F.F. Assaad.

MPI-Stuttgart. Universität-Stuttgart.

21.10.2002

Numerical approaches to the correlated electron problem:

Quantum Monte Carlo.

The Monte Carlo method. Basic.

Spin Systems. World-lines, loops and stochastic series expansions.

The auxiliary field method I

The auxiliary filed method II

Special topics

Magnetic impurities Kondo lattices.

Metal-Insulator transition

One magnetic impurity. Cu with Fe as impurity.

Fe: 3d 64s 2 Hunds rule: S=2

Invers

e su

scepti

bili

ty.

T

1T

T

Free spin.

Screened spin.

Temperature.Temperature

Resi

stiv

ity.

Resistivity minimum. (Normal: a + bT 2)

)0(

Kondo problem: crosover from free to screened impurity spin. Many body non-perturbative problem.

The Kondo problem is a many-body problem

Impurity spin.

Electrons.

p

k P´ k Spin-flip

scattering of p

P´ k

Spin of p isconserved

The scattering of electron kwill depend on how electronp scattered.

Thus, the impurity spin is a source of correlations between conduction electrons.

+H cc ss

s ,,

,)( k

kk

k SIf cc ssssssN

J',,',',,2 pkk,p σ

k can spin-flip scatter

k cannot spin-flip scatter

Ground state at J/t >>1

+ S IcJH cct

,

,,,

, j

Ijiji

i

S I

f

2

1

.H.c,,

, cct

δI

δI

)(||1

)( ||2

,EEmn

ZS mn

f

I

En

mnf Se

)(S f

t/

T <TK

T >TK

Dynamical f-spin structure factor

Ground state:

Spin singlet

J/t = is relevant fixpoint. Wilson (1975)

00

SSf

I

f

IId

Numerical (Hirsch-Fye impurity algorithm):

T/TK

J/t = 1.2J/t = 1.6J/t = 2.0

T

TK/t 0.21

TK/t 0.06TK/t 0.12

is the only low energy scaleeTJt

K

/

T>>J: Essentially free impurity spin. TI1

Lattices of magnetic impurities.

SS iii

jiji,

fcJcctH

,,

),(

))(( 21

,21

,,,,,,

,,),(

nnUcffcVcctH ff

f iii

iiiii

jiji,

Periodic Anderson model (PAM).

Kondo lattice model (KLM).

Charge fluctuations on f-sites.

Charge fluctuations on f-sites frozen.

UVJ f/2

Conduction orbitals:

Impurity orbitals:

c,i

f

,i

Simulations of the Kondo lattice.

Consider:

cffccckH iiiiJ

ik

kk ,,,,4

2

,,

,)(

We can simulate this Hamiltonian for all band fillings. No constraint on Hilbert space.

How does H relate to the Kondo lattice?

SS fi

ci

ik

kk JcckH

,,

,)( )( H.c.,,,,,

ffccJ iiiii

nnnnJ fi

ci

fcii

i

,

Conservation law: 0)1()1( ][,,,,

, nnnnH ffff

iiiii

TTP ||0

T

HT

H ee KLM ||Chose so that

0)1()1( ,,,, nnnn ffff

iiiii

Let P0 be projection on Hilbert space with :

Then: HHP KLM0

Mean-field for Kondo lattice.

Order parameters:

r

SS

,,,,,,,,

,,,

fccffccf

mm fzczc f

iiiiiiii

iiii

2

4

1

2

4

1,,

,,,,

,,,,SS

fccf

fccfzzcc ff

iiii

iiiiiiii SS

Decoupling:Two energy scales:

J/t = 4, <nc>=0.5

Cv

T/tTK

Below TK r>0.Same as forsingle impurity.

Tcoh

Below Tcoh

Fermi liquid

Tcoh TK e Jt /

(S. Burdin, A. Georges, D.R. Grempel et al. PRL 01)

4r2

2r

1

ε

,,

,,,,,

,,,

JN

J

ff

fccfccHMF

i σ σiσi

σiσiσiσiσi

σkσkσkk

Mean field Hamiltonian (paramagnetic)

0r

HH MFMF

Saddle point.Exact for theSU(N) modelat N

(Ce x La 1-x ) Pb 3

X=1

X=0.6

Crossover to HF state.

Finite Temperature:

Coherence. Single impurity like

),( ππ ),( π0 ),( 00 ),( ππ

E(k

)5.0,2/ ntJ c

k

Ground state (Mean-field).

Luttinger volume: nc+1

Zk

),( ππ ),( π0 ),( 00 ),( ππ

Fermi liquid with large mass or small coherence temperature.

0 fcMean-Field Problems: , magnetism, finite T.

J/t=2 J/t = 4

t

mnDZm 1

Periodic table of elements

I. Coherence. (FFA. PRB 02)

Note:Conduction band is half-filled and particle hole symmetry is present. Allows sign-free QMC simulations but leads to nesting. At T=0 magnetic insulator.

