Fermions without fermion fields

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Author(s): R. C. Ball

Article Title: Fermions without Fermion Fields

Year of publication: 2005

Link to published article:

http://dx.doi.org/10.1103/PhysRevLett.95.176407

arXiv:cond-mat/0409485v3 [cond-mat.str-el] 8 Sep 2005

R.C. Ball

∗

Department of Physis, University of Warwik, Coventry CV4 7AL, UK

ItisshownthatanarbitraryFermionhoppinghamiltoniananbemappedintoasystemwithno

fermionelds,generalisinganearliermodelbyLevin&Wen[1,2℄. Alloperatorsinthehamiltonianof

theresultingdesriptionommute(ratherthanantiommute)whenatingatdierentsites,despite

thesystemhavingexitationsobeyingFermistatistis. Whilst extraonserveddegreesoffreedom

are introdued, theyare all loally identiedinthe representation obtained. The same methods

apply to Majorana (half) fermions, whih for artesian latties mitigate the Fermion Doubling

Problem. Thegenerality of these results suggests that the observation of Fermion exitations in

naturedoesnotdemandthatantiommutingFermioneldsbefundamental.

Asfundamental entities,fermioneldsappearin

on-it with the priniple of loality: all fermion reation

andannihilation operatorsantiommutenomatterhow

far apart are the points in spae at whih they at

(without restrition by ausal onnetion). This

fea-ture is built into Quantum Field Theory through the

use of antiommuting Grassman elds, and

supersym-metristringtheorieslikewisehaveexpliit

antiommut-ingoordinates. LoalityisonlypreservedinthePhysis

by onservation offermion number,foring thefermion

operators to appear in pairs. In reent years there has

beensustainedinterest in howpartile statistisanbe

manipulated[3℄,andunderstandingtheFrational

Quan-tumHallEet[4 ℄haswidennedtheappreiationthatthe

statistisoftheelementary exitationsofasystemneed

notsimplyreetthestatistisofitsomponents. Thus

weaskwhetheritisreallyneessarytoputfermionelds

intophysis`byhand',orwhethertheyanalwaysbe

un-derstood asexitationsemergingfromquantumsystems

builtofoperatorswhoseationisstritlyloal.

It is well known how fermions relate to hard ore

bosons in one dimension[5℄ and how they an be built

out of bosons in two dimensions with attahed

mag-neti ux[6℄, and that neither approah extends

natu-rally to three dimensions of spae. The present letter

builds on the reent work of Levin & Wen[1, 2℄, who

showedthat partiularmodelsof pairwise fermion

hop-ping ould be represented in terms of operators

obey-ing loality, whilst theelementary exitations remained

stritlyfermioni. Theirmehanismisessentiallythe

re-verse of how Kitaev[7℄ showed that a partiular

hexag-onallattie spinmodel has fermioni exitations. They

showedthattheirmehanismworkedonsimple

hyperu-bi latties, in twoand three dimensionsexpliitly, and

interpretitintermsofstring-netondensation[2,8℄.

Here I show that fermion hopping on an arbitrary

graphanbemappedinto theexitations ofa

Hamilto-nian devoid of expliitfermioni operators,in whih all

operatorsatingatdierentsites ommute. The

dimen-sionalityofspaewhihthegraphmightapproximate is

∗

Eletroniaddress: r..ballwarwik.a.uk ;URL:http://go.

warwik.a.uk/theory;

irrelevanttothemapping,whihissensitiveonlytoloal

oordination numbers (presumed nite). This strongly

suggeststhat quitearbitrary fermion exitation spetra

anbe represented, and perhapsunderstood, as arising

fromtheexitationofsystemswhosefundamental

oper-atorsobeyloalityin theirommutationproperties.

