Download Quantum Anti-ferromagnet from Fuzzy Super

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 (YITP)
Daniel P. Arovas (UCSD) Xiaoliang Qi
Shoucheng Zhang (Stanford)
(Stanford)
Topological State of Matter
Theoretical (2005, 2006) and Experimental
Discoveries of QSHE (2007)
and
subsequent
discoveries
of
TIs
Order
 Local Order parameter (SSB)
Wen
 Topological Order
QAFM, QHE, QSHE, TI, TSC
Topological order is becoming a crucial idea in
cond. mat., hopefully will be a fund. concept.
Main topic of this talk:
How does SUSY affect toplogical state of matter ?
2
Physical Similarities
Quantum Hall Effect
QHE: 2D
 Gapful bulk excitations
 Gapless chiral edge modes
``Featureless’’ quantum liquid
: No local order parameter
Valence Bond Solid State
Spin-singlet bond = Valence bond
``locked’’
QAFM: 1D
 Gapful bulk excitations
 Gapless edge spin motion
＝
or
``Disordered’’ quantum spin liquid
: No local order parameter
Math. Web
Fuzzy Geometry
Schwinger
formalism
Hopf map
Quantum Hall Effect
Spin-coherent
state
Valence Bond Solid State
Simplest Concrete Example
Fuzzy Sphere
Haldane’s sphere
Monopole charge :
Radius :
Local spin of VBS state
Spin magnitude :
＝ or
Fuzzy and Haldane’s spheres
Fuzzy Sphere
Berezin (75),Hoppe (82), Madore (92)
Schwinger formalism
Haldane’s Sphere
Hopf map
: monopole gauge field
6
One-particle Basis
Fuzzy Sphere
States on a fuzzy sphere
Haldane’s sphere
Wu & Yang (76)
LLL basis
Haldane (83)
Translation
Fuzzy sphere
Schwiger operator
LLL
Hopf spinor
Simply, the correspondence comes from the Hopf map:
The Schwinger boson operator and its coherent state.
Laughlin-Haldane wavefunction
Haldane (83)
Stereographic
projection
: index of electron
SU(2) singlet
Simplest Concrete Example
Fuzzy Sphere
Haldane’s sphere
Monopole charge :
Radius :
Local spin of VBS state
Spin magnitude :
＝ or
Translation to internal spin space
LLL states
SU(2) spin states
External space
Internal space
Cyclotron motion of electron
Precession of spin
1/2
1/2
-1/2
-1/2
Haldane’s sphere
Bloch sphere
Interpret as spin coherent state
Correspondence
Arovas, Auerbach, Haldane (88)
Laughlin-Haldane wavefunction
Valence bond solid state
Affleck, Kennedy, Lieb, Tasaki
(87,88)
Particle index
Lattice-site index
Filling factor
Two-site VB number
Total particle number
Lattice coordination number
Monopole charge
Spin magnitude
Examples of VBS states (I)
VBS chain
Spin-singlet bond = Valence bond
``locked’’
＝
VBS chain
or
Examples of VBS states (II)
Honeycomb-lattice
Square-lattice
Particular Feature of VBS states
Neel state
Ground-state
Gap
(bulk)
 Order
parameter
SSB
Gapless
Local
15
Valence bond solid state
No SSB
Gapful (Haldane gap)
Exponential decay
of spin-spin correlation
Disordered spin liquid
Non-local
 VBS models are ``solvable’’ in any high dimension.
(Not possible for AFM Heisenberg model)
Hidden Order
den Nijs, Rommelse (89), Tasaki (91)
Classical Antiferromagnets
+1
-1
-1
+1
-1
+1
-1
Hidden (non-local) Order
VBS chain
-1
+1
Neel (local) Order
0
+1
-1
No sequence such as +1
0
+1
-1
0
0
0
-1
-1
+1 0
Generalized Relations
Fuzzy Geometry
Quantum Hall Effect
 Fuzzy two-sphere
 2D-QHE
 4D-
 Fuzzy four-
 2n-
 Fuzzy 2n q-deformed
 q-deformed CPn-
 Fuzzy CPn
Valence Bond Solid State
 SU(2)-VBS  SO(5)-  SO(2n+1)-
 q-deformed-  SU(n+1)-
Mathematics of higher D. fuzzy geometry and QHE
can be applied to construct various VBS models.
Related References of Higher D. QHE
1983
2D QHE
Laughlin, Haldane
2001  4D Extension of QHE : From S2 to S4
Zhang, Hu (01)
 Even Higher Dimensions: CPn, fuzzy sphere, ….
Karabali, Nair (02-06), Bernevig et al. (03),
Bellucci, Casteill, Nersessian(03)
Kimura, KH (04), …..
Super  Landau models on supermanifolds
manifolds Ivanov, Mezincescu,Townsend et al. (03-09),
……
Bellucci, Beylin, Krivonos, Nersessian, Orazi (05)...
