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Interaction effect on topological insulators Studies based on interacting Green’s functions
Lei Wang Institute of Physics, Chinese Academy of Sciences
2011.8.6
Xincheng Xie, ICQM, Peking University
Xi Dai, IOP-CAS
Shun-Li Yu & Jianxin Li, Nanjing University
Hao Shi & Shiwei Zhang, College of William & Mary
Acknowledgment
Shun-Li Yu, X. C. Xie and Jianxin Li, PRL, 107, 010401 (2011)Lei Wang, Hao Shi, Shiwei Zhang, Xiaoqun Wang, Xi Dai and X. C. Xie, arXiv:1012.5163Lei Wang, Xi Dai and X. C. Xie, arXiv:1107.4403Lei Wang, Hua Jiang, Xi Dai and X. C. Xie, in preparation
OutlineInteraction effect on topological insulators
Solve interacting models
Characterize interacting topological phases
Characterization based on interacting Green’s functions: Local self-energy approximation
Frequency domain winding number
(optional) Pole-expansion of self-energies
Discussions
Kane-Mele-Hubbard model
H1 = UX
i
(ni" �1
2)(ni# �
1
2)
H = H0 +H1
Preserve PH & TR symmetry
only t1: Graphene model
t1 and t2: Kane-Mele QSHE
t1 and U: Semimetal, spin liquid and AFI from small to large U
H0 = �t1X
hi,ji,�
c†i�cj� + it2X
hhi,jii,�
�⌫ijc†i�cj�
Phase diagrams
Hohenadler, Lang and Assaad, PRL, (2011)Zheng,Wu and Zhang, Arxiv, (2010)Yu, Xie and Li, PRL, (2011)
Interacting Haldane model
Spinless: single copy of Kane-Mele model
QHE without Landau levels
Add staggered onsite energies: topological transition, Chern # changes, gap closes at phase boundary
F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)
H0 = �t1X
hi,ji
c†i cj + it2X
hhi,jii
⌫ijc†i cj
H1 = VX
hi,ji
(ni �1
2)(nj �
1
2)
H = H0 +H1
Mean-field phase diagram
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.00
1
2
3
4
5
V
HFGL∆
0 1 2 3 4 5 6 70.0
0.2
0.4
0.6
0.8
1.0
V
t 2
CDWTopological
Interaction will generate the mass term and break the topological phase
V ninj ! V hniinj + V nihnji
! V �(ni � nj)
hnii =12
+ (�1)⌘i�
t2 = 0.5
Chern number and Gap (ED)C =
i
2⇡
Z Zd✓
x
d✓y
[h @ @✓
x
| @ @✓
y
i � h @ @✓
y
| @ @✓
x
i]
Topological transition point decreases with decreasing t2
CDW transition remains finite with vanishing t2
Topological transition occurs before CDW transition for small t2
Gap close links to the topological transition
CDW structure factor and Gap (CPQMC)
t2 = 0.1
t2 = 1.0
CDW structure factor and Gap (CPQMC)
t2 = 0.1
t2 = 1.0
Many-body phase diagram
Studied the interplay of topological order and CDW long range order
Mean-field picture: large CDW order destroys the topological phase
Mean-body calculations: topological transition could occur before the CDW one
Summary on numerical findings
Both uncover gapped featureless phase
NO direct characterization: correlation functions, excitation gaps, edge currents, spectral functions ...
Characterization beyond single particle basis:
Chern/Z2 # with twisted boundary conditions
Entanglement entropy/spectrum
Topological indices expressed by Green’s function
Ishikawa formula for 4D QHE
Appears as the coefficient of the Chern-Simons term. Describes Hall conductance physically.
Mathematically:
Single particle Green’s function involved, taken into account the effect of interaction, disorder as well as finite temperature.
