Topological pumps and topological quasicrystals

Transcription

1 Topological pumps and topological quasicrstals PRL 109, (01); PRL 109, (01); PRL 110, (013); PRL 111, 6401 (013); PRB 91, (015); PRL 115, (015), PRA 93, (016), PRB 93, (016), Nat. Phs. 1, 350 (016); Nat. Phs. 1, 64 (016); Nature 553, 55 (018), Nature 553, 59 (018) arxiv: , arxiv: , arxiv: & arxiv:

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

3 Quantum Hall effect Landau gauge: p 1 m k mac a ma c ip ( pl ) p,(, n ) e (...) e Quantized Hall conductance e h I B E E nk n = 4 Chern number d p d p (p,p ) (p,p ) Tr, P P P i p p μ ω c n = 3 n = n = 1 p P n n n P

4 Quantum Hall effect Landau gauge: p 1 m ip ( pl ) p,(, n ) e (...) e mac a ma c k B Confining potential: Chiral edge modes bulk E nk n = 4 edge Chern number Number of chiral edge modes ω c μ n = 3 n = n = 1 See Yasuhiro s talk and references therein p

5 Laughlin/Halperin s argument Landau gauge: p 1 m ip ( pl ) p,(, n ) e (...) e k mac a ma c B φ/ t Chern number Number of charges moved over a pump ccle μ bulk E nk p

6 Laughlin/Halperin s argument Landau gauge: p 1 m ip ( pl ) p,(, n ) e (...) e k mac a ma c B φ/ t Chern number Number of charges moved over a pump ccle μ bulk E nk p

7 Hofstadter model Hamiltonian:, Spectrum: i b, 1,..,,1.. t c c h c t e c c h c, k.. cos( ) t c c h c t b k c c, k 1, k, k, k Harper, PPSL A 68, 874 (1955) Azbel, JETP 19, 634 (1964) Hofstadter, PRB 14, 39 (1976) b p q q-1 gaps ik E,, k k c e c.5 E b = 8/ π kb b t t

8 Hofstadter model Hamiltonian: k, Quantized Hall conductance: TKNN, PRL 49, 405 (198) Chern numbers:.. cos( ) t c c h c t b k c c, k 1, k, k, k e h d p d p (p,p ) (p,p ) Tr, P P P i p p b μ.5 b = 8/5 3 E π 4 1 k t t P

9 Hofstadter model Hamiltonian: Quantized Hall conductance: TKNN, PRL 49, 405 (198) Chern numbers: b r p qm r r q p q r.. cos( ) t c c h c t b k c c k,, k 1, k, k, k e r h E μ 1 D. Osadch and J. E. Avron, J. Math. Phs. 4, 5665 (001).5 E kb b b = 8/ π t P t

10 Topological pump (1+1) Harper model as a topological pump: D. J. Thouless, Phs. Rev. B 7, 6083 (1983) H ( ) t c c h. c. t cos( b ) c c t 1 / 7 Spectrum:.5 b = 8/5 Topological edge states of a D crstal Boundar states of a 1D topological pump E π

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

12 Photonic waveguide arra A waveguide: i V z z A photonic crstal: Lahini et al., PRL 103, (009) i t t V z tight binding model propagation replaces time z

13 Photonic waveguide arra Diagonal modulation: i t( ) V z 1 1 z Off-diagonal modulation: i t t z z

14 Eperiment Setup: input: single site output: distribution Evolution: overlap with eigenstates z epansion according to eigenstates No chemical potential

15 Generalizations Other 1D models Off-diagonal Harper model: H t t b c c h c ( ) cos( ) 1.. n b t.5 E t ϕ π Phs. Rev. Lett. 109, (01)

16 intensit densit densit densit Eperiment 1: adiabatic pumping (1D+1) Pumping:.5 E Adiabatic pumping: off-diagonal ϕ π H ( z) z z H t t b c c h c ( ) cos( ) 1.. n Phs. Rev. Lett. 109, (01)

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

18 Quasiperiodic models Sstems with long-range order but no translation smmetr. 1D Fibonacci chain: L LS S L L LS LSL LSLLS LSLLSLSL D Penrose tiling: 1 1 dn ( n ) ( n 1) 1 1 1

