The realization of Moiré flat bands inmonolayer graphene and the generalized nonHermitian 2D Nielsen-Ninomiya theorem

The Main Building in RIKEN Wako Campus

Ching-Kai Chi
RIKEN iTHEMS
Interdisciplinary Theoretical and Mathematical Science Programu@Yukawa InstituteTwo Questions related to twisted bilayer
grapheneFlat bands stems from the coupling of two Dirac cones indifferent graphene layers<latexit sha1_base64="C6XvfM0l8N29gkbjeUYM9aIxzoA=">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</latexit>

K

0K<latexit sha1_base64="BaygwqNv2sCEd5AFu12B9ZLJ8vE=">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</latexit>

A. Vishwanath et al. PRL 122, 106405 (2019)
Y. Cao et al. Nature 556, 43–50 (2018)

The Dirac cones at K 0 and K form two identical Moiré spectra.
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Question 1:
Can the two Dirac cones in monolayer graphene form at bands?

?
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YES!!!

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Graphene on periodic potential induced by substrate

Main problem
Space-time inversion symmetry or chiral symmetry quantize winding number⌫<latexit sha1_base64="FTbcc4377HpfRZexoTEEZDXflAs=">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</latexit>

Koshino’s lecture No. 1

vF k ·= v F ( kx x +ky y )⌫ = +1

K0

vF k ·= v F ( kx x +ky y )⌫ = +10

vF k ·= v F ( kx x +ky y )⌫ = +1= v F ( kx x

ky y )⌫=<latexit 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The different orientations can lead to at bands?

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1Question 2:
In non-Hermitian physics, can two exceptional points be located at
K or K’ so that there is twisted physics exceptional points?
EP +1

+1

K
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K<latexit sha1_base64="RyLRINDoro9CSG9QqeNIIoMU51o=">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</latexit><latexit sha1_base64="PHjXe028yEJ4xfSoco8HJ76WijI=">AAAJ13iclVZbb9s2FFbbXeLs0nZ73As7ZVg6qEHkAusw7KHFkMxBhsBLk7RN5AUURcmEqUtIKrHBqnsb9jRgT9uf2Q/Zv9khZTuWlBWYAJvU+T4enjsUFpxJtb39z63bd955973313rrH3z40cd3793/5ETmpSD0mOQ8Fy9DLClnGT1WTHH6shAUpyGnL8LJ9wZ/cUmFZHl2pGYFHaU4yVjMCFZGtLH/5cb5PXd7a9s+qLvx5xvXmT/D8/trfwdRTsqUZopwLOWZv12okcZCMcJptR6UkhaYTHBCz2Cb4ZTKkbbGVugLkEQozgX8MoWsdPWExqmUszQEZorVWLYxI/TC9Cb4rFTxNyPNsqJUNCP1XXHJkcqRcR5FTFCi+Aw2mAgG5iIyxgITBSFqXKTGgsKLUAqHLZf0GFQJQeOmNASb1m80FmICKaCxN/fMrMZ36SUCF2NGpl5acsVEfuXZcIB7nqJTRfK08IwKledcNq8D27Cpj6ZUCsLVtJkBzLlHUsWKkZ7OmnSjW8hY3qTEgxrjnizDmCWlaIWgGR4vsv54dQjWAwjQGLJuKkw/+3H4w85B1RTunBx1hcNXR4O9ZyaGEY0DQV1fb8ISa9evHla1dBjrwBidF/UqUj2Mq2qOHokOeiSWqEyya9iEF2CQWTyjVxDsFGeRDggPK/g3idAhL2kXF0tc0KgLp0sYmg16BHcpyZJCZjhr4yG9qMx/wjJNL0obnjaHWg6F7X8xQAsWAs9WVGW1oKtqQaz13Uw7hIo/80ca1q03QZ6ZkUOYoiY/FUJN8nNKLBlWIM+z2GGBD4a0c7H1ZiXVbRYUoKXtsuQtuoAlFzT5Fl4oWZJi8DUN8ym85jwyRbERWPlG22t1WtVVo5Q+7YATABXjEQVFMZpUgWDJWEH48iv0qIGAHS1DCpPB4bk+eORXP+sAqhEEnSTuV7WNZnDkXO+3Celgbh/BXA8q23+r+CWYuDCh7oNMarjushjjTOWp/mrZIHRawHR0fbfvPoYzHGcJp8j10Wuw3e0js7qPUSAssDhFBsCFy9FgIRljDm0aw1jVfqX7y/4rVd13MtbPj6vN/sNrIG8hYOS5P4cnuzqYUDjrmSCh3Srw2kGAqWRTf2Y9HcaxevAtstkfteMpsGWueteiwF2W8tqAC2c7WiyrbxTRWKFGsCBUdR3cdDg41IvhhPRe8OCwlfLgoIkfdEvCVm/d0gUwBZt2OtpSTDNfE1pls79SNp2iCvfmKEx0yGELjRZoJGvwf03X4crFw87Fq9XajrmFUi26EYkqi4AhvDttw9OlM7aD1+GDx29/3nQ3J/0t/+ut/k999+l380+fNecz53Nn0/GdJ85TZ+AMnWOHOGPnd+dP56/eq94vvV97v9XU27fmZz51Gk/vj38BZLSfDA==</latexit>

K

Graphene0-1

K0<latexit sha1_base64="PHjXe028yEJ4xfSoco8HJ76WijI=">AAAJ13iclVZbb9s2FFbbXeLs0nZ73As7ZVg6qEHkAusw7KHFkMxBhsBLk7RN5AUURcmEqUtIKrHBqnsb9jRgT9uf2Q/Zv9khZTuWlBWYAJvU+T4enjsUFpxJtb39z63bd955973313rrH3z40cd3793/5ETmpSD0mOQ8Fy9DLClnGT1WTHH6shAUpyGnL8LJ9wZ/cUmFZHl2pGYFHaU4yVjMCFZGtLH/5cb5PXd7a9s+qLvx5xvXmT/D8/trfwdRTsqUZopwLOWZv12okcZCMcJptR6UkhaYTHBCz2Cb4ZTKkbbGVugLkEQozgX8MoWsdPWExqmUszQEZorVWLYxI/TC9Cb4rFTxNyPNsqJUNCP1XXHJkcqRcR5FTFCi+Aw2mAgG5iIyxgITBSFqXKTGgsKLUAqHLZf0GFQJQeOmNASb1m80FmICKaCxN/fMrMZ36SUCF2NGpl5acsVEfuXZcIB7nqJTRfK08IwKledcNq8D27Cpj6ZUCsLVtJkBzLlHUsWKkZ7OmnSjW8hY3qTEgxrjnizDmCWlaIWgGR4vsv54dQjWAwjQGLJuKkw/+3H4w85B1RTunBx1hcNXR4O9ZyaGEY0DQV1fb8ISa9evHla1dBjrwBidF/UqUj2Mq2qOHokOeiSWqEyya9iEF2CQWTyjVxDsFGeRDggPK/g3idAhL2kXF0tc0KgLp0sYmg16BHcpyZJCZjhr4yG9qMx/wjJNL0obnjaHWg6F7X8xQAsWAs9WVGW1oKtqQaz13Uw7hIo/80ca1q03QZ6ZkUOYoiY/FUJN8nNKLBlWIM+z2GGBD4a0c7H1ZiXVbRYUoKXtsuQtuoAlFzT5Fl4oWZJi8DUN8ym85jwyRbERWPlG22t1WtVVo5Q+7YATABXjEQVFMZpUgWDJWEH48iv0qIGAHS1DCpPB4bk+eORXP+sAqhEEnSTuV7WNZnDkXO+3Celgbh/BXA8q23+r+CWYuDCh7oNMarjushjjTOWp/mrZIHRawHR0fbfvPoYzHGcJp8j10Wuw3e0js7qPUSAssDhFBsCFy9FgIRljDm0aw1jVfqX7y/4rVd13MtbPj6vN/sNrIG8hYOS5P4cnuzqYUDjrmSCh3Srw2kGAqWRTf2Y9HcaxevAtstkfteMpsGWueteiwF2W8tqAC2c7WiyrbxTRWKFGsCBUdR3cdDg41IvhhPRe8OCwlfLgoIkfdEvCVm/d0gUwBZt2OtpSTDNfE1pls79SNp2iCvfmKEx0yGELjRZoJGvwf03X4crFw87Fq9XajrmFUi26EYkqi4AhvDttw9OlM7aD1+GDx29/3nQ3J/0t/+ut/k999+l380+fNecz53Nn0/GdJ85TZ+AMnWOHOGPnd+dP56/eq94vvV97v9XU27fmZz51Gk/vj38BZLSfDA==</latexit>

EP -1

Non-Hermitian

Focus: The generalized non-Hermitian 2D Nielsen-Ninomiya theorem
No. There are some constraints.

Outline
1. Monolayer graphene -> at bands
I.

