Novel Quantum States in Condensed Matter (NQS2022)
Kyoto Univ, Nov. 1-4, 2022

Physics of Graphene&moiré materials

Mikito Koshino (Osaka University)

Lecture plan
1. Graphene (day 1)
Dirac electrons in graphene
Pseudospins & topology
Landau levels
Singular orbital diamagnetism

2. Moiré 2D materials (day 2 & 3)
Twisted bilayer graphene
Flat bands & superconductivity
Continuum model
Hofstadter butterfly
Twisted quasicrystals

Family of 2D materials
Graphenehexagonal BN

Transition metal
dicharcogenides

Phosphorene

2D superconductors, 2D ferromagnets, etc…

Van der Waals heterostructure

A. Geim, Nature 499, 419 (2013)

Overlooked aspects in “LEGO picture”
Lattice mismatch

Rotational degree of freedom… Does the interference patternaffect the physical properties?

Twisted bilayer graphene (TBG)

Moiré patterns
in atomic lattices

Magic angle and flat bands

Energy (meV)

Flat bands appear
at certain “magic angles”
R. Bistritzer and A. MacDonald, PNAS
108, 12233 (2011).

Correlated Insulating states andsupercondutivity in flat bands
Y. Cao et al, Nature 556, 43 (2018)
Y. Cao et al, Nature 556, 80 (2018) →M. Yankowitz et al., Science 363, 1059 (2019).

Family of moiré 2D materials
Twisted bilayer graphene

Twisted multilayer graphenes

Park et al, Nature 590, 249 (2021)

Graphene-hBN

Dean, et al, Nature 497, 598 (2013)

ABC-trilayer graphene-hBN

TMDC-TMDC
C. Jin et al,Nat. Phys. 15, 1140 (2019)

Chen et al Nature 572, 215 (2019)

Twisted bilayer graphene (TBG)

Moiré period

𝜃 = 2∘

𝜃 = 5∘

L = 7.04nm

2.82nm𝜃 = 10∘1.41nmMoiré period
cos 𝑘𝑥cos 𝑘′𝑥cos 𝑘𝑥 + cos 𝑘 ! 𝑥

cos (𝑘 ! − 𝑘)𝑥=
moiré wave number

Moiré period in TBG
𝐴 = 𝑅 (rotation matrix)𝒃"

Moiré wave vector

𝑮"# = 𝒃" − 𝒃"!

𝜃
𝒃"! = 𝑅 𝒃"
𝑅 ! "# = 𝑅

Moiré period (real space)

|𝑮#
" |=

𝐿# =

𝜃
2 sin 𝑏
2

𝑎
2 sin

𝜃
2

Continuum model
Huge number of atoms(10,000 at 𝜃 = 1∘ )Coarse-graining
to extract the long-wave components

Continuum model

Lopes dos Santos et al, PRL 99, 256802 (2007)
Bistritzer and MacDonald, PNAS 108, 12233 (2011)
Kindermann and First, PRB 83, 045425 (2011)
Moon and Koshino, PRB 87, 205404 (2013)graphene’s Dirac Hamiltonian

𝐻$ = 𝑣𝝈 ⋅ 𝒑

Moiré interlayer coupling:
𝑈$$
𝑈%$

𝑈$%
= 𝑢&
𝑈%%

1
1

1 '𝒒!⋅𝒓
1
𝑒
+
1
𝜔

1
𝜔∗ '𝒒"⋅𝒓
𝑒
+ ∗
𝜔
1

𝒒%
Brillouin zone of graphene 1graphene 2𝒒'𝒒&

𝜔 '𝒒#⋅𝒓𝑒
1

Brillouin zone of graphene 1graphene 2

Band structure of TBG
K valley“magic angle”

Flat band

K’ valleyFlat band physics
Total energy of electrons = kinetic energy + potential energy(band energy)E(electron-electron interaction)Eflat band

ky

ky
kx

kx
No kinetic energy→ physics is governed byelectron-electron interaction!

