Topological Mechanical Metamaterials Sebastian Huber Nature - - PowerPoint PPT Presentation

SLIDE 1

| | Sebastian Huber Condensed matter theory and quantum optics


 Nature Materials (2018) Nature (2018) Nature Physics (2016)

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Topological Mechanical Metamaterials

Sebastian Huber

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A mechanical metamaterial: tailoring the phonons to our liking

momentum frequency Electronic bands momentum frequency ~ THz Phonon bands

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A mechanical metamaterial: tailoring the phonons to our liking

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momentum frequency ~ THz

frequency by design

~ Hz—GHz

SLIDE 4

| | Sebastian Huber Condensed matter theory and quantum optics

Flavours of topological mechanics

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Mass-spring models

m∂2

t uα i = −Dαβ ij uβ j

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Acoustic waves

∂2

t ρ(x) = c2r2ρ(x)

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Elastic waves

τmk(x) = ckmij ⇥ ∂xiuj(x) + ∂xjui(x) ⇤

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ρ∂2

t um(x) = ∂τmk

∂xk

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SDH Nature Phys. (2016)

SLIDE 5

| | Sebastian Huber Condensed matter theory and quantum optics

From electrons to phonons

5

i~∂tψ = Hψ

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ψgs(θx, θy)

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momentum energy Electronic bands momentum frequency

∂2

t x = −Dx

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SLIDE 6

| | Sebastian Huber Condensed matter theory and quantum optics

A more concrete mapping

6

¨ xi(t) = −Dij xj(t) + Γij ˙ xj(t)

A conservative discrete system

d dt x(t) ˙ x(t)

• =

 0 1 −D Γ x(t) ˙ x(t)

• First order time derivative

DT = D, D > 0, ΓT = −Γ ψ =  QT i1  x(t) ˙ x(t)

• D = QTQ

Formally: a BdG-type problem

Kane & Lubensky, Nature Phys (2013) Süsstrunk and SDH, PNAS 2016

i d dtψ =  QT Q iΓ

• |

{z }

H

ψ

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UC = 1 −1

• K

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HUC + UCH = 0

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x y z

SLIDE 7

| | Sebastian Huber Condensed matter theory and quantum optics

Half the dynamical matrix

7

Isostatic lattices Generic lattices

Qij xj+1 xj ei D = QTQ

D > 0 ⇒ Q = √ D QT 6= Q QT ≡ Q

unit cell

C

unit cell

B

unit cell

y x

D

f=f0 f=f0e

2 i 3

f=f0e

4 i 3

A

x y s r s r

Kane & Lubensky, Nature Phys (2013) Süsstrunk and SDH, PNAS 2016

SLIDE 8

| | Sebastian Huber Condensed matter theory and quantum optics

Generic dispersion: flavors of TI’s

8

• 1

1 Wave vector k Frequency in Hz Wave vector k Wave vector k −π π

• 1
• 0.5

0.5 1 Wave vector k Periodic γ = 0 Frequency in Hz Wave vector k Periodic 0.5 1 Open

• Fig. 3.

i d dtψ =  QT Q iΓ

• ψ

D = QTQ

• Isostatic lattices
• Zero-frequency: static

properties

• Floppy modes
• States of self-stress
• Generic lattices
• High-frequency properties
• wave guides
• vibration isolation
• heat-diodes

Kane & Lubensky, Nature Phys (2013) Lubensky et al., Rep. Prog. Phys. (2015) Süsstrunk and SDH, PNAS 2016

SLIDE 9

| | Sebastian Huber Condensed matter theory and quantum optics

A selection of topological effects

10

Bulk Edge Edge Edge Bulk

Separate channels in real space

Edge Bulk Edge of Edge

Separate them even “further”

SLIDE 10

| | Sebastian Huber Condensed matter theory and quantum optics

Multipole expansion

11

ϕ(r) = 1 4π Z dr0 ρ(r0) |r − r0| Pi = Z dr riρ(r) Q = Z drρ(r) Monopole Dipole Quadrupole Qij = Z dr rirjρ(r) ϕ(r) = 1 4π Q r + 1 4π Pi ri r3 + 1 4π Qij 3rirj − r2δij 2r5 + · · ·

SLIDE 11

| | Sebastian Huber Condensed matter theory and quantum optics

▪ 
 
 
 
 ▪ 
 
 ▪

Dipole density

12

P = Z dr rρ(r) ) P = hΨ|r|Ψi |WR(r)i = 1 2π Z π

−π

dk eik(R−r)|φki p = Z dr hW0(r)|r|W0(r)i = 1 (2π)2 Z Z π

−π

Z π

−π

drdkdq eir(k−q)hφk|i∂q|φqi = i 2π Z π

−π

dk hφk|∂k|φki = 1 2π Z π

−π

dk Ax(k)