Westartfromagenerifermionhoppinghamiltonian,

H

hop

=

X

i

c

+

i

V

i

c

i

+

X

<ij>

c

+

i

t

ij

c

j

(1)

wherethefermioneld

c

i

hasstandardantiommutation properties

{

c

i

, c

j

}

=

c

+

_i

, c

+

_j

= 0

and

c

+

_i

, c

j

=

δ

ij

, the

V

i

are simple `on-site' potentials (relative to the fermionhemialpotential)and the

t

ij

=

t

∗

ji

are simple `inter-site'hoppingmatrixelements. Theonnetivityof

thegraph (or lattie) is enoded by whih elements

t

ij

arenon-zero,butbelowwewillneedtoexpliitlyrestrit

the sum over links

< ij >

to those ases. We will also

exploitthegaugeinvarianeofthehamiltonian,thattwo

modelsrelatedby

t

(2)

jk

=

t

(1)

jk

e

iϑ

jk

areequivalent(through adjustmentofphaseofthe

c

k

)providedtherelativephase fatorsmultiplytounityaroundalllosedloops.

Wenowintroduenewoperators

S

ij

=

−

S

ji

modulat-ingthehoppingsarosseahlink,suhthatthese

opera-torsommutewitheahotherandtheoriginalfermions,

andwitheigenvalues

s

ij

=

−

s

ji

=

±

1

. Thehamiltonian isthen generalisedto

H

gauge

=

X

i

c

+

_i

V

i

c

i

−

X

<ij>

i S

ij

c

+

i

u

ij

c

j

+

X

<ij..z>

g

ij..z

S

ij

S

j.

..S

.z

S

zi

.

(2)

Here

u

ij

=

is

0

ij

t

ij

so that the rst two terms reover thehopping hamiltonian (1)for apartiular set of

out-omes

s

0

ij

=

−

s

0

ji

oftheoperators

S

ij

. Weaddextra ou-plings

g

ij..z

to(produtsround)`Wilsonloops'ofthe

S

ij

operators;setting

g

ij..z

/ s

0

ij

s

0

j.

..s

0

.z

s

0

zi

suiently

nega-tiveensuresthatonlyombinationsofthe

S

ij

eigenvalues whiharegaugetransformationsoftheoriginalhopping

hamiltonian ontribute to the low energy states of the

new hamiltonian. Beause the operators

S

ij

ommute

witheahotherandthehamiltonian,theyouldbe

onstantsofthemotion,andwethenreovertheoriginal

tightbindinghamiltonian(uptogaugesymmetry). Thus

thesystemdoesstillhaveitsoriginalfermionexitations.

Thekeyto obtainingarepresentationwithloality is

now to fatoriseeah link operator into apairof

Majo-rana[half-℄fermions,

S

jk

=

i m

jk

m

kj

where the Majorana half-fermion operators are

Hermi-tian (for simpliity) and haveantiommutation

proper-ties

{

m

jk

, m

j

′

,k

′

}

= 2

δ

jj

′

δ

kk

′

aswell asantiommuting with allthe originalstandard fermionoperators

c

i

. We thenassoiateeahnewhalf-fermionwith thesiteof its

rst index, motivating us to rewrite the hopping terms

inthehamiltonianas

b

+

ij

t

ij

b

ji

,where

b

ij

=

m

ij

c

i

.

Cruiallythenewoperators

b

ij

ommute,

[

b

ij

, b

i

′

j

′

] = 0

and

b

+

ij

, b

i

′

j

′

= 0

, when their left (or site) indies

are unequal,

i

6

=

i

′

. The loop terms an also be

ex-pressed in terms of operators onforming to loality in

this way. Regrouping the fators in eah loop givesus

S

ij

S

jk

S

k.

..S

.z

S

zi

=

B

i,zj

B

j,ik

B

k,j.

..B

z,.i

where

B

j,ik

=

i m

ji

m

jk

.

Weannallyeliminateallfermioninotationbywriting

n

i

=

c

+

i

c

i

andheneexpressthehamiltonianas

H

gauge

=

X

i

V

i

n

i

+

X

<ij>

b

+

_ij

u

ij

b

ji

+

X

<ij..z>

g

ij..z

B

i,zj

B

j,i.

..B

z,.i

.