Review of twisted bilayer graphene continuum mode

II. Experiment: Monolayer graphene on SiC substrat
III. The low-energy physics of monolayer graphene on substrate

2. Exceptional points in the hexagonal Brillouin zoneI.

The comparison of the 2D Nielsen-Ninomiya theorem between
Hermitian and non-Hermitian system

l

e

s

fl

II. The generalized Non-Hermitian Nielsen-Ninomiya theorem forthe 17 wallpaper groups

Review of twisted bilayer graphene
The bilayer of the small twisted angle forms the Moire
pattern in the real space
(a)

(b)

q3

Kb

y

q3Kb

q2

q1

q1

θ

q2

C3z

Γ

Kt

Kt

Moire BZ
Courtesy of ICFO/Xiaobo Lu

(c)

x

(d)

|qi | = 2kD sin(✓/2)<latexit sha1_base64="L5Ek5KFtFCgLo5gefDBLnb8fJCg=">AAAKYHiclVZbb9s2FFa7W+Jdmmxv2x7YuQOSQc0id9gNGNBiSZYgQ+ClSdol8gyKomzC1CUkldhg1If9mr1u/2av+yU7pGTHkrICE2CTOt/Hw3OHgowzqba3/753/40333r7nZXVzrvvvf/Bg7X1D89kmgtCT0nKU/EywJJyltBTxRSnLzNBcRxw+iKY/GjwF1dUSJYmJ2qW0UGMRwmLGMEKRMO1T2/84HLIbn7oTYY7yJcsQRu+GlOFv+xtDte621vb9kHtjVdtuk719Ifrq64fpiSPaaIIx1JeeNuZGmgsFCOcFh0/lzTDZIJH9AK2CY6pHGjrRoE+B0mIolTAL1HISpdPaBxLOYsDYMZYjWUTM0I3iO+CL3IVfTvQLMlyRRNS3hXlHKkUmbCgkAlKFJ/BBhPBwFxExlhgoiB4tYvUWFB4EUrhoOGSHoMqIWhUlwZgU+dOYyEmkBwauZVnZjW+S3ckcDZmZOrGOVdMpNeuDQe45yo6VSSNM9eoUGnKZf06sA2byqlLpSBcTesZwJy7JFYsG+jprE43uoWM5F1KXKg+7so8iNgoF40Q1MPjhtYftwxBx4cAjSHrpvb0s5/7P+0eFXXh7tlJW9j/9WT/4JmJYUgjX9CupzdgiXTXKzaLUtqPtG+MTrNyFbHuR0VRoSeihZ6IBSpHyS1swgswyCye0GsIdoyTUPuEBwX8m0TogOe0jYsFLmjYhuMFDG0IPYLblNGCQmY4aeIBvSzM/4glml7mNjxNDrUcCtv/YoAWLASeLalKSkFb1ZxY6rubdgwVf+ENNKxbr/w0McOIMEVNfgqE6uTnlFgyrECusthigQ+GtHu59Wop1U0WFKCl7bHRa3QBS85p8jW8QLJRjMHXOEin8Jry0BTFI9/KHzW9VudFWTVK6fMWOAFQMR5SUBShSeELNhorCF96jR7XELCjYUhmMtgf6qPHXvGb9qEaQdBK4mFR2mgGR8r1YZMQ71f2Ecz1fmH7bxm/AhPnJpR9kEgN111lY5yoNNZfLBqETjOYjl2v2+s+gTMcJyNOUddDN2B7t4fM2n2CfGGB+SmyD1y4HO3PJWPMoU0jGKvaK3Rv0X+5KvtORvr5abHR27wF0gYCRg69Cp7saX9C4axrgoT2Ct9tBgGmkk39hfW0H0Xq4ffIZn/QjKfAlrnsXYMCd1nKjQHnzra0WFbPKKKRQrVgQajKOrjrsH+s58MJ6QP/4XEj5f5RHT9ql4St3rKlM2AKNm11tKWYZr4lNMrmcKlsWkUVHFQoTHTIYQMN52goS/B/Tdf+0sX91sXL1dqMuYViLdoRCQuLgCG8PW2D84UztoPr6I5p4R1zDhkdZZvUjmflvShrIUGFBC3kskKgoTvwfeU1v6bam7Pelvf1Vu+Xr7pPv6u+tFacT5zPnA3Hc75xnjr7Tt85dYjzu/OH86fz1+o/nZXOg856Sb1/rzrzkVN7Oh//CwV50is=</latexit>y

R. Bistritzer and A. H. MacDonald, Proc. Natl. Acad. Sci. 108, 12233 (2011).

The interlayer coupling, which is complicated, can be
simpli ed in the momentum space — Continuum Modelθp3 p2 Mt Koshino’s lectures
x
fi

C4z

(b)

y

Continuum Modelq3

Kb

The low-energy
physics is describedby momentum space hopping between theθ
q3Kb
q 2 two Dirac cones in thex different layers

q2

q1

q1

C3z
Γ (a)

(b)

Kt

Kt

Bottom layer

p2

q3

Mt

Mb
p4 p1 Kbq3

yTop(d)layer(b)

p3

y

C4z q

θx

3

q1

2

q1

q1

C3z

θ

q2
Γ

x

C3z

Kt

Kt

θ

q Kb

q2

yKb

q3Kb

q2

x

q1 , q2 , q3 y

momentumhoppingΓ
q1 1st nearest neighbor
(c)
(d)
g2
0
1
Htd ( k)
wTM
wT
wT
0 t
K
p
2
B wT0
C
pH
g1 q1 )
00
3 d( k +BCHTBG ( k) = @
M
b
A
wT
0
H
(
k
+
q
)
0
d
2
p4 p1
wT
0p3 p2 Mt
0
H d ( k + q3 )
C4z
y
M
(d)
Hd ( k) = vFp( kxb x + ky y )p1
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Kt

θ

x

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T = cos x + sin y , where = 2⇡/3
Interlayer coupling
AB basis hoppingp2 Mtθp3 p2 Mt
p3
g1
x is suppressed
AA basis hopping
0
C4z
Mb
Mb
p p
Stephen Carr, Shiang Fang, Ziyan Zhu, Efthimios Kaxiras, Phys. Rev. Research 1, 013001 (2019)<latexit sha1_base64="5+BP5gzTzuclE42f4J3dCiAEbgY=">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</latexit><latexit sha1_base64="GZ8xbO7ri20La8LsPZdgiIBuDb8=">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</latexit>

p

p

4

1

(a)

(b)

q3

Kb

y

q3Kb

q2

q1

q1

θ

q2

x

C3z

Γ

Kt

Kt

(c)

y

(d)

±(q1
p3 p2 Mt
Mb
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2nd nearest neighbor momentum hoppingq2 ), ±(q2q3 ), ±(q3

θ

q1 )
x

C4z

p4 p1

Forbidden:
Same Dirac cone in the same layer

g2

p3

p2

Mb

 

p4 p1

Mt
g1

(a)

(b)

q3

Kb

q3Kb

q2

q1

q1

(c)
3rd nearest neighbor momentum
hopping|2qi |

θ

q2
Γ

x

C3z

Kt

Kt

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2q1(d), 2q2 ,

2q3

y

Hopping strengthθp3 Hopping
p2 Mt strength decreases as momentum
distance increasesC4z

x

Mb

p4 p1

Approximation:

g2

p2

Mt

p3
g1
We keep the nearest momentum hoppingMb
and neglect the remaining hopping of thep4 p1long distance.

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Momentum distance

R. Bistritzer and A. H. MacDonald, Proc. Natl. Acad. Sci. 108, 12233 (2011).

0
Hd ( k)B wT0
HTBG ( k) = B@ wTwT<latexit sha1_base64="tp80vazxw9XHI9cT0wXj4W3ZE2g=">AAAK/XiclVZbb9s2FJa7W6vd0u1xD2Pnbkg2JbCcYRsGDGi7JHWQIfNaJ+0SeQYlUbJg6hKSSmyw6n7N3oa9Dvsp+zc7pGTHkrICNWCTOt93Ds+VlpvRiIte79/OrTfefOvtd27fMd997/0PPty4+9EpT3PmkRMvpSl77mJOaJSQExEJSp5njODYpeSZO/tR4c8uCeNRmozEIiPjGIdJFEQeFiCabPwzmEiHxXL06HFRbDp7s60fTMeNMTIHE18/oy/Q1WjSKxe57WQ8KqqHcu84ZkWoVL5y3IuJrRR7+lsSrlWb1P5WRUPmtdWlcp26u6VpDonxZKPb2+npD2pv7GrTNarPcHL3juX4qZfHJBEexZyf271MjCVmIvIoKUwn5yTD3gyH5By2CY4JH0ud4wJ9DhIfBSmDbyKQlq5rSBxzvohdYMZYTHkTU0LLjW+Cz3MRfDeWUZLlgiReeVaQUyRSpGqG/IgRT9AFbLDHInAXeVPMsCegsrWDxJQReGBCYLcRkpyCKcZIUJe64JN5o7OQE+gcElhVZGpVsXMrZDibRt7cinMqIpZeWTodEJ4lyFx4aZxZyoRIU8rrx4FvWLV1XcqZR8W8XgFMqeXFIsrGcr6o05VtxgN+kxELRoNaPHeDKMxZIwX19Fi+jscqU2A6kKApVF0Nhnz40/Dx/nFRF+6fjtrC4a+jweFDlUOfBA4jXVtuwhLIrl1sFaV0GEhHOZ1m5QoDNwyKokJHrIWO2ArlYXINq/QCDDKNJ+QKkh3jxJeOR90CflUhpEtz0sbZCmfEb8PxCoY7AmYEtynhiuItcNLEXXJRqN8wSiS5yHV6mhyiOQS2/8cAK5gxvFgzlZSCtqklsbR3M+0JdPy5PZaw7rx00kTdlF4kiKpPgVCd/JR4mgwrkKsqtlgQgyLtX+y8XCt1kwUNqGkHUfgKW8DiSxp/Bc/lURhjiDV20zk8ptRXTXHf0fL7zajFWVF2jRDyrAXOABQR9QkYCtCscFgUTgWkL71C2zUE/Gg4kqkKDifyeNsuftP/HSBoFfGoKH1UF0dK5VGTEA8q/zxM5aDQ87eOX4KLSxfKOUi4hOMusylORBrLL1cDQuYZ3I5du9vv7oIOxUlICera6AX43u0jtXZ3kcM0sNTyBsCFw9FgKZliCmMawLUq7UL2V/OXi3LueCCfnhSb/a1rIG0g4OTEruDZgXRmBHQtlSR0UDhWMwlwK+nSn+tIh0Eg7n2PdPXHzXwyrJnr0TUocJamvFDgMtiWFc3qK0MkEKiWLEhV2Qc3KTtP5PJyQvLQufekUXLnuI4ft1tCd2850hkwWTRvTbSmqGG+JjTa5mitbVpN5R5WKNzoUMMG6i9Rn5fga92uw7WDh62D17u1mXMNxZK1M+IXGgFHaPu2dc9WwegJrqN7aoT3lB5SNsoxqaln5bkoayFuhbgt5KJCYKBNeL+ym29T7c1pf8f+Zqf/y9fdB4+qN63bxifGZ8amYRvfGg+MgTE0Tgyv82lnv3Pc+dn83fzD/NP8q6Te6lQ6Hxu1j/n3fxA0/r8=</latexit>

