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Kazuki Hasebe

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

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Kazuki Hasebe

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  15. Neel state Valence bond solid state • Ground-state SSB No SSB • Gap • (bulk) Gapful (Haldane gap) Gapless Exponential decay of spin-spin correlation Disordered spin liquid • Order parameter Local Non-local Particular Feature of VBS states • VBS models are ``solvable’’ in any high dimension. (Not possible for AFM Heisenberg model)

  16. Classical Antiferromagnets +1 -1 +1 -1 -1 -1 -1 -1 -1 +1 +1 +1 +1 VBS chain Hidden Order den Nijs, Rommelse(89),Tasaki(91) Neel (local) Order Hidden (non-local) Order 0 0 0 No sequence such as +1 -1 0 0 -1 +1 0

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

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

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

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

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

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

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

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

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

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

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

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

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

  30. Behaviors of Schmidt coefficients sSBulk= 1 sSBulk= 2 Totsuka & KH (12) 3 Schmidt coeff.  2+1 5 Schmidt coeff.  3+2 The double degeneracy is robust under ‘’any’’ perturbations (if a discete sym. is respected).

  31. sSEdge = 1/2 sSEdge = 1 Origin of the double degeneracy sSBulk= 1 SEdge= 1/2 Double deg.(robust) SEdge= 0 Non-deg. ``edge’’ B A sSBulk= 2 SEdge= 1 Triple deg. (fragile) SEdge= 1/2 Double deg. (robust)

  32. Understanding the degeneracy via edge spins Edge spin Edge spin Bulk-(super)spin S=2 1 S/2 Bulk (super)spin : general S SUSY SUSY 1/2 S/2-1/2 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. SUSY brings stability to topological phase.

  33. Summary 1. Math. of fuzzy geometry and QHE can be applied to construct novel QAFM. • 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. SUSY 2. SUSY plays a cucial role in the stability of topological phase. Edge spin : integer half-integer First realization of susy topological phase in the context of noval QAFM!

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

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