Group theoretic approach to disordered systems

Volume122, number6,7 PHYSICSLETTERSA 22 June1987

GROUP THEORETIC APPROACH TO DISORDERED SYSTEMS

Y.S. PRAHALADDepartmentofMathematics,GoaUniversity,Bambolim,SantaCruzP.O.Goa403005, india

Received14 October1986; acceptedfor publication27 March1987Communicatedby A.R. Bishop

Theproblemof a quantumparticlein a gaussianhomogeneous,randompotential is shownto admit a stochasticsymmetrygroupanditsunitary representationsarefound.It is pointedoutthat thetheoryof grouprepresentationplaysarole in thetheoryofamorphoussolidsanalogousto its role in thetheoryof crystals.

1. Introduction Let (~l,~,P)bea probabilityspaceandJthegroupof its measurepreserving Borel automorphisms;

The universalfeaturesobservedin the electronic G= 11” (respectively1”) be given its usual groupspectraof regularcrystallinesolids owe their origin structure.T: G—J bea homomorphismof G into J.tothepresenceof a largesymmetrygroup[1,21. The U actson ~, asFloquet—Bloch [3] representation succinctly — — U 2expressesthis. Althoughamorphoussystemsexhibit ~ — g euniversalfeatures[4], no singleunifying principle TheG actionis assumedto beergodic.Let V: ~—+Rhasbeenisolatedto accountfor this. By their very bea randomvariableinL2(P). SetV(x,w)= V(T~w).definition, symmetryconsiderationsseemtoberuled Definition 1. A randomfield V: Gx 11—401 is saidtooutat first sight.This noteis to indicatethat this is be a stochastic flow, if it is of the formfar from the caseandalmostall the modelsconsid- V(x,w)= V

0(T~co),V0eL2 (P). Clearly a stochastic

eredin the literatureadmit a stochasticsymmetry. flow field is strictly homogeneous.ConverselyanyThis work has beeninspired by some remarksof strictlyhomogeneousrandomfield canberealizedasMackey [5,6] anda paperby Ogura[71.Evena cur- a stochasticflow field.sory look at the paperby Kirsch andMartinelli [81 We now assumethat V is a stochasticflow field.showsthatsymmetryconsiderationis at the root of V

0eL2(P) implies that VeL~~(G) with probability

all their results. one. Kirsch and Martinelli [8] have treatedtheoperatortheoreticaspectof thesesystems.Here theapproachis slightly different. We are looking for

2. Models symmetries.Themodel hamiltonianwill be treatedasan operatoron L2 ( Gx f~)ratherthana random

Themodel consideredhereis that of a particlein operatoronL2(G), i.e. weareadaptingadifferentiala randompotential,assumedto be a homogeneous, equationapproachina way. TheHilbert spaceof themetrically transitive,gaussianrandomfield on R” systemis .~° = L2 (G x il).(respectivelyZr). The hamiltonianisH(x,w)=—V2+V(x,co), (1)

3. InvariancewhereV2 is the laplacianin RV (respectivelydiscreteversionin

1P) and V(x,w),xell” or l~,w eL~is the Definition 2. Let ~ve~° definethestochastictrans-randomfield. lationon ~a by

0375-9601/87/$03.50© ElsevierSciencePublishersB.V. 335(North-HollandPhysicsPublishingDivision)

Volume122, number6,7 PHYSICSLETTERSA 22June1987

(~aW)(x,w)= W(x+a,Ta ~)) where2 is the Haarmeasureon ~ the dualof U. Inthe conventionalsolid statephysicist’slanguage,thewavefunctionis a planewave modulatedby a ~

x,acG and coe~l. (3) invariant randomfield.

Proposition 1. ~i~’~’is a unitary representationof Uon ~°, stronglycontinuousif U = R~.

Proposition 2. ~2”~is a symmetry group of the 4. Decompositionof ~“

hamiltonian.Proof With the assumptionthat V is a stochastic Contraryto its appearance(9) is not a complete

flow field, clearly, decompositionof .i~”into a directintegralor sumof

~ H] —0 theirreducibleone-dimensionalrepresentationof ~.

