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
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
Oct. 26, 2005 KIAS 8
Tam, Tsai, & Campbell, cond-mat/0505396
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
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
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
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
Oct. 26, 2005 KIAS 16
...4cos2cos 43 FkggV
4
1K
umklapp scatteringFk4
becomes relevant for
5.55 tU
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
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 ?
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
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
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
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
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
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
2
1,
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
LxLRxRSU
viH
2eff
)2( 2
2LRg
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
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)