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Renormalization-group studies of the 2D Hubbard model. A.A. Katanin a,b and A.P. Kampf a. a Theoretische Physik III, Institut f ü r Physik, Universit ät Augsburg, Germany b Institute of Metal Physics, Ekaterinburg, Russia. 2003. Content. The model The weak-coupling regime: - PowerPoint PPT Presentation
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1
A.A. Katanina,b and A.P. Kampfa
2003
aTheoretische Physik III, Institut für Physik, Universität Augsburg, Germanyb Institute of Metal Physics, Ekaterinburg, Russia
Renormalization-group studies
of the 2D Hubbard model
2
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
ContentContent
3
The 2D Hubbard modelThe 2D Hubbard model
iii nnUccH
,kkkk
0',
)1coscos('4)coscos(2
tt
kktkkt yxyx k
Cuprates (Bi2212)AB
B
La2-x SrxCuO4 Bi2212
Experimental relevance: cuprates
Ruthenate Sr2RuO4
A. Ino et al., Journ. Phys. Soc. Jpn, 68, 1496 (1999).
D.L. Feng et al., Phys. Rev. B 65, 220501 (2002)
A. Damascelli et al,J. Electron Spectr. Relat. Phenom. 114, 641 (2001).
4
The weak coupling regimeThe weak coupling regime
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. ?
Why it is interesting:
• Non-trivial• Gives the possibility of rigorous numerical and semi analytical RG treatment.
• U < W/2
However, 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'
Interaction alone is not enough to produce magnetic or superconducting instabilities in the weak-coupling regime
,1 0,
0,
ph
phph
U q
,
1 0,
0,
pp
pppp
U q
5
The parameter spaceThe parameter space
0.01.00.0
0.5
t'/t
n
The line of van Hove singularities
Nesting
The simplest mean-field (RPA) approachbecomes inapplicable close to the line due to “the interference” of different channels of electron scattering:
pp-scattering ph-scattering
6
Theoretical approachesTheoretical 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)
7
The two-patch approachThe 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 topology of the Fermi surface is different !
2
B
A
Possible types of vertices
There is no separation of the channels: each vertex is renormalized by all the channels
8
The two patch equations at T » |The two patch equations at T » |
T 2ln
T ln
T ln
T
ln
ph,00
ph,0Q
)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
pp,00
pp,0Q
9
The vertices: scale dependenceThe 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
Many-patch renormalization groupMany-patch renormalization group
),,(),(),,(),,(
3212121321 pppppVppppLpppdpV
dT
pppdVTppT
),(),,(),-,(
),()],,(),,(
),,(),,(2[
3232321
31231311
3123123131
ppppLpppVpppppVdp
ppppLpppppVpppppV
,p)pp,p(p)V,p,p(pVpppppVpppVdp
phTT
phTT
TTTT
11
The phase diagram: vH band fillingsThe phase diagram: vH band fillings
32 - patchRG approach
T=0, =0
12
MF: W. Hofstetter and D. Vollhardt, Ann. Phys. 7, 48 (1998)
The vicinity of half fillingThe vicinity of half filling
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 RG approach:
t'=0; n<1
13
The flat-band ferromagnetismThe flat-band ferromagnetism
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) (projected QMC)
kx
ky
~1/1/2
t’/t=1/2
U>0
Mielke and Tasaki (1993. 1994)
14
Ferromagnetism and RGFerromagnetism and RG
k k
k
k kqk
qkkq
)(
)()(0
f
ff
Momentum cutoff: noTemperature cutoff: yes
FS
15
The flat-band ferromagnetismThe flat-band ferromagnetism
T-matrix result for FM instability by Hlubina et al.
16
Ferromagnetism due to vHSFerromagnetism 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)
17
Possible order parametersPossible order parameters
k,Qk-k
k,Qk-k
k,k-kk
k,k-k
k,q,kk
k,k,kk
k,k,kk
k,k,k
k,Q,kkk
k,Q,kkk
k,Q,kk
k,Q,kk
,-k
,-
,-dSC
,-sSC
PS
BS
BC
F
SF
CF
SDW
CDW
ccfO
ccO
ccfO
ccO
ccO
ccfO
ccfO
ccO
ccfO
ccfO
ccO
ccOCharge-density wave
Spin-density wave
Charge-flux
Spin-flux
Phase separation
Ferromagnetism
Bond-charge order (PI)
Bond-spin order (A)
s - wave supercond.
d - wave supercond.
- Pairing
- Pairing
ph,q=Q
ph,q=0
pp,q=0
pp,q=Q
yx kkf coscos k
18
The phase diagram at U=2t
SDW spin-density wave; CDW charge-density wavedSC d - wave superconductivityCF charge flux; SF – spin flux; PS phase separation
(nVH=1)
19
The phase diagram at U=2tThe phase diagram at U=2t
(nVH=0.92)
20
The phase diagram at U=2tThe phase diagram at U=2t
(nVH=0.73)
21
ConclusionsConclusions
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 symmetryb) To go away from the van Hove band fillingc) 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)
22
The patching schemeThe patching scheme
23
From: J.V. Alvarez et al., J. Phys. Soc. Jpn., 67, 1868 (1998)
From: J.V. Alvarez et al., J. Phys. Soc. Jpn., 67, 1868 (1998)