Oct. 26, 2005 KIAS 1

Competing insulating phases in one-dimensional extended Hubbard models

Akira Furusaki (RIKEN)Collaborator: M. Tsuchiizu (Nagoya)

M.T. & A.F., PRL 88, 056402 (2002) PRB 66, 245106 (2002) PRB 69, 035103 (2004)

Oct. 26, 2005 KIAS 2

Contents

Extended Hubbard model Ionic Hubbard model Generalized Hubbard ladder

Various types of insulators: Mott insulator, Charge-Density Wave, Peierls insulator, Band insulator, staggered-flux state, ….

One-dimensional models of interacting electrons at half filling

Weak-coupling approach, Bosonization

Oct. 26, 2005 KIAS 3

Extended Hubbard model at half filling

)( ,,1,1,

,

jjjj

j cccctH

1,, j

jjj

jj nnVnnU

,,, jjj ccn ,, jjj nnn

0, VU

tU V

Oct. 26, 2005 KIAS 4

Standard phase diagram (before 1999)

Emery (1979)Hirsch (1984)Cannon, Scalettar, Fradkin (1991)……….

Oct. 26, 2005 KIAS 5

Weak-coupling theory (g-ology)

1-loop RG

charge sector

Charge gap if VU 2

Spin sector

Spin gap if VU 2

spin sector

RL

Oct. 26, 2005 KIAS 6

Phase diagram since 1999Discovery of Bond-charge-density wave (BCDW) phase

Nakamura (1999, 2000)Sengupta, Sandvik, Campbell (2002)…..

or Bond-Order-Wave (BOW)

Found numerically

Oct. 26, 2005 KIAS 7

Vertex correction

2213 )6(

41)2( V

tCVU

tCVUg

2211 )2(

41)2( V

tCVU

tCVUg

)]2/log[cot(2)(1 C cos2)(2C

Separate transitions in charge & spin sectors

In the strong-coupling regime 1st order SDW-CDW transition||3g

Degeneracy of zeros of and are lifted3g1g

Oct. 26, 2005 KIAS 8

Tam, Tsai, & Campbell, cond-mat/0505396

Oct. 26, 2005 KIAS 9

Bosonization

,/,//

,/,//

,/,/

2121

)(exp2

)(

LRLRLR

LRLRLR

LRFLR xixika

x

sincos1 ,, jjj

SDW nn

cossin1 ,, jjj

CDW nn

charge

spin

Order parameters

coscos..1 ,1,,1,

chcccc jjjj

jBCDW

sinsin..1 ,1,,1,

chcccc jjjj

jBSDW

LR

LR

Oct. 26, 2005 KIAS 10

Bosonized form of the Hamiltonian density

2222

22 LxRxLxRxvv

H

LxRxLxRxgg

22 22

2cos

22cos

2 22 ag

ag sc

LxRxLxRx

cs ag

ag

2

22 22cos2cos

2

2cos2

2cos2 22 LxRx

cLxRx

s gg

kinetic energy

marginal perturbation

relevant perturbation

irrelevantperturbation

3ggc 1gg s

SU(2) symmetry

sgg etc

Oct. 26, 2005 KIAS 11

Order parameters

cossinCDW

sincosSDW

coscosBCDW

sinsinBSDW

Classical analysis

cos2cos2cos2cos, cssc gggV 1ggs

||0,2 csscCDW gggVV

||2

,0 csscSDW gggVV

||0,0 csscBCDW gggVV ||2

,2 csscBSDW gggVV

3ggc

02||3 Vggcs

0csg 0csg

Oct. 26, 2005 KIAS 12

cos2cos2cos2cos, cssc gggV Phase transitions

SDW-BCDW transition: 2nd order CDW-BCDW transition: 2nd order CDW-SDW transition:1st order

Oct. 26, 2005 KIAS 13

Ground-state phase diagram from bosonization approach

1-loop RG + classical analysis

ttVU cc 3.2,0.5,

M. Tsuchiizu and A.F., Phys. Rev. Lett. 88, 056402 (2002)

Oct. 26, 2005 KIAS 14

Numerical Results

ttVU tt 5.2,7.4,

Quantum Monte Carlo

Sengupta, Sandvik, & Campbell,Phys. Rev. B 65, 155113 (2002)

ttVU bb 746.3,2.7,

DMRG

Y.G. Zhang, PRL 92, 246404 (2004)

Oct. 26, 2005 KIAS 15

Tricritical point on the CDW-BCDW phase boundary

SSE QMC

LLS

K

nnnneL

qSkj

kkjjkjiq

/2/2

1

//

,,,,,

)(/

Luttinger liquid parameterat the continuous transition

Sandvik, Balents & Campbell, PRL 92, 236401 (2004)

141

K

1K

Oct. 26, 2005 KIAS 16

...4cos2cos 43 FkggV

41

K

umklapp scatteringFk4

becomes relevant for

5.55 tU

Oct. 26, 2005 KIAS 17

Phase diagram (schematic)

5.5/5 tU t

9/ tUc

1st order transition

CDW-BCDW c=1 Gaussian

SDW-BCDW c=1 SU(2)1

Oct. 26, 2005 KIAS 18

Extended Ionic Hubbard model at half filling Ionic Hubbard model Nagaosa & Takimoto (1986), Egami, Ishihara, & Tachiki (1993)

Fabrizio, Gogolin, & Nersesyan, PRL 83, 2014 (1999)

,,,,,1,1,ionic 1

j j jj

jjjjjjj nnnUcccctH

and Mott insulator0U 0

0 and Band insulator0U

Quantum Phase Transition ?

