Topological Phonons, Edge States, and Guest Modes. 5/17/2016 Zanjan School

Transcription

1 Topological Phonons, Edge States, and Guest Modes

2 Topological Defects Closed path in real space Path in order parameter space: topological invariant Periodic Brillouin Zone Function on torus Torus It no defects on surface, topology is completely determined by two winding numbers, i.e., by a lattice vector Closed path of function

3 Topology and Adiabatic Continuity Insulators are topologically equivalent if they can be continuously deformed into one another without closing the energy gap genus = 0 Are there topological phases that are not adiabatically connected to the trivial insulator (ie the vacuum)? genus = 1

4 Topological Electronic States and Topological Insulators A major theme in theory or electronic states: Berry s phases Polyacetylene Quantum Hall effect Topological Insulators surface conducting states (Graphene + 3d): Charlie Kane, Gene Mele, Liang Fu, Shoucheng Zhang Kane and Mele, PRL 95, ; PRL (2005) Hasan and Kane, Rev. Mod. Phys. 82, 3046 (2012) S. C. Zhang and B. Yan, Rept. Prog. Phys. 75, (2012) Gapped states can have different topological classifications leading to different surface states. Charlie Kane Gene Mele

5 Topology in hbar physics Polyacetalene: 1D chain with dimerization u = ± u 0 Su, Schrieffer, Heeger 79 u<0 A phase u>0 B phase A B 0 (x) E Two Dimensions: Quantum spin Hall insulator Kane and Mele 05 QSHI conduction band Conducting edge Insulating interior u(x) valence band E F /a 0 /a

6 Outline Topological index and edge states in the SSH model A Mechanical SSH model Kagome based lattices with distinct topological properties Topological count and edge states in kagomebased lattices Guest elastic modes Weyl modes and lines 3D Lattices Jamming

7 Su Schrieffer Heeger Model Two types of sites: 1 and 2: 1 connects to 2 but not 1 to 1 or 2 to 2 t 1 t 2 H = å ét + t l ê y y ë y y 1 l,1 l,2 2 l,2 l+,1 + hc.. ù úû Q = t d + t d ll 1 ll 2 l-1, l ; å = [ y Q y + y Q y ] ll,' l,1 ll l,2 l,2 ll' l',1 Q T ll ' = t d + t d = C 1 ll 2 l + 1, l ll å = Y () khk () Y() k; Y = ( y, y ) k 1 2 * Ck () = Q() k = t + te 1 2 ika Hk ( ) t 0 t e 1 2 ik t t e ik 0 * Q k ( ) Qk ( ) 0

8 Gapped Spectrum 2 2 E = Q( k) = t + t + 2tt coska t = t = t 1 2 ¾¾¾¾2 t cos( ka/2) t ka=p ¾¾¾¾ t Spectrum has zeros at ka = p when t = t and a gap of E = 2 t - t when t ¹ t. g

9 Zero modes localized at endpoints H(k) 0 Q(k) C(k) 0 t1 t t1 t2 0 Ck ( ) 0 0 t1 t Ey = t y + t y ; E = 0 l2 1 l1 2 l+ 1,1 y =-( t / t ) y ; l+ 1,1 1 2 l,1 y = -( t / t ) y ; y = 0 l-1,1 2 1 l,1 l,2 Ey = t y + t y ; E = 0 l1 1 l2 2 l-1,2 y =-( t / t ) y ; l-1,2 1 2 l,2 y = -( t / t ) y ; y = 0 l+ 1,2 2 1 l,2 l,1 Decaying sol'n on sites 1 from right if t < t ; 1 2 from left if t > t 1 2

10 Topological Charge of Gapped States Ck () = t + et º t+ zt = Cz () ( = 0, at z=-t / t) ik Cauchy s Argument Principle 2p 1 g'( z) 1 d n = ò dz = ò dk 2 pi gz ( ) 2pi dk 0 = # zeros - # poles = integer ik gz () = 0; z = e < 0; no poles Im k >0: decaying soln ik ln g( e ) Phase of C t 1 <t 2 P d dk dk ln Ck ( ) 1,0 t 2 <t 1

11 Domain wall states Tie right and left decaying states together to create a zeroenergy mode localized at the domain wall s Hs =-H z z For every state with energy E, there is a state with energy E ( particle hole symmetry ): There is a topologically protected zero mode at the boundary between the two distinct topological phases.