Real space picture
Electrons attracted to AA regions

-e
-e

-e
-e

Gigantic “atoms”

Experimental facts

Tear-and-stack technique

Kim et al,
Nano Lett. 16, 1989 (2016)

Correlated insulating states & superconductivity
q = 1.10°

nBand filling(number of the occupied minibands)
q = 1.05°
K, K’
n=+4

n=-4

𝜅

𝛾

𝜇

𝜅′

8 minibands(including spin and valleys)
Lu et al.,Nature 574, 653 (2019).

Band insulator
Superconducting

Correlated insulator

Analogy to quantum Hall ferromagnet
Monolayer graphene in B-field
0th Landau level

Spontaneous symmetry breaking

Nomura and MacDonald, Phys. Rev. Lett. 96, 256602 (2006)

Analogy to quantum Hall ferromagnet
In TBG (zero B-field)

Ochi, Koshino and Kuroki, PRB 98, 081102 (2018)
Xie & MacDonald, Phys. Rev. Lett. 124, 097601 (2020).
Kang & Vafek, Phys. Rev. Lett. 122, 246401 (2019).
Liu et al, A. Phys. Rev. Research 3, 013033 (2021)
Liu & Dai, Phys. Rev. B 103, 035427 (2021).
Wu & Das Sarma. Phys. Rev. Lett. 124, 046403 (2020).
Zhang et al, Phys. Rev. B 102, 035136 (2020)
Bultinck. et al. Phys. Rev. X 10, 031034 (2020)

Quantum anomalous Hall insulator!

Intrinsic quantized anomalous Hall effect in TBG
Serlin et al., Science 367, 900(2020)alignedC2T extrinsically broken by aligned hBN substrate

Correlated insulating states & superconductivity
q = 1.10°

nBand filling(number of the occupied minibands)
q = 1.05°
K, K’
n=+4

n=-4

8 minibands(including spin and valleys)
Lu et al.,Nature 574, 653 (2019).

Band insulator
Superconducting

Correlated insulator

Twisted trilayer graphene

Layer3 : 𝜽
Layer2 : 𝟎
Layer1 : 𝜽Large Pauli limit in twisted trilayer graphene
Cao, et al. Nature 595, 526 (2021)

Derivation of the continuum model

Continuum model
Coarse graining to extract the long-wave components

J. Lopes dos Santos, N. Peres, and A. Castro Neto, PRL 99, 256802 (2007),
R. Bistritzer and A. MacDonald, PNAS 108, 12233 (2011),
M. Kindermann and P. First, PRB 83, 045425 (2011).
Moon and Koshino, PRB 87, 205404 (2013).

Continuum model

Lopes dos Santos et al, PRL 99, 256802 (2007)
Bistritzer and MacDonald, PNAS 108, 12233 (2011)
Kindermann and First, PRB 83, 045425 (2011)
Moon and Koshino, PRB 87, 205404 (2013)graphene’s Dirac Hamiltonian

𝐻$ = 𝑣𝝈 ⋅ 𝒑

Moiré interlayer coupling:
𝑈$$
𝑈%$

𝑈$%
= 𝑢&
𝑈%%

1
1

1 '𝒒!⋅𝒓
1
𝑒
+
1
𝜔

1
𝜔∗ '𝒒"⋅𝒓
𝑒
+ ∗
𝜔
1

𝒒%
Brillouin zone of graphene 1graphene 2𝒒'𝒒&

𝜔 '𝒒#⋅𝒓𝑒
1

graphene’s Dirac Hamiltonian

𝐻$ = 𝑣𝝈 ⋅ 𝒑

Moiré interlayer coupling:
𝑈""
𝑈#"

𝑈"#
= 𝑢$
𝑈##

1
1

1 %𝒒!⋅𝒓
1
𝑒
+
1
𝜔

1
𝜔∗ %𝒒"⋅𝒓𝑒+ ∗𝜔1

𝜔 %𝒒#⋅𝒓𝑒1

Derivation of interlayer Hamiltonian (1D version)
M. Koshino, New J. Phys. 17 015014 (2015)

2D version:
Bloch state of layer 1
Bloch state of layer 2Interlayer coupling:Coupling condition

