Kazuki Hasebe

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

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 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) Hasebe, Kimura (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 Hasebe, Kimura (04-09) Non-compact manifolds • Hyperboloids, …. 2010 Jellal (05-07) Hasebe (10)

4. Relations to QHE ``Solvable’’ Model of Quantum Antiferromagnets 1987-88 Valene bond solid models Affleck, Kennedy, Lieb, Tasaki (AKLT) Arovas, Auerbach, Haldane (88) • q-SU(2) Klumper, Schadschneider, Zittartz (91,92) Totsuka, Suzuki (94) 200X • SU(N) Greiter, Rachel, Schuricht (07), Greiter, Rachel (07), Arovas (08) Higher- Bosonic symmetry • Sp(N) Schuricht, Rachel (08) • SO(N) Tu, Zhang, Xiang (08) Tu, Zhang, Xiang, Liu, Ng (09) Super- symmetry • OSp(1|2) , SU(2|1) Arovas, Hasebe, Qi, Zhang (09) Hasebe, Totsuka (10) 2010

5. Fuzzy Geometry, Landau modeland Supersymmetry

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

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

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

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

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

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

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

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

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

15. Many-body level Correspondence

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

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

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

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

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

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

23. Examples of VBS states VBS chain VBS chain

24. Examples of VBS states Honeycomb-lattice Square-lattice

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

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

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

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

29. 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

30. SUSY VBS (SVBS) states

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

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

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

34. Construction of the Parent Hamiltonian

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

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

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

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

39. Superconducting order parameter

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

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

42. String Order Parameter

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

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