12 Graphene Various perturbations break the symmetry and lead to gaps at the Dirac points. Note similarity to gapping of phonon states in the twisted kagome lattice. [Graphene: rotational symmetry about vertical axis; kagome pattern for every q x.]

13 1D Topological Mechanical Model Blue SSH site > site: Red SSH site > bond

14 C, s s C c ( ) c ( ) C c E, s 1 s, 2, s1, s 1 2 1(2) ( q) c ( ) c ( ) e Mechanical SSH Model ( a 2rsin ) rcos k k a s,, s 4r cos l C 2 C iqa T s s,, s s k c c s T D 1 s 2 s1 2 Same spectrum as SSH: Zero mode on left (right) if c c c c Non linear mode B. G. G. Chen, N. Upadhyaya, V. Vitelli PNAS 111, (2014).

15 Topological Mechanics Show Conductor movie Bryan Chen, Nitin Upadhyaya, Vincenzo Vitelli (Leiden) PNAS, 111,13004,(2014)

16 2 D Maxwell Lattices: z=2d=4 Maxwell Lattice one that under periodic boundary conditions have z=2d exactly, i.e., z=4 in two dimensions and z=6 in three dimension (Isostatic later) Kagome Lattices

17 Topological States in Isostatic Lattices H 1 2 kqqt 1 2 kq(k)q (k) Not clear how to extract topological information from this: Introduce a model coupling sites to bonds. Remember that in the periodic case, N B = dn, and Q is a square matrix. H k 0 Q Q T 0 QQ T 0 ; H 2 k 2 0 Q T Q H 2 has the same spectrum as H, except for zero modes Topological information is contained in Q. To get nontrivial topological states, constraint of equal length (but not z=4) bonds must be relaxed.

18 Transitions between Topological States P T = 0 P T = 0 Different topological states,characterized by a polarization vector P T classes are separated by gapless states produced by states of self stress Transition state with states of self stress P =-a T 1 P =-a T 1

19 T 1 V cell P T 1 V BZ Choose symmetric unit cell; Calculate Q S dsp T Topological Polarization Polarizations i d 2 k BZ k Im Tr lnq R T n i a i ; b i a j 2 ij T N cell GR T /2; G k i b i i Place charge +2 at each site bond: zero net charge in unit cells; polarization is zero in symmetric cell L N cell GP L /2 P L R L Polarization of surface cell = d r s r b sites s bonds b L T n 0 s 1 1 : charge 2 a2 4 4 : charge -1 a2 6 6 : charge -1 a P 2a a a a L

20 b 2 Mode Count 1 1 First recall dn=n +2 B. Assign charge +2 to each site and 1 to each bond (like Pebble game Thorpe). The periodic lattice is charge neutral, but charge distribution within cells can have a dipole moment. b b b a 2 b 1 Spectrum is gapless with modes with frequency ~ q 2 a 3 a 1 Define: N S dn N L T L T 0 local count = surface charge Topological count = "Polarization" charge B

21 Symmetric and Surface Gauges w n e iqa n ; S n w n 1 detc sym g 1 S 2 S 3 g 2 S 3 S 2 g 3 S 1 S 2 g 0 S 1 S 2 S 3 a 1 b n w n c n w n n1 z x e iq x x ; z y e i 3q y y/2 3 3 n1 w 1 z x ; w 2 z x 1/2 z y ; w 2 z x 1/2 z y 1 One pole in z y detc surf e iqr L detc sym No poles, only zeros: Topological charge of surface compatible unit cells simply counts the totalnumberofzero modes atthesurface!

22 Twisted Kagome Surface States P T R T 0

23 Topological Surface States Non zero R T moves zero modes from one surface to the opposite one.

24 G b1 G b1 Domain Wall l G ( RT R 2 1 zero modes T r T ) l r G ( RT RT) 2 1 States of Self Stres T Modes of full H with zero modes from Q ( States of self stress) and form Q T (zero modes)

25 Stresses and zero modes at defects J. Paulose, B. G. G. Chen, V. Vitelli, Nature Physics 11, (2015). Stress is concentrated at selfstress domain wall causing mechanical failure Dislocations localize soft modes and self stress: J. Paulose, A. S. Meeussen, V. Vitelli, PNAS 112, (2015)

26 Guest Modes K æ G K K K ö æ xxxx xxyy xxxy G G u ö xx G u u æ ö xx xy K K K ; G = xxyy yyyy yyxy G G u u = u u yy K K K xy yy G ç è ø xxxy yyxy xyxy u èç ø èç xy ø Zeros to linear order in q qx G G q = ( u -detu ) y G xy u xx if