wT0
H d ( k + q1 )
0
0

wT
0
H d ( k + q2 )0<latexit sha1_base64="5+BP5gzTzuclE42f4J3dCiAEbgY=">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</latexit>

T = cosx + siny , where

1

wT
C
0
C
A
0
H d ( k + q3 )= 2⇡/3<latexit sha1_base64="dTv4mE6p0DdztlWsuvUmCMZkn1k=">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</latexit>

Hd ( k) = vF ( kx x +ky y )

q2<latexit sha1_base64="R6WMtHSSxKvg7K+Dg1GXVbTunKM=">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</latexit>

q3 q<latexit sha1_base64="rfIT9D9d0y6j3E7G/Mua8JkM7vs=">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</latexit><latexit sha1_base64="PrEhbAaupuerCLIeOVxqcBgTA7U=">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</latexit>

1

Extend to the momentum lattice and compute the energy spectrum

Parameter normalization and at bands0
Hd ( k)B wT0
HTBG ( k) = B@ wTwT<latexit sha1_base64="tp80vazxw9XHI9cT0wXj4W3ZE2g=">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</latexit>

wT0
H d ( k + q1 )
0
0

wT
0
H d ( k + q2 )0<latexit sha1_base64="dTv4mE6p0DdztlWsuvUmCMZkn1k=">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</latexit>

Hd ( k) = vF ( kx x +ky y )

1

wT
C
0
C
A
0
H d ( k + q3 )↵ ⌘ w/vF |q1 |<latexit sha1_base64="ide0C5+iDkP0Pmam4ZfHLadN5ZY=">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</latexit>

The two-fold degenerate bands are absolutely at due to the holomorphic wave functions.
fl

fl

PHYSICAL REVIEW LETTERS 122, 106405 (2019)

Outline

2.

Experiment: Monolayer graphene on SiC substrat

3.

The low-energy physics of monolayer graphene on
substratee

Review of twisted bilayer graphene continuum model1.

Experiment: Monolayer graphene on SiC substrate

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The angle between the two sets of the reciprocal lattice vectors (Graphene:
B1 , B2 , SiC: b1 , b2 ) is ⇡/2.
<latexit sha1_base64="Zpt7+NXB7/JbIQOupzmyplYGFY0=">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</latexit>

Graphene lattice constant aGr = 2.46Å
<latexit sha1_base64="LPApNwv6EZq8r74fVFBXa5l0uI8=">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</latexit>

SiC constant aSiC = 3.07ÅGuang Bia
University of Missouri

Congcong L
RIKEN iTHEMS

Advanced Materials 2200625 (2022)

en

Qiangsheng Lu, Congcong Le, Xiaoqian Zhang, Jacob Cook, Xiaoqing He, Mohammad Zarenia, Mitchel Vaninger, Paul F. Miceli,David J. Singh, Chang Liu, Hailang Qin, Tai-Chang Chiang, Ching-Kai Chiu,* Giovanni Vignale,* and Guang Bian*

ARPES measurement
Dirac Cone Cloning
Graphene Brillouin Zone

Advanced Materials 2200625 (2022)

Theoretical Understanding
Simple Graphene tight-binding model (nearest neighbor hopping) withperiodic potential<latexit sha1_base64="CFu3IAjFi5UceFYjOIJZzLHBQ2k=">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</latexit>

V (r) = w cos(b1 · r) + cos(b2 · r) + cos( (b1 + b2 ) · r)
SiC: b1 , b2
Spectra from simulation<latexit sha1_base64="Ktf5+1YfD7Nkubk96kLjNOWCyIQ=">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</latexit>

Experiment

Perturbation theory<latexit sha1_base64="CFu3IAjFi5UceFYjOIJZzLHBQ2k=">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</latexit>

V (r) = w cos(b1 · r) + cos(b2 · r) + cos( (b1 + b2 ) · r)<latexit sha1_base64="1QhY9NdRyYtRflwtcYZsdCxqzrA=">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</latexit>|

k i =|

0
ki +

X

p6=k

h
0
| pi

0
0
X
|V
|
p
ki
+
|
0
0
E k Epp6=k,l6=k

0
0
0
0
h
|V
|
ih
|V
|
p
l
l
ki
0
+ ···
pi
0
0
0
0
(Ek Ep )(Ek El )1st orderp = k ± b1 , k ± b2 , k ± (b1 + b2 )First clones

One reciprocal vector shift<latexit sha1_base64="aZCiv4KQZOhz4nEMfkY0CYK/8uI=">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</latexit>

Perturbation theory<latexit sha1_base64="1QhY9NdRyYtRflwtcYZsdCxqzrA=">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</latexit>|

k i =|

0
ki +

X

p6=k

2nd orderh0| pi<latexit sha1_base64="cxceZwuu30doCe652F5ukDf9wUs=">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</latexit>

0
0
X
|V
|
p
ki
+
|
0
0
E k Epp6=k,l6=k

p =k ± 2b1 , k ± 2b2 , k ± 2(b1 + b2 ),k ± (b1

Second clone

0
0
0
0
h
|V
|
ih
|V
|
p
l
l
ki
0
+ ···
pi
0
0
0
0
(Ek Ep )(Ek El )b2 ), k ± (2b1 + b2 ), k ± (b1 + 2b2 )Two reciprocal vector shift

Periodic potential + Graphene model can faithfully capture
the low-energy physics of graphene on the substrate

Tight-binding Model<latexit sha1_base64="CFu3IAjFi5UceFYjOIJZzLHBQ2k=">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</latexit>

V (r) = w cos(b1 · r) + cos(b2 · r) + cos( (b1 + b2 ) · r)<latexit sha1_base64="691TrTsmjlyYJuJvm/4v3nkwJwk=">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</latexit>

HGr (k) =<latexit sha1_base64="V8GPiuvo5YsbZGNvmNbKhYlQ/7c=">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</latexit>

hk = tep

ikx aGr / 3✓(1 + e

0
h⇤k

p

hk
0

i 3kx aGr /2◆

ky aGrcos)
2

Change substrate lattice constant<latexit sha1_base64="LPApNwv6EZq8r74fVFBXa5l0uI8=">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</latexit><latexit sha1_base64="uf9FfLKYcE/q6xTnSXv3eJN1cvw=">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</latexit>

SiC constant aSiC = 3.07Å

asub = 3.8Å

bsub<latexit sha1_base64="CqekRFpr5lMuBn5bzWfbFvBZ/Xk=">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</latexit>

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bsub

The periodic potential brings the coupling between the two Dirac cones.

VHS: Van Hove Singularity

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asub !

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3aGr = 4.26Å

bsub

The rst-ordered Dirac cones overlap
the main Dirac cone<latexit sha1_base64="f9CQj7mrSBjwWnk+jSykyjzMftI=">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</latexit>

asub = 4.368Å ⇡bsub<latexit sha1_base64="CqekRFpr5lMuBn5bzWfbFvBZ/Xk=">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</latexit><latexit sha1_base64="P5l3dJYndasqbT+hlWJoGsnrAIw=">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</latexit>

p

3aGr

bsub

A zero-bias peak in a gap

flfi

Hint of at bands at zero energy

Outline

2.

Experiment: Monolayer graphene on SiC substrat

3.

The low-energy physics of monolayer graphene on
substratel

Review of twisted bilayer graphene continuum modee1.

