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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)
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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
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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
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Oct. 26, 2005 KIAS 4
Standard phase diagram (before 1999)
Emery (1979)Hirsch (1984)Cannon, Scalettar, Fradkin (1991)……….
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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
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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
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Oct. 26, 2005 KIAS 7
Vertex correction
2213 )6(
41)2( V
t
CVU
t
CVUg
2211 )2(
41)2( V
t
CVU
t
CVUg
)]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
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Oct. 26, 2005 KIAS 8
Tam, Tsai, & Campbell, cond-mat/0505396
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Oct. 26, 2005 KIAS 9
Bosonization
,/,//
,/,//
,/,/
2
12
1
)(exp2
)(
LRLRLR
LRLRLR
LRFLR xixika
x
sincos1,, jj
jSDW nn
cossin1 ,, jj
jCDW nn
charge
spin
Order parameters
coscos..1,1,,1,
chcccc
jjjj
jBCDW
sinsin..1,1,,1,
chcccc
jjjj
jBSDW
LR
LR
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Oct. 26, 2005 KIAS 10
Bosonized form of the Hamiltonian density
2222
22 LxRxLxRx
vvH
LxRxLxRx
gg
22 22
2cos
22cos
2 22 a
g
a
g sc
LxRxLxRx
cs ag
a
g
222 2
2cos2cos2
2cos2
2cos2 22 LxRx
cLxRx
s gg
kinetic energy
marginal perturbation
relevant perturbation
irrelevantperturbation
3ggc 1ggs
SU(2) symmetry
sgg etc
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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
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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
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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)
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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)
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Oct. 26, 2005 KIAS 15
Tricritical point on the CDW-BCDW phase boundary
SSE QMC
L
LSK
nnnneL
qSkj
kkjjkjiq
/2
/2
1
//
,,,,,
)(/
Luttinger liquid parameterat the continuous transition
Sandvik, Balents & Campbell, PRL 92, 236401 (2004)
14
1 K
1K
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Oct. 26, 2005 KIAS 16
...4cos2cos 43 FkggV
4
1K
umklapp scatteringFk4
becomes relevant for
5.55 tU
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Oct. 26, 2005 KIAS 17
Phase diagram (schematic)
5.5/5 tU t
9/ tU c
1st order transition
CDW-BCDW c=1 Gaussian
SDW-BCDW c=1 SU(2)1
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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
j
jjjjjj nnnUcccctH
and Mott insulator0U 0
0 and Band insulator0U
Quantum Phase Transition ?
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Oct. 26, 2005 KIAS 19
Spontaneously Dimerized Insulating Phase (SDI)
(= BCDW Phase)
0 U1cU
0c0
0
s
c
0
0
s
c
0
0
s
c
2cU
0jD 0jD0jD
Ising KT
BI SDI MI
Fabrizio, Gogolin, & Nersesyan, PRL 83, 2014 (1999)
,,1,1, jjjjj ccccD
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Oct. 26, 2005 KIAS 20
Extended ionic Hubbard model nearest-neighbor repulsion V
,,,,,1,1, 1
j j jj
j
jjjjjj nnnUcccctH
j
jjnnV 1
Bosonization
perturbative RG + classical analysis
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Oct. 26, 2005 KIAS 21
Bosonized form of the Hamiltonian density
2222
22 LxRxLxRx
vvH
LxRxLxRx
gg
22 22
2cos
22cos
2 22 a
g
a
g sc
LxRxLxRx
cs ag
a
g
222 2
2cos2cos2
2cos2
2cos2 22 LxRx
cLxRx
s gg
Kinetic energy
marginal perturbation
irrelevantperturbation
3ggc 1gg s
cossin
2 2a
gW relevant perturbationagW 4
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Oct. 26, 2005 KIAS 22
Classical analysis cossin2cos2cos, Wsc gggV
cossinCDW
sincosSDW
coscosBCDW
sinsinBSDW
0Wg 0Wg
Gaussian Ising
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Oct. 26, 2005 KIAS 23
0||3 ggcs
Ground-state phase diagramcf. 0
1st order transition
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Oct. 26, 2005 KIAS 24
0V
Schematic phase diagrams
tV tV
tU tU
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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
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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)
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Oct. 26, 2005 KIAS 27
Strong-coupling approach
4 basis states
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Oct. 26, 2005 KIAS 28
VU HHH0
|||| Vtt HHHH
degenerate perturbation theory
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Oct. 26, 2005 KIAS 29
CDW—S-Mott transition
D-Mott—S-Mott transition
||
2|| 2
2
2V
UV
tK
j
xj
zj
zj hKH 1eff
Ising model in a transverse field
UJV
th
4/3
2 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 U
tJ
2||3
3
5 thx 4 VUhz
0zh gapless (c=1)
Gaussian transition
0zh
0zh
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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
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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
2
1,
Bosonization
Hamiltonian density
charge spin
Order parameters
sssccV ,,,,pinning potential
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Oct. 26, 2005 KIAS 32
Ising transitions
cossincoscossincosMott-S sscc
cossinsinsincoscosMott-S sscc
cossincoscoscoscosMott-D sscc
cossinsinsinsincosMott-D sscc
order
disorder
Disorder parameters
ss
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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
LxLRxRSU
viH
2eff
)2( 2
2LRg
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Oct. 26, 2005 KIAS 34
Duality transformation Momoi & Hikihara, PRL (2003)
,2,,1,,,2
1jjj ccd
,
,,,,2exp
jjj dd
iU ,,
1,,
jj idUUd
DDWCDW OUUO 1
FDWPDW OUUO 1PDWFDW OUUO 1
CDWDDW OUUO 1
MottMott SD
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Oct. 26, 2005 KIAS 35
model JVUt
V’
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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)