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2008
Renormalization-group investigation
of the 2D Hubbard model
A. A. Katanina,b
a Institute of Metal Physics, Ekaterinburg, Russiab 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
iii nnUccH
,kkkk
0',
)1coscos('4)coscos(2
tt
kktkkt yxyx k
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)
La2-x SrxCuO4 Bi2212
Experimental relevance: high-Tc cuprates
A. Ino et al., Journ. Phys. Soc. Jpn, 68, 1496 (1999).
D.L. Feng et al., Phys. Rev. B 65, 220501 (2002)
U
tt'
• Provides a prototype model of interacting fermionic systems leading to nontrivial physics
The case of general Fermi surface
lnmax( , )Fv q T
)( FEN
Possible types of instabilities: Superconducting (only for U<0) Ferro- and antiferromagnetic instabilities are not in the
weak-coupling regime
,1 0
0
q
U
k qkk
qkkq
ff0
k kqk
kqkq
ff10
0
0
1 q
U
k1,
k2 ,'
k3,
k4,'
k1= k2; k3= k4: BCS channelk1= k3; k2= k4: ZS channelk1= k4; k2= k3: ZS' channel
There is no‘interference’ between different channels (channel separation)
kF
The Fermi liquid
k1+k2=k3 + k4
=
=
k
k+qk
q-k
5
The parameter space
0.01.00.0
0.5
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 (kk+Q) n=1; t'=0; • van Hove singularities (k=0) n=nVH; 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
)cossin(2 2222yx
Ak kkt
)/'2arccos()2/1( tt
)sincos(2 2222yx
Bk kkt
Similar to the “left” and “right” moving particles in 1D
But the geometry of the Fermi surface and the dispersion are different !
2
B
A
8
The two patch equations at T » |
)22(g))(g(/
)2()g(2)g(2/
)(g)g(g2))(g(/
22)(g)g(2/
24
2221
212
24
2304
12314303
22
2134212
23
2212
21341212111
ggggdgdddg
ggdgdddg
gdgdgdddg
ggdggdgdddg
RRRd
Rd
RRd
Rd
pp
ph
ph
pp
)/1/(tan2)(
;12/)(
]);)/11[(ln,min(2)()(
;1/2)()(
21',3
2'0,2
2',1
2',0
Q
Q
0 )/ln(
/'2
T
ttR
Possible types of vertices
There is no separation of the channels: each vertex is renormalized by all the channels
9
The vertices: scale dependence
g1
g2
(inter-patch direct)
g3 (umklapp)
g4
g1
(inter-patch exchange)
g2
g3
g4
(intra-patch)
()
()
U=2t, t'/t=0.45 (nVH=0.47)
U=2t, t'/t=0.1 (nVH=0.92)
10
Phase diagram: vH band fillings
32 - patchfRG approach
T=0, =0
Functional renormalization group
o Projecting momenta to the Fermi surfaceo Projecting frequencies to zeroo 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
12
2
( )1 1 1( ) ( ) ( , ) [ ]
2 2 2Tr C Q Q Tr Q
(T. Morris, 1994; M. Salmhofer and C. Honerkamp, 2001)
• Expands in fields
( )1
1( ) ( ) ( ).... ( )
!m m
mm
d X X X Xm
( )
• Obtains the equations for the coefficients of the expansion
(2) (2)
(4) (4) (2) (2)
1 1( ) ( , ) [ ( )]
2 21 1
( ) [ ( )] [ ( ) ( )]2 2
Q Tr S
Tr S Tr S G
%
%%%
( ) ( 2)1
1( ; ) ' ( , ') ( ').... ( ')
!m m m
mm
X d X X X X Xm
%
where
…
• Truncates the hierarchy of equations, e.g. (4) ( ) 0 %
13
1PI scheme
T
2
1 1
2 ( )n
k kn k n k
iG S
i i
k
k k
Temperature cutoff
14
Phase diagram: vH band fillings
32 - patchfRG approach
T=0, =0
15
Ward identitiesWard identities
( 2 ) k k qk k k q k q k k k q k q k k qk
q G G
dV S
d
( )dV
V G S S G Vd
is fulfilled up to the order V2 only
Ward identity:
Replacement:d
S Gd
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, 2005-2008)
(A. Katanin, Phys. Rev. B 70, 115109 (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. ImadaJourn. Phys. Soc. Jpn 70, 3052 (2001).
48-patch fRG approach:
MF
17
Angular dependence of the order parameterHole-doped sc, A. A. Katanin and A. P. Kampf, Phys. Rev. 2005
Electron-doped sc, A. A. Katanin, Phys. Rev. 2006
J. Mesot et al., Phys. Rev. Lett. 83, 840 (1999).
H. Matsui et al., Phys. Rev.Lett. 95, 017003 (2005).
Pr0.89LaCe0.11CuO4
Hot spot
Max
18
Taking 6-point vertex into account
0.1t
0.1t
U 2.5t
dV S
d
( )dV
V G S S G Vd
0
S d V G V G V S
A. Katanin, arXiv:2008
19
Scattering rates
From spin-fermion theory:
(R. Haslinger,Ar. Abanov, andA. Chubukov,2001)
FL
NFL
FL
NFL
0.1t
0.1t
0.1t
Landau 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 T2 dependence at low T and T1-dependenceat 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
3D anis2 i ii
JH H H
S S
La2CuO4
field-theor. RG + 1/N expansion, V. Yu. Irkhin, A. Katanin et al., PRB 1997
QPT
*0T
QD
QC
RC
vs.
similarity to CSB in QCD ?
O(N) or O(N)/O(N-2) NL-model
Frustration and quantum criticality