<latexit sha1_base64="3+/PgfAow7gNOhBThoSyxSHn/W8=">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</latexit>

The Continuum Model for asub ⇡

p

3aGr

Periodic potential form the substrate<latexit sha1_base64="CFu3IAjFi5UceFYjOIJZzLHBQ2k=">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</latexit>

V (r) = w cos(b1 · r) + cos(b2 · r) + cos( (b1 + b2 ) · r)<latexit sha1_base64="Jfz/cVdS4wGbOaQxJ+T4bxoFMhE=">AAALpniclVZbb9s2FJa7W5vs0myPe2HnLkgWJbDcYTdgQIs1WYIMgdfm0sVyPYqiZMHUJSSVOGDV37e/sD+x1+1xh5TtWJLTdQJiUuf7eHjukZexSMhO58/WnXfefe/9D+7eW1n98KOPP7m/9umpSHNO6AlJWcpfeFhQFiX0REaS0RcZpzj2GD3zxj9p/OySchGlybG8zuggxmESBRHBEkTDtRZ2R1iq0+LHdVfk8VC5Hi8Qeen6OAwpH8IrQqcbsGwiMkVdd8XNeJrJFK2jq9mxMQBeFKL+wmkt3nI9b+gUrojCGA87pRYgby3jdf+bt230bd/CBtvW3Ysc++UPQlvLTr/lLW9hTendLbabeAyG99udnY55UHPjTDdta/r0hmv3bNdPSR7TRBKGheg7nUwOFOYyIowWK24uaIbJGIe0D9sEx1QMlCmGAn0JEh8FKYe/RCIjXTyhcCzEdewBM8ZyJOqYFtpevAzu5zL4bqCiJMslTUh5V5AzBJWgiwv5EadEsmvYYMIjMBeREeaYSCjBykVyxCm8cCmxV3NJjUAV5zSoSj2waWWpsRATKHEa2FPP9Kp9F3bIcTaKyMSOcyYjnl7ZJhzgni3pRJI0zmytQqYpE9XrwDas+68qFZwwOalmADNmk1hG2UBNrqt0rZuLQCxTYkMPM1vkXhCFOa+FoBoe2zf+2GUIVlwI0AiyrjtYPfml9/PuUVEV7p4eN4W93473D57oGPo0cDltO2oDlkC1nWKzKKW9QLna6DQrVx6rXlAUU/SYN9BjPkdFmNzAOrwAg8zgCb2CYMc48ZVLmFfAr06E8lhOmzif45z6TTiewzDMoEdwkxLOKeQaJ3XcoxeF/g2jRFGYEzo8dQ41HArb2xigBXOOrxdUJaWgqWpGLPUtpz2Diu87AwXrzms3TfRIJ5GkOj8FQlXyc0oMGVYgT7PYYIEPmrR7sfN6IdV1FhSgoe1F4Rt0AUvMaOINPM8MQfA19tIJvKbM10XxsByOD+tey/OirBop1XkDHAMoI+ZTUBQgmKg8CkcSwpdeoe0KAnbUDMl0BntDdbTtFC+VC9UIgkYSD4vSRj04UqYO64R4f2ofwUztF6b/FvFLMHFmQtkHiVBw3WU2wolMY/XVvEHoJIPp2Hba3fYjOMNwEjKK2g56Bba3u0iv7UfI5QaYnSL7wIXL0f5MMsIM2jSAsaqcQnXn/ZfLsu9EoJ6fFBvdzRsgrSFgJPwXLOHxnnLHFM7aOkhor3DtehBgKpnU942nvSCQD35AJvuDejw5NsxF72oUuMtQXmlw5mxDi2F1tSIaSFQJFoSqrINlh91najackDpwHzyrpdw9quJHzZIw1Vu2dAZMHk0aHW0ouplvCLWyOVwom0ZReQdTFCY65LCG+jPUFyX4v6Zrb+HiXuPixWqtx9xAseLNiPiFQcAQ1py23vncGdPBVfSpbuGn+hzSOso2qRzPyntR1kC8KeI1kIspAg29At9XTv1rqrk57e443+x0f/26/fj76ZfWXetz6wtrw3Ksb63H1r7Vs04s0vqj9Vfr79Y/qxurR6snq2cl9U5reuYzq/Ks/v4vopBIOQ==</latexit>

V̂ =

X

Six momentum hopping termsc†r V (r)cr

r

X⇥ †
/w
ck+b1 0 ck + c†k+b2 0 ck + c†k b1 b2 0 ck
k

⇤
†
†
†
+ ck b1 0 ck + ck b2 0 ck + ck+b1 +b2 0 ck

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b1<latexit sha1_base64="ZTHgDYRTUyuz1vp38lPite/i/Ys=">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</latexit>

b2<latexit sha1_base64="50QBpA071xy1U9l0lzsZfAL54nM=">AAAKSHiclVbbbts2GFazU+2d2u5yN+zUAemgBJFbdMOuWqzJHGQIvDRJu1heQFGUrJk6hKQSG6z6DLvdHmlvsLfY3bC7/aRkx5KyAhNgk/q/jz//M+TnLBZyZ+fPWxvvvPve+x/c7vU//OjjTz69c/feqcgKTugJyVjGX/lYUBan9ETGktFXOac48Rl96c++0/jLS8pFnKXHcpHTSYKjNA5jgiWITrc83z8fnN+xd7Z3zIO6G7fe2Fb9jM7v9hwvyEiR0FQShoUYuzu5nCjMZUwYLfteIWiOyQxHdAzbFCdUTJQxt0RfgiRAYcbhl0pkpOsnFE6EWCQ+MBMsp6KNaaHjJzfB40KG30xUnOaFpCmp7goLhmSGtPsoiDklki1ggwmPwVxEpphjIiFIjYvklFN44VJiv+WSmoIqzmnYlPpgU/9GYyEmkAQaOrVnetW+CyfiOJ/GZO4kBZMxz64cEw5wz5F0LkmW5I5WIbOMieZ1YBvWFdKUCk6YnDczgBlzSCLjfKLmiyZd6+YiFDcpcaDKmCMKP4yjgrdC0AyPExh/nCoEfQ8CNIWs6xpTz34Yfb97WDaFu6fHXeHop+Ph/jMdw4CGHqe2qzZhCZXtlg/LSjoKlaeNzvJq5YkahWVZo8e8gx7zFSqi9BrW4QUYZAZP6RUEO8FpoDzC/BL+dSKUzwraxfkK5zTowskKhnaDHsFdSrSikAVO27hPL0r9H8WpoheFCU+bQw2Hwva/GKAFc44Xa6rSStBVtSRW+m6mHUHFj92JgnX7jZeleuiQWFKdnxKhJvkFJYYMK5DrLHZY4IMm7V5sv1lLdZsFBWhoe3H0Fl3AEkuaeAvPF3GUYPA18bM5vGYs0EXxwDPyB22v5VlZVY2U6qwDzgCUMQsoKArRrPR4HE0lhC+7QlsNBOxoGZLrDI7O1eGWW/6sPKhGEHSSeFBWNurBkTF10CYkw9o+gpkalqb/1vFLMHFpQtUHqVBw3WU+xanMEvXVqkHoPIfpaLv2wH4EZxhOI0aR7aLXYLs9QHq1HyGPG2B5igyBC5ej4VIyxQzaNISxqtxSDVb9V8iq70SoXpyUm4OH10DWQsDIc7eGZ3vKm1E46+ggob3Sc9pBgKlkUj82no7CUN7/FpnsT9rx5Ngw171rUeAuQ3mtwaWzHS2GNdCKaChRI1gQqqoObjrsHanlcEJq37t/1Eq5d9jED7slYaq3aukcmDyedzraUHQzXxNaZXOwVjadovL3axQmOuSwhQZLNBAV+L+m62jt4lHn4vVqbcfcQIni3YgEpUHAENadtv7ZyhnTwU30uW7h5/oc0jqqNmkcz6t7Ud5B/BrxO8hFjUBD9+H7ym1/TXU3p4Nt98n24MfH9tPH9ZfWbetz6wtr03Ktr62n1tAaWScWsX6xfrV+s37v/dH7q/d375+KunGrPvOZ1Xj6G/8CjLnMfg==</latexit>(1)b2<latexit sha1_base64="XpqFtN8TLizhOgBDzb9i6VTcbU4=">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</latexit>

Momentum hoppings between the two Dirac conesb1<latexit sha1_base64="KbvEfd6ejVU3Ee4nnlKjJ68Os1k=">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</latexit>

b1

b2<latexit sha1_base64="DvntSZDw1l59CDPKPB66JaY88IM=">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</latexit><latexit sha1_base64="bHpaTtecKTVjN6+qVzenhw7uR5g=">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</latexit>(2)
The Dirac cone couples to the high energy bands<latexit sha1_base64="yNoRDQ7CoGYCN6k5OVgXd800rhM=">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</latexit>

b1 + b2<latexit sha1_base64="7Rp3/gQhuaIKlpEZOwHKP0h+VGc=">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</latexit>

Approximation: keep (1), neglect (2)<latexit sha1_base64="3+/PgfAow7gNOhBThoSyxSHn/W8=">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</latexit>

The Continuum Model for asub ⇡0<latexit sha1_base64="XSl8y5vsh1tijgEya41UHOdKbvM=">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</latexit>

Hd ( k)
B w 0
HMG ( k) = B
@ w 0
w 0

w 0
⇤
HD
(0, k + q1 )00<latexit sha1_base64="+dAGwr2Ng4TIUM9+n5Mw0YykiKE=">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</latexit>

Off-diagonal:

p

3aGr
1

w 0
0
⇤
HD
( , k + q2 )
0

w 0
C
0
C
A
0
, k + q3 )

⇤
HD
(

X⇥ †
⇤
†
†
w
ck+b1 0 ck + ck+b2 0 ck + ck b1 b2 0 ck
kDiagonal:

Dirac cones with adjustment
KK 0
qi = b i
2π /3
q1 , q2 , q3
<latexit sha1_base64="DJVYwGVR3oTBXGjnKTTTDH+6iuc=">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</latexit>

Momentum shifts<latexit sha1_base64="gO4PEMCWgW4UA+l73Ik2Viq4Yvw=">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</latexit>−2π /3Form Moire BZ

Y

⇤
HD
( ,
<latexit sha1_base64="1Iu7HSzGL6+PDZXoOdiEePjjRGc=">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</latexit>k) = R vF ( kx x

<latexit sha1_base64="Tk0L/P/wRm3upJuQbUUBZPKWAOo=">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</latexit>

R = cos /2 0i sin /2 z

2π /3

Γ
2π /3

0
B2

<latexit sha1_base64="xheAa4hab297z+Bur7Q/5xyFNMw=">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</latexit>

q2<latexit sha1_base64="d4Py34u3UzUKyyyKF28aY3iMyNM=">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</latexit>xΓ

ky y )R 1

B3

q3
A1 −2π /3<latexit sha1_base64="dTv4mE6p0DdztlWsuvUmCMZkn1k=">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</latexit>

ky y )

2π /3<latexit sha1_base64="9dX/JEwSi+2AbAvTTWB2R29KeMs=">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</latexit>−2π /3

Hd ( k) = vF ( kx x +−2π /3

q1 0
ΓPhase shifts

bsub
i

B1

2π /3
0

−2π /3

The Tight-binding model and the continuum model
are in agreement<latexit sha1_base64="XSl8y5vsh1tijgEya41UHOdKbvM=">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</latexit>0

Hd ( k)
B w 0
HMG ( k) = B
@ w 0
w 0

w 0
⇤
HD
(0, k + q1 )
0
0

w 0
0
⇤
HD
( , k + q2 )
0

1

⇤
HD
(

w 0
C
0
C
A
0
, k + q3 )↵ ⌘ w/vF |q1 |<latexit sha1_base64="ide0C5+iDkP0Pmam4ZfHLadN5ZY=">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</latexit>

q2<latexit sha1_base64="R6WMtHSSxKvg7K+Dg1GXVbTunKM=">AAAKR3iclVbbbts2GFa6U+2d2u1yN+zcAemgGpE7bMOuWqzJHGQIvDRpu0ReQFGUTJg6hKQSG6z6CrvdHmmPsKfY3bDL/aRkx5KyAhNgk/q/jz//MxTknEm1s/Pn1q233n7n3fdu9/rvf/DhRx/fufvJc5kVgtATkvFMvAywpJyl9EQxxenLXFCcBJy+CObfG/zFJRWSZemxWuZ0muA4ZREjWIHoxA8uzkfndwY7wx37oO7GqzcDp34m53d7rh9mpEhoqgjHUp55O7maaiwUI5yWfb+QNMdkjmN6BtsUJ1ROtbW2RF+AJERRJuCXKmSlmyc0TqRcJgEwE6xmso0ZoRskN8FnhYq+nWqW5oWiKanuigqOVIaM9yhkghLFl7DBRDAwF5EZFpgoiFHjIjUTFF6EUjhouaRnoEoIGjWlAdjUv9FYiAnkgEZu7ZlZje/SjQXOZ4ws3KTgionsyrXhAPdcRReKZEnuGhUqy7hsXge2YVMgTakUhKtFMwOYc5ckiuVTvVg26Ua3kJG8SYkLRcZdWQQRiwvRCkEzPG5o/XGrEPR9CNAMsm5KTD/5cfLD7mHZFO4+P+4KJz8fj/efmBiGNPIFHXh6G5ZID7zyQVlJJ5H2jdFZXq0i0ZOoLGv0WHTQY7FGZZxewya8AIPM4im9gmAnOA21T3hQwr9JhA54Qbu4WOOChl04WcPQbdAjuEuJ1xSyxGkbD+hFaf5jlmp6UdjwtDnUcihs/4sBWrAQeLmhKq0EXVUrYqXvZtoRVPyZN9WwDl/7WWpmDmGKmvyUCDXJzyixZFiBXGexwwIfDGn3Yvh6I9VtFhSgpe2x+A26gCVXNPkGXiBZnGDwNQmyBbxmPDRFcd+38vttr9VpWVWNUvq0A84BVIyHFBRFaF76gsUzBeHLrtDDBgJ2tAzJTQYn5/rwoVf+on2oRhB0knhQVjaawZFxfdAmJOPaPoK5Hpe2/zbxSzBxZULVB6nUcN1lPsOpyhL95bpB6CKH6TjwBqPBIzjDcRpzigYeegW2D0bIrINHyBcWWJ0iY+DC5Wi8kswwhzaNYKxqr9Sjdf8Vquo7GelnJ+X26ME1kLUQMPLcq+H5nvbnFM66Jkhor/TddhBgKtnUn1lPJ1Gk7n2HbPan7XgKbJmb3rUocJelvDLgytmOFssaGUU0UqgRLAhVVQc3HfaP9Go4Ib3v3ztqpdw/bOKH3ZKw1Vu1dA5MwRadjrYU08zXhFbZHGyUTaeogv0ahYkOOWyh4QoNZQX+r+k62bh40rl4s1rbMbdQokU3ImFpETCEd6dtcLp2xnZwE31qWvipOYeMjqpNGsfz6l6Ud5CgRoIOclEj0NB9+L7y2l9T3c3z0dD7ejj66avB42H9pXXb+cz53Nl2POcb57EzdibOiUMc5vzq/Ob83vuj91fv794/FfXWVn3mU6fx9Lf+BTSwzFA=</latexit>

q3 q<latexit sha1_base64="rfIT9D9d0y6j3E7G/Mua8JkM7vs=">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</latexit><latexit sha1_base64="PrEhbAaupuerCLIeOVxqcBgTA7U=">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</latexit>

1

The approximation is good

The low-energy physics in Twisted Bilayer Graphene is equivalent
to monolayer graphene on the substrate0
Hd ( k)B wT0
HTBG ( k) = B
@ wT
wT

1

<latexit sha1_base64="tp80vazxw9XHI9cT0wXj4W3ZE2g=">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</latexit>

wT0
H d ( k + q1 )
0
0

wT
0
H d ( k + q2 )0<latexit sha1_base64="5+BP5gzTzuclE42f4J3dCiAEbgY=">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</latexit>

T = cosx + siny , where

wT
C
0
C
A
0
H d ( k + q3 )= 2⇡/3<latexit sha1_base64="dTv4mE6p0DdztlWsuvUmCMZkn1k=">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</latexit>

Hd ( k) = vF ( kx x +<latexit sha1_base64="XSl8y5vsh1tijgEya41UHOdKbvM=">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</latexit>0

Hd ( k)
B w 0
HMG ( k) = B
@ w 0
w 0

w 0
⇤
HD
(0, k + q1 )
0
0

⇤
HD
( ,

ky y )w 00⇤HD( , k + q2 )0<latexit sha1_base64="1Iu7HSzGL6+PDZXoOdiEePjjRGc=">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</latexit>k) = R vF ( kx x⇤
HD(ky y )R 1

Unitary Transformation
HTBG ( k) = U HMG U †<latexit sha1_base64="QZSqlcthIun0BxOuCJ1pf4FonA8=">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</latexit><latexit sha1_base64="NqfEXYYEDx4U7+S9jMnbQeqqClU=">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</latexit>

where U = diag( 0 , T0 , T

1

,T )

w 0
C
0
C
A
0
, k + q3 )Comparison
Twisted Bilayer
Graphene

MonolayeGraphene on substrate

Coupling

Interlayer coupling

Periodic potential

Size oMoire BZ

Twisted Angle

Substrate lattice constant(Magic Angle)(Magic lattice constant)

Coupled
Dirac cones

Two identical
Dirac cones

Two Dirac cones with
opposite orientations

Approximation

Neglect longer
momentum hopping

Neglect hopping between a Diraccone and high energy bands

 

 

 

r

 

f

Advanced Materials 2200625 (2022)

Question 2:
In non-Hermitian physics, can two exceptional points be located at
K or K’ so that there is twisted physics exceptional points?
EP +1

+1

K
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K<latexit sha1_base64="RyLRINDoro9CSG9QqeNIIoMU51o=">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</latexit><latexit sha1_base64="PHjXe028yEJ4xfSoco8HJ76WijI=">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</latexit>

K

Graphene0-1

K0<latexit sha1_base64="PHjXe028yEJ4xfSoco8HJ76WijI=">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</latexit>

EP -1

Non-Hermitian

Focus: The generalized non-Hermitian 2D Nielsen-Ninomiya theorem

Nielsen-Ninomiya Theorem for non-Hermitian exceptional points
In 2D non-Hermitian lattices
Integer discriminant numbers quantizes exceptional points
Y⌫(kld ) =i
2⇡<latexit sha1_base64="M0zy21L3LcALKVZzdKuba0uCezQ=">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</latexit>