G = n b : Layer 1’s reciprocal vector
G’ = n’b’: Layer 2’s reciprocal vector

Ganaralized Umklapp condition
Brillouin zone (Extended)b
Layer 1kk+b = k’+b’k’

Layer 2b’t(q)Coupling strengtht(k+b)

Fourier trnsfm.
of T(r)0k+b= k’+b’qCoupling condition

G = n b : Chain 1’s reciprocal vector
G’ = n’b’: Chain 2’s reciprocal vector

Coupling condition:b
Chain 1kk+b = k’+b’ ... ”meeting point”

k’

Chain 2b’t(q)Coupling strengtht(k+b)

Fourier trnsfm.
of T(r)0k+b= k’+b’qCoupling strength

G = n b : Chain 1’s reciprocal vector
G’ = n’b’: Chain 2’s reciprocal vector

Coupling condition:kb

Chain 1bbk’
Chain 2

b’
b’
b’

b’
b’

b’

t(q)
Fourier trnsfm.
of T(r)0q

Long-wavelength limit
When |b-b’| << b, b’

k’ = k + n GM

kb

Chain 1

GM = b – b’ (moire reciprocal vector)

b

b

k’
Chain 2

b’
b’
b’

b’
b’

GMGMGM
GM << b, b’

LM >> a, a’(moire period >> atomic scale)b’Twisted bilayer graphene~
A

A

Bloch basis of layer 1

Bloch basis of layer 2
B~
BInterlayer couplingt(q): Fourier transform of T(R)
M. Koshino, New J. Phys. 17 015014 (2015)
See also: R. Bistritzer and A. MacDonald, PNAS 108, 12233 (2011)

Twisted bilayer graphene𝒕(𝒒)Continuum model for TBG
Interlayer coupling:

𝑈""
𝑈#"

𝑈"#
= 𝑢$
𝑈##

1
1

1 %𝒒!⋅𝒓
1
𝑒
+
1
𝜔

1
𝜔∗ %𝒒"⋅𝒓𝑒+ ∗𝜔1

𝜔 %𝒒#⋅𝒓𝑒
1

Continuum model vs. lattice modelq = 3.89°

q = 2.65°

lattice (tight-binding)Moon and Koshino,
Phys. Rev. B 87,
205404 (2013)

q = 1.47°

continuumGraphene on hBN

Graphene + hexagonal-BN
Grapheneh-BN
1.8 % lattice constant mismatchC

B N

Moon and Koshino,
Phys. Rev. B 90, 155406 (2014)

Continuum model

G

𝑈
hBN

Reduced 2×2 Hamiltonian (second order perturbation):

𝐻()) = 𝐻* + 𝑈 + 𝐸 − 𝐻,-. /' 𝑈“effective potential”

Graphene + hexagonal-BNBand structure

Moon and Koshino,
Phys. Rev. B 90, 155406 (2014)

See also
Kindermann et al, PRB 86, 115415 (2012).
Wallbank, et al, PRB 87, 245408 (2013).
Jung, et al, PRB 89, 205414 (2014).

Capacitance spectroscopy
G. Yu, et al., Nat. Phys. 10, 525 (2014).

Double walled carbon nanotube

Double-walled carbon nanotube (DWNT)
period (inner)period (outer)--- can be treated as “1D moire system”
Iijima, Nature 354, 56 (1991).
R. Saito, et al , J. Appl. Phys. 73, 494 (1993).
J.-C. Charlier and J.-P. Michenaud, Phys. Rev. Lett. 70, 1858(1993).
S. Okada and A. Oshiyama, Phys. Rev. Lett. 91, 216801 (2003).
P. Lambin, V. Meunier, and A. Rubio, Phys. Rev. B 62, 5129 (2000).
K.-H. Ahn, et al, Phys. Rev. Lett. 90, 026601 (2003).
S. Roche, et al, Phys. Rev. B 64,121401 (2001).
Y.-G. Yoon, et al, Phys. Rev. B 66, 073407 (2002).
S. Uryu, Phys. Rev. B 69, 075402 (2004).…