Oct. 26, 2005 KIAS 19

Spontaneously Dimerized Insulating Phase (SDI) (= BCDW Phase)

0 U1cU

0c00

s

c

00

s

c

00

s

c

2cU

0jD 0jD0jD

Ising KT

BI SDI MI

Fabrizio, Gogolin, & Nersesyan, PRL 83, 2014 (1999)

,,1,1, jjjjj ccccD

Oct. 26, 2005 KIAS 20

Extended ionic Hubbard model nearest-neighbor repulsion V

,,,,,1,1, 1

j j jj

jjjjjjj nnnUcccctH

j

jjnnV 1

Bosonization

perturbative RG + classical analysis

Oct. 26, 2005 KIAS 21

Bosonized form of the Hamiltonian density

2222

22 LxRxLxRxvv

H

LxRxLxRxgg

22 22

2cos

22cos

2 22 ag

ag sc

LxRxLxRx

cs ag

ag

2

22 22cos2cos

2

2cos2

2cos2 22 LxRx

cLxRx

s gg

Kinetic energy

marginal perturbation

irrelevantperturbation

3ggc 1gg s

cossin

2 2agW relevant perturbationagW 4

Oct. 26, 2005 KIAS 22

Classical analysis cossin2cos2cos, Wsc gggV

cossinCDW

sincosSDW

coscosBCDW

sinsinBSDW

0Wg 0Wg

Gaussian Ising

Oct. 26, 2005 KIAS 23

0||3 ggcs

Ground-state phase diagramcf. 0

1st order transition

Oct. 26, 2005 KIAS 24

0V

Schematic phase diagrams

tV tV

tU tU

Oct. 26, 2005 KIAS 25

Generalized Hubbard ladder at half filling

j njjnjnj cctcctH

, 2,1,2,,1,,,1,,|| h.c.

j njjjjnjnj SSJnnVnnU

2,1 ,2,1,,2,,1,,,,,

t⊥V⊥, J⊥

tpair

h.c.,2,,2,,1,,1,pair

jjjj cccct

Oct. 26, 2005 KIAS 26

rung singlet state (D-Mott)

charge density wave (CDW)

Various Insulating Ground States that can appear in half-filled ladders

singlet paring state (S-Mott)

staggered flux state (SF)d-density waveorbital antiferromagnet

Ex. SO(5) ladder model

VUJ 4

ground-state phase diagram

U

V

Lin, Balents & Fisher (1998)Fjaerestad & Marston (2002)

Oct. 26, 2005 KIAS 27

Strong-coupling approach

4 basis states

Oct. 26, 2005 KIAS 28

VU HHH0

|||| Vtt HHHH

degenerate perturbation theory

Oct. 26, 2005 KIAS 29

CDW—S-Mott transition

D-Mott—S-Mott transition

||

2|| 2

22

VUV

tK

j

xj

zj

zj hKH 1eff

Ising model in a transverse field

UJVth

4/32 2

hK

hK

ordered state

disordered state

XXZ model in a magnetic field

j

zjz

xjx

j

zj

zj

yj

yj

xj

xj ShShSSSSSSJH 111eff

UV Ut

J2||3

35

thx 4 VUhz

0zh gapless (c=1)

Gaussian transition

0zh

0zh

Oct. 26, 2005 KIAS 30

)()()(,

QkckckfOk

AA

),( Q

1sf

density wave order

s-wave

p-wave

||sin kf p staggered dimerization

d-wave kkfd coscos ||

d-density-wave =SF

f-wave

kkf f cossin ||

Weak-coupling approach

,

||||0 coscos2k

ktktH ,, kk cc

s-wave

p-wave

d-wave

f-wave

These states break Z2 symmetry

Oct. 26, 2005 KIAS 31

sc r

Lr

Rr

r

sc r

Lr

Rr

gvH,

,,2,

,

2,

2, 2

scsccc ggga

2cos2cos2cos2cos2cos2cos2

132122

scscsc ggg 2cos2cos2cos2cos2cos2cos 654

sssssc ggg 2cos2cos2cos2cos2cos2cos 987

:c :s

cossincoscossincosCDW sscc

cossincoscoscoscosDDW sscc

cossinsinsincoscosPDW sscc

cossinsinsinsincosFDW sscc

yxiyx y

21,Bosonization

Hamiltonian density

charge spin

Order parameters

sssccV ,,,,pinning potential

Oct. 26, 2005 KIAS 32

Ising transitions

cossincoscossincosMott-S sscc

cossinsinsincoscosMott-S sscc

cossincoscoscoscosMott-D sscc

cossinsinsinsincosMott-D sscc

order

disorder

Disorder parameters

ss

Oct. 26, 2005 KIAS 33

Universality class of quantum phase transitions

Gaussian transition (c=1)Ising transition (c=1/2)

SU(2) criticality (c=3/2)2or 1st order transition

M. Tsuchiizu and A. FurusakiPhys. Rev. B 66, 245106 (2002)

LxLRxRiH effIsing

22effGaussian 2 cxcx

vH

LxLRxRSUviH

2eff

)2( 2

2LRg

Oct. 26, 2005 KIAS 34

Duality transformation Momoi & Hikihara, PRL (2003)

,2,,1,,, 21

jjj ccd

,

,,,,2exp

jjj ddiU ,,

1,,

jj idUUd

DDWCDW OUUO 1

FDWPDW OUUO 1PDWFDW OUUO 1

CDWDDW OUUO 1

MottMott SD

Oct. 26, 2005 KIAS 35

model JVUt

V’

Oct. 26, 2005 KIAS 36

Summary Competing interactions competing phases exotic order Various (density) ordered phases Various Mott insulating phases

2D systems ? M.T. & A.F., PRL 88, 056402 (2002)

PRB 66, 245106 (2002) PRB 69, 035103 (2004)