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On a factor associated with the unordered phase of λ-model on a cayley tree
Vol. 53 (2004)
REPORTS ON MATHEMATICAL PHYSICS
No. 1
ON A FACTOR ASSOCIATED W I T H THE UNORDERED PHASE OF X-MODEL ON A CAYLEY TREE FARRUH MUKHAMEDOV Department of Mechanics and Mathematics, National University of Uzbekistan Vuzgorodok, 700095 Tashkent, Uzbekistan (e-mail: [email protected]) (Received October 30, 2001 - - Revised October 4, 2003)
In this paper we consider nearest-neighbours models, where the spin takes values in the set e~ = {01, 02 . . . . . 0q} and is assigned to the vertices of the Cayley tree 1'k. The Hamiltonian is defined by some given X-function. We find a condition for the function ~. to determine a type of yon Neumann algebra generated by the GNS construction associated with the unordered phase of the ),-model. We give also some physical applications of the obtained result. 2000 Mathematical Subject Classification: 47A67, 47L90, 47N55, 82B20. Keywords: von Neumann algebra, )~-model, Cayley tree, unordered phase, Gibbs measure, GNS construction.
1.
Introduction
It is known that in quantum statistical mechanics specific systems are identified with states on corresponding algebras. In many cases the algebra can be chosen to be a quasi-local algebra of observables. States on these algebras, satisfying KMS condition, describe equiLibrium states of a quantum system. On the other hand, for classical systems with finite radius of interactions, limiting Gibbs measures are known to be Markov random fields. In connection with this, there arises a problem o f constructing analogues of noncommutative Markov chains. Accardi explored this problem in [1], where he introduced and studied noncommutative Markov states on the algebra of quasi-local observables, which were in agreement with the classical Markov chains. Modular properties of the noncommutative Markov states were studied in [2, 3]. The type analysis of quasi-free factors (i.e. factors generated by quasi-free representations) has been an interesting problem since the appearance of the pioneering work by Araki and Wyss [4]. A family of representations of uniformly hyperfinite algebras, which can be treated as a free quantum lattice system, was constructed in [5]. In this case, factors corresponding to these representations had type IIIx, ~. e (0, 1). More general constructions of product states were considered in [6]. In the case of CAR-algebra, the rough classification into types Ioo, II1, Iloo and III was obtained in the 60's by several authors (see e.g. [7, 8]). The classification IH
2
E MUKHAMEDOV
of type III quasi-free factors in terms of spectral properties of positive operators parameterizing quasi-free states was described in [9]. It is known (see e.g. [9, 10]) that every quasi-free state on CAR-algebra can be regarded as the product state on A = ®n_>IM2(C). Observe that the product states can be viewed as the Gibbs states of a Hamiltonian system in which interactions between particles of the system are absent, i.e. the system is a free lattice quantum spin system. So, it is interesting to consider quantum lattice systems with nontrivial interactions, which leads us, as mentioned above, to the consideration of the Markov states. Simple examples of such systems are the Ising and Potts models, which have been studied in many papers (see, for example, [11]). We note that all Gibbs states corresponding to these models are Markov random fields. The full type analysis of von Neumann algebras associated with the Markov states is still an open problem. In [12], for the Ising model on a Cayley tree the types of factors corresponding to the translation-invariant Gibbs states were found. In [13] it was proved for a class of nonhomogeneous Potts model, that a von Neumann algebra associated with an unordered phase of this model is a factor of type III1. Some specific cases of the Markov states were considered in [14]. The present paper is devoted to the type analysis of one class of Markov states, which correspond to certain Hamiltonian systems with nearest-neighbour interactions. More precisely, X-model on the Cayley tree will be considered, in which spin variables take their values in a set q~ = {r/1. . . . . r/q}, where r/k ~ ~q-1, k = 1, q. Observe that the considered model generalizes the notion of Z-model introduced in [15], where the spin variables take their values 4-1. The paper is organized as follows: in Section 2 we give some preliminary definitions of the ~.-model on the Cayley tree and corresponding Gibbs measures. We also recall a few definitions from the theory of von Neumann algebras. In Section 3 we reduce the problem of describing Gibbs measures to the problem of solving a nonlinear functional equation. In Section 4 we determine the types of von Neumann algebras generated by GNS representation associated with a diagonal state corresponding to the unordered phase of the ~-model. We find a condition for a function )~ which guarantees that the corresponding von Neumann algebra is a factor of type 1118. Section 5 is devoted to some applications of the obtained result to physical models and analysis of types of factors corresponding to Markov random fields associated with stochastic matrix.
2.
Definitions and preliminary results
The Cayley tree (or Bethe lattice) F ~ (see [16]) of order k > 1 is an infinite tree, i.e. a graph without cycles, from each vertex of which exactly k + 1 edges go out. Let F k = (V, A, i), where V is the set of vertices of F k, A is the set of edges of F k and i is the incidence function associating each edge l ~ A with its endpoints x, y E V. If i(l) = {x, y}, then x and y are called neighbouring vertices, and we write l = (x, y). The distance d(x, y), x, y ~ V on the Cayley tree defined by the
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
3
formula
d(x, y) = min{d I 3x --- x0, xl . . . . , Xd-1, Xd = y e V such that the pairs (x0, Xl) . . . . . (Xd-1, Xd) are neighbouring vertices}. We set
Wn = {x e V l d ( x , x °) = n}, vn = u ~ = l wm = [x e v I d(x, x °) < n}, Ln = { l = ( x , y ) e L I x, y e V,}, for an arbitrary point x ° e V. Denote Ixl = d(x, x0), x e V. A collection of pairs (x, Xl) . . . . . (Xd-1, y) is called path from the point x to the point y. We write x < y if the path from x ° to y goes through x. Vertex y we call a direct successor of x if y > x and x, y are the nearest neighbours. Denote by S(x) the set of direct successors, i.e.
S(x) = {y e Wn+l I d(x, y) = 1},
x e Wn.
Observe that any vertex x ¢ x ° has k direct successors and x ° has k + 1. PROPOSITION 2.1 [17]. There exists a one-to-one correspondence between the set V of vertices of the Cayley tree of order k > 1 and the group Gk+l of the free products of k + 1 cyclic groups of the second order with generators al, a2 . . . . . ak+l. Let us define a group structure on the group Gk+l a s follows. Vertices which correspond to the "words" g, h e Gk+l are called the nearest neighbours and are connected by an edge if either g = ha i or h = gaj for some i or j . The graph thus defined is a Cayley tree o f order k. Consider a left (resp. right) transformation shift on Gk+l defined as: for go e Gk+l we set Tgoh = goh (resp. Tgoh = hgo, ) Vh e Gk+l. It is easy to see that the set of all left (resp. right) shifts on Gk+l is isomorphic to the group Gk+a. Let • = {r/a, 72 . . . . . /'/q}, where ~1, r/2 . . . . . rlq are vectors in ~q-1, be such that 1,
OiOj =
if i = j , 1
q-l'
ifis~j.
(2.1)
We consider models where the spin takes values in the set ~ = {r/l, r/2. . . . . 0q} and is assigned to the vertices of the tree. A configuration tr on V is then defined as a function x e V --+ or(x) e ~ ; the set of all configurations coincides with f2 = ~ rk. The Hamiltonian is of a X-model form, Hx(tr) = ~ X(tr(x), or(y); J ) , (x,y)
(2.2)
F. MUKHAMEDOV
4
where J 6 11~n is a coupling constant and the sum is taken over all pairs of neighbouring vertices (x, y), a 6 f2. Here and below 3. : ~ x • x ]I{n 7> ]~ is some given function. From a physical point of view the interactions between particles do not depend on their locations, therefore from now on we will assume that )~ is a symmetric function, i.e. L(u, v; J ) = ~.(v, u; J) for all u, V e I~q-1. We note that ~.-model of this type can be considered as a generalization of the Ising model. The Ising model corresponds to the case q = 2 and ~.(x, y; J ) =
- J x y , x, y, J ~ R. We consider a standard a-algebra )v of subsets of f2 generated by cylinder subsets, all probability measures are considered on (f2, Y). A probability measure /z is called a Gibbs measure (with Hamiltonian Hz) if it satisfies the DLR equation: Yn = 1,2 . . . . and a n ~ V " :
f
(an), where Vwlw,,+l v, is the conditional probability
vV~ wlwn+! (On) = Z-1 (°')l Wn+!) e x p ( - f l H ( o ' n IIo91w,+, )), where fl > 0. Here air, and Wlw,+~ denote the restrictions of a , w ~ [2 to Vn and Wn+l, respectively. Next, an : x ~ Vn -+ an(x) is a configuration in Vn and H(anlloglw~+,) is defined as the sum H(an) + U(an, ~olw,+~) where
H(an) = U(an, wlw.+,) =
Finally, Z(O)lWn+l) stands
Z ).(an(X), an(y); J), (x,y)ELn
(2.3)
Z ~.(an(X), w(y); J). (x,y):x~Vn,yeWn+l
for the partition function in V~ with the boundary condi-
tion colw,+~
Z(WlWn+~) = E
exp(-flH(anllwlw,+~).
