Emergent topological phenomena in antiferromagnets with noncoplanar spins

Transcription

1 Emergent topological phenomena in antiferromagnets with noncoplanar spins - Surface quantum Hall effect - Dimensional crossover Bohm-Jung Yang (RIKEN, Center for Emergent Matter Science (CEMS), Japan)

2 Collaborators Naoto Nagaosa (U. Tokyo and RIKEN) Mohammad Saeed Bahramy (RIKEN)

3 Outline 0. Introduction Quantum Hall effect without magnetic field Topological phenomena in noncoplanar magnets 1. Surface quantum Hall effect Noncoplanar spin ordering in pyrochlore antiferromagnet Topologically protected bound state in a continuum 2. Dimensional crossover in thin films of pyrochlore iridates Giant anomalous Hall effect New topological phase: anti-chern insulator

4 Quantum Hall effect under magnetic field Quantized Hall conductivity Boundary gapless modes µ Berry phase induced topological response TKNN invariant : Thouless, Kohmoto, Nighingale, Nijs (1982)

5 Quantum Hall effect without magnetic field t 2 exp(iφ) Current loop (imaginary hopping) can generate quantum Hall insulator!

6 Quantum anomalous Hall effect in TI On the surface of 3D topological insulator with bulk ferromagnetism C.-Z. Chang, Q.-K. Xue, Science (2013) 2D massless Dirac particle on surface

7 Berry phase due to non-coplanar spin gauge flux F S k c i > c j > Conduction electron S i Localized spin S j acquire a phase factor = Berry phase Fictitious flux (in a continuum limit) scalar spin chirality

8 Quantum Hall effect in non-coplanar magnets Ohgushi, Murakami, Nagaosa Spin Berry phase can generate quantum Hall effects

9 Anomalous Hall effect in pyrochlore magnets Localized Itinerant Taguchi, Nagaosa, Tokura (2001)

10 Spin Berry phase in pyrochlore lattice Noncoplanar magnetic ordering (spin anisotropy) R 2 Mo 2 O 7 (R=Nd, Sm, Gd): 2-in/2-out (A=Nd, Sm, Eu) : all-in/all- Cd 2 Os 2 O 7, A 2 Ir 2 O 7 out 3-in/1-out (FM) 2-in/2-out (FM) all-in/all-out (AFM) Systems with broken lattice symmetry : Surface, film, distortion New emergent topological phenomena Φ 1 Φ 2 Φ 3 Φ 4

11 1. Surface quantum Hall states Spin Berry phase in antiferromagnets with non-coplanar spins Topologically protected bound state in a continuum B. -J. Yang, M. S. Bahramy, and N. Nagaosa, Nature Communications 4, 1524 (2013).

12 Surface QHE in pyrochlore all-in/all-out AFM Kagome layer on the surface Finite effective magnetic field on the surface! Surface QH (T. Arima) Normal Insulator Gardner et al. RMP (2010) Is it stable?

13 Stability of surface quantum Hall states Localization of wave functions of QHI? Chern number dilution?

14 Band structure of the heterostructure Energy dispersion E(k) (1) Decoupled bands (2) Fully coupled bands? 1D edge and 2D bulk states of QHI can be localized on the top layer Spatial distribution of 1D edge and 2D bulk states of QHI?

15 Stacked honeycomb lattice model Simplest model with two-bands Complex 2 nd neighbor hopping (Haldane model) 2D QHI Stacked 2D NI Staggered chemical potential µ S -µ S

16 Layer-resolved wave function distribution Ψ loc (kx,ky,l z =0) 2 Substrate 2D QHI Substrate

17 Ψ loc (kx,ky,l z =0) 2 t =pt t t Ψ loc (z) 2 Ψ loc (z) 2

18 Description of the localized bound state Use Green s function of quantum Hall insulator Valence band Conduction band Energy Localized state (a pole of G R ) Extended state Two-band Fano model : Integrate out substrate

19 Topological protection of the bound state Bound state pole : 3D bulk valence band 3D bulk conduction band Anti-parallel Parallel The localized bound state in the continuum is protected by the topology of h and a!