Ishikawa and Matsuyama, Z Phys C Part Fields, 33, 41 (1986),Ishikawa and Matsuyama, Nucl Phys B 280, 523 (1987) ,Qi, Hughes, and Zhang, PRB, 78, 195424 (2008)
n =⇤µ�⇥⇤⌅15�2
Tr
Zd4kd⇥
(2�)5G⌅µG
�1G⌅�G�1G⌅⇥G
�1G⌅⇤G�1G⌅⌅G
�1
�5(GL(M,C)) = Z
Noninteracting Case
G�1(�,k) = i� �Hk �!G�1 = i
4D TKNN formula
�kiG�1 = ��kiHk
n =⇤µ�⇥⇤⌅15�2
Tr
Zd4kd⇥
(2�)5G⌅µG
�1G⌅�G�1G⌅⇥G
�1G⌅⇤G�1G⌅⌅G
�1
Integrated ω
Hk =5X
a=1
hak�
a hak � ha
k/|hk|
n =3
8⇡2
Z
BZd4k "abcdeh
ak@k
x
hbk@k
y
hck@k
z
hdk@k
�
hek
Interacting Case: Mean-field self-energy
Raghu, Qi, Honerkamp, and Zhang, PRL (2008)Dzero, Sun, Galitski, and Coleman, PRL (2010)
G�1(�,k) = i� �Hk � �(�,k)
Li, Chu, Jain, and Shen, PRL (2009)Groth, Wimmer, Akhmerov, and Beenakker, PRL (2009)
Interacting Case: Mean-field self-energy
Raghu, Qi, Honerkamp, and Zhang, PRL (2008)Dzero, Sun, Galitski, and Coleman, PRL (2010)
G�1(�,k) = i� �Hk � �(�,k)= i! � Hk
Li, Chu, Jain, and Shen, PRL (2009)Groth, Wimmer, Akhmerov, and Beenakker, PRL (2009)
×
Interacting Case: Mean-field self-energy
Topological “Mott” insulator: mean-field bond-order wave states
Topological “Kondo” insulator: renormalized-hybridization Hamiltonian
Topological “Anderson” insulator: self-consistent Born approximation
Raghu, Qi, Honerkamp, and Zhang, PRL (2008)Dzero, Sun, Galitski, and Coleman, PRL (2010)
G�1(�,k) = i� �Hk � �(�,k)= i! � Hk
Li, Chu, Jain, and Shen, PRL (2009)Groth, Wimmer, Akhmerov, and Beenakker, PRL (2009)
×
Interacting Case: Local self-energy approximation
G�1(�,k) = i� �Hk � �(�,k) ⇡ i� �Hk � �(�)
�(�,k) ⇡ �(�)
Interacting Case: Local self-energy approximation
G�1(�,k) = i� �Hk � �(�,k) ⇡ i� �Hk � �(�)
�(�,k) ⇡ �(�)
⌘ G�1
atom
(�)�Hk
Interacting Case: Local self-energy approximation
G�1(�,k) = i� �Hk � �(�,k) ⇡ i� �Hk � �(�)
�(�,k) ⇡ �(�)
⌘ G�1
atom
(�)�Hk
Set to QH state with TKNN number equals to 1
Set several trial forms of the self-energy
Finish the frequency-momentum integration, see what happens to n.
Self-energy vs. Chern number
⌃(!) n
0 1
1
i!0
1
(i!)32
1
i! � 1+
1
i! + 11
Hk
⌃(!) n =G�1
atom
(!) = ! �=⌃(!) =G�1
atom
(!)
0 1 !
1
i!0 ! +
1
!
1
(i!)32 ! � 1
!3
1
i! � 1+
1
i! + 11 ! +
2!
!2 + 1
Conjecture
Stereographic projection
deg{⇥G�1
atom
}�TKNN
=G�1
atom
!R
S1
Interacting Chern number =
=G�1
atom
= � �=�(�) : R 7! R
Proof
�kiG�1 = ��kiHk
n =⇤µ�⇥⇤⌅15�2
Tr
Zd4kd⇥
(2�)5G⌅µG
�1G⌅�G�1G⌅⇥G
�1G⌅⇤G�1G⌅⌅G
�1
Separate integrations
G�1 = G�1
atom
� hak�
a @!G�1 = @!G
�1
atom
Hk =5X
a=1
hak�
a hak � ha
k/|hk|
n =2
⇡2
Z
BZ
d4k
Zd!
2⇡i
@!G�1
atom
(G�2
atom
� |hk|2)3"abcdeh
ak@kx
hbk@ky
hck@kz
hdk@k
�
hek
Proof (cont.)Z 1
�1
d!
2⇡i
@!G�1
atom
(G�2
atom
� |hk|2)3=
Z z(1)
z(�1)
dz
2⇡i
1
(z2 � |hk|2)3
ââââââââââââââââââ ReGatom-1
ImGatom-1
Proof (cont.)Z 1
�1
d!