~,, ee v. Since we consider the nearest-neighbour interaction, the Gibbs measures of the ),-model possess a Markov property: given a configuration wn on Wn, random configurations in Vn-1 and in V \ Vn+] are conditionally independent. It is known (see [18]) that for any sequence ~o(n) ~ f2, any limiting point of measures fly, is a
o)fn)[Wn+l
Gibbs measure. Here ~v.
o){n)lWn+l
is a measure on f2 such that ¥ n ' > n"
~,v. { v v" ,o("qw.+,({a ~ ~2 I air., = an,]) = ,o(")lw.+~(a,,iv~) 0
if a,,Iv.,\v. = eo(')lv~,\v~, otherwise.
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
We now recall some facts from the theory of von Neumann algebra. Let B ( ~ ) be an algebra of all bounded linear operators on a Hilbert space ~ (over the field of complex, numbers C). A weak (operator) closed ,-subalgebra A/" in B(7-() is called von Neumann algebra if it contains the identity operator I. By Proj(A/) we denote the set of all projections in A/'. Von Neumann algebra is a factor if its center Z ( A ; ) ( = {x • A; I x y = yx, Y y • A/'}) is trivial, i.e. Z(AF) = {LI I ~- • C}. Von Neumann algebras are split into the classes I (In, n < o0, Ioo), II (111, 1Ioo) and HI. An element x • A/" is called positive if there is an element y • A/" such that x = y*y. A linear functional co on .A/" is called a state if co(x'x) > 0 for all x e A/" and co(I) = 1. A state co is said to be normal if co(supxe) = supco(x~) for any bounded increasing net {xa} of positive elements of 3f. A state co is called trace (resp. faithful) if the condition co(xy) = co(yx) holds for all x, y 6 A/" (resp. if the equality c o ( x ' x ) = 0 implies x = 0). Let .M be a factor, co be a faithful normal state on AF and tr~ be the modular group associated with co (see Definition 2.5.15 in [19]). We let I ' ( a °)) denote the Connes spectrum of the modular group cr~ (see Definition 2.2.1 in [20]). It is said [20] that the factor A/" is of type (a) Ili1, if F(tr °~) = IR, (b) IIIx, if F(cr ~°) = {nlogL, n • Z}, ~. ~ (0, 1), (c) III0, if F(cr °)) = {0}. (See e.g. [19, 21] for details on von Neumann algebras and the modular theory of operator algebras.)
3.
Construction of Gibbs states for the L-model In this section we give a construction of a special class of limiting Gibbs measures for the ;.-model on the Cayley tree. Let h : x --> hx = (hi,x, h2,x . . . . . hq-l,x) • l~q-1 be a real vector-valued function of x • V. Given n = 1, 2 . . . . . consider the probability measure/~(n) on ~v, defined
by /~(n)(~n) = Z~-I exp{-/~H(~n) + E
hxcy(x)}.
(3.1)
xEWn
Here, as before, crn : x • Vn -'+ crn(x) and Z~ is the corresponding partition function,
z. = ~ ~v~
exp{-/~n(6n) + ~ hx6(x)}. xcWn
The consistency conditions for /z(n)(Crn), n > 1, are Z / z ( n ) ( o ' n - - l ' O'(n)) = Jtz(n--1)(O'n--1)' trIn)
where a (n) = {or(x), x • Wn}.
(3.2)
F. MUKHAMEDOV
6
= V and /zl,/zz . . . . be a sequence of probability Let V1 c V2 c ... t0n=lVn ~ measures on ~Vn, ~v2 . . . . satisfying the consistency condition (3.2). Then, according to the Kolmogorov theorem (see, for example [22]) there exists a unique limit Gibbs measure /zh on f2 such that for every n = 1, 2 . . . . and an 6 dPv" holds the equality (3.3)
]J~({alVn = fin}) = ]-£(n)(cTn)•
Further we set the basis in ~q-1 to be r/l, r/2. . . . . r/q-1. The following statement describes conditions on hx guaranteeing the consistency condition of the measures /x(n)(crn). THEOREM 3.1. The measures tz(n~(an), n -- 1, 2 . . . . . satisfy the consistency condition (3.2) if and only if for any x ~ V the following equation I htx --~ Z F ( h y , )~) yESfx)
(3.4)
holds. Here and below h'x stands for the vector qqlhx and F : II~q-1 ~ I~q-1 is a function F(h; ~.) = (Fl(h; 3.) . . . . . Fq-l(h; )~)), with Fi(hl, hz . . . . . hq-1; ~.) Y]~q-~ exp{-fl)~(r/i, r/j; J)} exphj + exp{-fl~.(r/i, r/q; J)} = log ~q_~ exp{-3Z(r/q, r/j; J)} exphj + exp{-/~.(aq, r/q; J)}'
(3.4a)
i = 1,2, . . . , q - 1, h = (hi . . . . . hq-1).
Proof'. Necessity. According to the consistency condition (3.2) we have Zn-1 Z e x p { - / 3 Z
a~n)
Z Z )~(a(x)a(y); J ) + Z hxa(X)} xEWn-j yES(x) x~Wn = exp{ Z hx~r(x)}, xEWn-1
(3.5)
where we have used (3.1). From Eq. (3.5) we find Y~tn) exp{-fl)~(r/i, or(y); J) + hy~r(y)} 1--I ~-~o'~)exp[--flZ(r/q, if(y); J) q- hycr(y)} yES(x) = exp{hx(0i - r/q)},
i = 1, q - 1.
(3.6)
Now, set r/j = o r ( y ) = r/1 . . . . . r/q--l, then it follows from (3.6) that Y]~jq__~exp{-fl~.(r/i, r/j; J)} eXp{q~_lhj,y} q- exp{-fl~.(r/i, r/q; J)} q-1 yES,x) Y~j=I exp{--fl)~(r/q, r/j; J)}eXp[q-~_lhj,y} + exp{-C3~.(r/q, r/q; J)} 7
/
(3.7)
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
7
where i = 1 , q - 1. Here we have used Eq. (2.1). Denoting h'i, x = u-I q hi ,x, f r o m the last equation we get the required equation (3.4). Sufficiency. From (3.7) we get (3.6), (3.5) and hence we obtain (3.2). The proof is complete. [] Denote
D= {h=(hxE~q-a'xE
V) l h x =
E f(hy,~.), V x E V ] . yES(x)
According to Theorem 3.1 for any h = (hx, x ~ V ) ~ 79 there exists a unique Gibbs measure /Xh which satisfies Eq. (3.3). If the vector-valued function h ° = (hx = (0 . . . . . 0), x 6 V) is a solution, i.e. h ° 6 79 then the corresponding Gibbs measure /z~0z) is called the unordered p h a s e of the ~.-model. Since we are dealing with this unordered phase, we have to make an assumption which guarantees us the existence of the unordered phase. ASSUMPTION A. For the considered m o d e l the vector-valued f u n c t i o n h ° = (hx = (0, 0 . . . . .
0), x ~ V ) belongs to 79.
This means that Eq. (3.4) has the solution hx = ho = O, x ~ V. According to Proposition 2.1 any transformation S of the group Gk+l induces an automorphism S on V. By ~k+l we denote the left group of shifts of G k + l . Any T 6 ~k+l induces a shift automorphism 7" • £2 --+ £2 by (Tcr)(h) = tr(Th),
h ~ Gk+l, cr E £2.
It is easy to see that /z~0z)o T =/z(0~) for every I" ~ Gk+l. As mentioned above, (;~) the measure /z(0z) has a Markov property (see [23]). This measure /z0 enables a mixing property (see [23]), i.e. for any A, B 6 ~- one has lim lZ~oZ)(~'g(a) M B) =/z(0Z)(a)/z~0~)(B).