(3)

Fromtheirdenitionsitistrivialtohekthatanypair

drawnfromalltheoperators

n

i

,

b

ij

,

b

+

ij

,

B

i,jk

appearing in thehamiltonian(3) ommutewhentheirrstindies

aredistint. AsaresultweanfatortheoverallHilbert

spae(ofwavefuntions)onwhih theyatinto a

prod-utofsinglesiteHilbertspaes,andtheonlynon-trivial

ationofeahoperatoriswithintheorrespondingsingle

siteHilbertspae.

We now fous on eah site separately and note how

the required ommutation properties an be expliitly

onstruted. Firstnote that eah Dirafermionanbe

splitinto apairofhermitianMajoranahalf-fermions,

2

c

k

=

m

k

0

+

i m

k

−

1

,

(4)

in terms of whih

n

k

= (

im

k

0

m

k

−

1

+ 1)

/

2

. The om-plete set of operator properties then required on site

k

arenowjust theMajoranaantiommutationrelations

{

m

km

, m

kn

}

= 2

δ

mn

,

−

1

≤

m, n

≤

z

k

,

where

z

k

is theoordination numberof (i.e. number of links to)site

k

. These relationsareobeyedbystandard

(Eulidean)

4

×

4

Diramatriesfor

2 +

z

k

≤

5

andby their

2

s

_×

₂

s

generalisationsfor

2+

z

k

≤

2

s

+1

. Itisruial thatwedonot requiretorepresentfermion

antiommu-tationbetweenloalfermionoperatorsondierent sites,

beauseallthetermsinthehamiltonianontainaneven

numberof fermionfatorsfromeahsite. Writing

γ

k

(

i

)

forthe

k

'th Diramatrixating in theHilbert spaeof

the

i

'thsite,wethenhavealltheoperatorsinthebosoni

hamiltonian(3)expressedintermsofthese:

n

j

=

i

2

γ

0

(

j

)

γ

−

1

(

j

) +

1

2

,

b

jk

=

1

2

γ

k

′

(

j

) (

γ

0

(

j

) +

i γ

−

1

(

j

))

,

B

j,ik

=

i γ

i

′

(

j

)

γ

k

′

(

j

)

.

(5)

Hereonsite

j

,

1

≤

k

′

₍

_{j, k}

₎

_≤

_z

_j

denotes theloal index

assoiatedwithitslinktosite

k

.

Theend result is that all the operators appearing in

theHamiltonian areexpressed in terms of loal matrix

operators, whih in turn are all equivalent to bilinear

ombinations of Diramatries: these might loosely be

termed`spins'. Wheretheoriginal hoppinghamiltonian

hadlinkswithexpliitfermionhopping,thederived

gen-eralisationoftheLWmodel[1℄hasouplingbetweenthe

loalspins. Mostimportantly,thespinoperatorsfor

dif-ferent sites ommute - yet by onstrution the system

stillhastheoriginalfermioniexitations.

The loop produts of operators

B

i,jk

an be further simpliedintermsofprodutsofnewoperators

P

ij

=

γ

j

′

(

i,j

)

(

i

)

γ

i

′

(

j,i

)

(

j

)

(6)

in diret index orrespondene with the original

op-erators

S

ij

. These new operators are not equivalent to the

S

ij

operators: in partiular two operators

P

ij

antiommuteiftheyhaveonesite indexinommon.

Loality of all operators has been gained at the

ex-pense ofenlarging theHilbert spae. Theoriginal

hop-pinghamiltonian(1)atedonaHilbertspaeof

dimen-sion

2

N

where

N

is the number of sites, equivalent to

N

qbits,onefor eah site. Toonstrutthe generalised

WenHamiltonian (3 with 5) we rst added qbits equal

to the number of links,

A

. However in the nal loal

representationwearried fewernon-ommutations, and

multiplyingthedimensionofalltheloalHilbertspaes

leadstoadimensionequivalenttototalqbitount

C

=

N

E

+

1

2

N

O

+

A.

