THE GL(n,F,) ---INVARIANCE OF THE POTTS HAMILTONIAN

Internat. J. Math. & Math. Sci.

VOL. 20 NO. (1997) 33-36

33

THE GL(n,F,) ---INVARIANCE OF THE POTTS HAMILTONIAN

MIHAICARAGIU

Department

ofMathematics Penn StateUniversity 218McAllisterBuilding University Park,

PA

16802USA

MELLITA CARAGIU

Department

of Physics

"Babes-Bolyai"University Str. M. Kogalniceanu 3400Cluj-Napoca,

ROMANIA

(Received^]unc8, 1995and in revisedform

August

8,

1995)

ABSTRACT.

After defining^ameanfield byarithmeticmeans, using multiplicative characters offinite fields, its

Potts

Hamiltonian is exactly computed.

Moreover,

it provesto be invariant with respectto everychangeof basis in

Fq

over the primefield

KEY WORDS AND PHRASES:

Finitefields, characters,Hamiltonian invariance 1991AMS

SUBJECT CLASSIFICATION

CODES: 11L40, 82B20, 11T24,51E25.

1.

INTRODUCTION

Letus considerthe finite field

Fq,

q

pn,

as ann-dimensionalvectorspaceover itsprime subfield

Fp.

^Thisis, ofcourse, done by choosinga basise

(el,

^e2,

e2)m

^an

(ordered)

r-tuple of "vectors"

linearly independent over

Fp.

^{Then, we can see}

Fa

in a natural way as an n-dimensional hypercubic lattice with size p

Every

element z of this lattice can be expressed in a unique way as z zle ^/

+

zne,, the coefficientsz, satisfying periodic boundaryconditionsmodulop

(that

is, ifwe substitute

z

^{/p instead ofz,, the same point z is}

^obtained)

That means that the above defined hypercubiclattice canbeseen as a discrete modelofthe r-dimensionaltorus

T S

x

S

x x

S

weobtainthusafinite field

"wrapped"

^{on a}toms.

Suppose

now that r

>

1 is a natural number, and q I

(mod r).

^This happens, for example, if p^=_1

(mod r). We

mayconsider then a character(multiplicative) of

F,

^{that isa}surjective morphismof multiplicativegroups

f F

^-,

^Ur,

^where

^U,.

^isthegroup of complexr-throotsof unity Then,toevery pointzofour lattice weassociate an r-throotofunity,namely

f ^(z).

^Weobtain thus a discreteanalogue forasection ofa

U (1)-bundle

over the n-dimensionaltorus. Alsowemay see this asaphysicalfield,

f(z)

beingthe "spin" attached to site z.

Our

aim isto studythis field using some ideasofstatistical mechanics ([1J-basic

reference),

^{with the} hope of obtaining more information about such pure mathematical entitiesascharacters offinitefields.

3/ M CARAGIUAND M CARAGIU

2. NEAREST NEIGHBORS

Oncewehavechosenabasise

(el,

^e2,...,

e,)

^asabove,anatural concept ofnearestneighbor sites pops up namely, we saythatthesites z andw are e-nearest neighbors, andwe denotethisby z w

(mode),

ifthe differencez wbelongstotheset

{el,

en, et,

e,}

Ofcourse, this concept is stronglydependenton the choiceofthe basise Ifwechangethebasis, itmay happenthatweobtaina totallydifferentsetofnearest neighborpairs Obviously, if onechangesthe basisonly by permutingthe

e’s

between themselves, and/orby changingthesign of someof them, ^thesetofnearest neighborpairs remains the same. Ignoring the action ofthe symmetric group, this is just

Z/2Z

^x

Z/2Z-

invariance, but most ofthetransformations arenot so simple Weobtain thus amodel for a strange chaotic medium, in which the very idea ofnearest neighbor is strongly dependent on the action of

GL(n, Fp)

^thegroupwhichtakes care about all basechangesin

Fq

^over

F,

3.