I<latexit sha1_base64="w4Z6ZOg/WbeA2z9bH761jwkW9jg=">AAAKpniclVZbb9s2FFa7W+tdmm6Pe1HnbkgGJYjcYRuGDWi3OHOQIfDaXLpYrkdRlM2auoSkEhss+/v2F/Yn9ro97pCyHUvKCsyATfp8Hw/PXQpzRoXc3f3z1u233n7n3ffu3G29/8GHH93buP/xqcgKjskJzljGn4dIEEZTciKpZOR5zglKQkbOwulPBj+7JFzQLD2W85wMEzROaUwxkiAabYRBlhOOZMZTlBC1RwXWI9XVgyBBcoIRUz093AzCRE311g9fuEHOs2ik6PcvdcBILAfdEV3C293Ry+U+4HQ8kcMXqqNHG+3dnV37cZsbf7FpO4tPf3T/rhdEGS4SkkrMkBADfzeXQ4W4pJgR3QoKQXKEp2hMBrA1louhssHQ7ucgidw44/BNpWul6ycUSoSYJyEwjYuijhmhFyY3wYNCxt8OFU3zQpIUl3fFBXNl5prguhHlBEs2hw3CnIK5Lp4gjrCEFFQukhNO4A+XEoU1l9QEVHFO4qo0BJtaNxoLMYEUk9hbeGZW47vwxhzlE4pnXlIwSXl25dlwgHueJDOJsyT3jAqZZUxUrwPbkKm/qlRwzOSsmgHEmIcTSfOhms2rdKObi1jcpMSDGmaeKMKYjgteC0E1PF5k/fHKELQCCNAEsm4qWD35pf9z90hXhd3T46aw/9tx7+CJiWFE4oCTtq82YYlV29dbupT2Y2ULP8vLlSeqH2u9QI95Az3mK1SM02vYhBdgkFk8JVcQ7ASlkQowCzX8mkSokBWkifMVzknUhJMVDM0MPYKalPGKgucoreMhudDmd0xTRS4KG546h1gOge1/MUAL4hzN11SlpaCpakks9d1MewoVP/CHCtad10GWmpGGqSQmP9p1q+RnBFsyrEBeZLHBAh8MqXux83ot1XUWFKCl7dPxG3QBSyxp4g28UNBxgsDXJMxm8DdjkSmKh4GVP6x7Lc91WTVSqvMGOAVQUhYRUBS7U12OVQhfduVuVxCwo2ZIbjLYH6mjbV+/UAFUIwgaSTzUpY1mcGRMHdYJSW9hX/kssP23jl+CiUsTyj5IhYLrLvMJSmWWqC9XDUJmOUzHtt/utB/BGYbSMSNu23dfge3tjmvW9iM34BZYnsI94MLlbm8pmSAGbRrDWFW+hgfMsv8KWfadiNWzE73Z2boGshoCRo78BTzdV8GUwFnPBMnd14FXDwJMJZv6gfW0H8fywXeuzf6wHk+OLHPduxoF7rKUVwZcOtvQYlkdowiesm4lWBCqsg5uOhw8Vcvh5KqD4MHTWsqDoyp+1CwJW71lS+fA5HTW6GhLMc18TaiVzeFa2TSKKjxYoDDRIYc1NFqikSjB/zVd+2sX9xsXr1drPeYWShRvRiTSFgFDWHPahucrZ2wHV9E908J75pxrdJRtUjmel/e6eQMJF0jYQC4WCDR0C96v/PrbVHNz2tnxv97p/PpV+/GPizetO86nzmfOpuM73ziPnZ7Td04c7Pzh/OX87fzT2mwdtU5aZyX19q3FmU+cyqf1+78C2O/J</latexit>

DiscE [H](k) =(kld )

dk · rk ln DiscE [H](k)⌫(kjd ) =kjd 2BZ<latexit sha1_base64="aCBWRabK/DcTMwNV4J3oyMvnYYk=">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</latexit>




BZ

dk · rk ln DiscE [H](k) = 0









@BZ






BZ























BZ

Z Yang, AP Schnyder, J Hu, CK Chiu, Physical Review Letters 126 (8), 086401 (2021))

2

L. Fu et al, PRL 120, 146402 (2018
M. Sato et al, PRL 123, 066405 (2019)

Charge neutralization





i
2⇡

I

Ej (k)]

i<jYes, the Nielsen-Ninomiya theorem still holdsX[Ei (k)

✓

E
0

a
E

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Hermitian Dirac node◆

Non-Hermitian exceptional point (EP)S=
Chiral✓symmetry◆<latexit sha1_base64="FUOnMalVD0EZXBUZTz1lX8EfbQ0=">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</latexit>

Symmetryz

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H=

0
h

h
0

TopologicalinvariantWinding number w

Doubling
theoremYes Node #Discriminant number<latexit sha1_base64="aHd2kP1ewFFnwMu8uqwFdOvSK5U=">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</latexit><latexit sha1_base64="fFQM9duFoNT6+hUGcr+lv3C31xQ=">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</latexit>

Difference

Yes Node #2<latexit sha1_base64="fFQM9duFoNT6+hUGcr+lv3C31xQ=">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</latexit>⌫<latexit sha1_base64="ydhR+y3otfIcNP1W5fmRjO58XW4=">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</latexit>2

EPs for all energy levels

Nodes at zero energy<latexit sha1_base64="w4Z6ZOg/WbeA2z9bH761jwkW9jg=">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</latexit>

DiscE [H](k) =Y[Ei (k)Ej (k)]

2i<j<latexit sha1_base64="168+IhOndB1Dy0WZnhYFz9PLDQ4=">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</latexit>X

kjn 2BZ⌫ knj =

X

kjn 2BZi
2⇡I<latexit sha1_base64="EN79Z43yIWTSyaqNQ6vyvt1H4xs=">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</latexit>(kjn )

dk · rk ln det[h(k)] = 0

X

kjd 2BZi⌫(kjd ) =
2⇡

I

@BZ

dk · rk ln DiscE [H](k) = 0

Z Yang, AP Schnyder, J Hu, CK Chiu, Physical Review Letters 126 (8), 086401 (2021)

If crystalline symmetries are preserved, is the minimal number ofthe nodes greater than 2?
17 2D wallpaper groups

Crystalline symmetry constraints

The generalized Nielsen-Ninomiya Theorem for the 17 wallpaper group
Zero-energy (Dirac) nodes protected by chiral symmetry<latexit sha1_base64="+7bbEnp9rI/0pXkJlxll/LZllNM=">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</latexit>

Number=P

ki 2BZ |w(ki )| ⇠ number of Dirac nodes

The minimal number of the topological nodes<latexit sha1_base64="mscDDEi2OmS8exWYgEdtSPDAuPo=">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</latexit>

Graphene

C Le, Z Yang, F Cui, AP Schnyder, CK Chiu, PRB 106, 045126 (2022)s{C6 , S} = 0, {Mx , S} = 0

What about exceptional points in non-Hermitian systems?
non-Hermitian

Hermitian<latexit sha1_base64="168+IhOndB1Dy0WZnhYFz9PLDQ4=">AAAKd3iclVbrbts2FLa7W+rd0u3nfoydu8AZ1MByh20YUKDdkNRBisBLk7SL6RqURNmcqUtIKrHBqq+yV9lr7FH2b4eU7VhSVmACYjLn+3h47pKXciZVt/t3885773/w4Udbd1sff/LpZ59v3/viXCaZ8OmZn/BEvPKIpJzF9EwxxemrVFASeZy+9Ga/GvzlFRWSJfGpWqR0FJFJzELmEwWi8fZfWGbRWGNvNtZx/voPhFmMsIj0Lxd5jnCcIcxpqDrYS3ggFxEsepYXXCzYZKp2H++gdynBoSC+ZrnuIZwy0JmwWAH5GYki0lmf2c1RgL0IlCPsB4mCu4nHidFqhCDloDOgajjtFKLdEXrcRePtdnevax9U37jLTbuxfAbje1t/4iDxs4jGyudEyqHbTdVIE6GYz2newpmkKfFnZEKHsI1JROVI21Dn6FuQBChMBPzFClnp5glNImnCBMyIqKmsYkboeNFt8DBT4U8jzeI0UzT2i7vCjCOVIJM6FDBBfcUXsCG+YGAu8qcEoqsgwaWL1FRQ+EcoBTEsu6SnoEoIGpalHtjUutVYiAkUEA2dpWdmNb5LZyJIOmX+3IkyrphIrh0bDnDPUXSu/CRKHaNCJQmX5evANmKquyyVwudqXs4A4dzxI8XSkZ4vynSjW8hQ3qbEgQ7hjsy8kE0yUQlBOTxOYP1xihC0MARoClk3/aGfPh882z/Oy8L989O6cPD7af/wqYlhQEMsaNvVHVhC3Xbz3byQDkKNjdFJWqzQIYMwz5foqaihp2KNykl8A5vwAgwyi8f0GoIdkTjQ2OdeDr8mEdrjGa3jYo0LGtThaA3DqIAeIXXKZE3xFySu4h69zM3vhMWaXmY2PFUOtRwK2/9igBYiBFlsqIoLQV3Viljou512AhU/dEca1r23OInNwPSZoiY/OUJl8gvqWzKsQF5mscYCHwxp/3Lv7UaqqywoQEs7YJN36AKWXNHkO3ieZJOIgK8wiefaTmVTFA+wlT+oeq0u8qJqlNIXNXAGoGI8oKAoRLO8GOgQvuQaPSwhYEfFkNRkcDDWxw/d/LU2wx4EtSQe5XrzzXFUJUT9pX0+4bqf2/7bxK/AxJUJRR/EUsN1V+mUxCqJ9HfrBqHzFKZj22332o/gDCfxhFPUdtEbsL3dQ2ZtP4LXkgVWp/w+cOFy1F9JpoRDm9qXlgsvrXX/ZaroOxnqF2d5p7d7AyQVBIwcu0t4dqDxjMJZxwQJHeTYqQYBppJN/dB6OghDdf9nZLM/qsZTEMvc9K5Cgbss5Y0BV87WtFhWzyiClzsqBQtCVdTBbYfxiV4NJ6QP8f2TSsrxcRk/rpeErd6ipVNgCjavdbSlmGa+IVTK5mijbGpF5R0uUZjokMMKGqzQQBbg/5qug42LB7WLN6u1GnMLRVrUIxLkFgFDuJ228EnjVj9g6pvz3p77w17vt+/bT9zlx81W46vGN41Ow2382HjS6DcGjbOG39xpPm+eNc/v/tP6urXT6hTUO83lmS8bpafl/gs059m9</latexit>X

kjn 2BZ⌫ knj =

X

kjn 2BZi
2⇡I<latexit sha1_base64="EN79Z43yIWTSyaqNQ6vyvt1H4xs=">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</latexit>(kjn )

dk · rk ln det[h(k)] = 0

X

kjd 2BZi⌫(kjd ) =
2⇡

I

@BZ

dk · rk ln DiscE [H](k) = 0

They share the similar mathematical structure

But the discriminant number of an EP has an additional constraint.