Here

N

E

and

N

O

are the ountsof sites with even and oddoordinationnumberrespetively,theevenbeingless

eiently represented beause the number of

antiom-mutingDiramatriesisnaturallyodd.

Weshould nowexpet that there are

A

−

1

2

N

O

on-stants of the motion, and these and one more an be

foundasfollows. Firstfromeverylink

ij

,wehavethe

of

P

operators ommuteprovided they havenoends in

ommon, inluding losed strings whih have no ends.

Seondly, for every even oordinated site we have

her-mitian

Γ

k

=

i

(

z

+2)

/

2

Q

z

j

=

−

1

γ

j

(

k

)

antiommuting with

every

γ

on that site, and hene ommuting with every

termin thehamiltonian;however

Γ

k

antiommuteswith any

P

-stringending onsite

k

. Themaximalommuting

setofalltheseoperatorsthenappearstobetheunionof:

(a)allmutuallyinequivalentlosedloop

P

-strings,

equiv-alentto aminimalset of independent looptermsin the

hamiltonian(3),numbering

A

−

N

+ 1

; (b)the

Γ

k

forallevensites,numbering

N

E

;

() open

P

-strings ending on odd sites, with no ends

inommonandinequivalentunderprodutswith losed

loopstrings, all ofwhih enumerate to

N

O

/

2

by taking takingalltheoddsitesin (arbitrary)disjointpairs.

All oftheaboveommutewitheah otherandwith the

Hamiltonian, so we havein total

A

−

1

2

N

O

+ 1

expliit

onstants of the motion. Eah orresponding operator

haseigenvalues

±

1

,soitsonservationremovesoneqbit

fromthedynamis. Theauthoronjeturesthattheone

extraonservationlawrelatestoonservationoffermion

number(modulo

2

).

Canoneexploittheonstantsofthemotiontoredue

the size of the (quantum-mehanially oupled)Hilbert

spae? Trying this with

P

ij

operators indues sues-sivelylessloalantiommutationrelations,tendingto

re-build theoriginalfermioni representation. Howeverwe

aneliminatethesite-wiseloal

Γ

i

onevensites,leading usto adimension-independentgeneralisationofwhatin

onedimensionorrespondstotheAnisotropiHeisenberg

Modelrepresentationoffermions.

Letusfousonsomepartiular evensite

i

andin the

following drop referene to that index. We an

elim-inate

γ

−

1

=

i

(

z

+2)

/

2

γ

0

γ

1

..γ

z

Γ

, and then the operators

in the Hamilonian whih ontained

γ

−

1

take the forms

n

=

−

i

z/

2

2

γ

1

..γ

z

Γ + 1

/

2

,

b

k

=

1

2

γ

k

′

γ

0

1

−

i

z/

2

γ

1

..γ

z

Γ

,

b

+

_k

=

1

2

γ

0

γ

k

′

1 +

i

z/

2

γ

1

..γ

z

Γ

. Thenbeause

Γ

still

(evi-dently)ommuteswiththehamiltonianwean fouson

thesetorwitheigenvalue

Γ = 1

andmakethis replae-ment. Nowtheonlymatriesappearingare

γ

k

,

k

= 0

...z

, andweantakeaminimalantiommutingrepresentation

ofthese[10℄, intermsofwhihweobtain

n

=

1

2

(1 +

γ

0

)

,

b

k

=

γ

k

′

n

,

b

+

k

=

nγ

k

′

=

γ

k

′

(1

−

n

)

at eah even oordi-natedsite.

For an arbitrary graph of even oordinated sites, we

nowhavehamiltonian

H

even

=

X

i

V

i

n

i

+

X

<ij>

n

i

γ

j

′

(

i

)

u

ij

γ

i

′

(

j

)

n

j

+

X

<ij..z>

g

ij..z

P

ij

P

j.

..P

.z

P

zi

,

(7)

and the expliit form for the

P

ij

remains as per eqs.