Moire pattern in DWNT

From http://www2.chem.umd.edu/groups/wang/

Projection to radial direction

Moire pattern in DWNT

M. Koshino, P. Moon, Y-W. Son
submitted

M. Koshino, P. Moon, Y-W. Son,
Phys. Rev. B 91, 035405 (2015).

Flat bands in DWNT
decoupled

coupledVery long-period effective potential→ Bound states at the potential minima

Twisted interface of
3D topological insulator

Kawakami Koshino,
Twisted 3D topological insulator Fujimoto,arXiv:2206.13408

Interface state of twisted topological insulators
θ
stack

twist

Center of Dirac cones DOES matterΓgapped

1D bandM

M

M

M
dispersiveflatWhy is the Dirac cone center important?
(graphene)

Kawakami Koshino,
Twisted 3D topological insulator Fujimoto,arXiv:2206.13408

Wave function in real space
𝜃=8

𝜃=4
𝑦

𝜓$ +

𝑥
𝐿* ~

strongly coupled1
𝜃

tunable by 𝜽

weakly coupled

Twist 3D TI = Array of Tomonaga-Luttinger liquids
・exotic quantum Hall states C. Kane, et al., PRL(2002)
・topological phases
T. Neupert, et al., PRB (2014)
・quantum spin liquids
T. Meng, et al., PRB (2015)

Graphene on TMDC

Graphene on TMDC

MoS2

spin-orbit (SO) coupling induced on graphene
Z. Wang, et al, Nature communications 6, 8339 (2015).
M. Gmitra and J. Fabian, Physical Review B 92, 155403 (2015).
A. W. Cummings, J. H. Garcia, J. Fabian, and S. Roche, Phys.
Rev. Lett. 119, 206601 (2017).

… previous calculations mostly limited to 0 deg. rotation

Graphene on TMDC with twist

Y. Li and M. Koshino,
Phys. Rev. B 99, 075438 (2019)

Yang Li

Incommensurate, and no periodicity … how to calculate?

Effective SOC potential↓↑

Hamiltonian of graphene & TMDC:
graphene

Interlayer coupling

HT: TMDC

Hint: Interlayer
coupling

TMDC

HG: Graphene

Effective Hamiltonian of graphene:(second order perturbation)𝚪M(Effective SOC potential)K𝚪Induced spin-orbit potential as a function of q
Y. Li and M. Koshino,
Phys. Rev. B 99, 075438 (2019)
Zeeman-likeWS2

Rashba,(Zeeman)(Rashba)--- SO coupling strongly enhanced by rotation--- Zeeman vanishes at 30 deg (Rashba dominates)

Interlayer coupling strength[eV]AGraphene’s BZ

TMDC’s BZt(q) almost vanishes outside 1st BZ

Dominant coupling processes[eV]

Coupling condition

Graphene’s 1st BZ
TMDC’s 1st BZA=0t(white dot) = 0.14 eVGraphene’s K point
couples with only
3 points of TMDC

Calculation of the effective SOC potential(For K valley)𝜃-dependenceofHT

Second order perturbation:=(Effective potential)
EG : Energy level of graphene’s K

Numerical calculation
Extract spin-orbit parameters,
Zeeman-like

Rashba-likeInduced spin-orbit potential as a function of q
Y. Li and M. Koshino,
Phys. Rev. B 99, 075438 (2019)
Zeeman-likeWS2

Rashba,(Zeeman)(Rashba)--- SO coupling strongly enhanced by rotation--- Zeeman vanishes at 30 deg (Rashba dominates)

Why the proximity SOC
enhanced near 18 degree?
K

M
𝚪

K

M
𝚪

K
𝚪

Graphene’s K couples withthree k points of TMDC

M

WS2
At 17.9 deg:Graphene’s K couples
to Q valley of TMDC, where
the spin splitting is large.
→ large proximity SOC ingraphene
WS2

Different TMDCs

Y. Li and M. Koshino,
Phys. Rev. B 99, 075438 (2019)

WS2

MoS2
(Zeeman),,(Rashba)

MoSe2

,

,

WSe2

Selenides have larger Rashba-SOC at 30 deg.