20 Generalization Multi-band systems 2D QHI with Chern number 2 C=0 C=0 C=+1 C=-1 4 bound states Only bands with finite Chern number support localized bound states! Chern number of the 2D band = minimum number of localized bound states!

21 Localization of edge states Edge channel is exponentially localized along z-direction as long as the energy is within the 3D bulk gap

22 Conclusion of part 1 1.Surface quantum Hall state is stable against hybridization. 2. The minimal number of localized bulk states is given by the Chern number difference between 2D QHI and the effective 2D NI system. 3. The 1D chiral edge state is localized as along as it appears within a bulk band gap. 1D chiral edge or some 2D bulk of QHI Pyrochlore iridate does not show surface quantum Hall states. But instead we have found an even more interesting state there!

23 2. Dimensional crossover in pyrochlore iridate Emergent topological property in thin films of pyrochlore iridates B. -J. Yang and N. Nagaosa, submitted.

24 Pyrochlore iridates R 2 Ir 2 O 7,R=Nd, Sm, Eu, Y 1. Reduced bandwidth (B.J.Kim,J.Yu, T.W.Noh et al.) Ir Ir Ir Ir Ir 4+ : 5 electrons in t 2g with strong spin-orbit 2. Berry phase effect due to J=1/2 states Spin-orbit coupling bandwidth Coulomb interaction Correlated topological phase!

25 Phase diagram LDA+U calc. Wan, Vishwanath, Savrasov All-in/all-out AFM Dirac semi-metal (or Weyl semi-metal) : Topological Metal

26 What is the (topological) Weyl semimetal? Surface states : Fermi Arc +1-1 Topological invariant : chiral charge C= v 1 (v 2 v 3 ) v 1 (v 2 v 3 ) =±1 Witczak-Krempa Weyl fermion H(k)= (v 1 k)σ x + (v 2 k) σ y + (v 3 k) σ z

27 Quantum Hall effect of Weyl semi-metal H(k)= H kz (k x, k y ) =k x σ x + k y σ y + m σ z, m= (k z k 0 ) k y k x k z C(k z1 )=0 -k 0 C(k z2 )=1 C(k z3 )=0 k 0 k z Normal Quantum Hall Normal Surface spectrum (finite size along x-direction) k z 1D chiral Fermi Arc k z2 k y Surface metallic states exist! Each Weyl point has a topological charge ±1

28 Hall conductivity (σ xy ) in Weyl semi-metal π/a z -k 0 Quantum Hall k 0 +π/a z k z If k 0 =k 0 (m) σ xy 0 Trivial Insulator Weyl Semi-metal 3D quantum Hall insulator m k 0 (m)=0 k 0 (m)=π

29 Dimensional crossover in Weyl semi-metal 3D bulk Thin film G xy G xy 0 0 Trivial Insulator Weyl Semi-metal Dimensional crossover in pyrochlore iridates? 3D quantum Hall insulator Accidental gap-closing m m G xy 2D limit 0 Trivial Insulator 2D quantum Hall insulator m

30 3D bulk phase diagram of iridates Trivial =<S i > m=0.10 m=0.28 Weyl Critical Γ X W L Γ K X Γ X W L Γ K X [111] [111] 8 Weyl points Witczak-Krempa & YB Kim

31 Chern vectors and cubic symmetry For a pair of Weyl points Pyrochlore iridate :A system with multiple Weyl points (multiple Chern vectors) C: Chern veoctor Weyl semi-metal Trivial Anti-Chern insulator? insulator C? 1 C 4 C 1 +C 2 +C 3 +C 4 =0 due to cubic symmetry! C 2 C 3

32 [111] Surface states Weyl Semimetal Weyl Semimetal Critical Gapped insulator At the critical point, the Fermi surface shows Lifshitz transition Fully-gapped insulator has surface states at the Fermi energy (Stable?)

