Topological nonsymmorphic crystalline superconductors

Transcription

1 UIUC, 10/26/2015 Topological nonsymmorphic crystalline superconductors Chaoxing Liu Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Chao-Xing Liu, Rui-Xing Zhang, and Brian K. VanLeeuwen, Phys. Rev. B 90, (2014). Qing-Ze Wang and Chao-Xing Liu, arxiv: cond/ (2015).

2 Acknowledgement o Theory Qingze Wang (Penn State) Ruixing Zhang (Penn State) Brian VanLeeuwen (Penn State) Hsiu-chuan Hsu (Penn State) Xiaoyu Dong (Tsinghua) Haijun Zhang (Nanjing Uni.) Binghai Yan (Dresden) Xiao-Liang Qi (Stanford) Jainendra Jain (Penn State) Shou-Cheng Zhang (Stanford) Bjoern Trauzettle (Wuerzburg) o Experiment Cui-zu Chang (MIT) Weiwei Zhao (Penn State) Jagadeesh S. Moodera (MIT) Nitin Samarth (Penn State) Moses Chan (Penn State) Qikun Xue (Tsinghua) Yayu Wang (Tsinghua) Laurens Molenkamp (Wuerzburg) 2

3 Outline o Introduction, symmetry and topology in the classification of states of matter o Nonsymmorphic symmetry and its constraint on electronic states o Nonsymmorphic topological phases, mainly focusing on superconducting phase. o Conclusion and outlook 3

4 States of matter and Landau theory o Landau theory: states of matter are classified by symmetry E.g. a solid is different from a gas because it breaks translational symmetry. 4

5 Topological states of matter o Topological states for free fermions Topological states cannot be adiabatically connected to normal states without gap closing even though sharing the same symmetry. E.g. Quantum Hall state with Landau gaps. Topological states have edge/surface modes at the boundary (surface or interface). E.g. Chiral edge states of the QH effect 5

6 Quantum Hall Effect o The quantum Hall effect: the first example of topological states σ H = C e2 h C = 1,2,, n TKNN (1982) Klaus von Klitzing (1980) 6

7 Topological insulators o Topological insulators and time reversal symmetry Insulating bulk gap and helical edge states which are protected by time reversal symmetry. Molenkamp s group, Science (2007) Qi and Zhang, RMP (2011), Hasan and Kane, RMP (2010). 7

8 Symmetry and topological states o Symmetry protected topological states New topological phases can emerge when there is additional symmetry Can this idea be generalized to other types of symmetry? Particle-hole symmetry: topological superconductors Crystalline symmetry: topological crystalline insulators Other types of symmetry, Qi and Zhang, RMP (2011), Hasan and Kane, RMP (2010). 8

9 Example 1: Topological superconductors o BdG Hamiltonian for superconductors and particle-hole symmetry (redundancy). E Δ 0 k H = 1 2 ψ k + ψ k H BBB ψ k H BBB = k + ψ k h k μ Δ k Δ + k h k + μ Particle-hole symmetry CH BBB C 1 = H BBB C = K 9

10 Topological superconductors o Particle-hole symmetry and Majorana zero modes Kitaev (2003), Read & Green (2000), Fu & Kane (2008), Sau et al (2010), Alicea (2010), Lutchyn et al (2010), Particle-hole partner φ 0 = Cφ 0 γ 0 + = γ 0 φ E = Cφ E γ E + = γ E Majorana fermions: Particle=Antiparticle 10

11 Topological superconductors o Kitaev model for 1D p-wave superconductor: edge Majorana zero modes ψ i ψ i+1 Kitaev (2001) t 1 t 2 ψ i = γ 1i + iγ 2i, γ + αi = γ αi γ 1i γ 2i γ 1i+1 γ 2i+1 Su, Schrieffer and Heeger (1979) t 1 < t 2 Unpaired end Majorana zero modes 11

12 Topological superconductors o Topological superconductors Superconducting gap in the bulk Majorana zero modes at the boundary due to particle-hole symmetry 1d p wave TSC with 0d (bound) majorana end mode 2d p+ip TSC with 1d Majorana chiral edge mode (in analog to quantum Hall state) p+ip TSC C = 1,2,, n Read and Green (2000) 12