2⇡i
@!G�1
atom
(G�2
atom
� |hk|2)3=
Z z(1)
z(�1)
dz
2⇡i
1
(z2 � |hk|2)3
=
I
C
dz
2⇡i
1
(z2 � |hk|2)3
ââââââââââââââââââ ReGatom-1
ImGatom-1
Proof (cont.)
γ = # left - # right
Z 1
�1
d!
2⇡i
@!G�1
atom
(G�2
atom
� |hk|2)3=
Z z(1)
z(�1)
dz
2⇡i
1
(z2 � |hk|2)3
=
I
C
dz
2⇡i
1
(z2 � |hk|2)3
=3
16|hk|5�
ââââââââââââââââââ ReGatom-1
ImGatom-1
Proof (cont.)
γ = # left - # right
Z 1
�1
d!
2⇡i
@!G�1
atom
(G�2
atom
� |hk|2)3=
Z z(1)
z(�1)
dz
2⇡i
1
(z2 � |hk|2)3
=
I
C
dz
2⇡i
1
(z2 � |hk|2)3
=3
16|hk|5�
ââââââââââââââââââ ReGatom-1
ImGatom-1
Proof (cont.)
Interacting Chern number = FDWN⇥TKNN
n =2
⇡2
Z
BZ
d4k
Zd!
2⇡i
@!G�1
atom
(G�2
atom
� |hk|2)3"abcdeh
ak@kx
hbk@ky
hck@kz
hdk@k
�
hek
n = �⇥ 3
8⇡2
Z
BZd4k"abcdeh
ak@k
x
hbk@k
y
hck@k
z
hdk@k
�
hek
3D TI: dimension reduction
Interacting Z2 = FDWN⇥Z2
n ="µ⌫⇢�⌧15⇡2
Tr
Z
BZ
d3k
(2⇡)3
Z 1
0
dk�2⇡
Z 1
�1
d!
2⇡G@µG
�1G@⌫G�1 . . .
= �⇥ 1
8⇡2
Z
BZd3k
2|hk|+ h4k
(|hk|+ h4k)
2|hk|3"abcdh
ak@k
x
hbk@k
y
hck@k
z
hdk
Qi, Hughes, Zhang, PRB (2008), Wang, Qi, Zhang, PRL (2010), Li, Wang, Qi, Zhang, NatPhys (2010)
Hk + k��4 ! Hk
3D TI: dimension reduction
Interacting Z2 = FDWN⇥Z2
n ="µ⌫⇢�⌧15⇡2
Tr
Z
BZ
d3k
(2⇡)3
Z 1
0
dk�2⇡
Z 1
�1
d!
2⇡G@µG
�1G@⌫G�1 . . .
= �⇥ 1
8⇡2
Z
BZd3k
2|hk|+ h4k
(|hk|+ h4k)
2|hk|3"abcdh
ak@k
x
hbk@k
y
hck@k
z
hdk
Qi, Hughes, Zhang, PRB (2008), Wang, Qi, Zhang, PRL (2010), Li, Wang, Qi, Zhang, NatPhys (2010)
Hk + k��4 ! Hk
⌃(!) =G�1
atom
(!) ! 7! G�1
atom
(!)
0 1
=G
�1
atom
<G�1
atom
1
i!0
<G�1
atom
=G
�1
atom
1
(i!)32
<G�1
atom
=G
�1
atom
FDWN
⌃(!) =G�1
atom
(!) ! 7! G�1
atom
(!)
1
i! � 1+
1
i! + 11
=G
�1
atom
<G�1
atom
1
i!0
<G�1
atom
=G
�1
atom
1
(i!)32
<G�1
atom
=G
�1
atom
FDWN
⌃(!) =G�1
atom
(!) ! 7! G�1
atom
(!)
1
i! � 1+
1
i! + 11
=G
�1
atom
<G�1
atom
1
i!0
<G�1
atom
=G
�1
atom
1
(i!)32
<G�1
atom
=G
�1
atom
FDWN
Assumptions
(1). Momentum-independent self-energy
(2). Diagonal and orbital-independent self-energy
(3). Hamiltonian expanded by Gamma matrix
Assumptions
(1). Momentum-independent self-energy
(2). Diagonal and orbital-independent self-energy
(3). Hamiltonian expanded by Gamma matrix
Qi, Hughes, and Zhang, PRB, 78, 195424 (2008)
Band flatten
Assumptions
(1). Momentum-independent self-energy
(2). Diagonal and orbital-independent self-energy
(3). Hamiltonian expanded by Gamma matrix
Savrasov, Haule and Kotliar, PRL, 96, 036406 (2006)
Pole-expansion
Pole expansion and Pseudo-Hamiltonian
⌃(!) = ⌃(1) + V †(i! � P )�1V
G =(i! � Hk)�1
=
✓i! �Hk � ⌃(1) �V †
�V i! � P
◆�1
=
[i! �Hk � ⌃(1)� V †(i! � P )�1V ]�1 . . .