Strong coupling.

Fermi line tJ /

Brillouin zone.

Technical constraint: Conduction band has to be half-filled. Otherwise sign problem.

SSf

R

c

RR

jji

i JcctH

,

),,(,

Model.

Conduction band:Half-filled.

T/t = 1/20T/t = 1/30T/t = 1/15

T/t = 1/60

T/t = 1/2T/t = 1/5

T/t = 1/10

t

T/tO

pti

cal co

nd

uct

ivit

y

R

esi

stiv

ity

Optical conductivity and resistivity. J/t = 1.6

Single impurity likeCoherence.

cm -1

Temperature

Resi

stiv

ity

Schlabitz et al. 86.

Degiorgi et al. 97

L=8L=6

(L=8: 320 orbitals.)

Resi

stiv

ity

T/t

Thermodynamics: J/t = 1.6

L=6

L=8

T/t

T/t

T S

T S

ss

~)(Q

T*

T*

Scales as a function of J/t.

J/t

T/t T*

Ts

Tmin/2

T

EV

nch

Ms C

h

0

s

c

CV

T/t

T/t

Speci

fic

heat.

c

s

L=6

L=4

T S ss

~)(Q

J/t = 0.8

1/81/10

1/151/20

1/30

1/50

1/80

T/t

Resistivity

t

Scales as a function of J/t.

J/t

T/t T*

Ts

Tmin/2

Comparison T* with TK of single impurity probem.

Ts

J/t

Depleted Kondo lattice.

TK

T*

T* TK

Crossover to the coherent heavy fermion state is set by the single impurity Kondo temperature.

Note: CexLa1-xCu6 T* ~ 5-12 K for x: 0.73-1.TK ~ 3K

(Sumiyawa et al. JPSJ 86)

T/t

Tcoh ?

No magnetic order-disorder transition since strong coupling metallic state is unstable towards magnetic ordering.

CePd2Si2 (J.D Mathur et al. Nature 98.)

[ See also CeCu6-xAux]

RKKY

J

II Magnetism : Order-disorder transitions

RKKY Interaction

Kondo Effect.

TK ~ e-t/JEnergy scale

)0,(2~)(eff qJqJEnergy scale

Spin susceptibility of conduction electrons.

Competition RKKY / Kondoleads to quantum phase transitions.

Half-filled Kondo lattice.

One conduction electron per impurity spin.(FFA PRL. 99)

ModelModel

Strong coupling limit. J/t >> 1

Spin Singlet

1) Spin gap

EnergyJ

s

2) Quasiparticle gap.

Energy3J/4

qp

QMC , T=0, L

SS R0R

QR ffe

Ni

N

34

lim

m > 0, Q=(,): long range antiferromagnetic order.

3) Magnetism.

( m f )2=

1D eJt

s

/~

4/~ Jpq

(Tsunetsugu et. al. RMP 97)

Spin Dynamics: S(q, )

Fit: Perturbation in t/J.

Fit: Spin waves.

Excitations of disordered phase condense to form the order of the ordered state. Bond mean-field of Kondo necklace (G.M. Zhang et. al. PRB 00).

(S. Capponi, FFA PRB 01)

Single particle spectral function. A(k,

Fit: Strong coupling.

Weak coupling ?

(S. Capponi, FFA PRB 01)

Jc

TK

ms

f-Spins are frozen.a)

ππ 0 k

E( k

)E

( k)

ππ 0 k

Magnetic BZ.

Magnetic BZ.

Jc

TKms

b) Partial Kondo screening,remnant magnetic momentorders.

(M. Feldbacher, C. Jureka, F.F.A., W. Brenig PRB submitted.)

J < 0No Kondoeffect.

Single particle spectral function. A(k,

Fit: Strong coupling.

In ordered phase impurity spins are partially screened. Remnant moment orders.

Mean-field interpretation:Coexistence of Kondo screening and magnetism.(Zhang and Yu PRB 00)

(S. Capponi, FFA PRB 01)

/t

k,

/t

k,/t=

t

)(Qs

L = 4L = 6L = 10

TS ~ J2

Origin of quasiparticle gap at weak couplings.

J/t = 0.8

Quasiparticle gap oforder J is of magnetic origin at J < Jc ~ 1.5 t

Conclusions.

QMC algorithm for Kondo lattices.

Restriction. Particle-hole symmetric conduction bands.

Depleted lattices.

T* TK

Half-filled Kondo lattice in 2D.

Pairing. No.

II.Doped Mott insulators.

MPI-Stuttgart. Universität-Stuttgart.

Metal Half filling: Insulator. Scale U

Charge is localized. Internal degree offreedom (spin) is still active.

t

Hubbard Model.

cctH jiji

,,,,

),(),( 2

12

1 nnU iii

U

UStrong coupling U/t >>1 (Half filling)

tU

tJ2

~t Magnetic scale:

SS jiji

JH ,

Heisenberg Model.

