excitonic mass generation in honeycomb lattice -...
TRANSCRIPT
Excitonic mass generationin Honeycomb lattice
G. Matsuno, A. Kobayashi and H. Kohno
Nagoya University, Japan
Outline1. Introduction
2. Honeycomb lattice model
3. Summary
Dirac electronsCoulomb interactionExcitonic order parameters(Previous works)
FormulationSelf-consistent equationResults
A. Geim, K. Novoselov Nature 438 (2005) 197.
P. Wallace, Physical Review 71: 622-634. (1947)
(Graphite)
Graphene
S. Katayama, A. Kobayashi, Y. Suzumura, JPSJ. 75 (2006) 054705
α-(BEDT-TTF)2I3
K. Kajita, et al., J. Phys. Soc. Jpn. 61, 23 (1992).
N. Tajima, et al., J. Phys.Soc. Jpn. 75, 051010 (2006).
Massless Dirac electron system in solid
3
0
~
kveffH
Effective Weyl
Hamiltonian
Weyl Hamiltonian
rrrrrr nnVH 0dd
q
eqV
0
2
0
2
sublattice-representation )( ,,,,
LkLkRkRkk BABA
k
ks
yxF
yxF
yxF
yxF
k
ikkv
ikkv
ikkv
ikkv
H
0)(00
)(000
000)(
00)(0
0
R
R
L
L
'0 HHH
L R
kKk R kKk L
yyxxk kkH LR ˆˆ,
,0
Normal metal
２Ｄ Dirac electron
22
2)(
TF
RPA
eqV
still diverse at q=0.
.)1(
2)(
constq
eqV RPA
Polarization function
2222
2
16,
qv
Nqq
q
e
qqV
qVqV
2
0
0screened 2
,1,
q
eqV
0
2
0
2
Coulomb interaction(2dimensional)
Electron correlation
DOS
Long-range Coulomb interactionis important in Dirac electron system.
ω
q B. Wunsch et al. New J. Phys. 8 318.
screening effects
Excitonic mass generation
Gorbar et.al. PRB 66, 045108 (2002)
D. V. Khveshchenko, et al., (2004) Nucl. Phys. B 687, 323
2Δ
F
e
v
k
)(
)/(ln k
2.2
2
v
e
p
p
p
p
kTk
E
EpkV
B2tanh
2
22
kFk kvE
)41(2
N
approximate solution by Gorbar
numerical calculation
Mean field order parameters sublattice:̂
valley:̂
spins :ˆ
D.V.Khveshchenko, J. Phys.: Condens. Matter 21 (2009) 075303
４2=16 degrees of freedomwhich contribute to mass generation
S. Raghu et al., Phys. Rev. Lett. 100 (2008) 156401
LR
kkkkkk BBAA,
03ˆˆ
LR
kkkkkk BBAAsign,
33 )(ˆˆ
..ˆˆ11 chABBA L
k
R
k
L
k
R
kkk
..ˆˆ21 chAiBBiA L
k
R
k
L
k
R
kkk
singlettriplet
Real Space Momentum Space
3,12
3,12
2
1
ae
ae
3
ac
5486.2a Å
21 eeR mnnm
ABt
Honeycomb lattice
0,0),( 21 kk
0,2 2,0
2,2
1st BZ
RKLK M
Q
3
2
3
4,
3
4
3
2 Q
sk sk
sk
k
k
skskB
A
z
zBAH
, ,
,*
,,00
0ˆ
)23exp()2cos(1 yxk kikz 1atAB )~~
(2
3yx kiks
Formulation
i
i
ji
jiij
ji
ijji nU
nnVcccctH 22
1
,,
iiiii ccccn
U
RR
e
V jiij
2ji
ji
Mean field theory
'0 HHH
k
kk
k
kkkk
k
kkMF BBBBAAAAUH'
''
'
''
'
qkk
kkqkqkkkqkqk BBBBAAAAqV,',
'')(
qkk
kkqkqkkkqkqk BAABABBAqV,',
'')(~
)0,0(),(22
)(2
',',
21
)(mn
mqnqi
ssss
mnmn
eeUqV
),(22
)(2
3/1~)(
~ 21
mn
mqnqi
nmmnmn
eeqV
)(', qVU lss
singlet・triplet
linearized self-consistent field equation
excitonic order parameters
q
qk
qk
qk
kT
qW2
tanh2
Self-consistent equation
qkvFqk
2..32
',,
41
',,',, ccsskssk
ssk
a
sqk
b
sqk
q
ab
ssk qWN
,',',, )(1
',',,
singlet
sssskk
',
',,',
triplet
,ˆ
ss
sskssiki s
22
',,
11
',,
0
',, sskssk
q
ssk
0
)(~
)()()(
qll qVqVUqVqW
UqVqW l )()(
2
)(~
)(~
)(qVqV
qW
q=0,singlet case
q=0,triplet case
q=Q case
Comes from neutrality condition.
Order parameters in momentum space
even odd
0q
Summary
We study about excitonic mass generation in Honeycomb lattice model.In continuum model, several excitonic orders are degenerated.We find that in lattice model, the degeneracy of excitonic orders is broken.For realistic parameter U=1.2t(graphene), BOW state is stable for α>0.8.
sublattice -representation band-representation
2]..)(~
)(~1
[,
~',
~,
~',
~',,
~
q
L
sqk
R
sqk
L
sqk
R
sqk
sskccABqVBAqV
N
q
L
sqk
R
sqk
L
sqk
R
sqkccabba
qVqV
N..