— . ‘ ‘ Thereasonis that thestochasticflow admitsfurtherAlthough a simpletranslationis not a symmetry,a reductioninto invariant subspaces.This reductiontranslationaccompaniedby a simultaneouschange of eachof thefibreswill now be doneusingthe the-of realizationis a symmetry. ory ofmultipleWienerintegralsdevelopedby Ito (see

The reduction of the hamiltonian requires the refs. [10,11]).decompositionof .*‘ into a direct sumor directinte- Definition 3. Let ~ be the set of complexfunc-gral with respectto 9’~,which will now be done. tions which are measurablewith respectto all the

The physical equivalence of we .*‘ and92’~vimplies random variables Vi~.,xcGandsquareintegrablewiththat there exists a function g: U xc�—~11isuchthat respectto P. Theseare calledthe squareintegrable

functionalsof V~.g(x+a,w)=g(x,w)+g(a,T~‘w) (5) Definition 4. A random field ~i is said to be sub-for almost all x,ae Gand w� ordinatedto V~if it is homogeneous and is of the

Eq. (5) is nothing but the cocycleidentity. This form qi(x,w) = ~ T~co)for some randomvariablecanbeassumedtoholdeverywhereunderfairly mild ~‘~e.rç. Clearly w~is subordinatedto V~.conditions. Further Theorem2. Let ~ be a randomflow field subor-

dinatedto V~,thenthereexist functions~ L2 ( cY’)I r2a \( \ — ig(a,o) I \ I1’~~. w,~x,w, —e Wkx,w, ~U) such that

One would normally expect that it would now be i rnecessary to determine the cocycle.But in our case ~i(x,w) ~ J ~ ...,x~)this is elementary. Wenote that in the samerepre-sentation ~ is alreadyin the diagonalform. Beingthe unitary representation of G on .*‘, g hasto be XXI (x)x

2(x)~linear in a [91.Using the fact that the only Borel

xh [d~(X1,cu)...d~z(X~,co)], (10)functions invariant under Ta, acU are constantse~”°~— (7 where u is the random spectral measure associated

X~. ‘‘ ‘ ‘ with V~ and the random measures

wherex is a characterof G. h~[dM(x1,w)...d~(x~,w)]are the Ito—Wiener—Soon eachfibre 92”~actsas follows, Hermitemeasureson O”. Furtherthis correspond-

(~a ‘,. ~ ~. T enceis unitary.~ —X~a)W~(xcv) (8) Putting everything together we have the

whereçv~is a stochasticflow field invariant under decomposition:~2)a We now have the following Theorem3. Let we.Jf thenthereexistsa sequence

Theorem1 (StochasticBloch‘s theorem).Let ~ve~ of elements~i,,eL2( c~l” + I) which are symmetric in

then the variablewith subscripts,suchthat

w(x~w)=Jx(x)w~(Txw)d2(x)~ (9)

336

Volume 122, number6,7 PHYSICSLETTERSA 22 June1987

i ,‘ [2] Ziman,Principlesofthetheoryofsolids(CambridgeUniv.W(x,ci))= —~ I ~(x; x~~ Press,Cambridge,1972).

.1 [3]F.Bloch,Z.Phys.52(1928)555;

F. OdehandJ.B.Keller, J.Math.Phys. 5 (1964) 1499.XX(X)X,(X)...X~(X) [4] E.N. Economuo,M.H. Cohen,K.F. Freedand E.S. Kirk-

patrick, in: Amorphousandliquid semiconductors,ed.J.Xh”[d/.t(Xi,W)...dJL(X~,W)]d2(X) (Il) Tauc(Plenum,NewYork, 1974).

The correspondenceW—~{ço~} ~.. ~ which ascribes an [5] G.W. Mackey,Unitary grouprepresentationsin physics,

elementin the Fockspaceis unitary andeachcorn- probability and number theory (Benjamin/Cummings,Reading,1978).ponent in the direct sum is invariant under 2~. [6] G.W. Mackey,Adv. Math. 12 (1974) 178.

It is a simple matter to write outthis theoremfor [7] H. Ogura,Phys.Rev.A 11(1975)942.

R~andZ~separately. [81W. Kirsch andF. Martinelli, J. ReineAngew. Math. 334With theseresultsin handit shouldbe possibleto (1982) 141.

treatthespectraofdisorderedsolidsin thesameway [9] I. SegalandR.A. Kunze,Integralsandoperators(McGraw-Hill, NewYork, 1968).asthose of crystals. Following a deviceof Mackey, [10] P. Major, Multiple Wienerintegrals,Springerlecturenotes

the relaxation of the condition of homogeneity is also in mathematics,Vol. 849 (Springer,Berlin).beingattempted(in P andZ). [11] T. Hida, Brownianmotion (Springer,Berlin, 1980).

References

[1] C. Kittel, Quantumtheory of solids (Wiley, New York,1963).

337