EP at a C2 rotation center<latexit sha1_base64="RDEaj770zi4YtxLYLQeG0nnEVy4=">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</latexit>i⌫disc (KEP ) =2⇡Z

1/2 (KEP )

i
=2⇥
2⇡

I



d ln DiscE [HnH ](k) + d ln DiscE [HnH ](Ĉ2 k)⇣

d ln DiscE [HnH ](k)1/2 (KEP )

= 2j

⌘

C2 rotation symmetry<latexit sha1_base64="KqEY/bEkJgladuiEONeBqJTFTcw=">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</latexit>ln DiscE [HnH ](k) = ln DiscE [HnH ](Ĉ2 k)

Brillouin Zone

<latexit sha1_base64="+7aHQ+aZjvs2OdIYG0nE3xL2Mm0=">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</latexit>

Ĉ2 k
=

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1/2 (KEP )KEP<latexit sha1_base64="LQxbGNixs+B3Tysye4kJM+PNG4c=">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</latexit><latexit sha1_base64="4CMymJbjL6FZ3MPQEVNsXHvzD7c=">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</latexit>

1/2 (KEP )<latexit sha1_base64="6vqB0BQi11vnDK8CbyjRhT/PqK4=">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</latexit>k<latexit sha1_base64="J0F9ACwr9ssb3Xv8yq0PcNsjUYg=">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</latexit><latexit sha1_base64="UKC+zmzjDh1rN84YG3kOsZxbx+s=">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</latexit>(KEP ) Integral path

Cn rotation center

For chiral symmetry<latexit sha1_base64="bn4YjVxDs7fwYRRynX+6kFQpcnM=">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</latexit>⌫disc (KEP ) = nj

<latexit sha1_base64="kAcTfNyveqS7O7TLvh+tJbwLUxs=">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</latexit>ln det[h(k)] 6= ln det[h(Ĉ2 k)]⌫ 6= nj<latexit sha1_base64="IQ6UVhNM1zyr74RCYAEEwUuXkog=">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</latexit>

The generalized non-Hermitian Nielsen-Ninomiya Theorem
Hermitian nodes with [Cn+ , S] = 0, [Mx+ , S] = 0
The minimal number of exceptional points<latexit sha1_base64="0a+NcsYVUIr2BL4MGlN+UvPEdr0=">AAAJ0XiclVZbb9s2FFbbXeLslnZv2ws7dUC6qYHlDrsBA1oUyRxkC7zc2tVyDYqiZMHUJSSV2GBVFHvdj9jbXre/s3+zQ8p2LCkrMAE2qfN9PDx3yM9ZLGS3+8+Nm7feevuddzc6m++9/8GHH23dvnMmsoITekoylvFnPhaUxSk9lbFk9FnOKU58Rp/60ycaf3pBuYiz9ETOczpKcJTGYUywBNF465Phk3H64kvnePRD10EeGv48ni1ex1t2d6drHtTeuIuNbS2ewfj2xh9ekJEioakkDAsxdLu5HCnMZUwYLTe9QtAckymO6BC2KU6oGCnjRIk+B0mAwozDL5XISNdPKJwIMU98YCZYTkQT00LHT66Dh4UMvx2pOM0LSVNS3RUWDMkM6aCgIOaUSDaHDSY8BnMRmWCOiYTQ1S6SE07hhUuJ/YZLagKqOKdhXeqDTZvXGgsxgdTQ0Fl4plftu3AijvNJTGZOUjAZ8+zSMeEA9xxJZ5JkSe5oFTLLmKhfB7ZhXTd1qeCEyVk9A5gxhyQyzkdqNq/TtW4uQnGdEgdqjzmi8MM4KngjBPXwOIHxx6lCsOlBgCaQdV156vFPgx93D8u6cPfspC0c/HrS33+sYxjQ0OPUdtU2LKGy3fJ+WUkHofK00VlerTxRg7AsF+gJb6EnfIWKKL2CdXgBBpnBU3oJwU5wGiiPML+Ef50I5bOCtnG+wjkN2nCygqEJoUdwmxKtKGSO0ybu0/NS/0dxquh5YcLT5FDDobD9LwZowZzj+ZqqtBK0VS2Jlb7raUdQ8UN3pGDdeeVlqR5FJJZU56dEqE4+psSQYQXyIostFvigSbvnO6/WUt1kQQEa2l4cvUEXsMSSJt7A80UcJRh8TfxsBq8ZC3RR3POM/F7Ta/m8rKpGSvW8BU4BlDELKCgK0bT0eBxNJIQvu0QPagjY0TAk1xkcjNXhA7d8oTyoRhC0knhQVjbqwZExddAkJP2FfQQz1S9N/63jF2Di0oSqD1Kh4LqLfIJTmSXqi1WD0FkO09F27Z79EM4wnEaMIttFL8F2u4f0aj9EHjfA8hTpAxcuR/2lZIIZtGkIY1W5peqt+q+QVd+JUB2fltu9+1dA1kDAyLG7gKd7yptSOOvoIKG90nOaQYCpZFI/NJ4OwlDe/R6Z7I+a8eTYMNe9a1DgLkN5qcGlsy0thtXTimgoUS1YEKqqDq477B2p5XBCat+7e9RIuXdYxw/bJWGqt2rpHJg8nrU62lB0M18RGmVzsFY2raLy9xcoTHTIYQMNlmggKvB/TdfB2sWD1sXr1dqMuYESxdsRCUqDgCHMTFv4pHGbHzDtzVlvx/16p/fLV/aj7xYfNxvWp9Zn1rblWt9Yj6y+NbBOLWK9tv60/rL+7hx35p3Xnd8q6s0bizMfW7Wn8/u/r2SaQw==</latexit><latexit sha1_base64="qYG5yITIBOu1NNlK3W1mVrZ3pEc=">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</latexit>

WG
#1
#2
#3,4,5#6,7,8,9#10#11,12
#13
#14
#15
#16
#17

Generators
C2
Mx
C2 and Mx
C4
C4 and Mx
C3
C3 and My
C3 and Mx
C6
C6 and Mx

Rotation centers
No centers( , X, Y, M ) : 2Z⇥⇥
( , M ) : 4Z & (X, Y ) : 2Z⇥
( , K, K ) : 3Z
0

(K, K 0 ) : 3Z(⇥⇥

) : 6Z & (K, K 0 ) : 3Z & (M

1/2/3 ) : 2Z⌫abs
2
4 (G, , X, Y, or M )2̃
4̃
8 (G, , X, Y, or M )8̃
6 (G, , K, K 0 , or M1/2/3 )6̃ or 6̃(K, K 0 )
6̃
12 (G, , K, K 0 , or M1/2/3 )˜
12<latexit sha1_base64="bn4YjVxDs7fwYRRynX+6kFQpcnM=">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</latexit>

Cn rotation constraint⌫disc (KEP ) = njK<latexit sha1_base64="RyLRINDoro9CSG9QqeNIIoMU51o=">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</latexit><latexit sha1_base64="ehWLHsWeyKbcKTGsPogtENNVNAY=">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</latexit><latexit sha1_base64="PHjXe028yEJ4xfSoco8HJ76WijI=">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</latexit>

K

0

Minimal EP #⌫abs =

X

ki 2BZ

Mx<latexit sha1_base64="dokcr2DOEma+pv+dUPT9+lcqDlM=">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</latexit>

Mx re ection symmetry|⌫disc (ki )|destroys exceptional points at K and K’fl

C Le, Z Yang, F Cui, AP Schnyder, CK Chiu, PRB 106, 045126 (2022)

K and K 0 are C3 rotation centers<latexit sha1_base64="tcWx7Mp6/xOIxm0ChLW6ci5e5WM=">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</latexit><latexit sha1_base64="mOCIge/uHDjNrHbmgq8AgjU6YPo=">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</latexit>⌫disc (KEP ) = 3j
EP +1

EP +3
EP -1

Non-HermitianEP -3

Non-Hermitian

No Mx re ection symmetryfl

Non-Hermitian EP system: no analogous twisted bilayer graphene

The extension of the non-Hermitian Nielsen-Ninomiya Theorem
Magnetic Wallpaper Group
Conjugation crystalline symmetries ḡ
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M̄i = Mi K