(6), exept that the dimension of the loal Dira

ma-tries hasbeenhalved. It is nowpartiularly learhow

the hopping terms onserve fermion numbers, and

in-deed one an further show that

n

i

γ

j

′

(

i

)

u

ij

γ

i

′

(

j

)

n

j

=

n

i

(1

−

n

j

)

γ

j

′

(

i

)

u

ij

γ

i

′

(

j

) (1

−

n

i

)

n

j

making the inter-onnetion between oupation numbers

1

j

0

i

and

0

j

1

i

totally expliit. The loop terms are also partile

on-servingandeah

P

ij

fatoranbereorganisedinsimilar manner. If onespeialisesto

z

= 2

orrespondingto a onedimensionalhainofsites,theDiramatriesredue

toPaulimatriesandthersttwotermsofthe

hamilto-nianareexatlyequivalenttotheAnisotropiHeisenberg

spinhain[9℄,wellknowntorepresentfermionsinone

di-mension. Theperiodiasehasone loop term,but this

reduestoaxedsalar. Itisgratifyingthatsuha

long-known fermion hamiltonian turns out to have natural

extensionto arbitrarydimensionsandeventoarbitrary

graphs(ofevenoordinationnumber).

Allofouranalysisanbegeneralisedtotheasewhere

westartfromhoppingof anarbitrarynumber

h

of

half-fermionsoneahsite. Hithertowestartedfromone

stan-dardfermionpersiteinthefermionhoppinghamiltonian

(1)and split that into twohalf fermions (4),

h

= 2

. In thegeneral aseanynatural number

h

is allowed, with

theonsite potentialsandhoppingmatrixelements

mak-ingarbitrary(hermitian)mixingsamongstthe

h

ompo-nents. For

h

= 1

there are no on-site potential terms, andhermitiityrestritsthehopping matrixelementsto

bepurereal,

u

ij

=

u

′

ij

,givingus

H

hop,

1

=

i

X

<ij>

u

′

ij

m

0

i

m

0

j

.

(8)

Thestandardfermionasedisussedealieris

h

= 2

,and itredues to thesumof two(ommuting)

h

= 1

hamil-toniansiftheonsitepotentialtermsarezeroandthe

u

ij

areallpurereal(towithinagaugetransformation). The

ase

h

= 3

ommandsinterestforpartilephysis. The disussion of onservation laws and redution of

Hilbertspae generalisesquitetrivially,with the

under-standingthatevenandoddsitesareidentiedbywhether

h

+

z

isevenorodd. Partiularlysimplespinmodelsare

obtainedfrom

h

= 1

, where the loal operators

b

ij

are hermitian and an be diretly represented assingle

lo-al Diramatries

γ

j

(

i

)

(rather than bilinearproduts) requiringonly

z

i

Dira matries at eah site, and with

B

i,jk

∝

iγ

j

(

i

)

γ

k

(

i

)

. Quite general Majorana fermion hoppinghamiltoniansanthereforeemergefrom the

ex-itationsof models builtoutof simpleloal spin

opera-tors. Thegaugesymmetryhangesnaturallywith

h

. For

h

= 1

wehaveonly

Z

2

(signhanges), orrespondingto

thetransformationsexploitedinthe`KLWtrik'of

intro-duingthe

S

operators,andfor

h

= 2

wealready noted

U

(1) =

O

(2)

gaugesymmetry. For

h

= 3

wewould have

gaugesymmetry

O

(3)

iftheonsite termsaresuiently symmetri,butquitearbitrarymixingofomponents

al-lowed in the onsite terms of the hamiltonian would in

generalreduethegaugesymmetryto

O

(2)

.

Thesimple(hyper-)ubilattiesareofspeialinterest

asasoureofinsightintotheontinuumlimit,and

parti-lephysisinterestfousespartiularlyontheasewhere

thefermion exitations are(almost) massless. The

byLevinandWen, whih wasforsimplesquareand

u-bilattieswith

h

= 2

,fallsinto thisategory. However thatworkalsosueredfromthestandard`Fermion

Dou-blingProblem' oflattiefermionmodels[11℄,yielding

2

d

masslessfermions in

d

dimensions.

