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

qq

,

1 0,

0,

pp

pppp

U q

qq

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)

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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

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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

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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

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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

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The phase diagram: vH band fillingsThe phase diagram: vH band fillings

32 - patchRG approach

T=0, =0

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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

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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)

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Ferromagnetism and RGFerromagnetism and RG

k k

k

k kqk

qkkq

)(

)()(0

f

ff

Momentum cutoff: noTemperature cutoff: yes

FS

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The flat-band ferromagnetismThe flat-band ferromagnetism

T-matrix result for FM instability by Hlubina et al.

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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)

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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

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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)

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The phase diagram at U=2tThe phase diagram at U=2t

(nVH=0.92)

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The phase diagram at U=2tThe phase diagram at U=2t

(nVH=0.73)

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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)

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The patching schemeThe patching scheme

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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)