.... . .
!
Hk =
✓Hk + ⌃(1) V †
V P
◆
Pole expansion and Pseudo-Hamiltonian
⌃(!) = ⌃(1) + V †(i! � P )�1V
G =(i! � Hk)�1
=
✓i! �Hk � ⌃(1) �V †
�V i! � P
◆�1
=
[i! �Hk � ⌃(1)� V †(i! � P )�1V ]�1 . . .
.... . .
!
Hk =
✓Hk + ⌃(1) V †
V P
◆
Pole expansion and Pseudo-Hamiltonian
= G
⌃(!) = ⌃(1) + V †(i! � P )�1V
G =(i! � Hk)�1
=
✓i! �Hk � ⌃(1) �V †
�V i! � P
◆�1
=
[i! �Hk � ⌃(1)� V †(i! � P )�1V ]�1 . . .
.... . .
!
Hk =
✓Hk + ⌃(1) V †
V P
◆
Links interaction to non-interaction
TrG@!G�1G@k
x
G�1G@ky
G�1G@kz
G�1G@k�
G�1
= TrG@!G�1G@k
x
G�1G@ky
G�1G@kz
G�1G@k�
G�1
G = P†GP
Links interaction to non-interaction
n[G] = n[G] = n[Hk]
TrG@!G�1G@k
x
G�1G@ky
G�1G@kz
G�1G@k�
G�1
= TrG@!G�1G@k
x
G�1G@ky
G�1G@kz
G�1G@k�
G�1
G = P†GP
Toy model: 3D TI
-2 0 2 4 6 8 m
1
2Z2 and Gap
Hk = sin kx
�1 + sin ky
�2 + sin kz
�3 +M(k)�5
M(k) = m� 3 + (cos kx
+ cos ky
+ cos kz
)
Hk+ Interaction ?
-G -D D Gw
DOS
FDWN and Z2 calculation
ââââââââââââââââââ ReGatom-1
ImGatom-1
-2 -1 1 P
-3
-2
-1
1
2
V
⌃(!) =V 2
i! � P
|P |� = V 2
|P |� = V 2
Hk =
✓Hk V I4V I4 P I4
◆! 7! G�1
atom
(!) = i! � ⌃(!)
� �
Gap closing of pseudo-Hamiltonian
-6 -4 -2 2 4 P
-6-4-2
24
!✓
EkI4 ⌦ �z V I4V I4 P I4
◆Hk =
✓Hk V I4V I4 P I4
◆
EkP = V 2
! (±Ek) + P
2±
r[(±Ek)� P
2]2 + V 2
Bonus: surface states
n[G] = n[G] = n[Hk]
TrG@!G�1G@k
x
G�1G@ky
G�1G@kz
G�1G@k�
G�1
= TrG@!G�1G@k
x
G�1G@ky
G�1G@kz
G�1G@k�
G�1
G = P†GP
Bonus: surface states
n[G] = n[G] = n[Hk]
TrG@!G�1G@k
x
G�1G@ky
G�1G@kz
G�1G@k�
G�1
= TrG@!G�1G@k
x
G�1G@ky
G�1G@kz
G�1G@k�
G�1
Gsurf = P†GsurfPG = P†GP
Practical calculations
Three and four dims: DMFT calculations
Only overall shape matters FDWN: simple IPT solver
Only Matsubara Green’s function needed: exact CTQMC solver
Practical calculations (cont.)
Two dim cases:
Spatial fluctuations: k dependence of self-energy
FLEX, DCA and diagrammatic MC
Numerical integration: VEGAS algorithm-by Dr. Peter Lepage has been used for multi-dimensional quadrature in high energy physics for more than 30 years.
SummarySelf-energy contains the topological signature of interacting TI
Local self-energy approximation:Frequency domain winding number
Pole expansion of self-energy
Breaking down the topological phases without developing LRO: relevant to “spin liquid” phase?
It's interesting to see the synergies of many-body numerical methods with the study of interacting topological insulators though the calculation of their Green's functions