Kazuki Hasebe(Kagawa N.C.T.)

Collaborators,

Keisuke Totsuka Daniel P. Arovas Xiaoliang Qi Shoucheng Zhang

Quantum Antiferromagnets from

Fuzzy Super-geometry

(Stanford)

(YITP)

(UCSD) (Stanford)

Supersymmetry in Integrable Systems – SIS’12 (27-30, Aug.2012, Yerevan, Armenia)

Based on the works, arXiv:120…, PRB 2011, PRB 2009

Topological State of Matter 2

TI, QHE,

Theoretical (2005, 2006) and Experimental Discoveries of QSHE (2007)

Local Order parameter (SSB) Topological Order

TSC

Topological order is becoming a crucial idea in cond. mat., hopefully will be a fund. concept.

How does SUSY affect toplogical state of matter ?

and subsequent discoveries of TIs

QAFM, QSHE,

Main topic of this talk:

Order

Wen

Physical Similarities

QHE: 2D

Gapful bulk excitations

Gapless edge spin motion

``Featureless’’ quantum liquid : No local order parameter

QAFM: 1D

``Disordered’’ quantum spin liquid : No local order parameter

Spin-singlet bond = Valence bond

Quantum Hall Effect

Valence Bond Solid StateGapful bulk excitations

Gapless chiral edge modes

``locked’’

or＝

Math. Web

Quantum Hall EffectFuzzy Geometry

Valence Bond Solid State

Schwinger formalism

Spin-coherent state

Hopf map

Simplest Concrete Example Fuzzy Sphere

or＝

Haldane’s sphere

Local spin of VBS state

Monopole charge :

Spin magnitude :

Radius :

Fuzzy SphereFuzzy and Haldane’s spheres

Schwinger formalism

Berezin (75),Hoppe (82), Madore (92)

6

Haldane’s Sphere

Hopf map

: monopole gauge field

One-particle Basis

LLL basisHaldane (83)Wu & Yang (76)

States on a fuzzy sphere

Fuzzy Sphere

Haldane’s sphere

Translation

LLL Fuzzy sphere

Simply, the correspondence comes from the Hopf map: The Schwinger boson operator and its coherent state.

Schwiger operator Hopf spinor

Laughlin-Haldane wavefunction Haldane (83)

SU(2) singlet

Stereographic projection

: index of electron

Simplest Concrete Example Fuzzy Sphere

or＝

Haldane’s sphere

Local spin of VBS state

Monopole charge :

Spin magnitude :

Radius :

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

Interpret as spin coherent state

Correspondence

Laughlin-Haldane wavefunction Valence bond solid state

Lattice coordination numberTotal particle number

Filling factor

Spin magnitudeMonopole charge

Two-site VB number

Arovas, Auerbach, Haldane (88)

Affleck, Kennedy, Lieb, Tasaki (87,88)

Particle index Lattice-site index

Examples of VBS states (I)VBS chain

VBS chain

Spin-singlet bond = Valence bond ``locked’’

or＝

Examples of VBS states (II)

Honeycomb-lattice Square-lattice

Particular Feature of VBS states

VBS models are ``solvable’’ in any high dimension. (Not possible for AFM Heisenberg model)

Gapful (Haldane gap)

Non-local

Disordered spin liquid

Exponential decay of spin-spin correlation

Ground-state

Gap (bulk)

Gapless

SSB No SSB

Order parameter Local

Neel state Valence bond solid state

15

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

Generalized Relations

Quantum Hall EffectFuzzy Geometry

Valence Bond Solid State

2D-QHE

SO(5)- q-deformed-SO(2n+1)-

Mathematics of higher D. fuzzy geometry and QHE can be applied to construct various VBS models.

4D- 2n- q-deformed-CPn-

Fuzzy four-

Fuzzy two-sphere

Fuzzy CPn

Fuzzy 2n-q-deformed

SU(n+1)-SU(2)-VBS

Related References of Higher D. QHE1983 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)

Kimura, KH (04), …..

