multiplicative and additive cluster expansions for the evolution of quantum spin systems in the...

Volume 86A, number8 PHYSICSLETTERS 7 December1981

MULTIPLICATIVE AND ADDITIVE CLUSTER EXPANSIONS

FOR THE EVOLUTION OF QUANTUM SPIN SYSTEMS IN THE GROUND STATE

V.A. MALYSHEV andR.A. MINLOSMoscowState University,Moscow, USSR

Received15 July 1981

It is shownthat for thev-dimensionalquantumIsing model in thehigh temperatureregione—t~in the GNS representa-tion admitsa “multiplicative” N-particleclusterexpansionandH admitsan“additive” N-particleclusterexpansion.

Let us considerthe (v + 1)-dimensionalIsing model where< = < ~ ~ ~2 E~C,U~.is thetranslationfromwith continuoustime. It is a randomfield on Z~’X R= theslice Z” X {0} ontothe slice Z~X {t}.

{(x, t): x E Zv, t E R} with values±1in eachpoint. To Onecanshowthat ~1t is the stronglycontinuous

defineit we considerfirst for anyx E Z’~the stationary semigroupof positiveself-adjointoperatorsandMarkovprocess~ with two values±1definedby II 9’~II= 1. Then ET~= etH whereH ispositiveself-thestochasticsemigroupexp(—tHo~),Ho~= (~ ~). adjoint. -

Fordifferentpointsx theprocesses~~(t) are mutually The unitarygroupe~tHcanthen beidentified [1]independent.We denoteby (~2,~, P~)theprobability with the evolutionof a v-dimensionalquantumspinmeasurespacewhereall ~~(t) are defined.~2canbe systemwith the formalhamiltonianidentifiedwith the setof all functionsf(x, t) with val-ues±1which are stepwiseconstantfor anyfixedx. formal — ~—X’=i ~7x ~‘x’ -

Let usconsidera newprobabilitymeasurei’tA,T On(~7,~)with density Then 1C is thespaceof the GNS representationof the

quasilocalC~’algebrain thegroundstate.d 7’ - - tH

= —1 ~ We shall obtain a clusterexpansionfor e andd~

0 ZA,T exp~3~~ f ~x(~) ~ dt1 Hon the specialbasis(first appearedin ref. [21) which— we shallnow define.

whereA is the cubein Z0,x,x’ E A, T> 0. Let T,,x E Zv,be the setof ally E Z~suchthatWe shall consideronly thecasewhen13 is sufficient- y <x in lexicographicorder.~ is the orthogonalpro-

ly small, 1131 ~ 130(X). It is a standardresultthen that jection inL2(&2, ~, j.z) ontoL2(~7,~T~’ ji). Let usthereexistsa weaklimit for At Z~,T-÷00: i = putlim,LA,T. =~ (0~=~(0~—P~ ‘0~ f ~ (p ~2’~—1/2

Let for anyA C Z” X R ~A bethe minimaloalge- SX “Xx 1 ~ / x’~x~1’ X xk X XI

bra with regardto which all ~ aremeasurablefor = U , ~ z~, ‘~<°~, f~= 1,(x, t) EA. ThephysicalHilbert spaceis definedas xEI X

= L2(f2, ~o, /1), ~o= ~z~x {o} f~(t)= ~ f1(t)= x~I f~(t).

Thestochasticsemigroup9’s: ~JC-+ 1C (transfermatrix)

is definedby its matrix elements Theorem1. ~f1}is a completeorthonormalbasis in

~IC.Moreover

(si’ ~t~2)1(~’~2)>,

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Volume86A, number8 PHYSICSLETTERS 7 December1981

(f1) = 0, 1 ~ 0, M(fx/~2T~)= 0, M(f~/~r~)= 1. Moreoverw’(I, I’; 0) dw(I, I’; 0)/d.t satisfiesthe(2) estimates(3) withd

2 = 0.Theorem2. The semi-invaniants Proof. Let uscalculatethe derivativefor t = 0 on

both sidesof (4). Let usnotethen thatw(1,J’;t)=(f~

1(0),...,f~(0),f~1(t),~w(I,I’;O)=l, if!1’{x},

1{xi,...,Xm}, I’={x’1,...,x~}, (6)= 0, in othercases.

areC in t, t ~ 0, and for I,!’ * 0In fact we have

Idkw(I,I’;t)~ ~(C13 (Ce~)d2, k=0, 1,2,.... (f

1f1>=0, I~I’, (7)

dt~’ (3)w(I, I’; t) = 0 if for i’ is empty. if we use(2) andM?7 = M(M(rl/XT0 ~ Also we have

(8)

HereI’(t) is thetranslationof!’ to theslice Z~X {t},d

1 (2) is theminimal sumof the lengthsof thebonds Usingthe Möbiusinversionformulawe get from (7)of any connectedtreewith verticesin I U !‘(t) in the that w(I,I’; 0) = 0 if! ‘�‘ I’. Thefirst part of (6) fol-metrics1y1 I + 1y21 + ‘-- Iy”~(in Z~X R, the Z’~direc- lows from (2). Finallytion (R direction)Cdependsonly uponv andk. w(I,!; 0) = ~ (—l)’~’~(k— I)! = 0,

The proofof this theoremis similar to theproof kof the sametheoremfor discretetime modelsin ref.[3] andwill appearin ref. [4]. wherethe sum is over all partitionsof {l, ..., n}, n =

J(~2.Multiplicative cluster expansionfor e This gives thecompleteN-particleclusterexpan-

sionsfor all N and1131 ~Theorem3. The matrix elementsof etH admit the

following representationfor!,!’ * 0: The presentresultcanbe extendedto other latticemodels inhigh- andlow-temperatureregions(seee.g.

(f1, e_Wfj) = �~w(11,!j; t) ... w(In, I,; t) (4) ref. [3]). This andapplicationsto thestudyof the

and are equalto 0 if! = 0 or!’ 0. The sum is over all spectralpropertiesof H will appearin forthcomingan-partitions tides.

We remarkthat all w canbe representedas an ex-!i U...UIn =!, !~U..U!h =1’. plicit serieswith exponentialconvergence.

Proof This follows from (1) andthe formulawhichexpressesmomentsthroughthe semi-invariants. References

Additiveclusterexpansionfor H [1] V.A. Malyshev,FunktionalnyiAnalyz i JegoPril, 13 (1979)31.

[21 R.A. Minlos andJaG.Sinai,Teor. Mat. Phys.2 (1970)230.Theorem4. Matrix elementof H are equalto (3] V.A. Malyshev,Usp.Mat. Nauk.35 (1980)3.

(f1,Hf1) = — �~w’(11 ,I~; 0), (5) [4] V.A. MalyshevandR.A. Minlos,Trudy PetrovskiiSeminar(Moscow),to bepublished.

wherethe sumis overall nonemptyI~C I, !~C 1’ suchthatl—11 =!‘—!~.

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