2
)(~
)(~
1,
~',
~,
~',
~
',',,,',',,,ˆˆˆˆˆˆ
sk
band
ssksksk
MF
ssksk HUUHUU L
sk
L
sk
R
sk
R
sksk abba,
~,
~,
~,
~,
conductionak
:,
~
22
22
2
1
ii
ii
R
ee
eeU
22
22
2
1
ii
ii
L
ee
eeU
k
k
k
k
H band
k
~000
0~
00
00~
0
000~
2
3)0(
band-representation
valencebk
:,
~
BOW(Honeycomb)
BOW in Honeycomb lattice correspond to the Kekulé distortion.
C. Y. Hou et al., Phys. Rev. Lett. 98 186809(2007)
Topological aspect of the quantum mechanics
nnn iA
nn AB
c. c.i2
nm mn
kkz
nEE
nHmmHnB
yx
,2,1,0d2
1
BZ
2 z
nn kB
Berry接続
XASASB dddCSS
Berry曲率
Chern number (2D)
n: band index
S
C
A1
Berry phase
)(kB
L R
0,0 RL
0,0 RL
1
0
1
0,0 RL
D. J. Thouless et al. PRL 49 (1982) 405
ホール伝導率
h
ekAkf
kdezkxy
2
2
2
)](ˆ[)()2(
TKNN公式
LR
02
h
exy 量子異常ホール状態（QAH）
S. Raghu, X. L. Qi, C. Honerkamp, and S. C. Zhang, PRL 100, 156401 (2008).
F. D. M. Haldane, PRL 61, 2015 (1988).
02
h
es
xy
)(kB
量子スピンホール状態（QSH）
zzyyyxxx kkH
,, yx kkXParameter space
The simplest Hamiltonian with non-zero Chern number
1,, zyx
Eigen energy XE
Eigen function
X
kk
XX z
yyxx
z
i
2
1
X
kk
XX z
yyxx
z
i
2
1
xxk Hx
yyk Hy
32
2c. c.i
XEE
HHB zyx
kk yx
ψψψψ
using
2
zyxCh
“Magnetic monopole”Lower (filled) band
A2
xk
yk
2
10 Ch
2
10 Ch
00
32X
XB ,, yx kkX
“magnetic monopole” and discontinuity of Chern number
12
4
2
1d
2
12
2
S
00 X
XChCh
SB
xk
yk
S
BZ
2d2
1nn BkCh
A3
Topological insulator with TRS (time-reversal symmetry)
= a combined system of two subsystems with opposite sign Chern numbers
KB
KA
KB
KA
KB
KA
KB
KA
vk
vk
vk
vk
vk
vk
vk
vk
H
000000
000000
000000
000000
000000
000000
000000
000000
Kane-Mele model (graphene + spin-orbit interaction) PRL 95, 226801 (2005)
zzzyyzxx skkvH
yx kkk i
,,,, zyx
Ch= 1/2
Ch= 1/2
Ch= -1/2
Ch= -1/2
14
i 2
kkkkkkk uuuukdSpCh
Topological insulator with TRS = Quantum spin Hall system
A4
Spin Chern number:
,,,, zyx
c. c.i2
nm mn
kk
nEE
nHmmHnB
yx
Direct calculation of Berry’s phase gauge field
using the Hamiltonian for alpha-(BEDT-TTF)2I3
c. c.i41 ~,
2
44332211
nm mn
kk
EE
nHmmHnyx
c. c.i41
44332211
~,2
**
nm mn
nkmmkn
EE
dHddHdyx
kkkkkk
1m,2n
4,3,2,1,,, 4321
k : in BZ
11
Fictitious “magnetic field” given by Berry’s phase gauge field
excitonic mass generation in graphene
Monte Carlo calculations:Hands, S., and C. Strouthos, 2008, PRB 78, 165423.Drut et al.,2009, PRB 79, 241405(R); RPL 102, 026802; PRB 79, 165425. 1.1c
Polarization function with Δ(k) and anomalous Green functionJ-R. Wang et al., J. Phys.: Condens. Matter 23 (2011) 155602
q
GG FF 01.0
0
vΔ増大 α=32では
Static RPA : Khveshchenko, D. V., and H. Leal, 2004, Nucl. Phys. B 687, 323
Dynamical RPA on-shell: Gamayun, O. V., E. V. Gorbar, and V. P. Gusynin, 2010, Phys. Rev. B 81, 075429.
vpp ,92.0c
13.1cStrong k-dependence of k
cccT
~1ln
41
~
Nmev4B cTk
001.0
0
v
1
kvkb Effect of renormalized velocity:Khveshchenko, D. V., 2009, J. Phys.: Condens. Matter 21, 075303.
面直磁場によりαc減少、Δ増大
F. Amet, J. R. Williams, K. Watanabe, T. Taniguchi, and D. Goldhaber-GordonPRL 110, 216601 (2013)
グラフェンにおける電気抵抗
シリコン基板上においたグラフェンの電気抵抗(右図インセット)
２次元ディラック電子系に普遍的な振る舞い？