9 levin gu model 2012
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9 1 Levin Gu model 2012
2D SPT protected by GX
Ising paramagnet
Ho E Op147 9 FEI
Op HptNp
EE 93
IDWconf
Dwconf
eveneven
Domain wall picturetf tft
On the planespin conf DW conf
I tt
I
On the torusspin conf DW conf even even
0 1 odd er
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spin conf I put confodd e
even odd
no 3Iyodd odd
Levin Gu state
He F Bp fiin siBp TpxEgg e tree
Ii
11 Icnn.eu
t awayDwarf
T i trod yDw crossing boundaryof III
Et 488
of of pqzgq.tt NoDWfor88I if 5895 1 pw forsqq's
i 1 if D crossing z mod
it 1 if so mad
ycharges of totalDW underBp
F Tpx
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Dw o4514ft ed ftp.t totalDw o l
Dw 2 fF t tf I I
In 4 FATTY t a 11
me EFFIE K a so
In summary in the continuum 4 O4 3E Y
141 Ep t DWrpt
Egg iwine o
Bp e pwincap wine
yapmod
Bp fi had to i win cta cap
Bp IG Mind c ItCowincap cap
I i Mind o
Bp 147 11
12 is the Gs of Hi
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14 E H PW GTnonlocalsign
differenceof nonlocalsigns localH FBIlocal term
9.2 Ganging a global symmetrySystem with onsite global symmetry G
Uf Vig acting on H Hi
Ug H 0 AgeG
fgange
G gaugetheory with local gauge symmetry redundancy
Gange
Iggygggonquaelauia.ME
4
Ho Ti to g tried
U Q TIX
U Ho 0
ganging procedure
Adding 24 gauge field Mig on link isEnforcing local gauge symmetry Gauss law on
the total Hilbert space
I
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É Tiesthe
tlocal gorgetransformation
Gauss law XXX EXIT MI 1
F É p
s tidIMi Eph
12pm
local gaugesymmtransf at site i
I ÉgIgMiIT U
in
globalsymm
aN
212infinite of local symm redundancies
Minimal couplingFt Tt TtMit Ftcity citeitij g
t
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I 1
tiedg myth tffrey
Adding zero flux condition and Gauss law to Hamiltonin
F Ete MI FIZ Gaa B
I Tif Mik XXXFor Ising paramagnet
Ho E TX JEFFf ganging
HT E TX J E F Mi Ft
I rig Mik Ete Miget to minimize E
I I Ia Mii F IseopMi
E x EEZtoric code model
Excitations Ho E ri U TrixGS rt ti
excitedstate I ri 1 for some éL
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Z charge
II EAT E EEZ change e 4212flux m
newexcitations afterganging
Newexcitations m e endpoints of domain wallsConsider conf at Il Mi I we cando genge
transformation TixIf Tj for ie with 82 1
xx fties
F Il Mif l É tl My I
DW 2 regionundersymmaction openDW
DW is closed loop endpoints m excitations
Ganging o expand Hilbert spare
promote globalsymm to local gauge Symon redundancyintroduce new excitations gauge flux
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913 Ganging Levin Gu model to doublesymionmodel
tristedquantumdoublemodelwithez
Q How to show that Levin Gu state is different
from Ising paramagnet
A Gage global 212 symmetry
Ising paramagnet toric code D a
Levin Gn double semion pug
e
he s 5
Ho 59x
dgangingto Erp Op Epp MpgmgrMrf Op Ty HMPFLimf
Hi F Bp Bp Op Egg i1 2 7
f gangingHT F Op Ear MpgMairMrf
BEE F Eg i E.ME
charge excitation e Jp 1 for toBp 1 for HT
flux excitation m MptMfrMrf I
Ribbon operator for e Eto
Flux excitations m
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Vi TIMECpg Ip
Pp Vivi ViviM is a boson
m
Ifi
p Mad
X p
Y Y
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m
p éiopf
panesm I
m is semion
9.4 Group cohomology model for G SPTas General SPT wavefunction
Triangulate space manifold
08 8 88
j j gig8eG
TYgigdomainwall
127 E Ik e
HIIIII
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8
Retriangulations Pachner move change the shapes of Dws
I Eff 418.8 82,9 I
Pentagon eqO O
o
o
dos gog g988.84 80,815,86 03180.8 gg
Sgp83,84 UsSo8,274
Symmetry condition08 47 12UH E 218.3 83 463 3
883 21493
03680 983 Us So Ss
Uz E Z G Va
Symmetric local unitary transf183 Tf Us So.si a
t 83
so 3 UsGo sss dueso gs
du sans ÉEUs Us doz
03 E H G 04 2 G un 133 G On
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2 Dijkgraaf Witten twistedgungetheory ganged SPT
2 1 D ZIMA Egg In 4103fgonging
É Ms Ig Fm 03103563
dg l
D
Spt DWwithfixedbackfield DW
dof site 9 EG linkSijEG linkSjEGtrivial bundle nontrivialbundle I nontrivialbundles
coupleshomogeneous Udlsso nigga
Josh ga Odingainhomogeneous
Ud 801,912 Solid
g gUgo8sn Mda aol.gr
Z In I 4s
146 Ting I 46Cj isfixed dga
ZMats 1 EUN GSDMdt