SLIDE 12

| | Sebastian Huber Condensed matter theory and quantum optics

Dipole based topological systems

13

ρ = ∂ipi σ = nipi

Bulk Edge Edge Bulk Edge

SLIDE 13

| | Sebastian Huber Condensed matter theory and quantum optics

A conventional topological system

14

Bulk Edge Edge Bulk Edge

SLIDE 14

| | Sebastian Huber Condensed matter theory and quantum optics

Two copies of a Chern insulator

!15

H = ✓HΦ H−Φ ◆ = ✓HΦ H∗

Φ

◆ ✓xr,s yr,s ◆ = 1 √ 2 ✓1 −i 1 i ◆ ✓ψ+

r,s

ψ−

r,s

◆ U †HU = D = ✓ReHΦ ImHΦ ImHΦ ReHΦ ◆

Süsstrunk & SDH, PNAS (2016)

SLIDE 15

| | Sebastian Huber Condensed matter theory and quantum optics

Our implementation

!16

= D = ✓ReHΦ ImHΦ ImHΦ ReHΦ ◆

x y x y

unit cell

C

unit cell

B

unit cell

y x

D

f=f0 f=f0e

2 i 3

f=f0e

4 i 3

A

x y s r s r

unit cell

C

unit cell

B

unit cell

y x

D

f=f0 f=f0e

2 i 3

f=f0e

4 i 3

A

x y s r s r

unit cell

C

unit cell

B

unit cell

y x

D

f=f0 f=f0e

2 i 3

f=f0e

4 i 3

A

x y s r s r ± f s i n ( 4 π / 3 ) ± f s i n ( 2 π / 3 )

unit cell

C

unit cell

B

unit cell

y x

D

f=f0 f=f0e

2 i 3

f=f0e

4 i 3

A

x y s r s r f f c

• s

( 4 π / 3 ) f c

• s

( 2 π / 3 )

Süsstrunk & SDH, Science (2015)

SLIDE 16

| | Sebastian Huber Condensed matter theory and quantum optics

Helical edge modes

!17

• π/2

π/2

• π

π 2.0 2.2 2.4 2.6 Wave vector k Frequency in Hz

Süsstrunk & SDH, Science (2015)

SLIDE 17

| | Sebastian Huber Condensed matter theory and quantum optics

Not “just” a whispering gallery mode

!18

Süsstrunk & SDH, Science (2015)

SLIDE 18

| | Sebastian Huber Condensed matter theory and quantum optics

• 19

!=6000 km

Wheeler, J. Atmos. Sci. (2000) Delplace, Science (2017)

ω±(k) = ± p f 2 + c2k2 ω0(k) = 0 f = ˆ n · Ω C = 2 sign(f)

SLIDE 19

| | Sebastian Huber Condensed matter theory and quantum optics !20

Benalcazar et al., Science (2017)

ρ = 1 2∂i∂jqij σα = −∂j(nα

i qij)

λαβ = nα

i nβ j qij

α β λαβ σα = ∂jpα

j

pα

j = nα i qij

Edge Bulk Edge of Edge

Quadrupole based topological system

SLIDE 20

| | Sebastian Huber Condensed matter theory and quantum optics

A higher-order topological system

21

Edge Bulk Edge of Edge

Benalcazar et al. Science (2017) Schindler et al. Sci. Adv. (2018) Trifunovic and Brouwer, arXiv (2018) Schindler et al. Nature Phys. (2018)

SLIDE 21

| | Sebastian Huber Condensed matter theory and quantum optics

From dipoles to quadrupoles

22

Benalcazar et al. Science (2017)

pα = 1 4π2 Z

K

dk Tr[Aα,k] Aα,k = ihum

k |∂kα|un ki

pα = ( trivial 1/2 non-trivial

With, e.g., inversion symmetry:

SLIDE 22

| | Sebastian Huber Condensed matter theory and quantum optics

Formalizing the quadrupole moment: Wilson loops

23

pα = 1 4π2 Z

K

dk Tr[Aα,k] Aα,k = ihum

k |∂kα|un ki

Benalcazar et al. Science (2017) Benalcazar et al. PRB 2017

Wx(ky) = TC exp ✓ i I

C

dkxhun

k|∂kx|um k i

◆ kx ky

Wx(ky)

SLIDE 23

| | Sebastian Huber Condensed matter theory and quantum optics

Wilson loops encode the surface physics

24

Wx(ky) = TC exp ✓ i I

C

dkxhun

k|∂kx|um k i

◆

kx ky

Wx(ky)

eigenvalues: exp[2πiν±

x (ky)],

eigenvectors: v±

Fidkowski et al. PRL (2011)

Chern insulator Topological Insulator Trivial insulator?

SLIDE 24

| | Sebastian Huber Condensed matter theory and quantum optics

Nested Wilson loops capture the quadrupole

25

| ˜ w±

x,ki =

X

m

v±

m|um k i

split the filled bands

˜ p±

α = 1

4π2 Z

K

dk Tr[ ˜ A±

α,k]

˜ A±

α,k = ih ˜

w±

k |∂kα| ˜

w±

k i

Fidkowski et al. PRL (2011) Benalcazar et al. Science (2017)

Wx(ky) = TC exp ✓ i I

C

dkxhun

k|∂kx|um k i

◆ eigenvalues: exp[2πiν±

x (ky)],

eigenvectors: v±

SLIDE 25

| | Sebastian Huber Condensed matter theory and quantum optics

Quantizing the quadrupole moment: symmetries

26

pα = 1 4π2 Z

K

dk Tr[Aα,k] Aα,k = ihum

k |∂kα|un ki

pα = ( trivial 1/2 non-trivial

With, e.g., inversion symmetry:

Benalcazar et al. Science (2017) Benalcazar et al. PRB 2017

Symmetries

[Mx, H] = [My, H] = 0 [Mx, My] 6= 0 (˜ p±

y , ˜

p±

x ) = (1/2, 1/2)

SLIDE 26

| | Sebastian Huber Condensed matter theory and quantum optics

A model

27

Benalcazar et al. Science (2017)

SLIDE 27

| | Sebastian Huber Condensed matter theory and quantum optics

Design principle: Perturbative metamaterials

28

Rigid body modes Frequency

Coupling Projection

Hij = X

m

hψ0

i |ψmiνmhψm|ψ0 j i

|ψ0

j i

|ψmi

Matlack, Garcia, Palermo, SDH and Daraio, Nature Mat. 2018

|ψni

dxy

dx2−y2

SLIDE 28

| | Sebastian Huber Condensed matter theory and quantum optics

Design and tight-binding approximation

29

Silicon wafer, [100], 350µm thickness, TTV < 1µm

+ +

• 5 mm

+ +

• +

+

• +

+

• Matlack, Serra-Garcia, Palermo, SDH and Daraio, Nature Mat. 2018

Serra-Garcia et al. Nature (2018)

SLIDE 29

| | Sebastian Huber Condensed matter theory and quantum optics

Assessment of design quality

30

(˜ p+

y , ˜

p−

y ) = (0.58, 0.57)

(˜ p+

x , ˜

p−

x ) = (0.50, 0.50)

(px, py) = (0.06, 0.01)

Serra-Garcia et al. Nature (2018)

SLIDE 30

| | Sebastian Huber Condensed matter theory and quantum optics

Rogue phonons

+ - +

• Embedding

31

Ultrasound transducer Laser interferometer

Energy ∝ ∆z2(ω) ∆z(ω) ∝ ψ2

i (ω) ∼ 20 pm − 5 nm

dx2−y2

dxy

SLIDE 31

| | Sebastian Huber Condensed matter theory and quantum optics

“Quantized” quadrupole phase

32

SLIDE 32

| | Sebastian Huber Condensed matter theory and quantum optics

Trivial phase

33

Serra-Garcia et al. Nature (2018) Peterson et al. Nature (2018)

SLIDE 33

| | Sebastian Huber Condensed matter theory and quantum optics

Potential for wave guiding

34

SLIDE 34

| | Sebastian Huber Condensed matter theory and quantum optics

An LC-version

35

SLIDE 35

| | Sebastian Huber Condensed matter theory and quantum optics

LC Quadrupole

36

47pF 12µH 47pF 1MΩ VB 1MΩ VB VX VY

PX PY QXQY PX PY QXQY PX PY QXQY PX PY QXQY

SLIDE 36

| | Sebastian Huber Condensed matter theory and quantum optics

Quadrupole phase transition

37

1.0 10.0 Bias voltage VB [V] 4.00 6.00 8.00 Frequency [MHz] bulk 1.0 10.0 Bias voltage VB [V] 4.00 6.00 8.00 Frequency [MHz] corner

SLIDE 37

| | Sebastian Huber Condensed matter theory and quantum optics

Localization length

38

10 20 Bias voltage VB [V] 2 4 6 8 10 Localization length ξ [sites] 5 10 15 Sites from corner −1 1 ψ VB=9.382 V 5 10 15 Sites from corner −1 1 ψ VB=14.92 V

a) c) b)

SLIDE 38

| | Sebastian Huber Condensed matter theory and quantum optics

Edge physics and Berry phase

39

k Left k Left k Right k Right

−π/2 π/2 6.8 7.0 7.2 7.4 f [MHz] −π/2 π/2 5.8 6.0 6.2 f [MHz] −π/2 π/2 6.8 7.0 7.2 7.4 f [MHz] −π/2 π/2 5.8 6.0 6.2 f [MHz]

a) b)

k Top k Top k Bottom k Bottom e)

A B

ϕk = arg(ψA

k /ψB k )

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| | Sebastian Huber Condensed matter theory and quantum optics

The electrons can do it too!

40

SLIDE 40

| | Sebastian Huber Condensed matter theory and quantum optics

Acknowledgments

!41

Guillermo Villanueva, EPFL Tom Larsen, EPFL

• Dr. Marc Serra Garcia

Katie Matlack ⇒ Prof @ UIUC Valerio Peri Osama R. Bilal ⇒ Postdoc @ Caltech Roman Süsstrunk ⇒ Consultant @ AWK Group Chiara Daraio, Caltech

SLIDE 41

| | Sebastian Huber Condensed matter theory and quantum optics 42

Bulk Edge Edge Bulk Edge Edge Bulk Edge of Edge

Science 349, 47 (2015) Nature Phys. 12, 621 (2016) PNAS 113, E4767 (2016) NJP 19, 015013 (2017)

• Adv. Mater. 1700540 (2017)

Nature (2018) Nature Materials (2018)