Higher order topological insulators A paradigm for topological - PowerPoint PPT Presentation

Titus Neupert Yukawa Institute, Kyoto, November 03, 2017 Higher order topological insulators

A paradigm for topological states of matter ∂ ( ∂ M ) = ∅ (the boundary of a boundary is empty) … works when things are su ffi ciently smooth. Crystals have no smooth surface! arXiv:1708.03636 Frank Schindler Maia Vergniory Zhijun Wang Ashley Cook Stuart Parkin Andrei Bernevig (U Zurich) (Donostia) (Princeton U) (U Zurich) (Max Planck Halle) (Princeton U)

Topological Insulators Bulk-boundary correspondence: E gapless Dirac cones, gapped bulk band structure k 300K 10K d 3 k  � A a @ b A c + i2 Z Topological invariant: ✓ = � ✏ abc (2 ⇡ ) 3 tr 3 A a A b A c A a ; n,n 0 = � i h u n | ∂ a | u n 0 i with time-reversal symmetry θ = 0 , π Topological invariant with inversion: k z Y product over inversion eigenvalues ξ k = ( − 1) ν k y at time-reversal invariant momenta k ∈ TRIM k x

Superconducting and magnetic TI surface G = U ( 1 ) o Z T non-interacting SPT phase: 2 Dirac cone + s-wave pairing: TR breaking: anomalous QHE U(1) breaking: TRS SC with fractional conductivity Majorana in vortex without fractionalization σ xy = 1 / 2 Fu Kane Mele PRL 07 Fu Kane PRL 08 TI TI

Domain wall modes a) chiral fermion b) chiral Majorana mode - - - -- - - - - - - - - - - - - - - - - - - SC - M = ↑ - 1 - - - - - - - - - - - - - 1 - - - - - - - - M = ↓ M = ↓ - - - - 2 - - - - - - - - - - - - 3 - 1 - - - - - - - - - - - - 1 - - - - - - - - - 3 - - - - - - - - - - - - - - - - - - - - 3 1 - - - - - - - - - - - - 1 - - - - - 3 - - - - - - - 1 - - - - - - - - 1 2 - -- - - N p = 12 1 1 - - N p = 11 1 -- M M 1 - - − 10 − 8 − 6 − 4 − 2 0 2 − 6 − 5 − 4 − 3 − 2 − 1 0 10 - 5 0

Topology from Wilson loops k z Z 2 π � d k z h u m ( ~ k ) | @ k z | u n ( ~ W nm ( k x , k y ) = exp k ) i k y 0 k x occupied band unitary operator in filled eigenstates band subspace Define ‘Wilson loop Hamiltonian’ W ( k x , k y ) = exp[ − i H W ( k x , k y )] λ ( W ) π k y Resembles surface Hamiltonian with qualitatively identical gapless spectrum (single Dirac cone) k x − π λ ( W ) π Equivalence: eigenvalues of Wilson loop and projected position operator ˆ x ˆ P ˆ P − π

Topological crystalline insulators Stabilize more than one Dirac cone by adding crystalline symmetries Mirror symmetry: eigenvalues +i and -i in spinful system eigenstates on the mirror invariant planes in momentum space Mirror Chern number: k z Chern number in +i/-i subspace on the plane Z C ± = 1 ∂ k y A ± z − ∂ k z A ± d 2 k tr ⇥ ⇤ k y y k x =0 / π 2 π ∈ Z C + = − C − Time-reversal symmetry: k x Number of Dirac cones crossing line in surface BZ k y +i k x -i [L. Fu, Phys. Rev. Lett., 2011] k y

Higher-order topological insulators Electric circuit realization arXiv:1708.03647 normalized impedance (d-m)-dimensional boundary components of a d-dimensional system are gapless for m = N , and are generically gapped for m < N 1 2 3 dimension E SSH k 1 TI QHE Mechanical realization, Huber group arXiv:1708.05015 Rest of 2 this talk order [Benalcazar, Bernevig, Hughes Can only happen with 3 Science 357, 61-66 (2017)] spatial symmetries: generalizations of TCIs

Construction of a 2nd order 3D TI Protecting symmetry: C 4 T (breaks T, C 4 individually) surface construction from 3D TI: decorate surfaces alternatingly with outward and inward pointing magnetization, gives chiral 1D channels at hinges Adding C 4 T respecting IQHE layers on surface can change number of hinge modes by multiples of 2 c) Odd number of hinge modes stable against any C 4 T respecting surface manipulation Chern Bulk topological property Z 2 insulator

Construction of a 2nd order 3D TI Protecting symmetry: C 4 T (breaks T, C 4 individually) Bulk construction TI band structure plus (su ffi ciently weak) triple-q ( 𝜌 , 𝜌 , 𝜌 ) magnetic order Toy model with only C 4 T in z -direction ! X X H 4 ( ~ k ) = M + cos k i ⌧ z � 0 + ∆ 1 sin k i ⌧ y � i + ∆ 2 (cos k x − cos k y ) ⌧ x � 0 i i 3D TI T, C 4 breaking term λ ( H C ) Spectrum of column geometry 0 k z 0 2 π π