THE

MEANFIELD AND ITS POTTS HAMILTONIAN

We

returnnow toourfield

f

^{defined on}

Fq,

viewed as ann-dimensionallatticewrappedon

T We need,

in oneway or

another,

toprove thatthis is arepresentative field from the randomness point of view, i.e., thatitisa"meanfield",inthephysicaluseoftheterm. Thisfollowsfrom a result

[2]

aboutthe distribution of thevalues takenby amultiplicativecharacter of

Fq.

Namely, letus consideraninteger k

<

^p.

To

everyzin

Fq,

weCanassociateafunction

hz {0,

1, k 1

}n Ur,

givenasfollows

hz(al, an):= f(z +

ale

+ + anen).

Thefunctions

hz

^{canbeseen as ameans}of testing thelocalstateof thefield inthe e-neighborhood ofzrepresented byahypercubeof size

k,

^withthepointzasoneofitscomers: namely,

hz

^isonlyone

out ofatotal ofr^k" functions h

{0,

1, k-

1}" Ur.

The main result in

[2]

states that all the functionsh havean"almostequal" probability ofoccurrence as an

hz.

Namely,ifwedenote by

Nh

^thenumber of thosezin

Fq

forwhich

h

^h

(k

isfixed),wehave the followingestimate, where we have denoted k byb:

Nh ^(q + 1)/r + O(b.ql/2)

when q islarge.

The

proof

usesthe Lang-Weilestimates appliedto anappropriate algebraic curve, whosegenusis computed usingthe Riemann-Hurwitzformula.

Moreover,

theconstantimplied by0issmallerthan1.

Now,

we can assume thatourfieldis indeed a mearffield,inthe most naturalsense Weare thus readytodefine itsPortsHamiltonian:

H E

,,,W(mode)

where

(,)

^{is if j and 0 otherwise}

^(we

assume that the

elementary

energy of interaction

[1

is

J 1). By

definition,it seemsthat

H

is

dependent

on boththe basise and thecharacterchoice Of course

GL(n, F)

^actsas well on the Hamiltonian

H,

byan action inducedbyitsactionon thebasise(on whichtheneighborsystemis strongly dependent).

However,

weshallseethat theactionof

GL(n, F)

on the Hamiltonianistrivial.

. ^T aZ(,,,’)-VAmANC

We

prove now the following theorem.

THEOREM.

H

doesnot

depend

on the choiceof thebasise,noron the character

f.

PROOF.

First, wemaynotethatonemayextendthedefinitionof the charactertothewholefield by

f (0)

^:=0

^As

^aphysical interpretation one mayconsider either thespin0atthe sitez 0,or, maybe better,onemayconsider infactafieldtheoryonapuncturedtoms.

INVARIANCE OFTHEPOTTSHAMILTONIAN 35

Theproofisactually very simple. First, let us consider anapparentlymoregeneralsituation ifais anarbitrary element of

F,

^{we countfirst}the number of thosezin

Fq

for which the character takes the same valuesatzandz

+

^a

If

f (z) f (z + a),

then obviouslyz, z

+

^{a are} non-zero, and

(z + a)/z

^{yis an r-th}powerin

F ^-{1}

^{Thus, y can take a number of}

^(q-

^r-

^1)/r

^{values, and for every such y, z is uniquely}

determined

Thuswehavefound

(q

^r

1)/r

possiblevaluesofz withthecharactertakingthe samevaluesat z and z/a. Take now a el,eg,...,e,

In

each of these n directions, the number ofe-nearest neighbor pairsto whichthe same spin is assigned is

(q-

r-

1)/r

It followsthat the total energyor PottsHamiltonian isexactly given by

H n(q

^r

1)/r.

Thiscomputation, in

fact,

proves the

GL(n, Fv)-invariance,

^{because, as one can}see, the aboverelation for

H

doesnot

depend

^{oneor on}

f.

REMARK

1. The same invariance property holds also (if one takes r

2)

for the Ising Hamiltonian

].

REMARK

2. One maywork also with more general extensions of finite fields and obtain in a similarwayaconcept ofe-nearestneighbor for everybasisof

Fq,

^over

Fq

butthe prime field

F

r,ismore suitablytobe taken as abase field, duetoitsinterpretationas a linearspatial array

REMARK

3.