Twist bilayer graphene on WSe2
J. Lin et al. arXiv:2102.06566

Y. Li and M. Koshino,
Phys. Rev. B 99, 075438 (2019)

,

WSe2

“Our sample features a twist angle of ∼16 °between tBLG and WSe2, falling in the range thatis predicted to induce the strongest SOC strength.”Moiré systems in magnetic fields

E (meV)

Twisted bilayer graphene in B-field
B
Upper layer

B (tesla)

E (meV)hybridized bymoire pattern

Lower layer
B (tesla)

q = 2.65°

Energy spectrum vs. B

E (meV)

4
Upper layer

0+
E (meV)switch on
moire coupling
Lower layer

Energy (eV)

-12

-4

4
12
20

-4
0
-4

B (tesla)magnetic field (1 = 169T)
Moon and Koshino, Phys. Rev. B 85, 195458 (2012),
See also: Bistritzer and MacDonald, Phys. Rev. B 84, 035440 (2011).

Graphene on hBN

Moon and Koshino, Phys. Rev. B 90, 155406 (2014)
X. Chen, et al., PRB 89, 075401 (2014),
G. L. Yu et al.,Nat. Phys. 10, 525 (2014).

Spectrum in B-fieldG
hBN

B-field
K, K’Hofstadter’s butterfly

Hofstadter, Phys. Rev. B 14, 2239 (1976).

aEnergyPeriodic system + B-field

Douglas Hofstadter

Magnetic flux
per unit squareaBloch quantization(periodic potential)# of electrons
for a Bloch bandLandau quantization(B-field)# of electrons
for a Landau level

Periodic potential onlyStart

B-field onlyStartPeriodic potential & B-field … you can use both

Ex.)

:

&

= 3:1

Start
only

:

= 3:1

Periodic potential & B-field … you can use both

:

= 3:1

30 : 11300 : 101&

Irrationalslightly change
B-field
Infinite bands (Cantor set)

Start

GCD( ,)

1/3 11/30

Observe Hofstadter’s butterfly?
Condition to observe butterfly:

Ordinary atomic latticea ~ 0.1nm

B ～ 100,000 Tesla!!

aMoire superlattice … huge unit cella ~ 10 nm

B ～ O(10) TeslaaButterfly observed!

Nature 497, 598 (2013)

See also : Ponomarenko et al, Nature 497, 594 (2013)
Hunt et al, Science 340, 1427 (2013).

Beyond moiré superlattice𝜃 = 2∘

𝜃 = 5∘

𝜃 = 15∘

𝜃 = 30∘TBG of 30 deg
12-fold quasicrystal
P. Stampfli, Helv. Phys. Acta 59, 1260 (1986).

Penrose tile
10-fold quasicrystal
R. Penrose, Bull. Inst. Math. Appl. 10, 266 (1974).

Quasicrystals
Alloy quasicrystals
Shechtman (1984)

Cold-atom optical lattice
Tsai, Cui (2015)

Viebahn et al, Phys. Rev. Lett. 122, 110404 (2019)

Photonic quasicrystals

Freedman et al, Nature 440, 1166 (2006)

Quasicrystals =Redundant lattice vectors
more than the spatial dimension

Twisted bilayer graphene as a quasicrystal

𝒃,

𝒃-

𝒃#
𝒃.

Redundant reciprocal vectors
more than the spatial dimension→ quasicrystal

Small angles … Continuum model to extract the moiré period(essentially Bloch physics)
𝜃 = 2∘
Lopes dos Santos et al,
PRL 99, 256802 (2007)
Bistritzer and MacDonald,
PNAS 108, 12233 (2011)
Kindermann and First,
PRB 83, 045425 (2011)
Moon and Koshino,
PRB 87, 205404 (2013)

Large angles … No effective periods (truly quasiperiodic)
𝜃 = 30∘“Twisted quasicrystals”
30-deg twist bilayer graphene

Trilayer systems(ex. hBN / graphene / hBN)

Graphene + MoS2

“Quasibands” in 30deg TBG

TBG of 30 deg
12-fold quasicrystal
P. Stampfli, Helv. Phys. Acta 59, 1260 (1986).

Penrose tile
10-fold quasicrystal
R. Penrose, Bull. Inst. Math. Appl. 10, 266 (1974).

