Topological pumps and topological quasicrystals PRL 109 , 106402 - - PowerPoint PPT Presentation

SLIDE 1

Topological pumps and topological quasicrystals

PRL 109, 106402 (2012); PRL 109, 116404 (2012); PRL 110, 076403 (2013); PRL 111, 226401 (2013); PRB 91, 064201 (2015); PRL 115, 195303 (2015), PRA 93, 043827 (2016), PRB 93, 245113 (2016),

• Nat. Phys. 12, 350 (2016); Nat. Phys. 12, 624 (2016); Nature 553, 55 (2018), Nature 553, 59 (2018)

arXiv:1802.04173, arXiv:1804.01871, arXiv:1805.05209 & arXiv:1805.10670

SLIDE 2

Outline

Photonic topological pump (1D) Quantum Hall effect Topological pumps Quasiperiodic systems Atomic pumps Four-dimensional QHE

SLIDE 3

Quantum Hall effect

2 2 2

1 2 2

y x a c a a c

k p m x m m           

B

• Landau gauge:
• Quantized Hall conductance
• Chern number

2

( ) p , ( , )

(...)

y y y

ip y x p l n x y

e e 

 



x y

ωc

py Enk

n = 1 n = 2 n = 3 n = 4

 

1 (p ,p ) Tr , 2

x y

x y p p

P P P i       

2 2

d p d p (p ,p )

x y x y  

  

 

2 xy

e h   

μ

P

| |

n n n

P    



E I

SLIDE 4

Quantum Hall effect

• Landau gauge:
• Confining potential:

Chiral edge modes

• Chern number

Number of chiral edge modes

See Yasuhiro’s talk and references therein

B

2

( ) p , ( , )

(...)

y y y

ip y x p l n x y

e e 

 



x y

ωc

py Enk

n = 1 n = 2 n = 3 n = 4

bulk

μ

edge

2 2 2

1 2 2

y x a c a a c

k p m x m m           

SLIDE 5

Laughlin/Halperin’s argument

• Landau gauge:
• Chern number

Number of charges moved over a pump cycle

2

( ) p , ( , )

(...)

y y y

ip y x p l n x y

e e 

 



py Enk

bulk

μ

2 2 2

1 2 2

y x a c a a c

k p m x m m           

B

𝜖𝜚/𝜖𝑢

SLIDE 6

Laughlin/Halperin’s argument

• Landau gauge:
• Chern number

Number of charges moved over a pump cycle

2

( ) p , ( , )

(...)

y y y

ip y x p l n x y

e e 

 



py Enk

bulk

μ

2 2 2

1 2 2

y x a c a a c

k p m x m m           

B

𝜖𝜚/𝜖𝑢

SLIDE 7

Hofstadter model

• Hamiltonian:
• Spectrum:

Harper, PPSL A 68, 874 (1955) Azbel, JETP 19, 634 (1964) Hofstadter, PRB 14, 2239 (1976)

q-1 gaps

x,y x,

y y y

ik y k k

c e c 

   

† 2 † x,y 1,y x,y x,y 1 ,

. . . .

i bx x x y x y

t c c h c t e c c h c

  

   



 

† † x, 1, x, x, x,

. . 2 cos(2 )

y y y y y

k x k y y k k k

t c c h c t bx k c c 



   



E ky

b = 8/5

2π

• 2.5

2.5

1 2 3 4

p b q 

E b

b tx ty x y

SLIDE 8

Hofstadter model

• Hamiltonian:
• Quantized Hall conductance:

TKNN, PRL 49, 405 (1982)

• Chern numbers:

 

† † x, 1, x, x, ,

. . 2 cos(2 )

y y y y y

x k x k y y k k x k

t c c h c t bx k c c 



   



E ky

b = 8/5

2π

• 2.5

2.5

1 2 3 4

μ

P

2 xy

e h   

b tx ty x y

 

1 (p ,p ) Tr , 2

x y

x y p p

P P P i       

2 2

d p d p (p ,p )

x y x y  

  