13 Topological classification o Schnyder-Ryu-Furusaki-Ludwig classification Schnyder, Ryu, Furusaki, Ludwig PRB (2009), Kitaev (2009) Topological insulators P-wave topological superconductors 13

14 Example 2: Topological mirror insulators o Crystalline symmetry can also protect new topological states. Example: topological mirror insulators Mirror invariant plane m z : x, y, z x, y, z z No coupling between two mirror subspaces Even C e = 1 C = C e + C o = 0 Odd C o = 1 C M = 1 2 C e C o 0 14

15 Topological mirror insulators o Theoretical prediction and experimental observation of topological mirror insulators in SnTe family of materials Fu (2011); Timothy, et al, (2012) SuYang Xu, et al (2012); Dziawa, et al (2012); Tanaka, et al (2012) 15

16 Example 3: Topological mirror superconductors o How about combining mirror symmetry with superconductors (particle-hole symmetry)? Can we get new topological states? o Key question: does particle-hole symmetry exist in one mirror subspace or not? z Even SC?? Odd m z : x, y, z x, y, z SC?? 16

17 Topological mirror Superconductors o What are the mirror parities for ψ and ψ = CC? o The answer depends on the symmetry of gap functions Spinful case in D class η =, odd gap function ψ: +i ( i) D m Δ k D T m = ηδ k, η = ± η = +, even gap function ψ: +i ( i) ψ : +i ( i) ψ : i (+i) 17

18 Topological mirror superconductors o Different topological classifications for different types of gap functions. D m Δ k D T m = ηδ k, η = ± Spinful case in D class η =, odd gap function η = +, even gap function Even Even Odd Odd Fu (2010), Zhang (2013), Ueno (2013) 18

19 Classifications of topological mirror Superconductors and insulators o Topological classification with mirror (reflection) symmetry Hong and Shinsei (2013) Shiozaki & Sato (2014) 19

20 Outline o Introduction, symmetry and topology in the classification of states of matter o Nonsymmorphic symmetry and its constraint on electronic states o Nonsymmorphic topological phases, mainly focusing on superconducting phase. o Conclusion and outlook 20

21 Non-symmorphic symmetry o What is non-symmorphic symmetry? A combination of point group symmetry and fractional translation o Glide reflection and screw axis Glide reflection m τ Screw axis C n τ 21

22 Non-symmorphic materials o Why is non-symmorphic symmetry interesting? 157 of 230 space groups are non-symmorphic Examples of non-symmorphic superconductors Iron-based superconductors LaOBiS 2 22

23 Electronic band structures o Why non-symmorphic symmetry? Non-symmorphic symmetry gives a strong constraint on electronic band structures D g H k D g 1 = H(k) g = m z τ H k ψ k = Eψ k Glide parity τ = a z 2, 0,0 D g ψ k = δ m e ik τ ψ k δ m = ±, Spinless case x Glide parity depends on momentum k 23

24 Electronic band structures Two subspaces with opposite glide parities are connected to each other. D g ψ kx +2π/a = δ m e ik xa 2 +ii ψ kx +2π/a = δ m e ik xa 2 ψ kx +2π/a E +e ik xa 2 Fang and Fu (2015) Shiozaki, Sato and Gomi PRB (2015) i +i e ik xa 2 i +i g = m z τ τ = a 2, 0,0 π/a π/a k x 24

25 Electronic band structures o Difference between mirror symmetry and glide symmetry Fang and Fu (2015) glide i E +e ik xa 2 i mirror Shiozaki, Sato and Gomi PRB (2015) E + +i π/a e ik xa 2 π/a +i k x π/a π/a k x 25

26 Outline o Introduction, symmetry and topology in the classification of states of matter o Nonsymmorphic symmetry and its constraint on electronic states o Nonsymmorphic topological phases, mainly focusing on superconducting phase. o Conclusion and outlook 26

27 Topological non-symmorphic insulators o Topological non-symmorphic phases can exist for both insulators and superconductors o Two types of topological non-symmorphic insulators in three dimensions. pmg, pgg and p4g: Liu s group (2014), (2015) pg: Fang and Fu (2015) 27