Kimura, KH (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

Related Refs. of Higher Sym. VBS States

2011

1987-88 Valene bond solid models

Sp(N)

Tu, Zhang, Xiang (08)

Arovas, Auerbach, Haldane (88)

Higher- Bosonic symmetry

　 UOSp(1|2) , UOSp(2|2), UOSp(1|4) …

Arovas, KH, 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)

Totsuka, KH (11,12)

q-SU(2) Klumper, Schadschneider, Zittartz (91,92)Totsuka, Suzuki (94) Motegi, Arita (10)

Takuma N.C.T.

Supersymmetric Valence Bond Solid Model

20

Fuzzy Supersphere Grosse & Reiter (98)

Balachandran et al. (02,05)

Fuzzy Super-Algebra

Supersphere odd Grassmann even

(UOSp(1|2) algebra)

Super-Schwinger operator

Intuitive Pic. of Fuzzy Supersphere

1

1/2

0

-1/2

-1

Haldane’s Supersphere One-particle Hamiltonian

UOSp(1|2) covariant angular momentum

Kimura & KH, KH (05)

SUSY Laughlin-Haldane wavefunction

Super monopole

LLL basis

: super-coherent state

Susy Valence Bond Solid States 24

Arovas, KH, Qi, Zhang (09)

Hole-doping parameter

Spin + Charge Supersymmetry

Manifest UOSp(1|2) (super)symmetry

At r=0, the original VBS state is reproduced. Math.

Physics

‘’Cooper-pair’’ doped VBS spin-sector : QAFM

charge-sector : SC

Exact many-body state of interaction Hamiltonian

hole

Exact calculations of physical quantities25

SC parameter spin-correlation length

Two Orders of SVBS chain

Insulator

Superconductor

Insulator

Spin-sector

Quantum-ordered anti-ferromagnet

Charge-sector

Hole doping

Order Superconducting

Sector

Topological order

26

Takuma N.C.T.

Entanglement of SVBS chain

27

Hidden Order in the SVBS State

+1/2 -1/2

0

-1

+1 +1

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

Totsuka & KH (11)

28

SVBS shows a generalized hidden order.

sSBulk = 1 : S =1+1/2

E.S. as the Hall mark 29

Li & Haldane proposal (06)

What is the ``order parameter’’ for topological order ?

BA

Entanglement spectrum (E.S.)

Robustness of degeneracy of E.S. under perturbation

Hall mark of the topological order

Schmidt coeffients

Spectrum of Schmidt coeffients

Behaviors of Schmidt coefficients

The double degeneracy is robust under ‘’any’’ perturbations (if a discete sym. is respected).

30

3 Schmidt coeff. 2+1 5 Schmidt coeff. 3+2

sSBulk = 1 sSBulk = 2 Totsuka & KH (12)

Origin of the double degeneracy31

A B

``edge’’

Double deg. (robust)

Double deg. (robust)

Non-deg.

Triple deg. (fragile) sSBulk = 2

sSEdge

= 1/2 sSBulk = 1

sSEdge

= 1

SEdge = 0

SEdge = 1/2

SEdge = 1

SEdge = 1/2

Understanding the degeneracy via edge spins

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.

Edge spin

1/2

Bulk (super)spin : general S

Bulk-(super)spin S=2 1

Edge spin

S/2

S/2-1/2

SUSY brings stability to topological phase.

SUSY SUSY

32

Summary

Edge spin : integer half-integerSUSY

33

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.

1. Math. of fuzzy geometry and QHE can be applied to construct novel QAFM.

First realization of susy topological phase in the context of noval QAFM!

2. SUSY plays a cucial role in the stability of topological phase.

Symmetry protected topological order 34

TRS

Odd-bulk S QAFM spin

Z2 * Z2 Unless all of the discrete symmetries are broken

Qualitative difference between even-bulk S and odd-bulk S VBSs

: Inversion

Even-bulk S QAFM spin: SU(2)

Sbulk=2n-1 Sedge=Sbulk/2=n-1/2 2Sedge+1=2n

Sbulk=2n Sedge=Sbulk/2=n 2Sedge+1=2n+1Odd deg. (fragile)

Double deg. of even deg. (robust)

Hallmark of topological order : Deg. of E.S. is robust under perturbation.

Pollmann et al. (09,10)