Igl~oo
4.
(3.8)
Diagonal states and corresponding von Neumann algebras
Consider a C*-algebra A = ®rkMq(C), where Mq (C) is the algebra of q x q matrices over the field of complex numbers C. By eij, i, j ~ { 1, 2 . . . . . q }, we denote the basis matrices of the algebra Mq(C). Let CMq(C) denote the commutative subalgebra of Mq(C) generated by the elements eli i = {1, 2 . . . . . q}. We set CA = @ r k C M q ( C ) . Elements of the commutative algebra CA are functions on the space £2 = {ell . . . . . eqq} rk . Given a measure /z on the measurable space (£2, B), where B is the ~-algebra generated by cylindrical subsets of £2, we construct a state cou on A as follows. We set c % ( x ) = 0 if the tensor monomial x of the basis matrices eij, i, j E {1, 2 . . . . . q } , contains at least one partial isometry. If x ~ C A , we set w ~ ( x ) = f x d l z . The state thus obtained was introduced in [24] and was said to be [2
8
E MUKHAMEDOV
diagonal. In other words, if P : A ---> C A is a conditional expectation, then the state 09~ can be defined by 09~(x) = lz(P(x)), x ~ A, where Iz(P(x)) means an integral of a function P ( x ) with respect to the measure /z, i.e. Iz(P(x)) = f~ P ( x ) ( s ) d # ( s ) (see [25]). (x) (;~) By 09o we denote the diagonal state generated by the unordered phase 1% . On a finite dimensional C*-subalgebra Avn = ®vnMq(C) C A we rewrite the state (x) 09o as follows 09~o~) (x) - tr(eh(V~)x) tr(eB(V~)) ,
x ~ Av~,
(4.1)
where tr is the trace on Av~. The term - # ) ~ ( a ( x ) a ( y ) ; J) in (3.1) (see also (2.3)) we represent as a diagonal element of Mq(C)® Mq(C) in the standard basis as follows: B <1) 0
0
0
B (2) 0
0
-fl~.(a(x), a(y); J) =
,
0
0
(4.2)
B (q)
where B (k) : ( b ij k) qi j=l' k = 1, q, are q × q diagonal matrices, and bij,k=
{ --fl~.(r/k, r/i; J), 0
i = j, i = 1, q, i s~ j.
(4.3)
Consequently, using the fact that the state 09o(z) is diagonal, with (3.1), (4.1)-(4.3) (cf. [14], Chapter 1, Section 1) we find that the form of the Hamiltonian ['I(Vn) in the standard basis of A vn (i.e. with respect to the basis matrices) is regarded as B (~) 0
0 B (2) 0
0 0
.
~-I(Vn) =
~ ¢~x,y, (x,y}ELn
di)x,y'~"
0
0
B (q)
Denote .M = zrO~oX)(A)", where :ro,~0z~ is the GNS representation associated with the state 09o(x) (see Definition 2.3.18 in [15]). Our goal in this section is to determine the type of .A4.
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
9
REMARK 4.1. General properties of a representation associated with diagonal state were studied in [25], but concrete constructions of states were not considered there. A deep classification of types of factors generated by quasi-free states was obtained in [10]. For translation-invariant Markov states the corresponding type analysis was obtained in [26]. Now we define translations of the C*-algebra A. Every T ~ ~k+l induces a translation automorphism rr : A --+ A defined by ®
®
rT-( H a x ) = H aT~x). x~Vn
x~V~
Since the measure/z~o~) satisfies a mixing property (see F.q. (3.8)), we can easily obtain that co~0 ~) also satisfies the mixing property under the translations {rr}r~k+~, i.e. for all a, b ~ A holds the equality lim Oa~oX)(rg(a)b) = coo~x)"" I,a)coo~) (b).
Ig[~oo
According to Theorem 2.6.10 in [19] the algebra .M is a factor. We note that the modular group of A4 associated with to(0x) is defined by
"#trT°~(x) = lim exp{itH(A)}xexp{-itH(A)}, A--+Fk
where /at(A) =
~
dikx,y.
x 6 .M,
(4.4)
In order for the last limit to exist, we need to show
{x,y}EA
that a norm of potential if/ is finite (see Theorem 6.2.4 in [27]). First of all we recall the definition of the norm of the potential qs = Y~xcrk qs(X) IlqSlld = E e a " (
sup
n>__O
where d > 0. Here qs(X) e Ax Now we compute IIHIId:
Ill/lid =
~
Ilqs(X)ll),
x~F~ x~X,IXl=n+l
= @xM2(C).
e2d(sup
E
lieu,vii)
x~I'k x~X,X={u,v}
=ke
TM
sup {u,v}eL
lib.,vii
= ke 2d max Ibij g[ < oo. i,j,k
'
Hence the norm of ~/ is finite and limit in (4.4) exists.
10
F. MUKHAMEDOV By M ~ one denotes the centralizer of w~0x) which is defined as w(z)
./t4a={xeMlat
° (x)----x, for all
te/~}.
Since W(ox) is a Gibbs state, then according to Proposition 5.3.28 [27] the centralizer .M ° coincides with the set
M,o~o~ = {x e M I CO~oX)(xy)= W(o~)(yx), for all
y e M}.
(4.5)
By I-l[n] we denote the group of all permutations y of the set Vn such that r(x) = x
Vx e W,.
Every y ~ H[n] defines an automorphism off • .M --+ M by
® ~×(Hax)= x~V.
® Uav(x) xeVn
(4.6)
Oly I@xCVnMq(C)= ~, where id is the identity mapping. Denote 80 = U{ot× [ma ~ Fl[n]},
n > 1.
Simply repeating the proof of a proposition in [28] we can prove the following. PROPOSITION 4.1. The group Go = {or ~ So 1%
acts ergodically on .M, i.e. the equality a(x) = x, rot • Go, implies x = 01, 0 E C. PROPOSITION 4.2. The centralizer M "
is a factor of type 111.
Proof: From the definition of automorphism off (see (4.6)) it is easy to see that every automorphism a 6 Go is inner, i.e. there exists a unitary u~ 6 .A4 such that or(x) = u,~xu,~, * x ~ M . From the condition coo(x) o ot = co~o ~) we find 0-)
*
% (u,~xu~,) = CO~oX)(x)
Vx • M .
It follows from (4.5) that u~ • M %~ . According to Proposition 4.1 the group Go acts ergodically, this means that the equality u,~x = xu,~ for every ot • Go implies x = 0 1 , 0 • C. Hence we obtain {u~]ot • G o } ' = CI. Since A 4 ° ' C {u,~}' we then get M ~ n M = cl. In particular M ~ ' n M '~ = e l . This means that M ° is a factor.
[]
ON A FACTOR ASSOCIATED W I T H THE U N O R D E R E D PHASE
11
Now we are able to prove the main result of the paper. THEOREM 4.3. If the fraction ~(Fli , l"]j; J) - X(rlrn, 17l; J )
3.(Ok, rlp; J) - ~.(rlu, Or; J) is rational for every i, j, m, l, k, p, u, v ~ 1, 2 . . . . . q, here we put o = 1, then the von Neumann algebra M corresponding to the unordered phase of the ~.-model (2.2) on a Cayley tree is a factor of type III0, 0 6 (0, 1), otherwise .A4 is a factor of type 1111. Proof: It is known (see Proposition 2.2.2 in [20]) that Connes' spectrum F(a) of the group of automorphisms t~ = {Otg}g~6 of von Neumann algebra A4 has the form F(ot) = A{Sp(t~ e) I e ~ Proj(Z(A4~)), e 5~ 0}, (4.7)
where ore(x) = a ( e x e ) , x ~ eA4e and Z(A4 ~) is the centre of subalgebra .M a = {x ~ .A4 l ctg(X) = X, Vg ~ G}.
Here Sp(o0 is the Arveson's spectrum of the group of automorphisms ot (for more details see [20, 21]). By virtue of Proposition 4.2 we have Z ( A C ) = CI. Eq. (4.7) implies F(cr '°~o~))= Sp(a% ). We now consider the operator H ( V n ) =
~
~x,y. Let Sp(~/(V~)) denote the
(x,y)ELn
spectrum of ~/(V,). Setting trt~°z"" (x) = exp{itH(V~)}x e x p { - i t H ( V n ) } ,
x ~ A4,
we obtain Sp(a% .n) = Sp(H(Vn)) - Sp(I-/(Vn)) = {3. - / z I 3., IX e Sp(J?-/(Vn))}.