28 Topological non-symmorphic superconductors o Topological non-symmorphic insulating phase is found in 2D when there is additional chiral symmetry. (AIII class) o Here we look at non-symmorphic symmetry in superconductors (particle-hole symmetry) Topological glide superconductors Shiozaki, Sato and Gomi PRB (2015) It is different from the classification of topological mirror superconductors 28

29 Glide symmetry and particle-hole symmetry o We consider a superconducting system with glide symmetry o How particle-hole symmetry act on the state in one subspace? z +e ii τ x g = m z τ e ii τ 29

30 Glide symmetry and particle-hole symmetry Key question: if we consider a state ψ with glide parity ±e ii τ, what is the glide parity of its particle-hole partner ψ = CC? The answer to the above question again depends on the symmetry of gap function T D k g Δ k D k g = ηδ k, η = ± G g ψ = δe ik τ ψ, G g ψ = ηδ e ik τ ψ, δ = ± Qing-Ze Wang and Chao-Xing Liu, arxiv: cond/ (2015). 30

31 Glide symmetry and particle-hole symmetry o Glide parities of ψ and ψ depend on the momentum k. This is quite different from mirror parity case. η = +, even gap function ψ: e ik τ ( e ik τ ) k τ = 0 (k x = 0) ψ: +, ψ : + Same k τ = π 2 k x = π a ψ: +i, ψ : i Opposite ψ : e ik τ ( e ik τ ) 31

32 Glide symmetry and particle-hole symmetry o Glide parities of ψ and ψ depend on the momentum k. η =, odd gap function ψ: e ik τ ( e ik τ ) k τ = 0 (k x = 0) ψ: +, ψ : Opposite k τ = π 2 k x = π a ψ : e ik τ (e ik τ ) ψ: +i, ψ : +i Same 32

33 Glide symmetry and particle-hole symmetry o Particle-hole symmetry only exists in one line in the 2D Brillouin zone for one glide parity subspace. o We can find topological invariants in 1D line, but not in 2D plane. η = + η = 33

34 Model for topological glide superconductors o There is a Z 2 classification for the D class in 1D, the Kitaev model. o Our model for topological glide superconductors in a distorted square lattice 34

35 Model for topological glide superconductors o Spinless p-wave superconducting model Glide symmetry σ is for A and B sublattices Gap function D k g h k x, k y D k 1 g = h k x, k y D k g Δ ± T k D k g = ±Δ ± k, Qing-Ze Wang and Chao-Xing Liu, arxiv: cond/ (2015). 35

36 Model for topological glide superconductors o Energy spectrum for a slab configuration Δ + Δ 36

37 Extended Brillouin zone o Define extended Brillouin zone by glide symmetry Glide parity G g ψ k G g ψ k = e ik τ ψ k = δ η e ik τ ψ k δ η = + k = k δ η = k = k + Q, Q τ = ±π 37

38 Extended Brillouin zone o In extended BZ, our model Hamiltonian can be viewed as a weak topological superconductor Weak topological superconductors η pairing in the extended BZ Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 Δ 0 y Qing-Ze Wang and Chao-Xing Liu, arxiv: cond/ (2015). 38

39 Spin and Time reversal symmetry o More topological superconducting phases with spin or with time reversal symmetry Δ Δ + Qing-Ze Wang and Chao-Xing Liu, arxiv: cond/ (2015). 39

40 Spin and Time reversal symmetry o Topological superconductors with nodes and Majorana flat bands (Δ + for DIII and BDI) Qing-Ze Wang and Chao-Xing Liu, arxiv: cond/ (2015). 40

41 Conclusion and outlook o We have shown the existence of topological superconducting phases which are protected by nonsymmorphic symmetry. o Possible theoretical generalization: other symmetry classes, three dimensions, general space groups, topological defects, interacting topological systems Teo and Hughes (2012, 2014) Fu s group (2015), Vishwanath s group (2013), Teo and Hughes (2014) o Possible material realization of non-symmorphic topological insulators/superconductors? Photonic crystals? Iron pnictide superconductors? Soljacic group (2015), Shen s group (2015) 41

42 Conclusion and outlook o Surface approach for the classification of topological crystalline insulators (Based on 17 2D space group) XY Dong and CX Liu arxiv: (2015) 42

43 Thanks for your attention! 43