
,
t
'

0
• Provides a prototype model of interacting fermionic
systems leading to nontrivial physics
Experimental relevance: high-Tc cuprates
Cuprates (Bi2212)
La2-x SrxCuO4
A. Ino et al., Journ. Phys. Soc.
Jpn, 68, 1496 (1999).
Bi2212
D.L. Feng et al.,
Phys. Rev. B 65, 220501 (2002)
• The weak-coupling regime U < W/2
Why it is interesting:
• Non-trivial
• Gives a possibility of rigorous numerical and semianalytical RG treatment.
3
The case of general Fermi surface
k1,
k3,
kF
k1+ k2=k3 + k4
k4,'
k2 ,'
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
 q0
q 
,
1  U q0
 0q
q 
1  U 0q
k
 q0  
k
f k  f k q
 k   k q
 N ( EF )
=
1  f k  f q k
 
=



k
k
q k
The Fermi liquid
k+q
k
 ln
0
q
q-k

max( vF q,T )
 Possible types of instabilities:
 Superconducting (only for U<0)
 Ferro- and antiferromagnetic instabilities are not in the
weak-coupling regime
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. ?
0.5
The line of
van Hove
singularities
t'/t


0.0
1.0
n
0.0
Nesting
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'5
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)
The field-theory (two-patch) approach



2
t
(
sin

k

cos

k
)





2
t
(
cos

k

sin

k
)


B
2
A
k
22
x
22
y
B
k
22
x
22
y
A
Similar to the “left” and “right”
moving particles in 1D


(
1
/
2
)
arccos(
2
t
'/
t
) But the geometry of the Fermi surface
and the dispersion are different !
7
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
dg
/
d


2
d
(

)g
(g

g
)

2
d
g
g

2
d
g
g
1
1
1
2
1
2
1
4
3
1
2
dg
/
d


d
(

)(g
g
)

2
d
(g

g
)g

d
(g
g
)
2
1
2
3
2
1
2
4
3
1
2
2
2
2
2
dg
/
d



2
d
(

)g
g

2
d
(

)g
(
2
g

g
)
3
0
3
4
1
3
2
1
dg
/
d



d
(

)(g
g
)

d
(g
2
g
g

2
g
g
)
4
0
3
4
2
1
1
2
2
4
2
2
2
2
   1

R;
d
( ) 'pp
( )
2/
0
,0
2
'
2
d
(
)

(
)
2
min(

,ln
[(
1
1

R
)/
R
]);
1
ph
,Q
2
R2t'/t
ln(
/T)
'
2
d


(

)

2/
1

R
;
2
ph
,0
'

1
2
d


(

)

2
tan
(
R
/
1

R
)/
R
3
pp
,Q
8
The vertices: scale dependence
U=2t, t'/t=0.1 (nVH=0.92)
g3
(umklapp)
g2
(inter-patch direct)
g1

()
g4
U=2t, t'/t=0.45 (nVH=0.47)
g1
(inter-patch exchange)
g4
(intra-patch)
g2
g3
()

9
Phase diagram: vH band fillings
T=0, =0
32 - patch
fRG approach
10
Functional renormalization group
o Projecting momenta to the Fermi surface
o Projecting frequencies to zero
o 32-48 patches on the Fermi surface
(after M. Salmhofer and C. Honerkamp, 2001)
1PI functional RG
• Considers the evolution of the 1PI generating functional
1
  2   ( ) 
1
1
1
  ( )  Tr (C Q )  ( , Q )  Tr[Q 
 ]
2
2
2
2
 

(T. Morris, 1994; M. Salmhofer and C. Honerkamp, 2001)
• Expands
 ( ) in fields
  ( )  
1
d m X  ( m ) ( X ) ( X 1 ).... ( X m )

m!
m
• Obtains the equations for the coefficients of the expansion
1
1
2
2
1
1
 (4) ( )  Tr[ S (4) ( )]  Tr[ S (2) ( )G (2) ( )]
2
2
…
 (2) ( )  ( , Q )  Tr[ S (2) ( )]
where
 ( m ) ( X ; )  
m
1
d m X '  ( m  2) ( X , X ') ( X 1 ').... ( X m ')

m!
• Truncates the hierarchy of equations, e.g.  (4) ( )  0
12
1PI scheme
 Temperature cutoff   T
Gk 
1


S

k

i n   k   k
i n   k
1

 2
2 (i n   k   k )
13
Phase diagram: vH band fillings
T=0, =0
32 - patch
fRG approach
14
Ward identities
d 
V  S
d
dV 
 V  (G  S   S  G  ) V  
d
Ward identity:
q  ( 2 k k  k  q k   q   k k  k   q k  q )Gk  Gk   q   k   k  q 
k
is fulfilled up to the order V2 only
Replacement: S  
d 
G
d
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)
15
Half filling, non-nested Fermi surface
MF
n=1
48-patch fRG approach:
antiferromagnetic
QMC: H.Q. Lin and J.E. Hirsch,
Phys. Rev. B 35, 3359 (1987).
d-wave superconducting
PIRG: T. Kashima and M. Imada
Journ. Phys. Soc. Jpn
70, 3052 (2001).
MF: W. Hofstetter and D. Vollhardt, Ann. Phys. 7, 48 (1998)
16
Angular dependence of the order parameter
Electron-doped sc, A. A. Katanin,
Hole-doped sc, A. A. Katanin
and A. P. Kampf, Phys. Rev. 2005 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).
17
Taking 6-point vertex into account
d 
 V S
d
dV
 V (G S  S G ) V
d
0
 S


d  V G V G V S
U  2.5t
 0.1t
A. Katanin, arXiv:2008
 0.1t
18
Scattering rates
 0.1t
Landau
FL
NFL
Landau
FL
NFL
 0.1t
 0.1t
From spin-fermion
theory:
(R. Haslinger,
Ar. Abanov, and
A. Chubukov,
2001)
19
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
Future perspectives

Detail description of quantum critical points
similarity to CSB in QCD ?
T
vs.
RC
QD
QPT

T *  0
QC

Application to the localized Heisenberg (e.g. frustrated)
magnets – bosonic (magnons) vs. fermionic (spinons)
excitations
H 
J
 SiSi   H3D  Hanis
2 i
O(N) or O(N)/O(N-2) NL-model
La2CuO4
field-theor. RG + 1/N
expansion, V. Yu. Irkhin,
A. Katanin et al., PRB 1997
Frustration and quantum criticality


Combination with other nonperturbative approaches
Including long-range interactions, gauge fields etc.