Kazuki Hasebe 14 – 19 Dec. Miami 2010, USA (Kagawa N.C.T.) Based on the works (2009,2010) with Daniel P. Arovas, Xiaoliang Qi, Shoucheng Zhang, Keisuke Totsuka Supersymmetric Valence Bond Solid States (Stanford) (YITP) (California)

Introduction  Recently, a new kind of order, called topological order has attracted much attention in solid state physics.  Topological order (quantum order) is a order which cannot be characterized by a local order parameter like SSB.  Examples: quantum Hall effect, quantum spin Hall effect, (resonating) valence bond state etc.

What is valence bond ? Terminology originally from chemistry N. Lewis (1916) Heitler, London (1927) ``Octet Rule’’ by ``Valence Bond Method’’ by FF Valence bond valence electron  Resonance of valence bonds L. Pauling (1946)  Stability of molecules  ＣＯＯ Benzene

VB in two-site antiferromagnets ``valence bond’’ = spin singlet bond Classical G.S. are infinitely degenerate by the SU(2) sym.: Q.M. G.S. is unique and SU(2) singlet valence bond : spin-unpolarized

Valence Bond Solid States Affleck, Kennedy, Lieb, Tasaki (87,88) SU(2) spinsTwo kinds of bosons Up-spin Down-spin SU(2) singlet state = spin unpolarized quantum state : spin lattice fully occupied by valence bonds

A simplest VBS state = = : superposition of spin 1 anti-ferromagnetic states Quantum AF state

Examples of VBS states (I) VBS chain

Examples of VBS states (II) Honeycomb-lattice Square-lattice VB state can be defined on any lattice in any dimension.

Why Valence Bond Solid States important ?

Reason I The resonating valence bond state may be proposed by Ph. Anderson (87) realized as high Tc ground-state. Relation to High Tc :

Reason II  VBS models are ``solvable’’ in any higher dimension. (Not possible for antiferromagnetic Heisenberg model) Gapful (Haldane gap) Non-local Disordered spin liquid Exponential decay of spin-spin correlation  Ground-state  Gap Gapless SSBNo SSB  Order parameter Local Neel stateValence bond solid state VBS states are disordered, but quantum-ordered.

As a ``solvable’’ model ``Think inversely’’ : Don’t solve Hamiltonians. Construct Hamiltonian for a given state ! Not the spin-singlet combination ! Bond-spins

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 zero-energy G.S. is VBS is

Hidden Quantum Order VBS chain den Nijs, Rommelse (89), Tasaki (91) Classical Antiferromagnets Neel (local) Order Non-local Order No sequence such as

SUSY VBS (SVBS) states

Supersymmetic VBS states Arovas, Hasebe, Qi, Zhang (09) SU(2) quantum number Physical interpretationOperators Up-spin Down-spin (spinless) hole OSp(1|2) singlet state = SUSY extension of VBS state

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

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

Construction of the Parent Hamiltonian

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

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. =>

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

Superconducting order parameter

r-dependence of the correlation lengths Spin-spin correlation Superconducting correlation

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

Quantum Order in SVBS States The SVBS states show Generalized Quantum Order VBS +1/2-1/2 0 SVBS +1 +1/2 Hasebe & Totsuka (10)

String Order Parameter

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

``Derivation’’ of quantum Hall wavefunction Stereographic projection Coherent state rep. Arovas, Auerbach, Haldane (88) R. Laughlin (83) F.D.M. Haldane (83)

Translations Valence Bond Solid States Fuzzy Geometry Quantum Hall Effect Generalized mathematics of fuzzy geometry and QHE can be applied to construct various VBS models. On higher D. spheres On superspheres etc, etc. Fuzzy spheres Fuzzy supersphere etc, etc. Fuzzy CPn Counterparts should be here.

Summary The SVBS states exhibit ``high Tc-like’’ various properties depending on the amount of hope-doping. VBS states are ``solvable’’ AF models with quantum order. Generalized formulations of QHE and fuzzy geometry can be applied to construct VBS models. SUSY is incorporated to deal with hole-doped VBS model (bosons = spin, fermions = hole).

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