(4.8) 0.) n
It is clear that bii,k 6 Sp(H(V,)), ¥i, k 6 1, q. Eq. (4.8) then implies that Sp(tr% ' ) is generated by elements of the form bii,k -- bjj,1,
i, j, k, 1 ~ 1, q.
~'(rli'rlJ;J)-)~(rlm'rll;J) is a rational number, then there is a number 0 ~ (0, 1) Since ~.(rlk,rlp;j)_~.(rlu,rlv;j ) and integers mi.j.kj ~ Z (i, j, k, l ~ {1, 2 . . . . . q}) such that bii,k -- bjj,l ~- m i , j , k , l
log0.
(4.9)
12
F. MUKHAMEDOV
Hence we find that for some positive integer m the equality Sp(H(A)) = {nlogO}m=_m is valid. It follows that Sp(a '°(°z)) = {n log 0 },ez. Hence we get F ( a ~°~)) = {n log0}n~z. This means that A/[ is a factor of type III0. Otherwise F(a'°~~)) = II~, and this implies that .A4 is a factor of type III1. The theorem is proved. [] 5. 5.1.
Applications and examples Potts model We consider a Potts model on the Cayley tree F k whose Hamiltonian is regarded
as
H(a) = - Z
JScr(x)a(y),
(x,y)
where J 6 I~ is a coupling constant, as usual (x, y) stands for the nearest neighbour vertices and as before cr(x) ~ • = {r/i, r/2. . . . . r/q}. Here 8 is the Kronecker symbol. Eq. (2.1) implies that ~,(x)~(y) = q - 1 ~ ( x ) ~ ( y ) + q q-1
for all x, y ~ V. The Hamiltonian /-/(e) is therefore H(~r) = -
~
J'~r(x)cr(y),
(5.1)
i
(x,y)EL
where J ' -
q - 1 J. q Hence the ~.-model is a generalization of the Potts model, namely in this case the function ~. : • x • x / ~ --+ l~ is defined by JL(x, y; J') = - J ' ( x , y),
where x, y ~ ~q-1 and (x, y) stands for the scalar product in IRq-1. From (2.1) it is easy to see that J, if i = j , )~(r/i, r/j; J ) = J if i ~ j. (5.2) q-l' From (5.2) and (3.4) we can check that the assumption A (see Section 3) is true for the Potts model. So, there exists the unordered phase.
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
13
From (5.2) we can find that the fraction ~(rlk,~p; ~'(Tli'Oj;Jt)-~'(Om'Ol;Jt) Jt)-L(Ou,rlv; jr) takes values -4-1 and 0. So, by Theorem 4,3, the von Neumann algebra 34 is a Ilia-factor. From (4.9) we can obtain that 0 = exp /[ (q-1)T ~ }. Hence we obtain the following theorem. THEOREM 5.1. A v o n Neumann algebra 34 corresponding to the unordered phase of the Potts model (5.1) on a Cayley tree is a factor of type Ilia, where 0 = exp { ( q-J'q I -1)T/" REMARK 5.1. If q = 2, the considered Potts model reduces to the Ising model, for this model analogous results were obtained in [12]. For a class of inhomogeneous Potts model similar result were obtained in [13]. 5.2.
SOS model
In this subsection we consider the SOS (Solid-on-Solid) model on a Cayley tree. In the case of cubic lattice it was analyzed in [29]. The interest in these models is motivated by applications, in particular in the theory of communication networks [30]. The Hamiltonian of the SOS model is written as H(cr) = - J E let(x) - a(y)l, (x,y)
(5.3)
where J ~ IR and as before tr(x) ~ ~ = {rll . . . . . 0q}. It is clear that for this model the function ~. has the form Z(x, y; J) = - J I x - Y l , where x, y ~ ~q-1. From (2.1) we get 3.(0/,r/j; J) =
{0, ~/q--~-I _j ,
if i = j , if i 5~ j.
(5.4)
We note that from (5.4) and (3.4) it is easy to see that for the SOS model our assumption A is satisfied. Therefore there exists an unordered phase. From (5.4) we infer that the fraction ~.(rli, Oj; J) - )'07m, Or; J) ),(rlk, rlp : J) - ),(~u, 0~; J) can take values +1 and 0. Hence, according to Theorem 4.3, the von Neumann algebra 34 is a IIIa-factor. From the proof of Theorem 4.3 (to be precise from (4.9)), we can find the value of 0, i.e.
0 =°xp{ Thus we can formulate the following result. THEOREM 5.2. The yon Neumann algebra 34 corresponding to the unordered phase of SOS model (5.3) on a Cayley tree is a factor of type IIIa, where 0 =
14
E MUKHAMEDOV
5.3.
An example
Consideration of the above models gives the rise to the question: will the von Neumann algebra corresponding to the unordered phase of every ~.-model be a factor of type III0 for some 0 e (0, 1)? The next example shows that the answer to this question is negative. EXAMPLE 5.1. We define a function ~.(x, y; J ) as follows: ~,(01, 01) = I n
J,
)~(01, 02) = •(02, 01) = l n ( J V ~ ) , )~(01, 03) = ~-(03, 01) = ln(1.5Jv/-2),
(5.5) )~(02, 03) = ln)~(03, 02) = ln((1 - 0 . 5 V ~ ) J ) , )~(02, 02) = l n ( 2 J v ~ ) , ~-(03, 03) = ln(1.5J~¢/2), where J > 0. For the function ~.(x, y; J ) thus defined we can consider the corresponding )~-model on the Cayley tree with spin values in ~ . From (5.5) one easily verifies that the assumption A is satisfied. Hence, we can consider the unordered phase. From (5.5) it is not hard to check that the fraction in Theorem 4.3 takes an irrational value, hence, again by this Theorem 4.3, the von Neumann algebra A4 is a factor of type III1.
5.4.
Markov random fields
In this subsection we consider a case when )~(x, y) is not a symmetric function and the corresponding Gibbs measure is a Markov random field. Recall that matrix (Pij)2i,j=l is called stochastic if Pij >_ 0 and Pil • PiE = 1, i = 1, 2. Now let P = (Pij)2,j=l be a stochastic matrix such that Pij > 0 for all i, j e 1, 2. Define a function )~(x, y) as follows:
{ ~-(01,01) ~----- logpal,
~-(01, 0"]2) = --
log P12,
logp21,
~,(02, 02) = --
log P22-
)~(02, 01) = --
From now on we will consider the case /~ = 1 and q = 2. In this case Eq. (3.4) can be written as
Zx =
-I
P l l Z v + P12
: , yeS(x) P21Zy -q- P22
(5.6)
with Zx = exp{2hx}. It is clear that the function Zx -- 1 is a solution of (5.6),
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
15
which means that Assumption A is satisfied. By /z we denote the corresponding unordered phase of the ~.-model. Observe that if k = 1 then the measure /z is a Markov measure, associated with the stochastic matrix P (see [18]). By w , one denotes the diagonal state corresponding to the measure /~ on the C*-algebra A = ®r, M2(C). Repeating the reasoninig of Section 4 we find a quantized Hamiltonian corresponding to the state w,:
H(Vn)=
~)x,y, ~)x,y=
~
(o11 o o ) log P12 O 0 logp21
(x v~L,
O
O
.
(5.7)
log P22
As before let A4 = zro~(A)'. Let aft be the modular group of A4 associated with w , and .A4°u be the centralizer of co,. LEMMA 5.3. If there exist integers m j, j = 1, 2, 3 and the smallest number ~. e (0, 1) such that Pll = ~ml, P12
P2__~l= ~m2, P22
Pll = ~m3, P21
(5.8)
then for the modular group ~rt~ and the number to = -2zr/log~., the equality tri0u = Id is valid, here and below Id stands for the identity mapping. Proof: From (5.8) we have ~ml
Pll
-
-
1
/~ml + 1'
P12 -- ~ml + 1
~ml--m3
~ml--m2--m3
P21 - - -
~ml- + 1'
P22 - -
(5.9)
~.ml + 1
It follows from (5.7) that
[
,,~
N
exptitWx,y)
=
(~ml
1 .~- 1)it
( )~itml 0
0
0
0
0
0
1
0 0 ~.it(ml-m3) 0
0
0
)
0 ~it (m I --m2--m3)
" = Id. This completes Then the last equality together with (4.5) implies that trto the proof. [] Let us prove the following useful result•
16
F. MUKHAMEDOV
PROPOSITION 5.4. Let A/" be a factor, ~o be a faithful normal state on A/" and let trf be its modular group. If the equality tr~ = Id is valid for the number to = - 2 z r / l o g ) ~ then Af cannot be a factor of type III1.