Bloch theory does NOT work
à How to describe the electronic structure?
S. J. Ahn et al, Science 361, 782 (2018).

General theory for incommensurate interfaces
Long-range moire superlattice

Arbitrary 2D materials

No effective periods

Interface Hamiltonian :M. Koshino, New J. Phys. 17 015014 (2015)
See also: Bistritzer and MacDonald, PNAS 108, 12233 (2011)

Coupling condition

G = n b : Layer 1’s reciprocal vector
G’ = n’b’: Layer 2’s reciprocal vector

Ganaralized Umklapp condition
Brillouin zone (Extended)b
Layer 1kk+b = k’+b’k’

Layer 2b’t(q)Coupling strengtht(k+b)

Fourier trnsfm.
of T(r)0k+b= k’+b’qInterlayer coupling in an incommensurate stackk(2nd-order) (0th-order)

Layer 1’s BZkLayer 2’s BZk’(1st-order)(3rd-order)

Quasicrystal: Infinite number of wave-points are hybridized!
But, actually…
Eigenstates well described by finite wave points (high orders can be cut off)
Eigen energies plotted against initial k = Quasi-band!!

“Quasi-band theory” for 30 deg. TBG
12-Dirac cones approximation

P. Moon, M. Koshino, and Y-.W. Son
Phys. Rev. B 99, 165430 (2019)“Quasicrystal flat bands”
Quasi-periodic patterns
in wavefunction

Topological invariants in quasicrystals

hBN / graphene / hBN trilayer

N. R. Finney et al,Nat. Nanotechnol. 14, 1029 (2019).
L. Wang, et al, Nano Lett. 19, 2371 (2019).
Z. Wang, et al., Sci. Adv. 5, eaay8897 (2019).
Y. Yang, et al., Sci. Adv. 6, eabd3655 (2020).

Double moiré pattern(quasiperiodic)

Theory:Andelkovic, et al,
Nano Lett. 20, 979 (2020).
Leconte and J. Jung,
2D Mater. 7, 031005 (2020).
Oka and Koshino,
Phys. Rev. B 104, 035306 (2021).

Energy spectrum of BN / graphene / BNInterference of the two periodicitiesà fractal gap structure
like Hofstadter’s butterfly
à Any topologial characterizationfor energy gaps??

Oka and Koshino,
Phys. Rev. B 104, 035306 (2021).

Topological numbers?
Hofstadter butterfly (2D lattice + B-field)
Hofstadter, Phys. Rev. B 14, 2239 (1976).

Double-moiré system(zero B-field)

Energy gaps characterized byquantized Hall conductivity (Chern number)Thouless, Kohmoto, Nightingale, and Nijs,
Phys. Rev. Lett. 49, 405 (1982).

???

Topological gap labeling in quasicrystals

Oka and Koshino,
Phys. Rev. B 104, 035306 (2021).
Koshino and Oka,
Phys. Rev. Res. 4, 013028 (2022)

Periodic system
1
∗
𝑛1 =
𝑁𝑆
2𝜋 %

Brillouin zone area
𝒃𝟏 ×𝒃𝟐

Number of occupied bands

𝒃,

𝑁=3

𝑆∗
𝒃#

𝑁=2
𝑁=1

Quasiperiodic system
1
∗
𝑛1 =
F
𝜈
𝑆
2"
2"
2𝜋 %
2"

Second Chern number

Brillouin zone area𝒃𝒊 ×𝒃𝒋Multiple Brillouin zones(𝑖, 𝑗) = (1,2)(1,3)(1,4)(2,3)(2,4)(3,4)

Topological gap labeling in hBN / graphene / hBN
𝜈$% , 𝜈$& , 𝜈$' , 𝜈%& , 𝜈%' , 𝜈&'= (−1, 0, 1, −1, 0, −1)
0.0Energy [eV](−1, 1, 1, 0, 1, −1)(3, −2, −4, 2, −2, 3) 3, −4, −2, −2, −4, 3(−4, 2, 4, −2, 2, −4)(1, −1, −2, 1, −1, 1)-0.2(6, −4, −8, 4, −4, 6)(1, −2, −1, −1, −2, 1)(2, −3, −1, −2, −3, 2)(−1, 0, 0, 0, 0, −1)
-0.4
0.0