The Mott Insulator. Half filling (2D,T=0)

Charge.

Quasiparticle gap > 0

F.F. Assaad M. Imada JPSJ 95.

N1

Spin.

Long range magnetic order.Goldstone mode: Spin-waves.

N1

Mott Insulator U/W

Bandwidth W.

-(BEDT-TTF)2 CU[N(CN)2]C (2D)

V 2 O 3 (3D)

F.F. Assaad, M. Imada und D.J. ScalapinoPhys. Rev. Lett. 77 , 4592, (1996)

The Metal-Insulator Transition.

Metal

Doping Cuprates. (2D)[ (La Nd) 2-x Sr x Cu O 4 ]Superconductivity-Stripes.

Titanates (3D)(La xSr 1-xTi O 3 )

F.F. Assaad und M. ImadaPhys. Rev. Lett. 76 , 3176, (1996).Phys. Rev. Lett. 74 , 3868, (1995) .

iN

U

ji

jiN iitH cλccc2

,

)( ,1, cc Niii c

λ ,

,

,

if N/2

if N/2

δ

δ

How can we avoid the sign problem?

N = 4 n. No sign problem irrespective of lattice topologyand doping.

N=2: HN=2 = Hubbard

N = Mean-field

N > 2 Symmetry: SU(N/2) SU(N/2)

2:

U

Nn n

, , )( zm n n

2:

z zU

Nm m

Orbital Picture.

ElementaryCell

N Z.

2

4

6

8

0

2

3

1

N N/2-1

U N

z zU m mH

iN

U

ji

jiN iitH cλccc2

,

,)( ,1, cc Niii c λ ,

N/2if

N/2if

,

,

δ

δ

N = : SDW Mean field. nn iii

S

,,thatso0)(

)(/2)(

)()()( tNtt

t

Sttt

etP SN )(),( so that

eDeSNHN )(

Tr

erTdU

S Hdii

0

)(2

0Tln

2

1)(

4)(

i iii

jiji nncc UtH ,,

,,,,

)()(

with

Langevin:

More Formal.

1/N

E(N)/E(N=2)

D()(N)/D()(N=2)

T=0, 4 X 4, U/t = 4, 2 Löcher.

Lanczos.

Mean-field.

F.F. Assaad et al.PRL submitted.

Test.

Note: <n>=1, U/t >> 1 24/aberMF N

c c tT TU U

Single particle: N=4, T=0, U/t=3. 2D.

()/t

N(

)

=0,30X8

t/U=0 1 12 2)) (( ,, iU ii

H U nn

N(

)

=0, = 0:

U/2-U/2

L=6LL

0

N(

)

=1/6, = -U/2:

U

L=6L-1L-1

2

0

()/t ()/tN

()

N(

)

=1/14,30X8 =1/5,30X8, 30x12

Spin, S(q), charge, N(q), Structure Factors.

Real space (caricature).

Disctance between walls: 1/

=1/5

=0

Phase-shift in Spin Structure.

One dimensional

/2

/2

S(q

)N

(q)

N=4, T=0, U/t =3, 60X1, =1/5

4kf

2kf

N=4, T=0, U/t =3, 30X8

(/2,) (,) (,/2)

S(q)= 0= 1/14= 1/7= 1/5

= 1/4(,)()(,)

N=4, T=0, U/t =3, 30X8N(q)= 1/14 = 1/7= 1/5 = 1/4

()

(x,) Spin.

(0,0)

Charge. (2x,0) x =

Spin-and charge-Dynamics at =0.2 (T=0,N=4,U/t=3)

qx qx

60X130X430X830X12

First charge-excitation at q=(qx,0)

First Spin-excitation at q=(qx,)

Optical conductivity:

30 X 8, =0.2

Ohne VertexMit Vertex

/t

]),/2([' xxx Lq

qqNq xxx2' /),(),(

N(q,): Dynamical charge Structure factor.

Transport

Ly >4: Particle-hole continuum.

S(q

)

N(q

)

()(x,)

(y)

(0,0)

(2x,0)

(0,2y)

Charge.

Spin.

Two dimensionsLy=10, Lx=30, = 0.2

Two-dimensional metallic with no quasiparticles.Elementary excitations: spin and charge collective modes.

qy qy

qxqx

Interpretation of collective modes.

1) Analogy to 1D ?

2) Goldstone Modes. a) SU(2) SU(2) Symmetry is not broken.

0 : 2 und 0.M s

Energy is invariant under Translation:

/ 3 : 0.00001 , 0.8U t A t B t

2 2 c cs s

0.2 : 10.M

s

c

0 ( , )HFE c s 2 20 ( , ) ( 2 ) ( 2 )cos cosHF A BE s sc sc c

0 : und 0.M A

( ) cos( ) und ( ) cos( )n r r S r rQ Q c sc ssc

2 ,GQ Q c s M GQ s

b) Phasons.