interaction effect on topological insulators · 2020. 2. 13. · single particle green’s function...

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)



G�1(�,k) = i� �Hk � �(�,k)= i! � Hk


×


Topological “Mott” insulator: mean-field bond-order wave states

Topological “Kondo” insulator: renormalized-hybridization Hamiltonian

Topological “Anderson” insulator: self-consistent Born approximation


G�1(�,k) = i� �Hk � �(�,k)= i! � Hk


×

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

Metal-Insulator transition

⌃(!) ⇠ 1

i!

⌃(!) ⇠ 1

i! � P+

1

i! + P

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


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�


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

⌃(!) =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

Are they related ?

1

i!0

<G�1

atom

=G

�1

atom

Assumptions

(1). Momentum-independent self-energy

(2). Diagonal and orbital-independent self-energy

(3). Hamiltonian expanded by Gamma matrix

Assumptions




Qi, Hughes, and Zhang, PRB, 78, 195424 (2008)

Band flatten

Assumptions




Savrasov, Haule and Kotliar, PRL, 96, 036406 (2006)

Pole-expansion

Pole expansion and Pseudo-Hamiltonian

⌃(!) = ⌃(1) + V †(i! � P )�1V


⌃(!) = ⌃(1) + V †(i! � P )�1V

Hk =

✓Hk + ⌃(1) V †

V P

◆


⌃(!) = ⌃(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

◆


= 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

G = P†GP


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


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


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

Surface spectral functions⌃(!) =

V 2

i! � P

P = 0.15 P = 2

V = 1

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

Thank you for your attention: )

interaction effect on topological insulators · 2020. 2. 13. · single particle green’s function...

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