1. Kazuki Hasebe (Kagawa N.C.T.) Collaborators, Keisuke Totsuka Daniel P. Arovas Xiaoliang Qi Shoucheng Zhang Quantum Antiferromagnets from Fuzzy Super-geometry (Stanford) (YITP) (UCSD)(Stanford) Supersymmetry in Integrable Systems – SIS’12 (27-30, Aug.2012, Yerevan, Armenia) Based on the works, arXiv:120…, PRB 2011, PRB 2009

2. Topological State of Matter 2 TI, QHE, Theoretical (2005, 2006) and Experimental Discoveries of QSHE (2007)  Local Order parameter (SSB)  Topological Order TSC Topological order is becoming a crucial idea in cond. mat., hopefully will be a fund. concept. How does SUSY affect toplogical state of matter ? and subsequent discoveries of TIs QAFM, QSHE, Main topic of this talk: Order Wen

3. Physical Similarities QHE: 2D  Gapful bulk excitations  Gapless edge spin motion ``Featureless’’ quantum liquid : No local order parameter QAFM: 1D ``Disordered’’ quantum spin liquid : No local order parameter Spin-singlet bond = Valence bond Quantum Hall Effect Valence Bond Solid State  Gapful bulk excitations  Gapless chiral edge modes ``locked’’ or ＝

4. Math. Web Quantum Hall Effect Fuzzy Geometry Valence Bond Solid State Schwinger formalism Spin-coherent state Hopf map

5. Simplest Concrete Example Fuzzy Sphere or ＝ Haldane’s sphereLocal spin of VBS state Monopole charge : Spin magnitude : Radius :

6. Fuzzy Sphere Fuzzy and Haldane’s spheres Schwinger formalism Berezin (75),Hoppe (82), Madore (92) 6 Haldane’s Sphere Hopf map : monopole gauge field

7. One-particle Basis LLL basis Haldane (83) Wu & Yang (76) States on a fuzzy sphere Fuzzy Sphere Haldane’s sphere

8. Translation LLL Fuzzy sphere Simply, the correspondence comes from the Hopf map: The Schwinger boson operator and its coherent state. Schwiger operator Hopf spinor

9. Laughlin-Haldane wavefunction Haldane (83) SU(2) singlet Stereographic projection : index of electron

10. Simplest Concrete Example Fuzzy Sphere or ＝ Haldane’s sphereLocal spin of VBS state Monopole charge : Spin magnitude : Radius :

11. Translation to internal spin space SU(2) spin states 1/2 -1/2 1/2 -1/2 Bloch sphere LLL states Haldane’s sphere Internal spaceExternal space Cyclotron motion of electron Precession of spin Interpret as spin coherent state

12. Correspondence Laughlin-Haldane wavefunctionValence bond solid state Lattice coordination numberTotal particle number Filling factor Spin magnitude Monopole charge Two-site VB number Arovas, Auerbach, Haldane (88) Affleck, Kennedy, Lieb, Tasaki (87,88) Particle index Lattice-site index

13. Examples of VBS states (I) VBS chain Spin-singlet bond = Valence bond ``locked’’ or ＝

14. Examples of VBS states (II) Honeycomb-latticeSquare-lattice

15. Particular Feature of VBS states  VBS models are ``solvable’’ in any high dimension. (Not possible for AFM Heisenberg model) Gapful (Haldane gap) Non-local Disordered spin liquid Exponential decay of spin-spin correlation  Ground-state  Gap (bulk) Gapless SSBNo SSB  Order parameter Local Neel stateValence bond solid state 15

16. Hidden Order 000 +1+1 VBS chain den Nijs, Rommelse (89), Tasaki (91) Classical Antiferromagnets Neel (local) Order Hidden (non-local) Order +1 +1 No sequence such as +100 +10

17. Generalized Relations Quantum Hall Effect Fuzzy Geometry Valence Bond Solid State 2D-QHE SO(5)- q-deformed- SO(2n+1)- Mathematics of higher D. fuzzy geometry and QHE can be applied to construct various VBS models. 4D- 2n- q-deformed- CPn- Fuzzy four- Fuzzy two-sphere Fuzzy CPn Fuzzy 2n- q-deformed SU(n+1)- SU(2)-VBS

18. Related References of Higher D. QHE 1983 2D QHE 4D Extension of QHE : From S2 to S4 Even Higher Dimensions: CPn, fuzzy sphere, …. QHE on supersphere and superplane Landau models on supermanifolds Zhang, Hu (01) Karabali, Nair (02-06), Bernevig et al. (03), Bellucci, Casteill, Nersessian(03) Kimura, KH (04), ….. Kimura, KH (04-09) Ivanov, Mezincescu,Townsend et al. (03-09), 2001 Bellucci, Beylin, Krivonos, Nersessian, Orazi (05)... Super manifolds …… Non-compact manifolds Hyperboloids, …. Hasebe (10)Jellal (05-07) Laughlin, Haldane

