Download - Kazuki Hasebe
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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
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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
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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=
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Math. Web
Quantum Hall EffectFuzzy Geometry
Valence Bond Solid State
Schwinger formalism
Spin-coherent state
Hopf map
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Simplest Concrete Example Fuzzy Sphere
or=
Haldane’s sphere
Local spin of VBS state
Monopole charge :
Spin magnitude :
Radius :
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Fuzzy SphereFuzzy and Haldane’s spheres
Schwinger formalism
Berezin (75),Hoppe (82), Madore (92)
6
Haldane’s Sphere
Hopf map
: monopole gauge field
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One-particle Basis
LLL basisHaldane (83)Wu & Yang (76)
States on a fuzzy sphere
Fuzzy Sphere
Haldane’s sphere
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Translation
LLL Fuzzy sphere
Simply, the correspondence comes from the Hopf map: The Schwinger boson operator and its coherent state.
Schwiger operator Hopf spinor
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Laughlin-Haldane wavefunction Haldane (83)
SU(2) singlet
Stereographic projection
: index of electron
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Simplest Concrete Example Fuzzy Sphere
or=
Haldane’s sphere
Local spin of VBS state
Monopole charge :
Spin magnitude :
Radius :
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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
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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
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Examples of VBS states (I)VBS chain
VBS chain
Spin-singlet bond = Valence bond ``locked’’
or=
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Examples of VBS states (II)
Honeycomb-lattice Square-lattice
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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
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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
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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
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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
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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)
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Takuma N.C.T.
Supersymmetric Valence Bond Solid Model
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Fuzzy Supersphere Grosse & Reiter (98)
Balachandran et al. (02,05)
Fuzzy Super-Algebra
Supersphere odd Grassmann even
(UOSp(1|2) algebra)
Super-Schwinger operator
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Intuitive Pic. of Fuzzy Supersphere
1
1/2
0
-1/2
-1
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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
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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
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Exact calculations of physical quantities25
SC parameter spin-correlation length
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Two Orders of SVBS chain
Insulator
Superconductor
Insulator
Spin-sector
Quantum-ordered anti-ferromagnet
Charge-sector
Hole doping
Order Superconducting
Sector
Topological order
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Takuma N.C.T.
Entanglement of SVBS chain
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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
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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
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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)
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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
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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
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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.
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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)