(−4, 4, 2, 2, 4, −4)(6, −8, −4, −4, −8, 6)

1.5
0.5
1.0
Twist angle 𝜃 ! = − 𝜃 " [°]Oka and Koshino,
Phys. Rev. B 104, 035306 (2021);
Koshino and Oka,
Phys. Rev. Rsearch 4, 013028 (2022)

2.0

Koshino and Oka,
Phys. Rev. Research
4, 013028 (2022)

𝝂𝒊𝒋 = second Chern number
Adiabatic pumping in interlayer sliding
M. Fujimoto, H. Koschke, and M. Koshino,
Phys. Rev. B 101, 041112 (2020).

4D Quantum Hall effect
Hall current

Second Chern number

Dimensional
reduction
movementof moiré pattern↓charge current

X.-L. Qi, T. L. Hughes, and S.-C. Zhang,Phys. Rev. B 78, 195424 (2008).
Y. E. Kraus, Z. Ringel, and O. Zilberberg,
Phys. Rev. Lett. 111, 226401 (2013).

Physical meaning of 𝜈JK
k-space

Koshino and Oka,
Phys. Rev. Research 4, 013028 (2022)

r-space

Real space unit cell

2𝜋 %
𝑆2" =
∗
𝑆2"

1
∗
𝑛1 =
F
𝜈
𝑆
2" 2"
2𝜋 %
2"

𝜈2"
𝑛1 = F𝑆2"2"

𝜈%2 electrons reside in every unit cell of 𝑆%2

Adiabatic charge pumping

Koshino and Oka,
Phys. Rev. Research 4, 013028 (2022)

Charge pumping by sliding a single potential by a unit period𝒃𝒊𝜈2"𝑛1 = F𝑆2"
2"

Total charge
pumping:

Δ𝑷% =1> 𝜈%2 (𝒃2 ×𝒆3 )2𝜋2Mapping to a high-dimensional system
2D system + a periodic potential

Koshino and Oka,
Phys. Rev. Research
4, 013028 (2022)

3D system + B fieldpotential slides!
𝑦

𝐵53

𝑥

𝒃=

𝑒𝑎3
𝑩
ℏ

𝑧

𝑩

𝐵43

𝑦
𝑬

𝑥

Adiabatic pumping

E-field𝑎3

Quantum Hall effectMapping to a high-dimensional system
2D system + N-periodic potentials

Koshino and Oka,
Phys. Rev. Research
4, 013028 (2022)(2+N)D system + B field
𝑩𝒊 = (𝐵5% , 𝐵4% )(𝑖 = 1, … , 𝑁)

Adiabatic charge pumping
in potential sliding

Electromagnetic responsein (𝑥, 𝑦, 𝑧2 , 𝑧" ) -subspace (4D QHE)

1
Δ𝑷% => 𝜈%2 (𝒃2 ×𝒆3 )2𝜋2Second Chern number
Qi, Hughes, and Zhang, PRB 78, 195424 (2008).

Topological gap labelling in 3D quasicrystals
Shechtman (1984)

Tsai, Cui (2015)

Topological gap labelling in 3D quasicrystals
Yamamoto,Koshino,
Phys. Rev. B 105, 115410 (2022)

Number of states below a gap:

Third Chern number(6D quantum Hall effect)
Brillouin zone volume
𝒃𝒊 L (𝒃𝒋 ×𝒃𝒌 )1D QC … 1st Chern numbers
2D QC … 2nd Chern numbers
3D QC … 3rd Chern numbers

Brillouin zones(𝑖, 𝑗, 𝑘) = (1,2,3)(1,2,4)(1,3,4)(2,3,4)
106/17

Summary: Modelling twisted 2D materials
Interlayer coupling:

Twisted bilayer graphene

Double-wall carbon nanotube

Graphene on hBN

30-degree bilayer graphene(Quasicrystalline system)

Graphene on TMDC... and many moreSummary: twisted quasicrystals-- “Quasiband theory”
30-deg twist bilayer graphene
Graphene / TMDC system-- Topological invariants in quasicrystals
hBN / graphene / hBN system-- Twisted 3D topological insulator
Acknowledgments

