[PPT] - TKNN formula for general Hamiltonian D. J. Thouless, M. Kohmoto, M. PowerPoint Presentation

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TKNN formula for general Hamiltonian

with H. Fukaya, S. Yamaguchi (Osaka U), X. Wu (Ariel U) April 22, 2019, FLQCD 2019 @ YITP

2019/4/22 1

T. Onogi (Osaka Univ.)

arXiv:1903.11852

D. J. Thouless, M. Kohmoto, M. P. NighYngale, and M. den Nijs

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Cau$on

Change of the topic Previous speakers (Fukaya, Furuta)

Anomaly in four dimensions with boundary

This talk

Topological insulator in odd dimensions without boundary (D=2+1, D=4+1)

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1. Introduction

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Topological insulator

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Ø Close rela7onship to domain-wall New knowledge of topological ma@er è new hints to laAce fermions by Domain-wall fermion

example: Gapped symmetric phase by 4-fermi interac7on (Talk by Kikukawa)

Figure from Tokura et al. Nature Reviews Physics vol 1, 126 (2019)

Ø Interesting physics from non-trivial topology Bulk: insulator Surface: metal

Topology guarantees edge modes (Bulk-Edge correspondence)

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Two approaches to topology

ØMicroscopic approach:

Study the wavefunc9on of the free electron Hamiltonian Applied to various different free systems ( higher dim, higher symmetry) Classifica9on of topology is highly developed Looks rather technical (at least to me) Applicable only to free fermion systems

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Ø Field theory approach:

Introduce gauge field and study the effec9ve ac9on Conceptually simple Applicable also to interac9ng fermion systems

Seff(A)

TKNN formula (D. J. Thouless, M. Kohmoto, M. P. Nigh9ngale, and M. den Nijs)

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What characterizes topology?

ØMicroscopic approach:

Topology of Berry connection of single particle wavefunctions

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Ø Field theory approach:

Top # = Chern-Simons level of the 3-dim effecLve gauge acLon

K. Ishikawa : ConducLvity ç è Top. #

X

n

Z d2p c1(A(n))

A(n)

µ

⌘ ihn, p| ∂ ∂pµ |n, pi

Top. # =

TKNN formula: ConducLvity çè Top. #

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Ques%on

Two topological characterizations are identical? In some specific cases, yes. How generally identical and why ?

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We try to answer this ques%on in this work.

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Outline

1. Introduc/on
2. Review of TKNN formula
3. Review of field theory approach
4. Equivalence for general Hamiltonian
1. Chern-Simons level à Winding number
2. Winding number à TKNN formula
5. Summary

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2. Review of TKNN formula

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Anomalous Hall effect

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Hall current perpendicular to Electric field

hjxiE = σxyEy

Hall conduc;vity can be expressed by topological quan;ty using 1) Kubo formula from perturba;on theory 2) Formulae in quantum mechanics

2+1 dim system with Parity Violation

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|ni : eigenstate in free theory |niE : perturbed state

Electron states under electric field (perturba9on theory) hjxiE ⌘ X

n,En<0

hn|E evx L2 |niE

|niE = |ni + X

m6=n

|mihm|eEyy|ni En Em

Hall current under the electric field Kubo formula

l Transla9onal invariance: l Heisenberg equa9on:

n ⇒ (a, ~ p)

a : band label, ~ p : bloch momentum

[y, H] = ivy

where we have used

xy = ie2 L2 X

~ p

X

a

X

b6=a

✏ij ha, ~ p|vi|b, ~ pihb, ~ p|vj|a, ~ pi (Ea(~ p) Eb(~ p))2

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Derivation of Useful formula from

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vi = @ @pi H(~ p) H(~ p)|a, ~ pi = Ea(~ p)|a, ~ pi ha, ~ p|b, ~ pi = 0 (a 6= b)

ha, ~ p|vi|b, ~ pi = (Ea(~ p) Eb(~ p)ha, ~ p| @ @pi |b, ~ pi (a 6= b)

Combining with Kubo formula and defining

xy = e2 L2 X

~ p

X

a

✏ij @ @pi A(a)

j (~

p) = e2 2⇡ Z d2p 2⇡ X

a

✏ij @ @pi A(a)

j (~

p)

TKNN formula

A(a)

i

(~ p) ⌘ iha, ~ p| @ @pi |a, ~ pi

Berry connecDon

Chern number c_1 !