27 Elastic Energy a a 1 1 > < 0 0 : Negative Poisson ratio and R = 0; Q = O k T 2 det ( ) near origin : Positive Poisson ratio and R ¹ 0; T 3 Q = O k ) k = a x 1 det ( along k ; w ~ T T y k 2 Show Mathematica animation! a 1 >0 a 1 <0

28 Bulk Modes of Topological Lattices k = k k ^ = + ^ n + k zˆ nˆ; sin q cos q i a i a1 i(3 + a nˆ = Surface cos q k sin q ) d 2( cos q i a sin q) k 2 normal a a > 0:Im k ~ k 1 ^ Opposite signs on opposite surfaces < 0 :Im k ~ k 2 1 ^ Number surface modes (1 or 2) depends on q

29 Strain shifted zero modes Rocklin, Zhou, Sun, Mao, arxiv:

30 Topological Square Lattices Zeb Rocklin Michigan Bryan Chen UMass Vincenzo Vitelli Leiden +Martin Falk D. Z. Rocklin, B. G. g. Chen, M. Falk, V. Vitelli, TCL, arxiv: (2015) Two q=0 states of self stress but no elasticity. Weyl modes have interesting and nontrivial effect nonlinear dynamics (Bryan Chen) Rule rather than exception for large unit cell lattices Topological lattice with zero (Weyl) modes at q \neq 0 (like graphene Dirac point) with non zero winding number: Topological Polarization depends on surface q s, and zero modes change surfaces with q s. Sample traversing state of self stress when q s corresponds to Weyl q.

31 Weyl Phase diagram and domain wall zero modes (b) (c) (d) (e) (b) In region without Weyl modes like kagome (c) Region with a pair of Weyl modes enter at origin and disappear at zone edge (d) Transition region line of zero modes and SSS (e) Double critical lines of zero modes and SSS in two diretions ( UL, ) n ( q) = No. zero modes at q on free upper 0 (lower) surface n = No. of "binding bonds" added B n D () q = n U () q + n L () q -n B = No. of zero modes in DW

32 Topological Pyrochlore Lattices Olaf Stenull, CL Kane, TCL: Work in Progress Topologically distinct 3D phases, constructed following procedures developed for kagome lattices Weyl lines instead of Weyl points with associated transition of zero modes from one side of sample to other as Weyl lines are crossed. 3 zero modes at q=0: and 3 soft and 3 hard elastic directions (like origami).

33 Weyl points at jamming (a) Jammed Maxwell Lattice (b) Randomized quasicrystal approximate Probability distribution of inverse penetration depth: + on right, on left No. of zero modes on two surfaces and positions of Weyl modes. Two independent states of self stress at domain wall dividing two jammed Maxwell lattices

34 Large antagonistic unit cells Red: wall of states of self stress Blue: Zero energy domain wall

35 Origami In Progress Thanks to Chris Santangelo and Bryan Chen d=3; n=4; n b =12; dn n b =0: A Maxwell lattice Twisted kagome and topological versions gapped apparently no topological distinction between two 3 states of self stress and 3 zero modes at q=0: 3 soft elastic modes that involve 2d elasticity and change in thickness of membrane (zzstrain). Relax latter: One soft mode of inplane distortion, like flat kagome. Effective elastic energy has Helfrich bending energy (up down symmetry is broken) and strain curvature coupling (suggestive of Ron Resch)

36 In Progress Nonlinear response: Weyl Modes and their interactions with Guest Modes; Elastic and periodic Weyl modes mix producing interesting instabilities. (B. Chen, Z. Rocklin, V. Vitelli, TCL. Extend to 3D Weyl lines (O. Stenull, TCL) Real systems: Perfect physical Maxwell lattices do not exist. They either have bending forces between bonds at vertices or effective further neighbor interactions, but these interactions can be small in engineered materials. How does turning on small NNN or bending forces modify elastic response and zero modes (X. Mao, O. Stenull, Shu Yang, TCL) : Long wavelength: (d 1) Rayleigh modes on each surface; Shorter wavelengths; crossover to behavior similar to Maxwell lattice behavior but with low frequency surface modes Domain wall modes develop a finite frequency and become localized propagating phonons: Domain wall acts like a wave guide. Can we control phonon propagation by controlled placement of domain walls Self stress domain walls concentrate stress [Paulose, Meeussen,Vitelli PNAS 112, (2015)]. Can we control these stresses and places at which material breakdown occurs?