P. MoonY.-W. Son
(NYU Shanghai) (KIAS)

H. Oka(Osaka)K. Yamamoto M. Fujimoto Nguyen Nam(Osaka)(Osaka)(Tohoku)Experiments: J. R. Ahn (SKKU) , C.-W. Yang (SKKU)

Yang Li(Tohoku)Lattice relaxation and moire phonons

Moiré superlattice effect in phonons?
Electronic band

Moon and Koshino, Phys. Rev. B 87, 205404 (2013)

Phonon band?

(Static) lattice relaxation in 1D moiré system
No distortion

Local binding energy

Relaxed(self-distorted)

Local binding energy
domainboundary

domainLocal binding energy in twisted bilayer graphene
18.9

AB / BA: lowest energy

Binding energy (meV/Atom)

AA: highest energy

0.00

Binding energy (meV/Atom)

AA
15

10

5

SP
0

SP
AB

BA

Theoretical method --- continuum approach
Nam and Koshino,
Phys. Rev. B 96, 075311 (2017)
N. N. T. Nam(Tohoku)

Minimize the total energy(Numerically solving Euler-Lagrange equation)

Nam and Koshino,
Phys. Rev. B 96, 075311 (2017)

Relaxed structure of TBG
Local binding energy
No relaxation

BA

AB
BA

AA
AB

LM

AB
BA

LM = 5.33nm

LM = 13.4nm

LM = 17.3nm

See also:
K. Uchida, S. Furuya, J.-I. Iwata, and A. Oshiyama, Phys. Rev. B 90, 155451 (2014).
M. van Wijk, A. Schuring, M. Katsnelson, and A. Fasolino, 2D Mater. 2, 034010 (2015).
S. Dai, Y. Xiang, and D. J. Srolovitz, Nano Lett. 16, 5923 (2016).
S. K. Jain, V. Jurivci’c, and G. T. Barkema, 2D Mater. 4, 015018 (2016).

LM = 25.8nm

“Moiré phonon” in TBG
no coupling

2.65°

1.05°

0.547°

M. Koshino, and Y-.W. Son,
Phys. Rev. B 100, 075416 (2019)

0.547°

Mini band structure& spectral gaps

Phonon wavefunctions in TBG

M. Koshino, and Y-.W. Son,
Phys. Rev. B 100, 075416 (2019)

3rd

2nd

1st

Vibration of the moire lattice with“atoms” … AA sites“bonds” … domain walls

Moiré phonons in graphene/hBN

Series of flat bands!!

localization ofphonon wavefunction

L. P. A. Krisna and M. Koshino,
arXiv:2210.03381

Moiré electron-phonon interaction
moiré electrons

moiré phonon el-ph coupling
operators

M. Koshino and NNT Nam,
Phys. Rev. B 101, 195425 (2020)

moiré phonons

Moiré electron-phonon interaction
moiré electrons

M. Koshino and NNT Nam,
Phys. Rev. B 101, 195425 (2020)

Phonon-mediated e-e interaction 𝑈9:9

Stongly enhancedin small θ

moiré phonons

At the flat band:𝑈(/( × DOS ∼ 𝑂(1)(Strong-coupling regime)Summary: Modelling twisted 2D materials
Interlayer coupling:

Twisted bilayer graphene

Double-wall carbon nanotube

Graphene on hBN

30-degree bilayer graphene(Quasicrystalline system)

Graphene on TMDC... and many moreSummary: twisted quasicrystals-- “Quasiband theory”
30-deg twist bilayer graphene
Graphene / TMDC system-- Topological invariants in quasicrystals
hBN / graphene / hBN system-- Twisted 3D topological insulator
Acknowledgments

P. MoonY.-W. Son
(NYU Shanghai) (KIAS)

H. Oka(Osaka)K. Yamamoto M. Fujimoto Nguyen Nam(Osaka)(Osaka)(Tohoku)Experiments: J. R. Ahn (SKKU) , C.-W. Yang (SKKU)

Yang Li(Tohoku)