Origin of Magic Angles in Twisted Bilayer Graphene Grisha - - PowerPoint PPT Presentation

Grisha Tarnopolsky Harvard University Talk at PCTS workshop ”Critical Phenomena in Statistical Mechanics and Quantum Field Theory’’ Princeton University October 4, 2018

Origin of Magic Angles in Twisted Bilayer Graphene

Alex Kruchkov Harvard Ashvin Vishwanath Harvard

Talk based on: GT, A. Kruchkov, A. Vishwanath arXiv:1808.05250

Plan

• Graphene
• Twisted Bilayer Graphene (TBG)
• Continuum model for TBG and Magic angles
• Origin of Magic angles

• Tight Binding Model (ignore spin)
• Easy to diagonalize in Fourier

space

Graphene

• Dirac Cones at K and K’ points

P.R.Wallace, 1946

Graphene

• Graphene Brillouin Zone (BZ)
• Low energy effective Hamiltonian is Dirac Hamiltonian

(Dirac see is filled in graphene)

• Near K point (K-valley) the wave function obeys

2D Dirac equation

G.W.Semenoff, 1984

Twisted Bilayer Graphene (TBG)

Moiré Pattern

AA stacking region BA stacking region AB stacking region

L ∼ a/θ

Twisted Bilayer Graphene (TBG)

• Brillouin Zone Folding: From two Graphene Brillouin zones to
• ne Moiré Brillouin Zone (MBZ)
• Without interaction between layers we pack two Dirac cones in

Moiré Brillouin Zone

Assume no interaction between layers!

H = ✓ iv0σθ/2(r K2) iv0σ−θ/2(r K1) ◆

Continuum Model for TBG

Santos et all 2007, R.Bistritzer&A.MacDonald, 2010 Po, Zou, Vishwanath, Senthil, 2018, Yuan, Fu, 2018

• Hamiltonian with interaction between two layers
• General interaction matrix coupling has two parameters and

where

• r Φ(r) = (ψ1, χ1, ψ2, χ2)T

Bistritzer MacDonald Model

• Bistritzer and MacDonald set
• Then dimensionless Hamiltonian has only
• ne parameter
• Bistritzer&MacDonald defined Magic angle

as the angle where the Fermi velocity at K point vanishes

First Magic angle

Experiments

Cao, et al, Nature 556, 43 (2018)

0. 1.0

LDOS S D SiO2/Si Hexagonal boron nitride Gate

a b d

~ A Vg –50 50 Ks Ks Γs Γs Γ Ms Ms Ks ′ Ks ′ K′2 K′1 E (meV) = 1.08°

• c

e

• K1

K2

h

Twisted bilayer graphene

• Cao, et al, Nature 556, 81 (2018)
• Superconductivity is found at the

Magic angle θ ≈ 1.08

Yankowitz, et all, arXiv:1808.07865

Chirally symmetric Continuum Model

• We consider a continuum model for bilayer graphene with

α1 = 0.586 α2 = 2.221 α3 = 3.751

Energy

1.5 1.0 0.5 A B C D A A B C D A A B C D A 0.1 0.2 0.3 0.4

• 0.4
• 0.3
• 0.2
• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

0.02 0.04 0.06 0.08

• 1.5
• 1.0
• 0.5

a. b. c.

α1 α2 α3

Bandwidth

0.4 0.8

Energy

exact zeros

1 2 3 4 5 6

α

0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6

α Energy Band gap

1.2 1.6 2.0

e. f.

• This model has perfectly flat bands

at series of magic angles

α1 α2 α3 α4 α5 CS-CM (here) 0.586 2.221 3.75 5.28 6.80 BM-CM (Ref. [35]) 0.606 1.27 1.82 2.65 3.18

Bistritzer&MacDonald This model

Exactly flat bands

What is the theory behind the exactly flat bands?

α1 = 0.586

Energy

1.5 1.0 0.5 A B C D A

• 1.5
• 1.0
• 0.5

a.

