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

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

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

Quantum Hall effect • Landau gauge: 2   2 k 1 B p     2 y   x m x  a c I 2 2   m m a a c y E   2   ( ) ip y x p l p , ( , ) (...) n x y e y e y y x • Quantized Hall conductance 2 e E nk    xy h n = 4 • n = 3 Chern number ω c   n = 2 2 2   μ    d p d p (p ,p ) n = 1 x y x y P 0 0   1   p y   (p ,p ) Tr , P P P     x y p p 2     i x y | | P n n n

Quantum Hall effect • Landau gauge: 2   2 k 1 B p     2 y   x m x  a c 2 2   m m a a c y   2   ( ) ip y x p l p , ( , ) (...) n x y e y e y y x bulk edge • Confining potential: E nk Chiral edge modes n = 4 n = 3 ω c • Chern number n = 2 μ Number of chiral edge modes n = 1 p y See Yasuhiro ’ s talk and references therein

Laughlin/Halperin ’ s argument B • Landau gauge: 2   2 k 1 p     2 y   x m x  a c 𝜖𝜚/𝜖𝑢 2 2   m m a a c   2   ( ) ip y x p l p , ( , ) (...) n x y e y e y y bulk • Chern number E nk Number of charges moved over a pump cycle μ p y

Laughlin/Halperin ’ s argument B • Landau gauge: 2   2 k 1 p     2 y   x m x  a c 𝜖𝜚/𝜖𝑢 2 2   m m a a c   2   ( ) ip y x p l p , ( , ) (...) n x y e y e y y bulk • Chern number E nk Number of charges moved over a pump cycle μ p y

Hofstadter model x y • Hamiltonian: b t y      t x      † 2 † i bx . . . . t c c h c t e c c h c   x,y 1,y x,y x,y 1 x x y , x y     †     † . . 2 cos(2 ) t c c h c t bx k c c  x, 1, x, x, k x k y y k k y y y y x, k y • Spectrum: b = 8/5 Harper, PPSL A 68 , 874 (1955) 2.5 4 Azbel, JETP 19 , 634 (1964) Hofstadter, PRB 14 , 2239 (1976) 3   ik y p E c e y E c  q- 1 gaps b x,y x, k y 2 q k y 1 -2.5 2 π 0 k y b

Hofstadter model x y • Hamiltonian: b t y    t x      † † . . 2 cos(2 ) t c c h c t bx k c c  x, 1, x, x, x k x k y y k k y y y y , x k y • Quantized Hall conductance: TKNN, PRL 49 , 405 (1982) 2 e    b = 8/5 xy h 2.5 4 • Chern numbers:   3 2 2      d p d p (p ,p ) E x y x y μ 2 0 0   P 1 1     (p ,p ) Tr , P P P -2.5    2 π x y p p 0 k y 2 i x y

Hofstadter model x y • Hamiltonian: b t y    t x      † † . . 2 cos(2 ) t c c h c t bx k c c  x, 1, x, x, x k x k y y k k y y y y , x k y • Quantized Hall conductance: TKNN, PRL 49 , 405 (1982) 2 e D. Osadchy and J. E. Avron, J. Math. Phys. 42 , 5665 (2001)    b = 8/5 xy r h 2.5    2 4 • Chern numbers: 4   p 1  3 b 3 q E E    1 μ 2    2 r p qm r r   P 2 1 1 q -2.5   | | 2 π 0 k y b r 2

Topological pump (1+1) • Harper model as a topological pump: D. J. Thouless, Phys. Rev. B 27 , 6083 (1983)           † † ( ) . . 2 cos(2 ) H t c c h c t bx c c  1 x x x y x x x       2 / 7 x t x • Spectrum: x b = 8/5 2.5 Topological edge states of a 2D crystal E Boundary states of a 1D topological pump -2.5  2 π 0

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

Photonic waveguide array • A waveguide:     i V z z • A photonic crystal: Lahini et al. , PRL 103, 013901 (2009)         i t t V    1 1 1 z x x x x x x x z tight binding model propagation replaces time x

Photonic waveguide array • Diagonal modulation: z         ( ) i t V   1 1 z x x x x x x • Off-diagonal modulation:       i t t    z 1 1 1 z x x x x x x

Experiment • Setup: input: single site output: distribution z • Evolution: overlap with eigenstates expansion according to eigenstates x • No chemical potential

Generalizations – Other 1D models • Off-diagonal Harper model:           † ( ) 2 cos(2 ) . . H t t bx c c h c    1 x xy x x x n 2.5 x E y b t xy -2.5 t x ϕ 2 π 0 Phys. Rev. Lett. 109 , 106402 (2012)

Experiment 1: adiabatic pumping (1D+1) • density Pumping: 2.5 x density E x density -2.5 ϕ 2 π 0 x    • Adiabatic pumping: ( ) H z z off-diagonal intensity z           † ( ) 2 cos(2 ) . . H t t bx c c h c   n  1 x xy x x x Phys. Rev. Lett. 109 , 106402 (2012)

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

Quasiperiodic models Systems with long-range order but no translation symmetry. • 1D Fibonacci chain: L L → LS LS S → L LSL LSLLS LSLLSLSL       1 1 • 2D Penrose tiling:      2  ( 2) ( 1)  1 d n n         n       1 1

Long-range order vs. translations • Harper (or Aubry-André) model:           † † ( ) . . 2 'cos(2 ) H t c c h c t bn c c  n n 1 n n n • Long-range order vs. translations n  1..30  cos(2 ) bn n  101..130 The order determines the system up to translations of the origin (which have no effect on bulk properties). Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ , Phys. Rev. Lett. 109 , 106402 (2012)

Long-range order vs. translations • Harper (or Aubry-André) model:           † † ( ) . . 2 'cos(2 ) H t c c h c t bn c c  n n 1 n n n • Translations vs. ϕ   1 2        b = p/q:   (2 )mod 2 0, 2 , 2 , bn   q q     irrational b: (2 )mod2 [0,2 ] bn For the latter ϕ has no effect on bulk properties Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ , Phys. Rev. Lett. 109 , 106402 (2012)

Long-range order vs. translations • Harper (or Aubry-André) model:           † † ( ) . . 2 'cos(2 ) H t c c h c t bn c c  n n 1 n n n • Translations vs. ϕ ϕ → ϕ + ε is equivalent to n → n + n ε      †      † ( ) ( ) ( ) T E l l T H T H T     l l    ( ) l E T l T l   l independent of ϕ → flat bulk bands E 2 π ϕ 0 Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ , Phys. Rev. Lett. 109 , 106402 (2012)

Long-range order vs. translations • Harper (or Aubry-André) model:           † † ( ) . . 2 'cos(2 ) H t c c h c t bn c c  n n 1 n n n • Projector P:             † † ( ) ( ) ( ) ( ) P ~ H P T P T P T P T       • Berry curvature     1 1                            ( ( , ) , ) Tr Tr ( , ) ( ), ( ) P P P P P P           2 2 i i μ     1 Tr 1 Tr                     † ( ) ( ) ( ), ( ), ( ) ( ) c( 0, ) T P P P P P P T P             2 2 i i 2 π ϕ 0 Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ , Phys. Rev. Lett. 109 , 106402 (2012)