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Outline

1. Introduction
2. Review of TKNN formula
3. Review of field theory approach
4. Equivalence for general Hamiltonian
1. Chern-Simons level à Winding number
2. Winding number à TKNN formula
5. Summary

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3. Review of field theory approach

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Effective gauge action

Parity anomaly S. Deser, R. Jackiw, S. Templeton 1982, N. Redlich 1984

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IntegraJng out massive fermions in 3-dimensions

Seff(A) ≡ ln Z DψD ¯ ψe−

R ¯ ψ(D+m)ψ

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Seff(A) = iccsScs(A) + · · ·

Scs(A) ≡ Z d3x ✏µνλAµ@νAλ

ccs = − 1 8π m |m|

Parity violaJon of fermion induces Chern-Simons acJon

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Anomalous Hall conduc-vity from Chern-Simons ac-on

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Seff(A) = iccs Z d3x ✏µνλAµ@νAλ

hjii ⌘ @ @Ai Seff(A) = 2ccs✏iνλ@νAλ = 2ccs✏ijEj

Hall conduc-vity is given by the Chern-Simons coupling c_cs

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Winding number expression of Chern-Simons coupling

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Assuming mul?-photon vertex does not contribute

True for con?nuum theory and Wilson fermion on the laEce

ccs

can be obtained from 2-point func?on with fermion 1-loop

ccs = ✏α0β1α1 2 · 3! ✓ @ @q1 ◆

β1

Z d3 p (2⇡)3 Tr h S(p)Γ(1)

α0 [q1; p − q1]S(p − q1)Γ(1) α1 (−q1; p)

i

Assuming deriva?ve of vertex func?on does not contribute

True for con?nuum theory and Wilson fermion on the laEce

Γ(1)

µ [q; p]: fermion-fermion-photon vertex

p: incoming fermion momentum, q: incoming photon momentum

ccs = −✏α0β1α1 2 · 3! Z d3 p (2⇡)3 Tr  S(p)Γ(1)

α0 [0; p]@S(p)

@pβ1 Γ(1)

α1 (0; p)

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Using Ward-Takahashi identity

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Γ(1)

µ [0; p] = −i∂S−1(p)

∂pµ

ccs = −✏α0β1α1 2 · 3! Z d3 p (2⇡)3 Tr ⇥ S(p)@α0S−1(p)S(p)@β1S−1(p)S(p)@α1S−1(p) ⇤

ØTopological in S(p) has no singularity (true for gapped system) ØWinding number of a map T 3 → S3 for Wilson fermion

“Winding number” expression of Chern-Simons coupling

K. Ishikawa 1984

Golterman, Jansen, Kaplan 1993

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Outline

1. Introduc/on
2. Review of TKNN formula
3. Review of field theory approach
4. Equivalence for general Hamiltonian
1. Chern-Simons level à Winding number
2. Winding number à TKNN formula
5. Summary

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4. Equivalence for general Hamiltonian

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Fukaya, T.O., Yamaguchi, Xi arXiv:1903.11852

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Gapped fermion system in D=2n+1 dimensions.

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Fermions on 2n dim la<ce with con@nuous @me in Euclidean space

ψ, ψ†

H( ~ A)

~

A=0

: transla@onal inv. è band structure : can have many internal DOF è Many bands

SE = Z dt X

~ r

†(t,~ r)  @ @t + iA0 + H( ~ A)

(t,~

r)

No par@cular structure is assumed such as rela@vis@c fermion, or Wilson fermion, …..

