9 Levin Gu model 2012

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

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

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

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

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

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

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

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

Vi TIMECpg Ip

Pp Vivi ViviM is a boson

m

Ifi

p Mad

X p

Y Y

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

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

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