33 Pyrochlore iridate thin film Two main questions 1. Dimensional crossover in a system with multiple Weyl points cf) A 3D Chern insulator with one pair of Weyl points 2. Fully-gapped AF insulator with topological property? Anti-Chern insulator?

34 Weyl semi-metal in pyrochlore iridates Model Hamiltonian 1. Use J=1/2 states as a basis 2. Assume all-in/all-out magnetic ground states 3. Tune the magnitude of local magnetic moment m=<s i > [111] thin film : change the number of bilayers 1 bilayer 1 bilayer

35 Dimensional Crossover in [111] film G xy ([111] // z-axis) Rich physics in thin films! Bulk is gapped Surface states carry G xy N layer Gapped Surface Hall currents!

36 Accidental gap-closing at a Weyl point Near a Weyl point : H(k)= k x σ x + k y σ y + k z σ z In a film : k z = discretized while k x, k y are good quantum number Gap-closing is possible only when k x = k y = k z = 0 simultaneously. Accidental gap-closing is possible if k z = k z (m) = 0 E(k z ) m 1 m 2 m 3 C=n k z C=n+1

37 Open boundary condition No gap-closing at a single Weyl point. Two coupled Weyl points required! Open B.C. E k z k z m E= fixed [111 ] V Surface Brillouin zone

38 How to release the Hall currents? Surface states mediated topological phase transition 1. Surface states of anti-chern insulator 2. Frustrated lattice geometry induced surface states

39 Surface states of anti-chern insulator (1) KK film K T K T K Top Bottom (2) KT film T K Top Bottom On the surface terminated by T layer there is additional geometrical surface states!

40 Anti-Chern insulator A new topological phase induced by surface states m=0.20 m=0.40 m=0.60 Weyl Anti-Chern insulator Trivial insulator Gapped ψ(z) 2 Gapped

41 Berry curvature distribution 1. Berry curvature is concentrated near the region where two surface states are crossing 2. Wave function overlap between two surface states control the Hall conductance of the bulk system!

42 Conclusion of part 2 Thin films of pyrochlore iridates have unexplored rich physics (Dimensinal Crossover) Accidental band crossing in an intermediate dimension Giant anomalous Hall effect in antiferromagnets Surface state induced topological phase transition Anti-Chern insulator : a new phase existing only in thin films

43 Summary Emergent topological phenomena in noncoplanar magnets 1. Surface quantum Hall effect Bound state in a continuum 2. Dimensional crossover in pyrochlore iridate thin films Modified accidental band crossing in thin films (Giant Hall currents) Surface states induced new topological phases (Anti-Chern insulator)

44 Quantum Hall effect in non-coplanar magnets Ohgushi, Murakami, Nagaosa Spin Berry phase can generate quantum Hall effects

45 Topological number of Weyl semi-metal H= (v 1 k)σ 1 +(v 2 k)σ 2 +(v 3 k)σ 3 C= v 1 (v 2 v 3 ) v 1 (v 2 v 3 ) =±1 Wan et al., PRB Weyl point has a topological number i.e., chiral charge!

46 Finite size effect on Weyl semimetal -π/a z C C 1 C +1-1 k 0 -k 0 Quantum Hall +π/a z C Number of discrete momenta in [-k 0,k 0 ] 3 Number of discrete momenta in [-k 0 /3,k 0 /3] C -C/3 z-component C -C/3 k z

47 Anomalous Hall conductance

48 Periodic boundary condition Periodic B.C. E k z E= f(m) C -C/3 Accidental gap-closing is possible at all 8 Weyl points!

49 Layer resolved Fermi-surface: 20bilayer film Red (Blue) line : states localized on the top (bottom) surface Weyl Semimetal Anti-Chern insulator Trivial AF m=0.05 m=0.15 m=0.40 m=1.0 Fermi Arc Localized state due to lattice structure

50 Surface state mediated phase transition Exponentially small overlap between two surfaces!