 

SLIDE 9

Hofstadter model

b tx ty x y

E ky

b = 8/5

2π

• 2.5

2.5

1 2 3 4 E

| | 2

r r r

r p qm q     

1

2  

2

1   

3

1  

4

2   

b

• Hamiltonian:
• Quantized Hall conductance:

TKNN, PRL 49, 405 (1982)

• Chern numbers:

2 xy r

e h   

μ

P

p b q 

 

† † x, 1, x, x, ,

. . 2 cos(2 )

y y y y y

x k x k y y k k x k

t c c h c t bx k c c 



   



• D. Osadchy and J. E. Avron, J. Math. Phys. 42, 5665 (2001)

SLIDE 10

Topological pump (1+1)

E

b = 8/5

2π

• 2.5

2.5

• Harper model as a topological pump:
• D. J. Thouless, Phys. Rev. B 27, 6083 (1983)
• Spectrum:

x tx x

 

† † 1

( ) . . 2 cos(2 )

x x x y x x x

H t c c h c t bx c c   



   



Topological edge states of a 2D crystal Boundary states of a 1D topological pump

2 / 7       

SLIDE 11

Outline

Photonic topological pump (1D) Quantum Hall effect Topological pumps Quasiperiodic systems Atomic pumps Four-dimensional QHE

SLIDE 12

Photonic waveguide array

• A waveguide:
• A photonic crystal:

Lahini et al., PRL 103, 013901 (2009)

tight binding model propagation replaces time

z

i V    

1 1 1 z x x x x x x x

i t t V    

  

   

z z x

SLIDE 13

Photonic waveguide array

• Diagonal modulation:
• Off-diagonal modulation:

1 1 1 z x x x x x

i t t   

  

  

1 1

( )

z x x x x x

i t V    

 

   

z x z x

SLIDE 14

Experiment

z x

• Setup:

input: single site

• utput: distribution
• Evolution:
• verlap with eigenstates

expansion according to eigenstates

• No chemical potential

SLIDE 15

Generalizations – Other 1D models

• Off-diagonal Harper model:

† 1

( ) 2 cos(2 ) . .

x xy x x x

H t t bx c c h c   



       



n b tx txy x y

E

ϕ

2π

• 2.5

2.5

• Phys. Rev. Lett. 109, 106402 (2012)

SLIDE 16

z

intensity

n

Experiment 1: adiabatic pumping (1D+1)

• Pumping:
• Adiabatic pumping:
• ff-diagonal

E

ϕ

2π

• 2.5

2.5

z

 

( ) H z 

x density x density x density

† 1

( ) 2 cos(2 ) . .

x xy x x x

H t t bx c c h c   



       



• Phys. Rev. Lett. 109, 106402 (2012)

SLIDE 17

Outline

Photonic topological pump (1D) Quantum Hall effect Topological pumps Quasiperiodic systems Atomic pumps Four-dimensional QHE

SLIDE 18

Quasiperiodic models

• 1D Fibonacci chain:

L → LS S → L

• 2D Penrose tiling:

Systems with long-range order but no translation symmetry.

L LS LSL LSLLS LSLLSLSL

1 1 2 ( 2) ( 1) 1 1 1

n

d n n                           

SLIDE 19

Long-range order vs. translations

• Harper (or Aubry-André) model:
• Long-range order vs. translations

The order determines the system up to translations of the origin (which have no effect on bulk properties).