19. Related Refs. of Higher Sym. VBS States 2011 1987-88 Valene bond solid models Sp(N) Tu, Zhang, Xiang (08) Arovas, Auerbach, Haldane (88) Higher- Bosonic symmetry UOSp(1|2), UOSp(2|2), UOSp(1|4) … Arovas, KH, Qi, Zhang (09) Relations to QHE SU(N) Affleck, Kennedy, Lieb, Tasaki (AKLT) SO(N) Greiter, Rachel, Schuricht (07), Greiter, Rachel (07), Arovas (08) Schuricht, Rachel (08) Super- symmetry 200X Tu, Zhang, Xiang, Liu, Ng (09) Totsuka, KH (11,12) q-SU(2)Klumper, Schadschneider, Zittartz (91,92) Totsuka, Suzuki (94)Motegi, Arita (10)

20. Takuma N.C.T. Supersymmetric Valence Bond Solid Model 20

21. Fuzzy Supersphere Grosse & Reiter (98) Balachandran et al. (02,05) Fuzzy Super-Algebra Supersphere oddGrassmann even (UOSp(1|2) algebra) Super-Schwinger operator

22. Intuitive Pic. of Fuzzy Supersphere 1 1/2 0 -1/2

23. Haldane’s Supersphere One-particle Hamiltonian UOSp(1|2) covariant angular momentum Kimura & KH, KH (05) SUSY Laughlin-Haldane wavefunction Super monopole LLL basis : super-coherent state

24. Susy Valence Bond Solid States 24 Arovas, KH, Qi, Zhang (09) Hole-doping parameter Spin+ Charge Supersymmetry Manifest UOSp(1|2) (super)symmetry At r=0, the original VBS state is reproduced.  Math.  Physics ‘’Cooper-pair’’ doped VBS spin-sector : QAFM charge-sector : SC Exact many-body state of interaction Hamiltonian hole

25. Exact calculations of physical quantities 25 SC parameter spin-correlation length

26. Two Orders of SVBS chain Insulator Superconductor Insulator Spin-sector Quantum-ordered anti-ferromagnet Charge-sector Hole doping Order Superconducting Sector Topological order 26

27. Takuma N.C.T. Entanglement of SVBS chain 27

28. Hidden Order in the SVBS State +1/2-1/2 0 +1 +1/2 Totsuka & KH (11) 28 SVBS shows a generalized hidden order. sS Bulk = 1 : S =1+1/2

29. E.S. as the Hall mark 29 Li & Haldane proposal (06) What is the ``order parameter’’ for topological order ? B A Entanglement spectrum (E.S.) Robustness of degeneracy of E.S. under perturbation Hall mark of the topological order Schmidt coeffients Spectrum of Schmidt coeffients 

30. Behaviors of Schmidt coefficients The double degeneracy is robust under ‘’any’’ perturbations (if a discete sym. is respected). 30 3 Schmidt coeff.  2+15 Schmidt coeff.  3+2 sS Bulk = 1 sS Bulk = 2 Totsuka & KH (12)

31. Origin of the double degeneracy 31 A B ``edge’’ Double deg. (robust) Non-deg. Triple deg. (fragile) sS Bulk = 2 sS Edge = 1/2 sS Bulk = 1 sS Edge = 1 S Edge = 0 S Edge = 1/2 S Edge = 1 S Edge = 1/2

32. Understanding the degeneracy via edge spins In the SVBS state, half-integer spin edge states always exist (this is not true in the original VBS) and such half-integer edge spins bring robust double deg. to E.S. Edge spin 1/2 Bulk (super)spin : general S Bulk-(super)spin S=2 1 Edge spin S/2 S/2-1/2 SUSY brings stability to topological phase. SUSY 32

33. Summary Edge spin : integer half-integer SUSY 33  SVBS is a hole-pair doped VBS, possessing all nice properties of the original VBS model.  SVBS exhibits various physical properties, depending on the amount of hole-doping. 1. Math. of fuzzy geometry and QHE can be applied to construct novel QAFM. First realization of susy topological phase in the context of noval QAFM! 2. SUSY plays a cucial role in the stability of topological phase.

34. Symmetry protected topological order 34 TRS  Odd-bulk S QAFM spin Z2 * Z2 Unless all of the discrete symmetries are broken Qualitative difference between even-bulk S and odd- bulk S VBSs : Inversion  Even-bulk S QAFM spin : SU(2) S bulk =2n-1 S edge =S bulk /2=n-1/2 2S edge +1=2n S bulk =2n S edge =S bulk /2=n 2S edge +1=2n+1 Odd deg. (fragile) Double deg. of even deg. (robust) Hallmark of topological order : Deg. of E.S. is robust under perturbation. Pollmann et al. (09,10)