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)

IntroductionThe 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.

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)...

Supermanifolds

……

Non-compactmanifolds 　 Hyperboloids, ….

Hasebe (10)Jellal (05-07)

Laughlin, Haldane

``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)

Fuzzy Geometry, Landau model

and Supersymmetry

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)

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 CPn Karabali & Nair (02)

Haldane (83)

Wu & Yang (76)

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.

Fuzzy Supersphere Grosse & Reiter (98)

Balachandran et al. (02,05)Symmetric Rep.

Fuzzy Algebra

Supersphere

Non-anticommutative geo.

odd Grassmann even

(OSp(1|2) algebra)

Symmetric Rep. of Fuzzy Supersphere

Bosonic d.o.f. Fermionic d.o.f.

- sym.rep.

Picture of the basis elements of fuzzy supersphere

1

1/2

0

-1/2

-1

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)

Fuzzy super-geometry & super LLLLLL basis Fuzzy supersphere basis

Super-coherent state (super-Hopf spinor)Schwinger super-operator

Up to now, the correspondence is at one-particle level.

How about many-body level ?

Many-body level Correspondence

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

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

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?

Translation to Internal spin spaceSU(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

Step 1: Local Hilbert space

i: index of a particle i: index of a lattice site

Charge of monopoleMagnitude of spin

LLL SU(2) spin

(Ex.) Square latticei: index of a lattice site

or

Step2: Valence Bond

Valence bond (=Spin singlet bond)

: Entangled state without spin polarization

: Quantum Antiferromagnets

Spin-singlet

Examples of VBS statesVBS chain

VBS chain

Examples of VBS states

Honeycomb-lattice Square-lattice

Correspondence

Laughlin-Haldane wavefunction Valence bond solid state

Lattice coordination numberTotal particle number

Filling factor

Spin magnitudeMonopole charge

Two-site VB number

: number of bosons

Arovas, Auerbach, Haldane (88)

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

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

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

Hidden Order

0 00-1 +1 -1 +1 -1

VBS chain

den Nijs, Rommelse (89), Tasaki (91)

Classical Antiferromagnets Neel (local) Order

Hidden (non-local) Order

+1-1 -1 -1 -1+1 +1 +1

No sequence such as +1 -1 0 0 -1 +1 0

SUSY VBS (SVBS) states

Basis Elements of SVBS statesArovas, Hasebe, Qi, Zhang (09)

SU(2) quantum number

Physical interpretationOperators

Up-spin

Down-spin

(spinless) hole

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

SVBS chain

Valence-bond Hole-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

The physical property of SVBS chain

Insulator

Superconductor

Insulator

spin

Disordered quantum anti-ferromagnets

chargeHole doping

Superconducting order parameter

r-dependence of the correlation lengthsSpin correlation

Superconducting correlation

Hidden Order in SVBS States

The SVBS states Show a Generalized Hidden Order.

0 00-1 +1 -1 +1 -1

VBS

+1/2 -1/2

0

SVBS

-1

+1 +1

-1+1/2 +1/2 +1/2 +1/2

Hasebe & Totsuka (10)

String Order Parameter

``Crackion’’ by Single Mode Approximation

triplet-bond triplet-bond

gapful excitation

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.