 

† † 1

( ) . . 2 'cos(2 )

n n n n n

H t c c h c t bn c c   



   



cos(2 ) bn  1..30 n  101..130 n 

• Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phys. Rev. Lett. 109, 106402 (2012)

SLIDE 20

Long-range order vs. translations

• Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phys. Rev. Lett. 109, 106402 (2012)
• Harper (or Aubry-André) model:
• Translations vs. ϕ

b = p/q: irrational b: For the latter ϕ has no effect on bulk properties

 

† † 1

( ) . . 2 'cos(2 )

n n n n n

H t c c h c t bn c c   



   



1 2 (2 )mod 2 0, 2 , 2 , bn q q              (2 )mod2 [0,2 ] bn    

SLIDE 21

Long-range order vs. translations

• Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phys. Rev. Lett. 109, 106402 (2012)
• Harper (or Aubry-André) model:
• Translations vs. ϕ

ϕ → ϕ +ε

is equivalent to n → n + nε independent of ϕ → flat bulk bands

 

† † 1

( ) . . 2 'cos(2 )

n n n n n

H t c c h c t bn c c   



   



†

( ) ( ) H T H T

 

    

 

†

( )

l l

T E l l T

 

 



( )

l l E

T l T l

 

 

E

ϕ

2π

SLIDE 22

 

1 Tr ( ) ( ), ( ) 2 P P P i

 

          

• Harper (or Aubry-André) model:
• Projector P:

P ~ H

• Berry curvature

 

1 ( , ) Tr ( ) ( ), ( ) 2 P P P i

 

                     

 

1 ( , ) Tr , 2 P P P i

 

          

Long-range order vs. translations

• Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phys. Rev. Lett. 109, 106402 (2012)

ϕ

2π

c( 0, )    

 

† † 1

( ) . . 2 'cos(2 )

n n n n n

H t c c h c t bn c c   



   



P

†

( ) ( ) P T P T

 

    

μ

†

( ) ( ) P T P T

   

      

 

†

1 Tr ( ) ( ), ( ) 2 T P P P T i

   

          

SLIDE 23

Long-range order vs. translations

• Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phys. Rev. Lett. 109, 106402 (2012)
• Harper (or Aubry-André) model:
• Berry curvature
• Chern number:
• Chern numbers can be associated with the whole

family {H(ϕ)}. Thouless, PRB 27, 6083 (1983)

 

† † 1

( ) . . 2 'cos(2 )

n n n n n

H t c c h c t bn c c   



   

  

1 ( , ) Tr , ( ) 2 P P P i

 

             

2 2

d d ( , )

 

      

 

2

2 d ( )



    



• We can associate Chern numbers with any H(ϕ)!

SLIDE 24

Fibonacci-like chains

• Y. E. Kraus and OZ, Phys. Rev. Lett. 109, 116404 (2012);
• Fibonacci quasicrystal:
• Off-diagonal Fibonacci
• Diagonal Fibonacci

1 1 2 ( 2) ( 1) 1 1 1 n n                            1

n

d  

 

† 1

' . .

n n n n

H t t d c c h c



  



† † 1

( . .) '

n n n n n n

H t c c h c t d c c



  



n

SLIDE 25

Fibonacci-like chains

• Y. E. Kraus and OZ, Phys. Rev. Lett. 109, 116404 (2012);
• Fibonacci quasicrystal:
• β → 0:
• β → ∞:

   

 

1 ( ) tanh cos 2 3 cos tanh( )

n

d bn b b               

   

( 0) cos 2 3 cos

n

d bn b b           ( ) 2 ( 2 ) ( 1 ) 1 2 2

n

d b n b n                                 1 1 2 ( 2) ( 1) 1 1 1

n

d n n                           

Harper Fibonacci

β = 0 β = 1 β = 5 β → ∞

SLIDE 26

Fibonacci-like chains

• Y. E. Kraus and OZ, Phys. Rev. Lett. 109, 116404 (2012);

E ϕ

2π

• 2.5

2.5 Zijlstra, Fasolino and Janssen, PRB 59, 302 (1999). El Hassouani et al., PRB 74, 035314 (2006). Pang, Dong and Wang, J. Opt. Soc. Am. B 27, 2009 (2010). Martínez and Molina, PRA 85, 013807 (2012).

β = 0 β = 1 β = 5 β = 10 β → ∞

Off-diagonal: Diagonal: Topological boundary states.