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Energy eigenstates

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Nv Valence bands Nc Conduc<on bands Δ: Gap

for fixed ~ p

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Effec%ve gauge ac%on

We consider effec%ve ac%on a2er integra%ng out fermion

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eSeff(A) = Z DψDψ†e−SE

Seff(A) = X

k

akSk(A)

Fermion system is gapped è expanded as the sum of gauge inv. local actions

ak Sk

: coefficients : Gauge inv ac%ons

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Chern-Simons ac/on

Topological ac/on è defined for any geometry of the la9ce

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Scs(A)

Scs(A) = Z d2n+1x✏α0β1α1···βnαnAα0@β1Aα1 · · · @βnAαn

Seff(A) = · · · + iccsScs(A) + · · ·

ccs = k (2π)n(n + 1)!, k ∈ Z

Coefficient is quan/zed due to gauge invariance We will see Chern-Simons level k is a toplogical invariant. k = Chern-Simons level

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ccs =(−i)n+1✏α0β1α1···βnαn (n + 1)!(2n + 1)! ✓ @ @q1 ◆

β1

· · · ✓ @ @qn ◆

βn

×

n

Y

i=1

Z d2n+1xieiqixi n+1Seff(A) Aα0(x0)Aα1(x1) · · · Aαn(xn)

A=0,qi=0

can be obtained by differen5a5ng the effec5ve ac5on as

ccs

Fermion 1-loop diagram with n+1 external photons

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4-1 Chern-Simons level à Winding number

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For general Hamiltonian, Feynman rule can have fermion-fermion-multiphoton vertices

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è 1-loop n-point func?on from several diagrams in general.

q1

p

Γ(n)[qN, αN; · · · ; q1, α1; p]

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D=2+1 case

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ccs = −(−i)2✏α0β1α1 2!3! Z d3p (2⇡)3 ✓ @ @q1 ◆

β1

n Tr h SF (p)Γ(2)[−q1, ↵0; q1, ↵1; p] i + Tr h SF (p − q1)Γ(1)[−q1, ↵0; p]SF (p)Γ(1)[q1, ↵1; p − q1] io

q1=0

✓ ∂ ∂q1 ◆

β1

q1

−q1

p

q1

−q1

p

p − q1

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Two new contribu-ons in general case

1. Mul--photon vertex contribu-on à non-zero 2. Momentum deriva-ve of the vertex func-on à non-zero

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These new contribu-ons can give correc-ons to the winding number expression. However, one can show that they cancel due to new Ward-Takahashi iden--es

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New Ward-Takahashi iden00es

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Gauge invariant la>ce ac0on can be formally expanded by infinite series of covariant deriva0ves. Example:

†(t, ~ x)ei

R ~

x+a~ µ ~ x

d~ r0· ~ A(~ r0) (t, ~

x + a~ µ) = †(t, ~ x)

∞

X

n=0

an n!

Dn

µ

(t, ~

x)

Therefore, formally action can be expressed as

S = Z dt X

~ x ∞

X

n=0

†(t, ~ x)Mµ1···µn(Dµ1 · · · Dµn )(t, ~ x)

Same coefficient M appear in propagator and vertices

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Formal expansions of propagator and ver3ces

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Using the coefficients M,

S−1

F (p) = ∞

X

n=0

Mµ1···µn

n

Y

i=1

(ipµi)

Γ(1)[k, µ; p] = −i

∞

X

n=1 n

X

a=1

Mµ1···µa−1µµa+1···µn

a−1

Y

i=1

(i(p + k)µi)

n

Y

i=a+1

(ipµi)

Γ(2)[k, µ; l, ν; p] = −i2

∞

X

n=1 n

X

a,b=1

a<b

Mµ1···µa−1µµa+1···µb−1νµb+1···µn

a−1

Y

i=1

(i(p + k + l)µi)

b−1

Y

i=a+1

(i(p + l)µi)

n

Y

i=b+1

(ipµi) − i2

∞

X

n=1 n

X

a,b=1

a<b

Mµ1···µa−1νµa+1···µb−1µµb+1···µn

a−1

Y

i=1

(i(p + k + l)µi)

b−1

Y

i=a+1

(i(p + k)µi)

n

Y

i=b+1

(ipµi)

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New Ward-Takahashi iden0ty

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∂2Γ(1)[k, µ; p] ∂kν∂pλ

k=0

= ∂Γ(2)[k, µ; 0, λ; p] ∂kν

k=0

= ∂Γ(2)[0, λ; l, µ; p] ∂lν

l=0

1st deriva0ve of the two-photon vertex with respect to momentum is related to 2nd deriva0ve of the single photon vertex. The formal expression reproduces usual Ward-Takahashi iden00es. In addi0on, one also obtains the following 2nd order W-T iden0ty Correc0on terms to 1-loop expression is shown to be total deriva0ves and vanish.