For

n 1,thechange ofbasis is, in

fact,

^arescaling. Thus,we get, in thisparticular case, the"scale-invariance"of the

Potts

Hamiltonian. If one chooses another scalee,the elements ofthe

Fr:

^are reordered as e, 2e...,

(p- 1)e,

and the worst that can happen is the interchange of quadratic residues andnonresidues.

5.

FURTHER COMMENTS

Themostnatural stepforwardwould betodefine apartition functionZ.

Itis known that all the information about astatistical system is hiddeninthepartitionfunction Our hopeistodefine anappropriate

Z

inordertoget, via itsanalytic properties,newinsights concerning this strange connection betweenthearithmeticoffinite fieldsandfieldtheory.

Recallthat inordertodefine

Z,

weneed asetofstates. Then,weattachaBoltzmann weight to everystatein thisset.

Z

willbe the sumofallthese weights. Hopefully,inthe caseofafield associated with a character

f

^of

F

^{theBoltzmannweight hasa}simple form:

w(f) exp(-/( )/)

whereris the orderof

f,

ⁿthe dimensionofthe space,

and/

aparameterplayingthe role ofinverse temperature. Note thatq is the latticevolume

In

ordertoobtain aproper

Z,

one hasto selectwithgreatcare asetofstatessatisfyingtwobasic requirements. First,^itshouldnotbe a"small"set

(otherwise

^theformalism of statistical mechanicswillbe

useless).

Second, the selected states mustbe meaningful from an arithmetical point of view, that is, closely relatedtothemultiplicative characters.

However,

the total number of multiplicative characters of

Fq

isq 1, which israthersmall evenif we fixr, thenumber of r-valued spin distributions^on

F

^{is rq-1.}

^In

^{ordertodealwithalargernumber}

ofstates, onemay choosetotakeinto accountthemeanfieldproperty

(see

^section

3).

Namely, for every divisor r ofq-1, we will select a character

fr

^{of order r}

^(we

shall call it a "basic

state"). By

^the meanfieldpropertywewill agreetoconsiderevery such basicstate

fr

^{as anaveragefieldoutofatotal}

number of^rq-1 r-valued random spin distributions on our

punctured

toms. Thus, in theexpression of

Z

we havetomultiplytheBoltzmann weight

W (f)

with acorresponding factor of^rq-1. Ifwerestrict, for

example,

the multiplicity of the spinto asingle valuer,thenforeveryq 1

(mod r)

we get

36 MCARAGIUANDM CARAGIU

Z rq-lw(f) exp((q-

1)logr-

Bn(q-

1

r)/r).

One may note that the function

G(/3)

^{:= -lira}

(log Z)/3q

that is, the so-called Gibbs free energy density (in thermodynamical

limit)

presents in this caseasign change

at/3 (r/n)log

r

Onthe otherhand,ifthemultiplicityofrof thespinisunrestricted,that is ifrcanbeeverydivisorof q 1, thepartitionfunction willbe given byasum over the divisorsof q 1,namely

Z Z rq-lW(fr)"

r[q-1

More

generally, one may consider some special class of divisors ofq- 1. Then, the partition function

Z

willbethe associated divisor sum. Thiscouldlead ustosomedeeparithmeticalproblems

For

an explicit selection ofnew states, without makinguse ofthe meanfield approach, onemay consider the set ofall

"GL(n, F,)-gauge

^invariantfunctions: g:

Fq* ^Ur,

^{that is, functions forwhich}

the

Potts

Hamiltonian iswell defined

(does

^{not dependon}the choice of the basisof

Fq

over

F,)

^The

partition function can be written then as the sum over all such g of the Boltzmann weights

exp( -/3H (g)),

where

H (g)

is the Hamiltonian associatedtothefunctiong

[1]

[21

REFERENCES

BAXTER, J.,

Exactly Solved Models in StatisticalMechanics, London,

New

York, Academic

Press,

1982.

CARAGIU,

M. and

MULLEN, G., L.,

The distributionofr-thpowers^{in finite}fields, Penn State University Preprim,

PM

^161/1993.