E ϕ

2π

• 2.2

2.7

SLIDE 27

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5

/

Eigenvalue

=50.0

Fibonacci pumping

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=25.1

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=12.6

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=6.3

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=3.2

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=1.6

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=0.8

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=0.4

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=0.2

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=0.1

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=0.1

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=0.2

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=0.4

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=0.8

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=1.6

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=3.2

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=6.3

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=12.6

.5 .4 .3 .2 .1 .1 .2 .3 .4 .5

 

=25.1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5

/

Eigenvalue

=50.0

n density n density

z

• M. Verbin , OZ, Y. Lahini, Y. E. Kraus, and Y. Silberberg, PRB 91, 064201 (2015)

In atoms M.. Lohse, Nat. Phys. 12, 350 (2016).

• S. Nakajima et al. Nat. Phys. 12, 296 (2016)

SLIDE 28

Further implications

• Topological phase transition:
• Topological phase transition in space:

A sharp edge breaks long-range order and the boundary modes may not appear!!

bulk bulk

boundary

1



2



n

E

2



1



n

E x  



1

1



2



n

E

2



1



n

E

( ) H 

phase transition

SLIDE 29

Bulk transitions

• Adiabatic deformation:
• LDOS:
• Measurement:

II

2 1 6.5 b  

I

2 1 5 b  

361

n E

–2.2 2.2 1

z n

221

intensity

localization 0.35

n

221 1

• M. Verbin, OZ, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013)

bulk bulk

boundary

1



2



n

E

2



1



n

E x  

Harper Harper

SLIDE 30

Bulk transitions

• M. Verbin, OZ, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013)
• Adiabatic deformation:
• LDOS:
• Measurement:

I

2 1 5 b  

z n

Harper

II

2 1 5 b  

Fibonacci

221

n E

–2.4 2.4 1

localization

0.35

n

221 1

SLIDE 31

What about 2D quasicrystals?

† x,y x,y 1

2 cos(2 ) . .

y z y y

t t b y c c h c  



       

 

† x,y 1,y x,y

( , ) 2 cos(2 )

x y x w x x x

H t t b x c c    



  



SLIDE 32

What about 2D quasicrystals?

† x,y x,y 1

2 cos(2 ) . .

y z y y

t t b y c c h c  



       

 

† x,y 1,y x,y

( , ) 2 cos(2 )

x y x w x x x

H t t b x c c    



  



SLIDE 33

Outline

Photonic topological pump (1D) Quantum Hall effect Topological pumps Quasiperiodic systems Atomic pumps Four-dimensional QHE

SLIDE 34

4D quantum Hall effect

2

1 d d d d (2 )

x y z w BZ k

k k k      



First derivations:

• J. E. Avron et al., Comm. Math. Phys. 124, 595 (1989).
• J. Fröhlich and B. Perdini, in Mathematical Physics 2000 (Imperial College

Press, London, United Kingdom).. S.-C. Zhang and J. Hu, Science

• Vol. 294, 823 (2001):

X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).

4

S

py Enk

n = 1 n = 2 n = 3 n = 4

μ

2

; B e I E h

   

    B A A

  

   

SLIDE 35

4D quantum Hall effect

B x z

2

1 d d d d (2 )

x y z w xz yw BZ k

k k k     



B y w

• K. Kraus, Z. Ringel, and OZ, PRL 111, 226401 (2013)

See also PRL 115, 195303 (2015), PRA 93, 043827 (2016), PRB 93 245113 (2016) 2

; B e I E h

   

   

SLIDE 36

B B

4D quantum Hall effect

y w I E

Lorentz-type response v w

I I  

2

;

x yw xz z

e I n E h  

2 xw y z

B e I E h   

Bxw

See also PRL 115, 195303 (2015), PRA 93, 043827 (2016), PRB 93 245113 (2016)

x z y w I F

2

;

y x

B e I E h

  

   

• K. Kraus, Z. Ringel, and OZ, PRL 111, 226401 (2013)