Starting from a single half-fermion (

h

= 1

) hopping model halves the Fermion Doubling to give just

2

d

−

1

massless fermions in

d

dimensions. From the point of

viewofthespinmodels,thisisjust asnaturalastarting

point as

h

= 2

. Forsimpliity we present the alula-tioninonedimension,whih suestodemonstratethe

novelty, asthe elaborationto higherdimensions simply

follows in the same manner as Levin & Wen[1, 2℄. A

simplestonedimensionalMajoranahoppinghamiltonian

is

H

1

=

−

iu

′

X

n

m

n

m

n

+1

and imposing periodi boundary onditions for suh a

systemof

N

sitesresultsinaspetrumofstandard

mass-lessfermions,

H

1

= 2

u

′

π

X

k

=0

2

e

c

+

_k

e

c

k

−

1

sin

k.

Here the wavevetorsspan half the rst Brillouin zone

of the lattie,

k

=

n

2

π/N

,

n

= 1

to

N/

2

, the fermion

operators are given by

˜

c

k

= 1

/

√

2

N

P

n

e

−

ikn

m

n

and the negative wavevetor Fourier omponents give their

onjugates. These

c

˜

k

arestandardfullfermionoperators, inpartiularobeying

˜

c

+

_k

,

c

˜

l

=

δ

kl

.

Theaboveisineet astaggeredfermion[12℄solution

totheFermionDoublingProblem[11℄. Thefullfermions

obtained by halving the Brillouin zone an be mapped

onto one full fermion per two original sites, but these

arenon-loally relatedtotheoriginalhalffermions. For

exampleassoiatingthefullfermionswithevensitesgives

c

2

n

=

r

2

N

π

X

k

=0

e

ik

2

n

c

˜

k

=

m

2

n

2

+

i

π

X

q

m

2

n

+(2

q

+1)

2

q

+ 1

.

Thisparallels anothersolutionto theFermionDoubling

Problem, in whih highly non-loal hopping with

am-plitudes similar to the above is used to fore a better

spetral approximation to the ontinuum[13℄. Here we

have loal hopping for the Majorana partiles and the

non-loalamplitudesenodedinhowtheymaponto

on-served full fermions. We are not fored to work with

these non-loalrealtionshipsexpliitly, evenin thetight

binding hamiltonian, beause we an always make full

fermions moreloally,forexampleoutof adjaentpairs

ofhalf fermions;theprieof doingthis isthat in terms

ofthelatterthehamiltonianthenontainspairreation

andannihilationterms.

One an introdue mass without doubling the

spe-trum, simply by modulating the bond strengths

u

′

n,n

+1

→

√

u

′

2

₊

_ǫ

2

₊

_ǫ

₍

₋

₁₎

n

to open up an energy

gap at

E

= 0

(orresponding to

k

= 0

, π

). One

thenndsdispersion relation

E

(

k

) =

±

4

p

u

′

2

_sin

2

_k

₊

_ǫ

2

with two states at eah wavevetor over the

quar-ter Brilliouin zone

0

≤

k < π/

2

. Adding more

artesian dimensions with suitably frustrated signs to

the ouplings leads (as per Levin and Wen[1, 2℄ for

the full fermion ase) to the natural generalisations

in higher dimensions, for example

E

(

k

x

, k

y

, k

z

)

=

±

4

q

u

′

2

x

sin

2

k

x

+

u

′

y

2

sin

2

k

y

+

u

′

z

2

sin

2

k

z

+

ǫ

2

inthree di-mensions. Eah inrease in dimension generates a

dou-blingofthespetrum,totwospeiesoratwoomponent

fermionin

d

= 2

,andafouromponentfermionin

d

= 3

, mathingstandardfreefermionelds.

In overall summary, we an now onstrut

hamilto-nians in terms of stritly loal operators (in the sense

oftheirommutation relations)whoseexitations

orre-spondto fermionhoppingonwhatevergraphisdesired.