Proof: Let us assume that ./V be a factor of type III1 and denote a = o't0 ~. According to Lemma 2.9 [31] the crossed product .IV"x,~ Z is a factor of type III4. On the other hand, according to Section 22.6 [21], every element XA = ~ zr~(a(n))Un, n
A = {a(n)} C A/',
of the crossed product A/" x~ Z belongs to the centre of this algebra if and only if the following relations hold: (i) a(n)otn(x) = xa(n) for all x ~ A/', n e Z, (ii) a n ( a ( m ) ) = a(m) for all m, n ~ Z. Here zr~ and Un stand for the generators of the crossed product algebra A/" x~ Z. Now define a sequence AD = {ao(n)} as follows
ao(n)=
1, 0,
if I n l < D , if Inl > D ,
where D is an arbitrary fixed positive integer. Then it is easy to see that the conditions (i) and (ii) for the element XAD are satisfied. So the element Xao belongs to the centre. But Xao q[ CI, and thus we have a contradiction with the fact that A r x~ Z is a factor. The proposition is proved. [] Since ~o, is invariant with respect to the group of translations Gk+l then according to Corollary 5.3 [32] the factor .At has either type III1 or 1118, 8 e (0, 1). Let us assume that the condition (5.8) is satisfied, then according to Lemma 5.3 and Proposition 5.4 we conclude that the factor .M has type II18 for some 8 e (0, 1). This means that the Connes' spectrum lV(tr°~,) of tr'%, i.e. the set {nlog8 I n ~ Z} is discrete. Then using Proposition 16.4 [21] we find that the centralizer .M*v is a factor. Now we are going to show that 8 = ~. To do this we compute the Connes' spectrum. We have just proved that .M *~ is a factor, this means Z ( M °) = C1. Then Eq. (4.8) implies F(tr °~) = Sp(tr'%). Now repeating the argument of the proof of Theorem 4.3 we can prove that the factor M has type 1114. Otherwise M is a factor of type 1111. So we can assert the following. THEOREM 5.5. Let P = (Pij)ij=l 2 be a stochastic matrix such that Pij > 0 for all i, j = 1, 2 and at least one element of this matrix is different from 1/2 and o9~ be the corresponding Markov state. If the condition (5.8) is satisfied then zr~ (A)" is a factor of type 1114, otherwise it is a factor of type III1.
17
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
REMARK 5.2. If all elements of the stochastic matrix P are equal to 1/2 then the corresponding Markov state wu is a trace and consequently ~r,o,(A)" is a factor of type lib REMARK 5.3. Theorem 5.5 can be proved by using an analogous proposition as Proposition 4.2. On the other hand while proving Theorem 5.5 we have used another Proposition 5.4. Conversely, Theorem 4.3 can also be proved by using Proposition 5.4. Let us give some examples of stochastic matrices which satisfy and do not satisfy (5.8). EXAMPLE 5.2. Let p ~ (0, 1). Consider a stochastic matrix defined by 1 l+p
P=
(1
p) 1 p
Then one easily sees that the condition (5.8) is satisfied with the parameters ). = p, ml = - 1 , me = - 1 , m3 = 0. Hence, according to Theorem 5.5, we conclude that the corresponding Markov state generates a factor of type Hip. Observe that in the considered case the Markov state is a product state, and the last result agrees with the Powers' result (see [51). EXa_MPLE 5.3. Consider the equation x 5 - x 3 - x + 1 = 0. It is clear that this equation is equivalent to ( x - 1)(x3(x + 1 ) - 1 ) = 0. Suppose that 8 is a solution of the equation x3(x + 1) - 1 = 0 belonging to the interval (0, 1). Now define a stochastic matrix as follows 1
1
Pll -- 1 + 85,
P21 - - 1 q- 8 2 ,
P12
1
PI1,
P22
1
P21.
Then (5.8) is satisfied with the parameters ~. = ~, ml = - 5 , mE ----- 2, m 3 ---- --1. Now let us show that 8 is the smallest number which satisfies (5.8). Assume that there exist a number v e (0, 1) and integers rhj e Z, j = 1, 3, such that v < / z and Pll
--
- , ])/'r/I
P12
P21 ~
] ) m- 2 ,
P22
Pll
~
] ) m- 3
P21
hence we obtain ])rhl = 8 - 5 ,
])th2 = 82,
Vrh3 = 8 -1.
(5.10)
It is clear that rhj < 0, j = 1, 3. Denoting n3 = --rn3 we get 8 = vn3. This contradicts our assumption. Hence, according to Theorem 5.5, we find that the Markov state corresponding to the matrix P generates a factor of type III~. EXAMPLE 5.4. Define a stochastic matrix as follows Pll = 1/3, P12 = 2/3, P21 = 1/4, P22 = 3/4, then it is easy to see that (5.8) is not satisfied. So, the von Neumann algebra zro,~(A)" is a factor of type 1Ill.
18
E MUKHAMEDOV Acknowledgements
The final part of this work was done within the scheme of Mathematical Fellowship at the Abdus Salam International Center for Theoretical Physics (ICTP). The author thanks ICTP and IMU/CDE-program for providing financial support and all facilities. The author is also grateful to the referees for useful comments and observations. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [153 [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
L. Accardi: Funkt. anal, i ego proloj. 8 (1975), 1. V. Ya. Golodets and G. N. Zholtkevich: Theor. Math. Phys. 56 (1983), 80. L. Accardi and G. Frigerio: Proc. R. Ir. Acad. 83A (1983), 245. H. Araki and W. Wyss: Helv. Phys. Acta 37 (1964), 136. R. Powers: Ann. Math. 81 (1967), 138. H. Araki and E. J. Woods: Publ. R.I.M.S. Kyoto Univ. 3 (1968), 51. R. Powers and E. StOrmer: Commun. Math. Phys. 16 (1970), 1. G. E Dell'Antonio: Commun. Math. Phys. 9 (1968), 81. T. Murakami and S. Yamagami: Publ. R.1.M.S., Kyoto Univ. 31 (1995), 33. H. Arald: Publ. R.I.M.S., Kyoto Univ. 6 (1970/71), 385. C. M. Series and Ya. G. Sinai: Commun. Math. Phys. 128 (1990), 63. E M. Mukhamedov: Theor. Math. Phys. 123 (2000), 489. F. M. Mukhamedov and U. A. Rozikov: Theor. Math. Phys. 126 (2001), 206. V. Ya. Golodets and S. V. Neshveyev: Commun. Math. Phys. 195 (1998), 213. U. A. Rozikov: Siberian Math. J. 39 (1998), 427. R. J. Baxter: Exactly Solved Models in Statistical Mechanics, Academic Press, London, New York 1982. N. Ganikhodjaev: Dokl. Akad. Nauk Rep. Uzb. 4 (1990), 3. Ya. G. Sinai: Theory of Phase Transitions: Rigorous Results, Pergamon Press, Oxford 1982. O. Bratteli and D. Robinson: Operator Algebras and Quantum Statistical Mechanics L Springer, Bedin 1979. A. Connes: Ann. Ec. Norm. Sup. 6 (1973), 133. S. Stratila: Modular Theory in Operator Algebras, Abacus Press, Bucuresti 1981. A.N. Shiryaev: Probability, Nauka, Moscow 1980. F. Spizer: Ann. Prob. 3 (1975), 386. A . G . Shukhov: Funct. Anal, Appl. 14 (1980), 163. S. Stratila and D. Voiculescu: Representations of AF-algebras and of the Group U(oo), Lecture Notes Math. 486 (1975), Springer, Berlin. N.N. Ganikhodjaev and F. M. Mukhamedov: Methods Funct. AnaL Topology, 4 (1998), 33. O. Bratteli and D. Robinson: Operator Algebras and Quantum Statistical Mechanics II, Springer, Berlin 1981. W. Kriger: Z. Wahr. verw. Geb. 65 (1983), 323. A.E. Mazel and Yu. M. Suhov: Jour. Stat. Phys. 64 (1991), 111. F.P. Kelly: J. Roy. Statist. Soc. Ser B. 47 (1985), 379; 415; K. Ramanan, A. Sengupta, I. Ziedins and P. Mitra: Adv. Appl. Probab. 34 (2002), 58. U. Haagerup, E. Winslow: J. Func. Anal. 171 (2000), 401. E. St~rmer: Proc. Sym. Pure Math. 38 (1982), 149.