Topological invariant of a 2nd order 3D TI Same quantization with C 4 T as with T alone: d 4 x E · B θ R Z top = e i is topological invariant θ = 0 , π 8 π 2 ( C 4 T ) 4 = − 1 Di ff erent from existing indices, because Case of additional inversion times TRS , IT, symmetry: k { e i ⇡ / 4 , e − i ⇡ / 4 } ξ ~ k = ± 1 ( IC 4 ) 4 = − 1 use with eigenvalues ξ ~ Due to ‘Kramers’ pairs with same are [ IC 4 , IT ] = 0 ξ ~ k = ± 1 degenerate. Band inversion formula for topological index à la Fu Kane for C 4 T invariant momenta k z ( − 1) ⌫ = Y ξ ~ k ~ k ∈ I ˆ k y Cz 4 ˆ T I ˆ T = { (0 , 0 , 0) , ( π , π , 0) , (0 , 0 , π ) , ( π , π , π ) } k x 4 ˆ C z

Wilson loop topology of a 2nd order 3D TI Wilson-loop based bulk topological characterization Z 2 π � d k z h u m ( ~ k ) | @ k z | u n ( ~ W nm ( k x , k y ) = exp k ) i 0 λ ( H W ) π C 4 T implies Kramers-like degeneracies in Wilson loop spectrum at C 4 T invariant momenta 0 ( k x , k y ) ∈ { (0 , 0) , ( π , π ) } Z 2 Wilson loop winding between these momenta is topological invariant k x , k y − π (0 , 0) (0 , π ) ( π , π ) (0 , 0) ✓Z ( ⇡ , ⇡ ) ◆ ⌫ = 1 X d ~ k � l ( ~ k ) − � l ( ⇡ , ⇡ ) + � i (0 , 0) mod 2 k · @ ~ 2 ⇡ (0 , 0) l

Boundary topology of a 2nd order 3D TI Nested entanglement spectrum Nested Wilson loop spectrum ρ A = Tr B | Ψ ih Ψ | ⌘ 1 x-direction Wilson loop spectrum gapped e − H e Z e k z λ ( H e ) 4 k y entanglement A spectrum is k x 0 B gapped …one more: W is 2x2 matrix with topological Wilson loop of gapped slab k z − 4 0 2 π lower band: log is π spectrum: gapless modes Define: entanglement spectrum of ‘Wilson loop entanglement Hamiltonian Hamiltonian’ 1 e − H e − e ρ e; A 1 = Tr A 2 | Ψ e ih Ψ e | ⌘ Define: Wilson loop of Wilson loop Z e − e [Benalcazar et al., arxiv:1611.07987] λ ( H e − e ) 4 gapless chiral gapless chiral A 1 A 2 hinge modes mode 0 B k z k z − 4 0 2 π π

Gapless surfaces? consider adiabatically inserting a hinge chiral gapless surface turns gapless hinge at some critical angle ! H 4 ( ~ X X k ) = M + cos k i ⌧ z � 0 + ∆ 1 sin k i ⌧ y � i + ∆ 2 (cos k x − cos k y + r sin k x sin k y ) ⌧ x � 0 i i Critical angle nonuniversal , not fixed to particular crystallographic direction. Di ff erent from gapless surfaces of TCIs.

Electromagnetic response Flux insertion in quantum Hall ≡ φ system creates quantized + — dipole ≡ φ + — Flux insertion in chiral higher- order TI creates quantized ≡ φ quadrupole — + + —

2nd order 3D topological superconductor ! X X H 4 ( ~ k ) = M + cos k i ⌧ z � 0 + ∆ 1 sin k i ⌧ y � i + ∆ 2 (cos k x − cos k y ) ⌧ x � 0 i i has a particle hole symmetry P = τ y σ y K Interpretation: Superconductor with generic dispersion and superposition of two order parameters k,i = i ∆ 1 sin k i d ~ spin triplet, p-wave ∆ 1 Balian-Werthamer state in superfluid Helium-3-B spin singlet d x ² -y ² -wave ∆ 2 superconductor with chiral p + i d Majorana hinge modes

Time-reversal symmetric 2nd order 3D TI Stabilized by mirror symmetries and TRS One Kramers pair of modes on each hinge, like quantum spin Hall edge

Time-reversal symmetric 2nd order 3D TI Can also be defined with C 4 and time- reversal symmetry: Im C 4 eigenvalues Re TRS Kramers pairs ξ = − ξ = + Define 3D Z 2 index independently in each subspace. ξ One nontrivial: 3D Z 2 TI, surfaces are gapless Both nontrivial: 3D higher-order TI, surfaces gapped, edges gapless Both trivial: trivial insulator

Time-reversal symmetric 2nd order 3D TI Surface perturbations: No mirror chiral modes allowed in 2D domain wall boundary boundary A B Mirror x → —x, leaves domain wall y invariant eigenvalue +i x c) eigenvalue —i Not allowed: ? ? ? ? c) Allowed 2D surface perturbations: (2) (1) (3) (2) Number of upmovers of both mirror eigenvalues are equal

Time-reversal symmetric 2nd order 3D TI k y Bending the surface of a topological crystalline insulator k x mirror Chern number = 2 +i -i (upmovers — downmovers) Z with mirror eigenvalue —i n − n + = with time-reversal Z (100) (010) (110) (upmovers — downmovers) Z R with mirror eigenvalue + i One upmover with mirror eigenvalue +i (=mirror Chern number/2) Requires 3D bulk. c) Allowed 2D surface perturbations: (2) (1) (3) (2) Number of upmovers of both mirror eigenvalues are equal