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

qq

U

k qkk

qkkq

ff0

k kqk

kqkq

ff10

0

0

1 q

qq

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