Ifoneaeptsinter-sitehopping asanapproximationto

ontinuum partile dynamis, then this means that

al-mostanyfermionproblemanbeonstrutedoutofthe

exitationsofwhatmightlooselyhavebeendesribedas

aBosesystem.

The method has been presented for free

(non-interating)fermionsystems. Howeveritistrivially

gen-eralisabletotheaseofanarbitraryinteration,provided

thisisafuntionofthestatenumberoperators(orother

bilinearombinationsofloalfermionvariables). In

par-tiular Coulomb interations between dierent sites are

allowed,asareHubbardinterations. Modifyinghopping

aordingto oupation numbers,as in

t

−

J

model, is

alsoreadilyinorporated.

The omplete and relatively loal enumeration of all

theonstantsof the motionmay haveonsequenesfor

string-netondensateinterpretations[2,8℄. By

introdu-ingtermsintheHamiltonianoupledtoalltheonserved

quantities, we are free to remove all the degeneray of

eventhegauge-equivalentombinationsof

S

link

opera-torsintrodued in eq. (2) from low energystates, in as

many ways asthe degeneray introdued. If string-net

ondensatestatessurvivethissplitting, theyhaveto be

orrespondinglydegenerateto startwith.

The `spin' hamiltonians onstruted to give exatly

the speied fermions are somewhat umbersome. Are

there simpler loal spin hamiltonians whih still give

fermionexitations? ThegeneralisedAnisotropi

Heisen-berghamiltonian isoneasein point: settingtheonsite

termstozeroandremovingthenumberoperatorfators

from the hopping terms turns out to give the same as

startingfromaMajoranahoppinghamiltonian(8). Also,

we ould by deliberate onstrution build hamiltonians

whih are supersymmetri in their exitation spetrum,

justbyaddinginthedesiredbosons. Aretherevariations

Aknowledgments

The author aknowledges disussions with R.

Roe-mer,J.B.Staunton,F.Pinski,B.MuzykantskiiandD.R.

Daniels. Theattentiontodetailbyananonymousreferee

ledtosigniantimprovementsinthepresentation.

[1℄ MLevin and X-G Wen, Fermions, strings, and guage

elds in lattie spin models, Phys Rev B 67, 245316

(2003).Alsoond-mat/0302460.

[2℄ X-G Wen, Quantum order from string ondensations

andorigin of light andmassless fermions,PhysRevD

68,065003(2003).Alsohep-th/0302201.

[3℄ F.Wilzek.Phys.Rev.Lett.49:957,1982.

[4℄ R.B. Laughlin, Phys. Rev. Lett. 50 (18): 1395-1398

(1983).

[5℄ P.JordanandE.Wigner,Z.Phys.47,631(1928).

[6℄ P.HassenfratzandG.'tHooft,Phys.Rev.Lett.36,1119

(1976).

[7℄ A.Y. Kitaev, unpublished talk

http://online.itp.usb.edu/online/glasses_03; also

Ann.Phys.303,2(2003).

[8℄ M A Levin and X-G Wen, String-net ondensation:

A physial mehanism for topologial phases,

ond-mat/0404617.

[9℄ TNiemeijer,Physia36377(1967).

[10℄ Oneanindeedverifydiretly,forexamplebytakingan

expliit representation of the

γk

,

k

= 0

...z

, that all the

produts of any even number of thosematries (whih

isall wehave)atinginthesetor

Γ = 1

doat inthis

manner.Inreduingtherepresentationwealsoinduea

relation

γ

0

=

±

i

z/

2

γ

1

..γz

,where thenegativehoieof

signhasbeenusedintheresultsfollowing.

[11℄ H. B.Nielsen andM. Ninomiya,Nul.Phys.B185, 20

(1981)[Erratum-ibid.B195,541(1982)℄.

[12℄ J.B.KogutandL.Susskind,Phys.Rev.D11(1975)395.

[13℄ S.D.Drell,M.WeinsteinandS.Yankielowiz,Phys.Rev.