19 Long-range order vs. translations Harper (or Aubr-André) model: H ( ) t c c h. c. t 'cos( bn ) c c n Long-range order vs. translations n n1 n n cos( bn) n n The order determines the sstem up to translations of the origin (which have no effect on bulk properties). Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phs. Rev. Lett. 109, (01)

20 Long-range order vs. translations Harper (or Aubr-André) model: Translations vs. ϕ H ( ) t c c h. c. t 'cos( bn ) c c n n n1 n n b = p/q: irrational b: 1 ( bn) mod 0,,, q q ( bn) mod [0, ] For the latter ϕ has no effect on bulk properties Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phs. Rev. Lett. 109, (01)

21 Long-range order vs. translations Harper (or Aubr-André) model: H ( ) t c c h. c. t 'cos( bn ) c c n Translations vs. ϕ ϕ ϕ +ε is equivalent to n n + nε independent of ϕ flat bulk bands n n1 n n H( ) T H( ) T E l ( ) T l T l l T E ( ) l l T l l E Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phs. Rev. Lett. 109, (01) 0 ϕ π

22 Long-range order vs. translations Harper (or Aubr-André) model: H ( ) t c c h. c. t 'cos( bn ) c c n n n1 n n Projector P: P ~ H P( ) T P( ) T P( ) T P( ) T Berr curvature 1 1 ( (,, ) ) Tr Tr P P( ) P( ), P( ) P, i P i 1 μ P ( ) P ( ), P ( ) Tr T P ( ) P ( ), P ( ) T c( 0, ) i 0 π ϕ P Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phs. Rev. Lett. 109, (01)

23 Long-range order vs. translations Harper (or Aubr-André) model: H ( ) t c c h. c. t 'cos( bn ) c c n n n1 n n Berr curvature 1 (, ) Tr P, P P ( ) i Chern number: 0 0 Chern numbers can be associated with the whole We can associate Chern numbers with an H(ϕ)! famil {H(ϕ)}. Thouless, PRB 7, 6083 (1983) d d (, ) d ( ) 0 Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and OZ, Phs. Rev. Lett. 109, (01)

24 Fibonacci-like chains Fibonacci quasicrstal: d 1 n 1 1 ( n ) ( n1) Off-diagonal Fibonacci 1 H t t ' d c c h. c. n n n n Diagonal Fibonacci H t( c c h. c.) t ' d c c n n1 n n n n n Y. E. Kraus and OZ, Phs. Rev. Lett. 109, (01);

25 Fibonacci-like chains Fibonacci quasicrstal: 11 ( ) tanh 1 d ( ) cos bn 3 b cos b n n tanh( ) ( n 1) β 0: n β : Harper d ( 0) cos bn 3 b cos b Fibonacci dn( ) b ( n ) b( n 1 ) 1 β = 01 5 Y. E. Kraus and OZ, Phs. Rev. Lett. 109, (01);

26 Fibonacci-like chains Off-diagonal:.5 Diagonal:.7 E E ϕ π β = Topological boundar states. Zijlstra, Fasolino and Janssen, PRB 59, 30 (1999). El Hassouani et al., PRB 74, (006). Pang, Dong and Wang, J. Opt. Soc. Am. B 7, 009 (010). Martínez and Molina, PRA 85, (01) ϕ π Y. E. Kraus and OZ, Phs. Rev. Lett. 109, (01);

27 Fibonacci pumping 0.5 =50.0 =5.1 =1.6 =0.1 =0. =0.4 =0.8 =1.6 =3. = densit densit Eigenvalue n n / In atoms M.. Lohse, Nat. Phs. 1, 350 (016). S. Nakajima et al. Nat. Phs. 1, 96 (016) M. Verbin, OZ, Y. Lahini, Y. E. Kraus, and Y. Silberberg, PRB 91, (015) z

28 Further implications Topological phase transition: H ( ) 1 0 E n phase transition Topological phase transition in space: E n E n bulk boundar bulk A sharp edge breaks long-range order and the boundar modes ma not appear!! E n 1

29 localization intensit Bulk transitions Adiabatic deformation: Harper bi 1 5 LDOS: 1. E E n bulk boundar E n bulk 1 Harper bii Measurement: n z 0 1 n 1 M. Verbin, OZ, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phs. Rev. Lett. 110, (013) n

30 localization Bulk transitions Adiabatic deformation: Harper bi 1 5 LDOS:.4 E Fibonacci bii Measurement: n z 1 0 n 1 M. Verbin, OZ, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phs. Rev. Lett. 110, (013) n