REPORTS ON MATHEMATICAL PHYSICS
No. 1
ON A FACTOR ASSOCIATED W I T H THE UNORDERED PHASE OF X-MODEL ON A CAYLEY TREE FARRUH MUKHAMEDOV Department of Mechanics and Mathematics, National University of Uzbekistan Vuzgorodok, 700095 Tashkent, Uzbekistan (e-mail: [email protected]) (Received October 30, 2001 - - Revised October 4, 2003)
In this paper we consider nearest-neighbours models, where the spin takes values in the set e~ = {01, 02 . . . . . 0q} and is assigned to the vertices of the Cayley tree 1'k. The Hamiltonian is defined by some given X-function. We find a condition for the function ~. to determine a type of yon Neumann algebra generated by the GNS construction associated with the unordered phase of the ),-model. We give also some physical applications of the obtained result. 2000 Mathematical Subject Classification: 47A67, 47L90, 47N55, 82B20. Keywords: von Neumann algebra, )~-model, Cayley tree, unordered phase, Gibbs measure, GNS construction.
1.
Introduction
It is known that in quantum statistical mechanics specific systems are identified with states on corresponding algebras. In many cases the algebra can be chosen to be a quasi-local algebra of observables. States on these algebras, satisfying KMS condition, describe equiLibrium states of a quantum system. On the other hand, for classical systems with finite radius of interactions, limiting Gibbs measures are known to be Markov random fields. In connection with this, there arises a problem o f constructing analogues of noncommutative Markov chains. Accardi explored this problem in [1], where he introduced and studied noncommutative Markov states on the algebra of quasi-local observables, which were in agreement with the classical Markov chains. Modular properties of the noncommutative Markov states were studied in [2, 3]. The type analysis of quasi-free factors (i.e. factors generated by quasi-free representations) has been an interesting problem since the appearance of the pioneering work by Araki and Wyss [4]. A family of representations of uniformly hyperfinite algebras, which can be treated as a free quantum lattice system, was constructed in [5]. In this case, factors corresponding to these representations had type IIIx, ~. e (0, 1). More general constructions of product states were considered in [6]. In the case of CAR-algebra, the rough classification into types Ioo, II1, Iloo and III was obtained in the 60's by several authors (see e.g. [7, 8]). The classification IH
2
E MUKHAMEDOV
of type III quasi-free factors in terms of spectral properties of positive operators parameterizing quasi-free states was described in [9]. It is known (see e.g. [9, 10]) that every quasi-free state on CAR-algebra can be regarded as the product state on A = ®n_>IM2(C). Observe that the product states can be viewed as the Gibbs states of a Hamiltonian system in which interactions between particles of the system are absent, i.e. the system is a free lattice quantum spin system. So, it is interesting to consider quantum lattice systems with nontrivial interactions, which leads us, as mentioned above, to the consideration of the Markov states. Simple examples of such systems are the Ising and Potts models, which have been studied in many papers (see, for example, [11]). We note that all Gibbs states corresponding to these models are Markov random fields. The full type analysis of von Neumann algebras associated with the Markov states is still an open problem. In [12], for the Ising model on a Cayley tree the types of factors corresponding to the translation-invariant Gibbs states were found. In [13] it was proved for a class of nonhomogeneous Potts model, that a von Neumann algebra associated with an unordered phase of this model is a factor of type III1. Some specific cases of the Markov states were considered in [14]. The present paper is devoted to the type analysis of one class of Markov states, which correspond to certain Hamiltonian systems with nearest-neighbour interactions. More precisely, X-model on the Cayley tree will be considered, in which spin variables take their values in a set q~ = {r/1. . . . . r/q}, where r/k ~ ~q-1, k = 1, q. Observe that the considered model generalizes the notion of Z-model introduced in [15], where the spin variables take their values 4-1. The paper is organized as follows: in Section 2 we give some preliminary definitions of the ~.-model on the Cayley tree and corresponding Gibbs measures. We also recall a few definitions from the theory of von Neumann algebras. In Section 3 we reduce the problem of describing Gibbs measures to the problem of solving a nonlinear functional equation. In Section 4 we determine the types of von Neumann algebras generated by GNS representation associated with a diagonal state corresponding to the unordered phase of the ~-model. We find a condition for a function )~ which guarantees that the corresponding von Neumann algebra is a factor of type 1118. Section 5 is devoted to some applications of the obtained result to physical models and analysis of types of factors corresponding to Markov random fields associated with stochastic matrix.
2.
Definitions and preliminary results
The Cayley tree (or Bethe lattice) F ~ (see [16]) of order k > 1 is an infinite tree, i.e. a graph without cycles, from each vertex of which exactly k + 1 edges go out. Let F k = (V, A, i), where V is the set of vertices of F k, A is the set of edges of F k and i is the incidence function associating each edge l ~ A with its endpoints x, y E V. If i(l) = {x, y}, then x and y are called neighbouring vertices, and we write l = (x, y). The distance d(x, y), x, y ~ V on the Cayley tree defined by the
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
3
formula
d(x, y) = min{d I 3x --- x0, xl . . . . , Xd-1, Xd = y e V such that the pairs (x0, Xl) . . . . . (Xd-1, Xd) are neighbouring vertices}. We set
Wn = {x e V l d ( x , x °) = n}, vn = u ~ = l wm = [x e v I d(x, x °) < n}, Ln = { l = ( x , y ) e L I x, y e V,}, for an arbitrary point x ° e V. Denote Ixl = d(x, x0), x e V. A collection of pairs (x, Xl) . . . . . (Xd-1, y) is called path from the point x to the point y. We write x < y if the path from x ° to y goes through x. Vertex y we call a direct successor of x if y > x and x, y are the nearest neighbours. Denote by S(x) the set of direct successors, i.e.
S(x) = {y e Wn+l I d(x, y) = 1},
x e Wn.
Observe that any vertex x ¢ x ° has k direct successors and x ° has k + 1. PROPOSITION 2.1 [17]. There exists a one-to-one correspondence between the set V of vertices of the Cayley tree of order k > 1 and the group Gk+l of the free products of k + 1 cyclic groups of the second order with generators al, a2 . . . . . ak+l. Let us define a group structure on the group Gk+l a s follows. Vertices which correspond to the "words" g, h e Gk+l are called the nearest neighbours and are connected by an edge if either g = ha i or h = gaj for some i or j . The graph thus defined is a Cayley tree o f order k. Consider a left (resp. right) transformation shift on Gk+l defined as: for go e Gk+l we set Tgoh = goh (resp. Tgoh = hgo, ) Vh e Gk+l. It is easy to see that the set of all left (resp. right) shifts on Gk+l is isomorphic to the group Gk+a. Let • = {r/a, 72 . . . . . /'/q}, where ~1, r/2 . . . . . rlq are vectors in ~q-1, be such that 1,
OiOj =
if i = j , 1
q-l'
ifis~j.
