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Quantum Antiferromagnets from Fuzzy Super-geometry. Supersymmetry in Integrable Systems – SIS’12 (27-30, Aug.2012, Yerevan, Armenia). Kazuki Hasebe. Based on the works, arXiv:120…, PRB 2011, PRB 2009. (Kagawa N.C.T.). Collaborators , Keisuke Totsuka - PowerPoint PPT Presentation
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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)