1. Kazuki Hasebe 24-28 Aug. 2010, @ Supersymmetry in Integrable Systems, Yerevan, Armenia (Kagawa N.C.T.) Based on the works (2005 ～ 2010) with Yusuke Kimura, Daniel P. Arovas, Xiaoliang Qi, Shoucheng Zhang, Keisuke Totsuka Fuzzy Geometry, Supersymmetry, and Many-body Physics (Stanford) (YITP) (Oviedo) (California)

2. Introduction The correspondence between fuzzy geometry and LLL physics has become much transparent in the developments of higher D. quantum Hall effect. Today, I would like to discuss applications of such correspondence to many-body physics, in particular, to ``solvable’’ quantum antiferromagnets.

3. Generalizations of QHE and Landau Model 2010 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) Hasebe, Kimura (04), ….. Hasebe, Kimura (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

4. ``Solvable’’ Model of Quantum Antiferromagnets 2010 1987-88 Valene bond solid models Sp(N) Tu, Zhang, Xiang (08) Arovas, Auerbach, Haldane (88) Higher- Bosonic symmetry OSp(1|2), SU(2|1) Arovas, Hasebe, 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) Hasebe, Totsuka (10) q-SU(2)Klumper, Schadschneider, Zittartz (91,92) Totsuka, Suzuki (94)

5. Fuzzy Geometry, Landau model and Supersymmetry

6. Fuzzy Sphere Fuzzy Geometry A convenient way :Schwinger boson Symmetric Rep. Algebra  Straightforwardly generalized to fuzzy CPn Basis elements Index: Balachandran et al. (02) Madore (02)

7. Landau problem on a two-sphere : symmetric products of the coherent state (Hopf spinor) One-particle Hamiltonian Lowest Landau level Coserved SU(2) angular momentum : Monopole charge Algebra LLL basis  generalized to Landau model on CPnKarabali & Nair (02) Haldane (83) Wu & Yang (76)

8. Correspondence: Fuzzy geometry & LLL LLL basis There is one-to-one correspondence, between basis of fuzzy geometry and LLL basis. Fuzzy sphere basis Simply, the correspondence stems from Schwinger boson operator and its coherent state.

9. Fuzzy Supersphere Grosse & Reiter (98) Balachandran et al. (02,05) Symmetric Rep. Fuzzy Algebra Supersphere Non-anticommutative geo. oddGrassmann even (OSp(1|2) algebra)

10. Symmetric Rep. of Fuzzy Supersphere Bosonic d.o.f.Fermionic d.o.f. - sym.rep.

11. Picture of the basis elements of fuzzy supersphere 1 1/2 0 -1/2

12. Landau Problem on a Supersphere Super monopole One-particle Hamiltonian Conserved OSp(1|2) angular momentum Hasebe & Kimura (05) In the LLL satisfy the fuzzy supersphere algebra.  SUSY Landau model on CP{n/m} Ivanov et al. (03-09)

13. Fuzzy super-geometry & super LLL LLL basis Fuzzy supersphere basis Super-coherent state (super-Hopf spinor)Schwinger super-operator

14. Up to now, the correspondence is at one-particle level. How about many-body level ?

15. Many-body level Correspondence

16. Many-body wave-function of QHE Haldane (83) SU(2) singlet Stereographic projection The Laughlin-Haldane wavefunction is SU(2) singlet. : index of electron Antisymmetric under the interchange between i and j, reflecting the fermionic statistics of electrons

17. Supersymmetric Quantum Hall Effect Antisymmetric under interchange between i and j Hasebe (05) Mathematically, the construction is straightforward. QHE on a super-manifold … Laughlin-Haldane function : SU(2) singlet of coherent states Does it have any physical application ??? SUSY version : OSp(1|2) singlet of super-coherent states

18. Apply the correspondence to many-body states ??? ???? Remember Do these states appear in a context of physics ? If so, what is the physical interpretation of these states?

19. 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

20. Step 1: Local Hilbert space i: index of a particle i: index of a lattice site Charge of monopole Magnitude of spin LLLSU(2) spin

21. (Ex.) Square lattice i: index of a lattice site or

22. Step2: Valence Bond Valence bond (=Spin singlet bond) : Entangled state without spin polarization : Quantum Antiferromagnets Spin-singlet

23. Examples of VBS states VBS chain

24. Examples of VBS states Honeycomb-latticeSquare-lattice

25. Correspondence Laughlin-Haldane wavefunctionValence bond solid state Lattice coordination numberTotal particle number Filling factor Spin magnitude Monopole charge Two-site VB number : number of bosons Arovas, Auerbach, Haldane (88)

26. Why VBS states important ?  ``Solvable’’ in any higher dimensions (Not possible for antiferromagnetic Heisenberg model) : A model for gapful quantum antiferromagnets Affleck, Kennedy, Lieb, Tasaki (AKLT) (87,88)  Haldane Gap (gapful excitation for S=integer QAF)  Hidden (non-local) Order : New concept of order (``topological order’’) Disordered spin liquid Exponential decay of spin correlation

27. Aspects of a ``solvable’’ model ``Think inversely’’ : Don’t solve Hamiltonians. Construct Hamiltonian for the given state !

28. The parent Hamiltonian Projection operator to the SU(2) bond-spin J=2 The VBS chain does not have J=2 component, so This construction can be generalized to higher dimensions. The Hamiltonian whose ground state is VBS chain is

29. 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

30. SUSY VBS (SVBS) states

31. Basis Elements of SVBS states Arovas, Hasebe, Qi, Zhang (09) SU(2) quantum number Physical interpretationOperators Up-spin Down-spin (spinless) hole

32. As hole doped anti-ferromagnetic states Valence-bond Hole-pair r: doping ratio of hole-pairs SUSY Bond Valence-bond Hole-pair (Ex.) Typical configuration on a square lattice

33. SVBS chain Valence-bondHole-pair No sequence such as Typical sequence

34. Construction of the Parent Hamiltonian

35. OSp(1|2)-type Parent Hamiltonian Hole-number non-conservation OSp(1|2) spin-spin interaction

36. Physical Meaning of the SVBS state Replacing ``operator’’ Simply rewritten as Replacing VB with hole-pair The SVBS chain in the (spin-hole) coherent state rep. =>

37. Expansion of the SVBS Chain + + + SVBS interpolate the original VBS and Dimer. SVBS is a superposition of hole-doped VBS states. Superconducting property Insulator

38. The physical property of SVBS chain Insulator Superconductor Insulator spin Disordered quantum anti-ferromagnets charge Hole doping

39. Superconducting order parameter

40. r-dependence of the correlation lengths Spin correlation Superconducting correlation

41. Hidden Order in SVBS States The SVBS states Show a Generalized Hidden Order. 000 +1+1 VBS +1/2-1/2 0 SVBS +1 +1/2 Hasebe & Totsuka (10)

42. String Order Parameter

43. ``Crackion’’ by Single Mode Approximation triplet-bond gapful excitation

44. Summary The SVBS states exhibit various physical properties depending on the amount of hope-doping. SUSY is successfully applied to the construction of a ``solvable’’ hole-doped antiferromagnetic model. Further generalizations may be straightforward, such as SU(N|M). One-particle level correspondence is generalized to many-body physics. Generalized Landau models and QHE find ``realistic’’ applications in ``solvable’’ quantum antiferromagnets.