• The Hamiltonian has chiral symmetry

so the spectrum is symmetric around 0

• Relative rotations on

can be removed

• The spectrum has two stable zero modes at arbitrary angle

at the points and

• The Hamiltonian can be represented as

San-Hose, et all, 2012

Theory behind the perfectly flat bands

• Dimensionless Hamiltonian can be represented as

where ¯ ∂ = 1

2(∂x + i∂y) iq r

• r Φ(r) = (ψ1, ψ2, χ1, χ2)T,

2i¯ ∂ αU(r) αU(r) 2i¯ ∂ ! ψK,1(r) ψK,2(r) ! = 0 D(r)ψK(r) = 0

• Zero mode equation reads

• This always has a solution due to

symmetry of the interaction

α1 = 0.586

Energy

1.5 1.0 0.5 A B C D A

• 1.5
• 1.0
• 0.5

a.

where is the zero-mode solution

• We can try the construction

and is needed for periodic boundary conditions

Theory behind the perfectly flat bands

• An arbitrary point of the exactly flat band must satisfy

and thus with Bloch periodic boundary conditions

• The key observation - kinetic term is antiholomorphic

• Problem! Periodic holomorphic function is a constant!

Theory behind the perfectly flat bands

α1 = 0.586

Energy

1.5 1.0 0.5 A B C D A

• 1.5
• 1.0
• 0.5

a.

• The only possibility is to take meromorphic function

• One can construct such a function as a ratio of two

Theta functions with rational characteristics where

• Theta function satisfies the relations
• Problem! Meromorphic function has poles!

Theory behind the perfectly flat bands

• Exactly at magic angles the zero mode

solution has zeros at AB or BA stacking points

α1 = 0.586

Energy

1.5 1.0 0.5 A B C D A

• 1.5
• 1.0
• 0.5

a.

ψk(r) = ϑ ka1

2π 1 6 , 1 6 ka2 2π (z/a1|ω)

ϑ 1

6 , 1 6 (z/a1|ω)

ψK(r)

AB AA BA

1 2 3

ρK(r)

First magic angle α1 = 0.586

zero r0

a2 a1

AB AA BA

• Rotational symmetry of the interaction implies
• Observation: Zero-mode operator

has an integral of motion

Flat band relation to Fermi velocity

Question: Why zero Fermi velocity at point implies perfectly flat band? (opposite is obvious)

• Fermi velocity is a derivative of spectrum at point

and can be calculated as

vF (α) = |hψ⇤

K(r)|ψK(r)i|

hψK|ψKi

where is the zero-mode solution

v(α) = ψK,1(r)ψK,1(r) + ψK,2(r)ψK,2(r)

at vF (α) ⇠ v(α).

• Fermi velocity

inates and from at ψK,2(±r0) = 0

vF (α) ⇠ ψK,1(r0)ψK,1(r0)

is BA stacking point

Numerical values of the magic angles

Question: How to obtain positions of the magic angles theoretically?

• For general interaction and higher angles it is an open question
• Luckily the first magic angle is captured by perturbation theory in

ψK(r) = ψK,1 ψK,2 ! = 1 + α2u2 + α4u4 + . . . αu1 + α3u3 + . . . !

vF (α) = 1 3α2 + α4 111α6

49

+ 143α8

294

+ . . . 1 + 3α2 + 2α4 + 6α6

7

+ 107α8

98

+ . . .

Physical relevance of Chirally Symmetric continuum model

BAND GAP EVOLUTION 0.2 0.4 0.6 0.8 1.0 0.1 0.2 0.3 0.4 0.5

w0/w1 Band gap

α1 α2

2.22 2.22 guide for eyes 2.89 2.53 2.14 0.58 0.57 0.58 0.58 0.58 0.58 1.59

• Interaction term can be written in the form

where interaction between AA regions

• Lattice relaxation effects shrink AA regions, thus effectively decrease

• S. Carr et all, 2018

• Possibility to observe the second magic angle in experiment near

Open questions

10-1 10-2 10-3 10-4 10-5 10-6 10-7 1 vF α1 α2 α3 α4 α5 ∆α 3/2

a.

MAGIC ANGLE RECURRENCE PERIOD SATURATION

∆α α1 α2 α3 α4 α5 α6 α7 α8 α9

b.

α

10 8 6 4 2 −0.02 −0.04 −0.06 −0.08 −0.10 −0.12 −0.14 3/2 − ∆α

3/2 1 2 3 4 5 6

α

2 4 6 8 10

α

• Magic angles periodicity
• Existence of the magics angle for a general interaction

• Is the any topological reasons for magic angles?
• Flat bands in non-abelian SU(3) fields - application to trilayer graphene

Thank you for your attention!