(2.1)
We consider models where the spin takes values in the set ~ = {r/l, r/2. . . . . 0q} and is assigned to the vertices of the tree. A configuration tr on V is then defined as a function x e V --+ or(x) e ~ ; the set of all configurations coincides with f2 = ~ rk. The Hamiltonian is of a X-model form, Hx(tr) = ~ X(tr(x), or(y); J ) , (x,y)
(2.2)
F. MUKHAMEDOV
4
where J 6 11~n is a coupling constant and the sum is taken over all pairs of neighbouring vertices (x, y), a 6 f2. Here and below 3. : ~ x • x ]I{n 7> ]~ is some given function. From a physical point of view the interactions between particles do not depend on their locations, therefore from now on we will assume that )~ is a symmetric function, i.e. L(u, v; J ) = ~.(v, u; J) for all u, V e I~q-1. We note that ~.-model of this type can be considered as a generalization of the Ising model. The Ising model corresponds to the case q = 2 and ~.(x, y; J ) =
- J x y , x, y, J ~ R. We consider a standard a-algebra )v of subsets of f2 generated by cylinder subsets, all probability measures are considered on (f2, Y). A probability measure /z is called a Gibbs measure (with Hamiltonian Hz) if it satisfies the DLR equation: Yn = 1,2 . . . . and a n ~ V " :
f
(an), where Vwlw,,+l v, is the conditional probability
vV~ wlwn+! (On) = Z-1 (°')l Wn+!) e x p ( - f l H ( o ' n IIo91w,+, )), where fl > 0. Here air, and Wlw,+~ denote the restrictions of a , w ~ [2 to Vn and Wn+l, respectively. Next, an : x ~ Vn -+ an(x) is a configuration in Vn and H(anlloglw~+,) is defined as the sum H(an) + U(an, ~olw,+~) where
H(an) = U(an, wlw.+,) =
Finally, Z(O)lWn+l) stands
Z ).(an(X), an(y); J), (x,y)ELn
(2.3)
Z ~.(an(X), w(y); J). (x,y):x~Vn,yeWn+l
for the partition function in V~ with the boundary condi-
tion colw,+~
Z(WlWn+~) = E
exp(-flH(anllwlw,+~).
~,, ee v. Since we consider the nearest-neighbour interaction, the Gibbs measures of the ),-model possess a Markov property: given a configuration wn on Wn, random configurations in Vn-1 and in V \ Vn+] are conditionally independent. It is known (see [18]) that for any sequence ~o(n) ~ f2, any limiting point of measures fly, is a
o)fn)[Wn+l
Gibbs measure. Here ~v.
o){n)lWn+l
is a measure on f2 such that ¥ n ' > n"
~,v. { v v" ,o("qw.+,({a ~ ~2 I air., = an,]) = ,o(")lw.+~(a,,iv~) 0
if a,,Iv.,\v. = eo(')lv~,\v~, otherwise.
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
We now recall some facts from the theory of von Neumann algebra. Let B ( ~ ) be an algebra of all bounded linear operators on a Hilbert space ~ (over the field of complex, numbers C). A weak (operator) closed ,-subalgebra A/" in B(7-() is called von Neumann algebra if it contains the identity operator I. By Proj(A/) we denote the set of all projections in A/'. Von Neumann algebra is a factor if its center Z ( A ; ) ( = {x • A; I x y = yx, Y y • A/'}) is trivial, i.e. Z(AF) = {LI I ~- • C}. Von Neumann algebras are split into the classes I (In, n < o0, Ioo), II (111, 1Ioo) and HI. An element x • A/" is called positive if there is an element y • A/" such that x = y*y. A linear functional co on .A/" is called a state if co(x'x) > 0 for all x e A/" and co(I) = 1. A state co is said to be normal if co(supxe) = supco(x~) for any bounded increasing net {xa} of positive elements of 3f. A state co is called trace (resp. faithful) if the condition co(xy) = co(yx) holds for all x, y 6 A/" (resp. if the equality c o ( x ' x ) = 0 implies x = 0). Let .M be a factor, co be a faithful normal state on AF and tr~ be the modular group associated with co (see Definition 2.5.15 in [19]). We let I ' ( a °)) denote the Connes spectrum of the modular group cr~ (see Definition 2.2.1 in [20]). It is said [20] that the factor A/" is of type (a) Ili1, if F(tr °~) = IR, (b) IIIx, if F(cr ~°) = {nlogL, n • Z}, ~. ~ (0, 1), (c) III0, if F(cr °)) = {0}. (See e.g. [19, 21] for details on von Neumann algebras and the modular theory of operator algebras.)
3.
Construction of Gibbs states for the L-model In this section we give a construction of a special class of limiting Gibbs measures for the ;.-model on the Cayley tree. Let h : x --> hx = (hi,x, h2,x . . . . . hq-l,x) • l~q-1 be a real vector-valued function of x • V. Given n = 1, 2 . . . . . consider the probability measure/~(n) on ~v, defined
by /~(n)(~n) = Z~-I exp{-/~H(~n) + E
hxcy(x)}.
(3.1)
xEWn
Here, as before, crn : x • Vn -'+ crn(x) and Z~ is the corresponding partition function,
z. = ~ ~v~
exp{-/~n(6n) + ~ hx6(x)}. xcWn
The consistency conditions for /z(n)(Crn), n > 1, are Z / z ( n ) ( o ' n - - l ' O'(n)) = Jtz(n--1)(O'n--1)' trIn)
where a (n) = {or(x), x • Wn}.
(3.2)
F. MUKHAMEDOV
6
= V and /zl,/zz . . . . be a sequence of probability Let V1 c V2 c ... t0n=lVn ~ measures on ~Vn, ~v2 . . . . satisfying the consistency condition (3.2). Then, according to the Kolmogorov theorem (see, for example [22]) there exists a unique limit Gibbs measure /zh on f2 such that for every n = 1, 2 . . . . and an 6 dPv" holds the equality (3.3)
]J~({alVn = fin}) = ]-£(n)(cTn)•
Further we set the basis in ~q-1 to be r/l, r/2. . . . . r/q-1. The following statement describes conditions on hx guaranteeing the consistency condition of the measures /x(n)(crn). THEOREM 3.1. The measures tz(n~(an), n -- 1, 2 . . . . . satisfy the consistency condition (3.2) if and only if for any x ~ V the following equation I htx --~ Z F ( h y , )~) yESfx)
(3.4)
holds. Here and below h'x stands for the vector qqlhx and F : II~q-1 ~ I~q-1 is a function F(h; ~.) = (Fl(h; 3.) . . . . . Fq-l(h; )~)), with Fi(hl, hz . . . . . hq-1; ~.) Y]~q-~ exp{-fl)~(r/i, r/j; J)} exphj + exp{-fl~.(r/i, r/q; J)} = log ~q_~ exp{-3Z(r/q, r/j; J)} exphj + exp{-/~.(aq, r/q; J)}'
(3.4a)
i = 1,2, . . . , q - 1, h = (hi . . . . . hq-1).
Proof'. Necessity. According to the consistency condition (3.2) we have Zn-1 Z e x p { - / 3 Z
a~n)
Z Z )~(a(x)a(y); J ) + Z hxa(X)} xEWn-j yES(x) x~Wn = exp{ Z hx~r(x)}, xEWn-1
(3.5)
where we have used (3.1). From Eq. (3.5) we find Y~tn) exp{-fl)~(r/i, or(y); J) + hy~r(y)} 1--I ~-~o'~)exp[--flZ(r/q, if(y); J) q- hycr(y)} yES(x) = exp{hx(0i - r/q)},
i = 1, q - 1.
(3.6)
Now, set r/j = o r ( y ) = r/1 . . . . . r/q--l, then it follows from (3.6) that Y]~jq__~exp{-fl~.(r/i, r/j; J)} eXp{q~_lhj,y} q- exp{-fl~.(r/i, r/q; J)} q-1 yES,x) Y~j=I exp{--fl)~(r/q, r/j; J)}eXp[q-~_lhj,y} + exp{-C3~.(r/q, r/q; J)} 7
/
(3.7)
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
7
where i = 1 , q - 1. Here we have used Eq. (2.1). Denoting h'i, x = u-I q hi ,x, f r o m the last equation we get the required equation (3.4). Sufficiency. From (3.7) we get (3.6), (3.5) and hence we obtain (3.2). The proof is complete. [] Denote
D= {h=(hxE~q-a'xE
V) l h x =
E f(hy,~.), V x E V ] . yES(x)
According to Theorem 3.1 for any h = (hx, x ~ V ) ~ 79 there exists a unique Gibbs measure /Xh which satisfies Eq. (3.3). If the vector-valued function h ° = (hx = (0 . . . . . 0), x 6 V) is a solution, i.e. h ° 6 79 then the corresponding Gibbs measure /z~0z) is called the unordered p h a s e of the ~.-model. Since we are dealing with this unordered phase, we have to make an assumption which guarantees us the existence of the unordered phase. ASSUMPTION A. For the considered m o d e l the vector-valued f u n c t i o n h ° = (hx = (0, 0 . . . . .
0), x ~ V ) belongs to 79.
This means that Eq. (3.4) has the solution hx = ho = O, x ~ V. According to Proposition 2.1 any transformation S of the group Gk+l induces an automorphism S on V. By ~k+l we denote the left group of shifts of G k + l . Any T 6 ~k+l induces a shift automorphism 7" • £2 --+ £2 by (Tcr)(h) = tr(Th),
h ~ Gk+l, cr E £2.