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Therefore, for general Hamiltonian we obtain

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ccs = (−i)2✏α0β1α1 2!3! Z dp0 2⇡ Z

BZ

d2p (2⇡)2 × Tr  SF (p)@S−1

F (p)

@pα0 SF (p)@S−1

F (p)

@pβ1 SF (p)@S−1

F (p)

@pα1

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D=4+1 case

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Chern-Simons coupling is obtained from 3-point 1-loop diagrams. For general Hamiltonian, mulE-photon couplings can contribute and momentum derivaEve of vertex funcEons do not vanish.

∂2Γ(3)[q, µ; r, ν; s, λ; p] ∂qα∂rβ

q,r,s=0

= ∂3Γ(2)[q, µ; r, ν; p] ∂qα∂rβ∂pλ

q,r=0

However, one can derive new 3rd order WT-idenEty from formal expansion of the acEon in terms of covariant derivaEves as

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One can show correc4on terms cancel and c_cs is given by winding number expressions as Using previous 2nd order WT-iden4ty and new 3rd order WT-iden4ty

ccs = −(−i)3 · 2 3!5! Z d5p (2⇡)5 ✏α0β1α1β2α2 × Tr  SF (p)@S−1

F (p)

@pα0 SF (p)@S−1

F (p)

@pβ1 SF (p)@S−1

F (p)

@pα1 SF (p)@S−1

F (p)

@pβ2 SF (p)@S−1

F (p)

@pα2

.

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4-2 Winding number à TKNN formula

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This part was essen-ally already given by

Qi, Hughes, Zhang , Phys. Rev. B78, 195424, 2008

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Idea : Evaluate the winding number expression as follows

1. Rewrite the fermion propagator using eigenstates

S(p) = X

α

|↵, ~ pi 1 ip0 + Eα(~ p)h↵, ~ p|

2. Con-nuously deform only the eigenvalues to degenerate flat band
3. Carry out momentum integral over p^0

Eα(~ p)(< 0) − → Ev = constant Eα(~ p)(> 0) − → Ec = constant

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Step 1

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ccs = −n! · (2n + 1)(−i)n+1✏i1i2···i2n (n + 1)!(2n + 1)! Z d2np (2⇡)2n Z dp0 2⇡ Tr " 1 ip0 + H i

2n

Y

k=1

✓ 1 ip0 + H (@ikH) ◆#

Winding number expression can be rewri=en as Inser?ng complete set of energies eigenstates, one obtains

ccs = n!(−i)n+2 (n + 1)!(2n)! Z d2np (2π)2n J

J = X

α1,··· ,α2n

✏i1i2···i2n Z dp0 2⇡ h↵1@i1H|↵2ih↵2|@i2H|↵3i · · · h↵2n|@lH|↵1i (ip0 + Eα1)2(ip0 + Eα2) · · · (ip0 + Eα2n)

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Step 2

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Con0nuously deform Hamiltonian by changing only the eigenvalues keeping the gap to degenerate flat band. H(~ p) ⌘

Nv

X

a=1

Ea(~ p)|a(~ p)iha(~ p)| +

Nc

X

˙ b=1

E˙

b(~

p)|˙ b(~ p)ih˙ b(~ p)|

Hnew(~ p) = Ev

Nv

X

a=1

|a(~ p)iha(~ p)| + Ec

Nc

X

˙ b=1

|˙ b(~ p)ih˙ b(~ p)|

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Step 2

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Con/nuously deform Hamiltonian by changing only the eigenvalues keeping the gap to degenerate flat band. H(~ p) ⌘