 QHE on supersphere and superplane
Kimura, KH (04-09)
Non-compact
manifolds  Hyperboloids, ….
Jellal (05-07)
Hasebe (10)
Related Refs. of Higher Sym. VBS States
1987-88
Valene bond solid models
Affleck, Kennedy, Lieb, Tasaki (AKLT)
 Relations to QHE
Arovas, Auerbach, Haldane (88)
 q-SU(2) Klumper, Schadschneider, Zittartz (91,92)
Totsuka, Suzuki (94) Motegi, Arita (10)
200X
 SU(N)
HigherBosonic
Greiter, Rachel, Schuricht (07), Greiter, Rachel (07),
symmetry Arovas (08)
 Sp(N) Schuricht, Rachel (08)
 SO(N)
Tu, Zhang, Xiang (08) Tu, Zhang, Xiang, Liu, Ng (09)
Super UOSp(1|2) , UOSp(2|2), UOSp(1|4) …
symmetry
2011
Arovas, KH, Qi, Zhang (09)
Totsuka, KH (11,12)
Supersymmetric
Valence Bond Solid Model
Takuma N.C.T.
20
Fuzzy Supersphere
Supersphere
Grassmann even odd
Fuzzy Super-Algebra
Super-Schwinger operator
Grosse & Reiter (98)
(UOSp(1|2) algebra)
Balachandran et al. (02,05)
Intuitive Pic. of Fuzzy Supersphere
1
1/2
0
-1/2
-1
Haldane’s Supersphere
One-particle Hamiltonian
Kimura & KH, KH (05)
UOSp(1|2) covariant angular momentum
Super monopole
LLL basis
: super-coherent state
SUSY Laughlin-Haldane wavefunction
Susy Valence Bond Solid States
24
hole
Spin
+ Charge
Supersymmetry
Hole-doping parameter
Arovas, KH, Qi, Zhang (09)
At r=0, the original VBS state is reproduced.
 Math.
Manifest UOSp(1|2) (super)symmetry
Exact many-body state of interaction Hamiltonian
 Physics
spin-sector : QAFM
‘’Cooper-pair’’ doped VBS
charge-sector : SC
25
Exact calculations of physical quantities
SC parameter
spin-correlation length
Two Orders of SVBS chain
Sector
Charge-sector
Order
Superconducting
Spin-sector
Topological order
Hole doping
Insulator
Superconductor
Insulator
Quantum-ordered
anti-ferromagnet
26
Entanglement
of SVBS chain
Takuma N.C.T.
27
Hidden Order in the SVBS State
sSBulk = 1 : S =1+1/2
-1
+1/2
+1
+1/2
Totsuka & KH (11)
-1
+1/2
0
-1/2
+1/2
+1/2
+1
SVBS shows a generalized hidden order.
28
E.S. as the Hall mark
29
What is the ``order parameter’’ for topological order ?
A
Schmidt coeffients
Li & Haldane proposal (06)
Spectrum of Schmidt coeffients 
Entanglement spectrum (E.S.)
Robustness of degeneracy of E.S. under
perturbation
Hall mark of the topological order
B
Behaviors of Schmidt coefficients
sSBulk = 1
3 Schmidt coeff.  2+1
sSBulk = 2
30
Totsuka & KH (12)
5 Schmidt coeff.  3+2
The double degeneracy is robust under
‘’any’’ perturbations (if a discete sym. is respected).
Origin of the double degeneracy
31
sSBulk = 1 sSEdge
= 1/2
SEdge = 1/2
SEdge = 0
Double deg. (robust)
Non-deg.
``edge’’
B
A
sSBulk = 2
sSEdge
=1
SEdge = 1
SEdge = 1/2
Triple deg. (fragile)
Double deg. (robust)
32
Understanding the degeneracy via edge spins
Bulk-(super)spin
S=2
Edge spin
1
SUSY
1/2
Edge spin
Bulk
(super)spin
: general S
S/2
SUSY
S/2-1/2
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.
SUSY brings stability to topological phase.
33
Summary
1. Math. of fuzzy geometry and QHE can be
applied to construct novel QAFM.
 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.
2. SUSY plays a cucial role in the stability of
topological phase.
Edge spin : integer
SUSY
half-integer
First realization of susy topological phase
in the context of noval QAFM!
Symmetry protected topological order
34
Pollmann et al. (09,10)
Qualitative difference between even-bulk S and oddbulk S VBSs
Hallmark of topological order :
Deg. of E.S. is robust under perturbation.
 Odd-bulk S QAFM spin
Unless all of the discrete symmetries are broken
: Inversion
Z2 * Z2
TRS
Sbulk=2n-1 Sedge=Sbulk/2=n-1/2
2Sedge+1=2n
Double deg. of even deg. (robust)
 Even-bulk S QAFM spin
: SU(2)
Sbulk=2n Sedge=Sbulk/2=n
2Sedge+1=2n+1
Odd deg. (fragile)