K Complex conjugation<latexit sha1_base64="Hg2QG4V5QLJ3nch9qDe74E8MC0Q=">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</latexit><latexit sha1_base64="Soz82l2iKOvu56sGIL0ZMmOIVnQ=">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</latexit>

WG
#1
#2
#3,4,5#6,7,8,9#10#11,12

#13
#14

#15

#16
#17

Generators
C2
C̄2
Mx
M̄x
C2 and Mx
C2 and M̄xC̄2 and Mx /M̄x
C4C̄4
C4 and Mx
C4 and M̄xC̄4 and Mx /M̄x
C3
C̄3
C3 and My
C3 and M̄yC̄3 and My /M̄y
C3 and Mx
C3 and M̄xC̄3 and Mx /M̄x
C6C̄6
C6 and Mx
C6 and M̄xC̄6 and Mx
C̄6 and M̄xRotation centers
No centers( , X, Y, M ) : 2Z⇥⇥
No centers⇥( , X, Y, M ) : 2Z⇥
( , M ) : 4Z & (X, Y ) : 2Z⇥⇥
( , M ) : 4Z & (X, Y ) : 2Z⇥
( , K, K ) : 3Z⇥
0

(K, K 0 ) : 3Z
( , K, K 0 ) : 3Z⇥⇥
( , K, K ) : 3Z⇥( ) : 6Z & (K, K ) : 3Z & (M00(K, K 0 ) : 3Z(⇥⇥
(K, K ) : 3Z

) : 6Z & (K, K 0 ) : 3Z & (M0

1/2/3 ) : 2Z

1/2/3 ) : 2Z⌫abs
2
4 (G, , X, Y, or M )2̃2̃
2 (MLs, , X, Y, or M ) or 44̃
4 (MLs, , X, Y, or M ) or 8
2̃ MLs or 4̃
8 (G, , X, Y, or M )4̃8̃
8 (MLs, , X, Y, or M ) or 16
4̃ MLs or 8̃
6 (G, , K, K 0 , or M1/2/3 )0
6̃ or 6̃(K, K 0 )
6 (MLs, , or M1/2/3 ) or 12 (G, K, or K 0 )
06̃
6 (MLs, , K, K 0 , or M1/2/3 ) or 120
12 (G, , K, K 0 , or M1/2/3 )6̃ or 6̃(K, K 0 )˜
1212 (MLs, , K, K 0 , or M1/2/3 ) or 24
˜
6̃ MLs or 12˜
6̃ MLs or 6̃(K, K 0 ) or 12

Summary•

Twisted bilayer graphene and monolayer graphene withspeci c periodic potential approximately share the identical
low energy physics, which can host at bands.

K0<latexit sha1_base64="C6XvfM0l8N29gkbjeUYM9aIxzoA=">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</latexit>

Periodic potential

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K0<latexit sha1_base64="C6XvfM0l8N29gkbjeUYM9aIxzoA=">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</latexit>

K<latexit sha1_base64="BaygwqNv2sCEd5AFu12B9ZLJ8vE=">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</latexit>

Advanced Materials 2200625 (2022)•

Having exceptional points at K and K’ is dif cult. The
discriminate number must be a multiple of 3 and the re ection
symmetry along K must be broken.
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fl

fi

fl

fi

PRB 106, 045126 (2022)

Thank you !

Questions
Is it possible to have exceptional points separately located at K point and K' point?
EP +1

+1

K
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K<latexit sha1_base64="RyLRINDoro9CSG9QqeNIIoMU51o=">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</latexit><latexit sha1_base64="PHjXe028yEJ4xfSoco8HJ76WijI=">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</latexit>

K

0

-1

K0
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Graphene

Non-Hermitian?

K and K 0 are C3 rotation centers
If so, it can lead to twisted non-Hermitian bilayer.
<latexit sha1_base64="tcWx7Mp6/xOIxm0ChLW6ci5e5WM=">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</latexit>

EP

EP

EP -1

<latexit sha1_base64="P/RgWgvt4FKk0yh6kUhXdyt4eB8=">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</latexit>⌫disc (Ĉn KEP ) = ⌫disc (KEP ), ⌫disc (M̂ KEP ) =<latexit sha1_base64="CrJwtm6pjyE7RO/zCpZwJZJVAEo=">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</latexit>

KEP<latexit sha1_base64="LQxbGNixs+B3Tysye4kJM+PNG4c=">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</latexit><latexit sha1_base64="fltSkljiSbaQksSmaZ0E8UpyFUA=">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</latexit>⌫disc (KEP )

gKEP

g

The generalized Nielsen-Ninomiya Theorem<latexit sha1_base64="qYG5yITIBOu1NNlK3W1mVrZ3pEc=">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</latexit>

WG
#1
#2
#3,4,5#6,7,8,9#10#11,12
#13
#14
#15
#16
#17

Generators
C2
Mx
C2 and Mx
C4
C4 and Mx
C3
C3 and My
C3 and Mx
C6
C6 and Mx

Rotation centers
No centers( , X, Y, M ) : 2Z⇥⇥
( , M ) : 4Z & (X, Y ) : 2Z⇥
( , K, K ) : 3Z
0

(K, K 0 ) : 3Z(⇥⇥

) : 6Z & (K, K 0 ) : 3Z & (M

1/2/3 ) : 2Z⌫abs
2
4 (G, , X, Y, or M )2̃
4̃
8 (G, , X, Y, or M )8̃
6 (G, , K, K 0 , or M1/2/3 )6̃ or 6̃(K, K 0 )
6̃
12 (G, , K, K 0 , or M1/2/3 )˜
12The generalized Nielsen-Ninomiya Theorem<latexit sha1_base64="qYG5yITIBOu1NNlK3W1mVrZ3pEc=">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</latexit>

WG
#1
#2
#3,4,5#6,7,8,9#10#11,12
#13
#14
#15
#16
#17

Generators

Rotation centers
No centers( , X, Y, M ) : 2Z

C2Mx
C2 and Mx
C4
C4 and Mx
C3
C3 and My
C3 and Mx
C6
C6 and Mx⇥⇥( , M ) : 4Z & (X, Y ) : 2Z⇥
( , K, K ) : 3Z
0

(K, K 0 ) : 3Z(⇥⇥

) : 6Z & (K, K 0 ) : 3Z & (M

1/2/3 ) : 2Z⌫abs
2
4 (G, , X, Y, or M )2̃
4̃
8 (G, , X, Y, or M )8̃
6 (G, , K, K 0 , or M1/2/3 )6̃ or 6̃(K, K 0 )
6̃
12 (G, , K, K 0 , or M1/2/3 )˜
12The simple rule for a wallpaper group with Cn rotation generator:⌫abs = 2n<latexit sha1_base64="gpP7MNo+MkEcvpf6VyODeDv531c=">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</latexit>

Cn rotation symmetry connects n EPs with ⌫disc = +1
connects n EPs with ⌫disc = 1<latexit sha1_base64="kkdJMPGUraCFCiWv+aM57CHm5w8=">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</latexit><latexit sha1_base64="moWzzMgpPc2UtOkuLyXvu54KI2A=">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</latexit><latexit sha1_base64="mNlqbgdiM1jh5PI+TtkkVyMUBuI=">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</latexit>

Application: A surface of a 3D non-Hermitian topological insulator can violate
the Nielsen-Ninomiya theorem. Without breaking Cn rotation symmetry, theanomalous surface has n EPs.
+1

3D topological invariant ⌫3D = nl
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C2<latexit sha1_base64="48cBXRwRTv420y147hx+1QwCUfw=">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</latexit>+1Summary
1. When crystalline symmetries are preserved, the minimal number ⌫abs of
exceptional points is more than two.
<latexit sha1_base64="sPHDMuRr5UTXJoAjZ+sWDF9+6Nw=">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</latexit>

2. At the Cn rotation center, the charge of the EP is limited to<latexit sha1_base64="bn4YjVxDs7fwYRRynX+6kFQpcnM=">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</latexit>⌫disc (KEP ) = nj
3. The simple rule for a wallpaper group with Cn rotation generator:⌫abs = 2n<latexit sha1_base64="gpP7MNo+MkEcvpf6VyODeDv531c=">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</latexit>

4. Application: Cn rotation symmetry leads to that the topological invariant of
the non-Hermitian 3D topological bulk must be a multiple of n.
#1

#2

#1
M

Y

#(3, 4, 5)⌫3DM#2 nlY =<latexit sha1_base64="BaqkrJXi7P0TkVwJlxDSwRYXjzU=">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</latexit>#(6, 7, 8, 9)

MY #(3, 4, 5)

M F Cui, AP Schnyder,
M PRB 105,Y
#2#(3,045126
4, 5) (2022)
K 0 #1 C Le,YZ Yang,YCK Chiu,+3** Y -3* Y X M
X M
MX
Y
K
*
*
*
X
X
<latexit sha1_base64="nEbjpIk+2MSgzhCyUcdpXOXwvgg=">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</latexit>

Y#(6
M

*
Y

<latexit sha1_base64="6RHusok5Hxznr+KvJtBzTMc0gUU=">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</latexit>(a)#(11, 12)
(a)

*

X
(a)

(b)
#13
(b)

*

*

X
(b)

X

(c)

(c)

#14
(c)

#(6, 7, 8, 9)

X
*

(e)

#15

#16
(e)