Nv

X

a=1

Ea(~ p)|a(~ p)iha(~ p)| +

Nc

X

˙ b=1

E˙

b(~

p)|˙ b(~ p)ih˙ b(~ p)|

Hnew(~ p) = Ev

Nv

X

a=1

|a(~ p)iha(~ p)| + Ec

Nc

X

˙ b=1

|˙ b(~ p)ih˙ b(~ p)|

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Con*nuously deform Hamiltonian by changing only the eigenvalues keeping the gap to degenerate flat band. H(~ p) ⌘

Nv

X

a=1

Ea(~ p)|a(~ p)iha(~ p)| +

Nc

X

˙ b=1

E˙

b(~

p)|˙ b(~ p)ih˙ b(~ p)|

Hnew(~ p) = Ev

Nv

X

a=1

|a(~ p)iha(~ p)| + Ec

Nc

X

˙ b=1

|˙ b(~ p)ih˙ b(~ p)|

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Useful formulae

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ha(~ p)|@µH(~ p)|b(~ p)i = 0, h˙ a(~ p)|@µH(~ p)|˙ b(~ p)i = 0, ha(~ p)|@µH(~ p)|˙ b(~ p)i = (Ec Ev)ha|@µ˙ bi, h˙ a(~ p)|@µH(~ p)|b(~ p)i = (Ec Ev)h˙ a|@µbi, (a, b = 1, · · · , Nv, ˙ a, ˙ b = 1, · · · , Nc).

shows that inserted states should be valence and conduc;on band appearing alternately.

J =

Nv

X

a1,··· ,an=1 Nc

X

˙ a1,··· ,˙ an=1

✏i1j1···i2nj2n Z dp0 2⇡ 1 (ip0 + Ev)n+1(ip0 + Ec)n ha1|@i1H|˙ a1ih˙ a1|@j1H|a2i ⇥ · · · ⇥ han|@inH|˙ anih˙ an|@jnH|a1i + Z dp0 2⇡ 1 (ip0 + Ec)n+1(ip0 + Ev)n h˙ a1|@i1H|a1iha1|@j1H|˙ a2i ⇥ · · · ⇥ h˙ an|@inH|anihan|@jnH|˙ a1i

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Step 3

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p^0 integra3on can be easily carried out by Cauchy integral

J =

Nv

X

a1,··· ,an=1 Nc

X

˙ a1,··· ,˙ an=1

✏i1j1···i2nj2n(1)n+1 (2n)! (n!)2 ⇥ ha1|@i1 ˙ a1ih˙ a1|@j1a2i ⇥ · · · ⇥ han|@in ˙ anih˙ an|@jna1i.

Inser3ng this expression into c_cs using J, and using the defini3on of the Berry curvature (skipping detail) one obtains

chn(A) = 1 n! 1 (2π)n tr(Fn)

ccs ≡ k (n + 1)!(2π)n = (−1)n (n + 1)!(2π)n Z

BZ

chn(A),

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This result shows that Chern-Simons level in field theory approach and Chern number in microscopic approach (TKNN) are iden>cal for general Hamiltonian bilinear in fermion for D=2+1, 4+1 dimensions.

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Outline

1. Introduc/on
2. Review of TKNN formula
3. Review of field theory approach
4. Equivalence for general Hamiltonian
1. Chern-Simons level à Winding number
2. Winding number à TKNN formula
5. Summary

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5. Summary
We have shown microscopic approach (TKNN) and field

theory approach give iden?cal topological number for general Hamiltonian bilinear in fermion.

A series of Ward-Takahashi iden??es are crucial to show the

equivalence.

One should note no other details beyond gauge symmetry

(such as existence of rela?vis?c field theory at low energy) is needed.

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In 4+1 dimensions, there are two independent Chern
numbers. However, only a par<cular Chern number appeared.
This means that topological classifica<on in microscopic

approach may be finer, or those detailed structure may not be robust.

It would be interes<ng to see similar equivalence holds or not

for other cases such as systems with higher symmetry or systems with interac<ng fermions.

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Back-up

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1. Introduc,on

Topological insulators & Domain-wall fermion in D=2n+1 dimension Very closely related

Characterized by topology
Mass Gap in the bulk
Bulk-Edge Correspondence çè Gauge symmetry

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