SLIDE 37

B B

4D quantum Hall effect

y w

Lorentz-type response v w

I I  

2

;

x yw xz z

e I n E h  

2 xw y z

B e I E h   

See also PRL 115, 195303 (2015), PRA 93, 043827 (2016), PRB 93 245113 (2016)

x z y w

2

;

y x

B e I E h

  

   

Density-type response z x w

I I I   

2 2

;

xz y w yw xz w

B e e I E n E h h     

I E + Bxz

• K. Kraus, Z. Ringel, and OZ, PRL 111, 226401 (2013)

SLIDE 38

4D QHE => 2D topological pumps

B x z I E B y w I F B

 

† † x,y x,y 1 x,y x,y

. . 2 cos(2 ( ) ) ]

y w yw yw y

t c c h c t B y c c  



     

 

† † x,y 1,y x,y x,y x,y

( , ) [ . . 2 cos(2 ( ) )

x y x x z xz yz x

H t c c h c t x B y c c    



     



• K. Kraus, Z. Ringel, and OZ, PRL 111, 226401 (2013)

SLIDE 39

2D photonic topological pump (2D + 2)

• Topological charge pump in each direction:

† x,y x,y 1

2 cos(2 ) . .

y z y y

t t b y c c h c  



       

 

† x,y 1,y x,y

( , ) 2 cos(2 )

x y x w x x x

H t t b x c c    



  



Nature 553, 59 (2018)

SLIDE 40

2D photonic topological pump (2D + 2)

• Topological charge pump in each direction:

Nature 553, 59 (2018)

SLIDE 41

2D photonic topological pump (2D + 2)

Nature 553, 59 (2018)

SLIDE 42

2D photonic topological pump (2D + 2)

Nature 553, 59 (2018) 2

;

xw y z

B e I E h   

SLIDE 43

Outline

Photonic topological pump (1D) Quantum Hall effect Topological pumps Quasiperiodic systems Atomic pumps Four-dimensional QHE

SLIDE 44

Atomic topological pumps

B b x y

Cool atoms into a Mott insulator state Homogeneous delocalization over first Brillouin zone

• Nat. Phys. 12, 350 (2016)

SLIDE 45

Atomic topological pumps

• Nat. Phys. 12, 350 (2016)

1

( , )

n x x n

x v k t dk dt C d  



SLIDE 46

Atomic topological pumps

1

1 C  

1

1 C  

• Nat. Phys. 12, 350 (2016)

SLIDE 47

2D topological charge pump

PRL 111, 226401 (2013), Nature 553, 55 (2018) & Nature 553, 59 (2018) 2

;

xw y z

B e I E h   

B B y w I E Bxw x z y w I F

SLIDE 48

2D topological charge pump

Nature 553, 55 (2018)

• An optical superlattice

SLIDE 49

2D topological charge pump

Nature 553, 55 (2018)

• An optical superlattice

∝ Β

SLIDE 50

2D topological charge pump

Nature 553, 55 (2018)

x y

SLIDE 51

Bulk response

Nature 553, 55 (2018) 2 xw y z

B e I E h    1.07(8)  

SLIDE 52

6D QHE and 3D topological pumps

• I. Petrides, H. M. Price, and OZ, arXiv:1804.01871

Induced motion after a cycle

SLIDE 53

Collaborators

Photonic topological pumps

Kobi Kraus (RIP) Zohar Ringel (HUJI) Yoav Lahini (Tel-Aviv) Yaron Silberberg (Weizmann)

Atomic topological pump

Monika Aidelsburger (LMU) Michael Lohse (LMU) Christian Schweizer (LMU) Immanuel Bloch (LMU) Hannah Price (Birmingham) Tomoki Ozawa (ULB) Nathan Goldman (ULB) Iacopo Carusotto (Trento) Mikael Rechtsman (Penn state) Jonathan Guglielmon (Penn state)

4D Topology

Ioannis Petrides (ETH)