31 What about D quasicrstals?, 1, H (, ) t t cos( b ) c c w, t tz cos( b ) c,c,1 h. c.

32 What about D quasicrstals?, 1, H (, ) t t cos( b ) c c w, t tz cos( b ) c,c,1 h. c.

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

34 4D quantum Hall effect E nk n = 4 S 4 μ n = 3 n = n = 1 I 1 ( ) E h 0 e BZ B dk dk dk dk ; z w p B A A First derivations: J. E. Avron et al., Comm. Math. Phs. 14, 595 (1989). J. Fröhlich and B. Perdini, in Mathematical Phsics 000 (Imperial College Press, London, United Kingdom).. S.-C. Zhang and J. Hu, Science Vol. 94, 83 (001): X.-L. Qi and S.-C. Zhang, Rev. Mod. Phs. 83, 1057 (011).

35 4D quantum Hall effect K. Kraus, Z. Ringel, and OZ, PRL 111, 6401 (013) B z B w I 1 ( ) E h 0 e BZ B dk dk dk dk z w z w ; See also PRL 115, (015), PRA 93, (016), PRB (016)

36 4D quantum Hall effect K. Kraus, Z. Ringel, and OZ, PRL 111, 6401 (013) Lorentz-tpe response z E I B I e h B F I B w B 0 E ; w I v I w 0 e I n E h e Bw I E h w ; z z 0 z See also PRL 115, (015), PRA 93, (016), PRB (016)

37 4D quantum Hall effect K. Kraus, Z. Ringel, and OZ, PRL 111, 6401 (013) Lorentz-tpe response z I B + B z e h B E I B 0 E ; w I v I w 0 e I n E h e Bw I E h w ; z z 0 Densit-tpe response z I z I Iw 0 e B e I E n E z w w z w h 0 h ; See also PRL 115, (015), PRA 93, (016), PRB (016)

38 4D QHE => D topological pumps z E B I B F B I w H (, ) [ t c c h. c. t cos( ( B ) ) c c, 1, z z z,,, t c,c, 1 h. c. tw cos( ( w Bw) ) c,c,] K. Kraus, Z. Ringel, and OZ, PRL 111, 6401 (013)

39 D photonic topological pump (D + ) Topological charge pump in each direction:, 1, H (, ) t t cos( b ) c c w, t tz cos( b ) c,c,1 h. c. Nature 553, 59 (018)

40 D photonic topological pump (D + ) Topological charge pump in each direction: Nature 553, 59 (018)

41 D photonic topological pump (D + ) Nature 553, 59 (018)

42 D photonic topological pump (D + ) I e h B w 0 E z ; Nature 553, 59 (018)

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

44 Atomic topological pumps B Cool atoms into a Mott insulator state Homogeneous delocalization over first Brillouin zone b Nat. Phs. 1, 350 (016)

45 Atomic topological pumps v ( k, t) dk dt C d n n 1 Nat. Phs. 1, 350 (016)

46 Atomic topological pumps C1 1 C1 1 Nat. Phs. 1, 350 (016)

47 D topological charge pump z E B I B F I B w w I e h B w 0 E z ; PRL 111, 6401 (013), Nature 553, 55 (018) & Nature 553, 59 (018)

48 D topological charge pump An optical superlattice Nature 553, 55 (018)

49 D topological charge pump An optical superlattice Β Nature 553, 55 (018)

50 D topological charge pump Nature 553, 55 (018)

51 Bulk response I e h B w 0 E z 1.07(8) Nature 553, 55 (018)

52 6D QHE and 3D topological pumps Induced motion after a ccle I. Petrides, H. M. Price, and OZ, arxiv:

53 Collaborators Photonic topological pumps Zohar Ringel (HUJI) Kobi Kraus (RIP) Yoav Lahini (Tel-Aviv) Yaron Silberberg (Weizmann) Jonathan Guglielmon (Penn state) Mikael Rechtsman (Penn state) Atomic topological pump 4D Topolog Monika Aidelsburger (LMU) Michael Lohse (LMU) Christian Schweizer (LMU) Immanuel Bloch (LMU) Hannah Price (Birmingham) Tomoki Ozawa (ULB) Ioannis Petrides (ETH) Nathan Goldman (ULB) Iacopo Carusotto (Trento)