It is easy to see that /z~0z)o T =/z(0~) for every I" ~ Gk+l. As mentioned above, (;~) the measure /z(0z) has a Markov property (see [23]). This measure /z0 enables a mixing property (see [23]), i.e. for any A, B 6 ~- one has lim lZ~oZ)(~'g(a) M B) =/z(0Z)(a)/z~0~)(B).
Igl~oo
4.
(3.8)
Diagonal states and corresponding von Neumann algebras
Consider a C*-algebra A = ®rkMq(C), where Mq (C) is the algebra of q x q matrices over the field of complex numbers C. By eij, i, j ~ { 1, 2 . . . . . q }, we denote the basis matrices of the algebra Mq(C). Let CMq(C) denote the commutative subalgebra of Mq(C) generated by the elements eli i = {1, 2 . . . . . q}. We set CA = @ r k C M q ( C ) . Elements of the commutative algebra CA are functions on the space £2 = {ell . . . . . eqq} rk . Given a measure /z on the measurable space (£2, B), where B is the ~-algebra generated by cylindrical subsets of £2, we construct a state cou on A as follows. We set c % ( x ) = 0 if the tensor monomial x of the basis matrices eij, i, j E {1, 2 . . . . . q } , contains at least one partial isometry. If x ~ C A , we set w ~ ( x ) = f x d l z . The state thus obtained was introduced in [24] and was said to be [2
8
E MUKHAMEDOV
diagonal. In other words, if P : A ---> C A is a conditional expectation, then the state 09~ can be defined by 09~(x) = lz(P(x)), x ~ A, where Iz(P(x)) means an integral of a function P ( x ) with respect to the measure /z, i.e. Iz(P(x)) = f~ P ( x ) ( s ) d # ( s ) (see [25]). (x) (;~) By 09o we denote the diagonal state generated by the unordered phase 1% . On a finite dimensional C*-subalgebra Avn = ®vnMq(C) C A we rewrite the state (x) 09o as follows 09~o~) (x) - tr(eh(V~)x) tr(eB(V~)) ,
x ~ Av~,
(4.1)
where tr is the trace on Av~. The term - # ) ~ ( a ( x ) a ( y ) ; J) in (3.1) (see also (2.3)) we represent as a diagonal element of Mq(C)® Mq(C) in the standard basis as follows: B <1) 0
0
0
B (2) 0
0
-fl~.(a(x), a(y); J) =
,
0
0
(4.2)
B (q)
where B (k) : ( b ij k) qi j=l' k = 1, q, are q × q diagonal matrices, and bij,k=
{ --fl~.(r/k, r/i; J), 0
i = j, i = 1, q, i s~ j.
(4.3)
Consequently, using the fact that the state 09o(z) is diagonal, with (3.1), (4.1)-(4.3) (cf. [14], Chapter 1, Section 1) we find that the form of the Hamiltonian ['I(Vn) in the standard basis of A vn (i.e. with respect to the basis matrices) is regarded as B (~) 0
0 B (2) 0
0 0
.
~-I(Vn) =
~ ¢~x,y, (x,y}ELn
di)x,y'~"
0
0
B (q)
Denote .M = zrO~oX)(A)", where :ro,~0z~ is the GNS representation associated with the state 09o(x) (see Definition 2.3.18 in [15]). Our goal in this section is to determine the type of .A4.
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
9
REMARK 4.1. General properties of a representation associated with diagonal state were studied in [25], but concrete constructions of states were not considered there. A deep classification of types of factors generated by quasi-free states was obtained in [10]. For translation-invariant Markov states the corresponding type analysis was obtained in [26]. Now we define translations of the C*-algebra A. Every T ~ ~k+l induces a translation automorphism rr : A --+ A defined by ®
®
rT-( H a x ) = H aT~x). x~Vn
x~V~
Since the measure/z~o~) satisfies a mixing property (see F.q. (3.8)), we can easily obtain that co~0 ~) also satisfies the mixing property under the translations {rr}r~k+~, i.e. for all a, b ~ A holds the equality lim Oa~oX)(rg(a)b) = coo~x)"" I,a)coo~) (b).
Ig[~oo
According to Theorem 2.6.10 in [19] the algebra .M is a factor. We note that the modular group of A4 associated with to(0x) is defined by
"#trT°~(x) = lim exp{itH(A)}xexp{-itH(A)}, A--+Fk
where /at(A) =
~
dikx,y.
x 6 .M,
(4.4)
In order for the last limit to exist, we need to show
{x,y}EA
that a norm of potential if/ is finite (see Theorem 6.2.4 in [27]). First of all we recall the definition of the norm of the potential qs = Y~xcrk qs(X) IlqSlld = E e a " (
sup
n>__O
where d > 0. Here qs(X) e Ax Now we compute IIHIId:
Ill/lid =
~
Ilqs(X)ll),
x~F~ x~X,IXl=n+l
= @xM2(C).
e2d(sup
E
lieu,vii)
x~I'k x~X,X={u,v}
=ke
TM
sup {u,v}eL
lib.,vii
= ke 2d max Ibij g[ < oo. i,j,k
'
Hence the norm of ~/ is finite and limit in (4.4) exists.
10
F. MUKHAMEDOV By M ~ one denotes the centralizer of w~0x) which is defined as w(z)
./t4a={xeMlat
° (x)----x, for all
te/~}.
Since W(ox) is a Gibbs state, then according to Proposition 5.3.28 [27] the centralizer .M ° coincides with the set
M,o~o~ = {x e M I CO~oX)(xy)= W(o~)(yx), for all
y e M}.
(4.5)
By I-l[n] we denote the group of all permutations y of the set Vn such that r(x) = x
Vx e W,.
Every y ~ H[n] defines an automorphism off • .M --+ M by
® ~×(Hax)= x~V.
® Uav(x) xeVn
(4.6)
Oly I@xCVnMq(C)= ~, where id is the identity mapping. Denote 80 = U{ot× [ma ~ Fl[n]},
n > 1.
Simply repeating the proof of a proposition in [28] we can prove the following. PROPOSITION 4.1. The group Go = {or ~ So 1%
acts ergodically on .M, i.e. the equality a(x) = x, rot • Go, implies x = 01, 0 E C. PROPOSITION 4.2. The centralizer M "
is a factor of type 111.
Proof: From the definition of automorphism off (see (4.6)) it is easy to see that every automorphism a 6 Go is inner, i.e. there exists a unitary u~ 6 .A4 such that or(x) = u,~xu,~, * x ~ M . From the condition coo(x) o ot = co~o ~) we find 0-)
*
% (u,~xu~,) = CO~oX)(x)
Vx • M .
It follows from (4.5) that u~ • M %~ . According to Proposition 4.1 the group Go acts ergodically, this means that the equality u,~x = xu,~ for every ot • Go implies x = 0 1 , 0 • C. Hence we obtain {u~]ot • G o } ' = CI. Since A 4 ° ' C {u,~}' we then get M ~ n M = cl. In particular M ~ ' n M '~ = e l . This means that M ° is a factor.
[]
ON A FACTOR ASSOCIATED W I T H THE U N O R D E R E D PHASE
11
Now we are able to prove the main result of the paper. THEOREM 4.3. If the fraction ~(Fli , l"]j; J) - X(rlrn, 17l; J )
3.(Ok, rlp; J) - ~.(rlu, Or; J) is rational for every i, j, m, l, k, p, u, v ~ 1, 2 . . . . . q, here we put o = 1, then the von Neumann algebra M corresponding to the unordered phase of the ~.-model (2.2) on a Cayley tree is a factor of type III0, 0 6 (0, 1), otherwise .A4 is a factor of type 1111. Proof: It is known (see Proposition 2.2.2 in [20]) that Connes' spectrum F(a) of the group of automorphisms t~ = {Otg}g~6 of von Neumann algebra A4 has the form F(ot) = A{Sp(t~ e) I e ~ Proj(Z(A4~)), e 5~ 0}, (4.7)
where ore(x) = a ( e x e ) , x ~ eA4e and Z(A4 ~) is the centre of subalgebra .M a = {x ~ .A4 l ctg(X) = X, Vg ~ G}.
Here Sp(o0 is the Arveson's spectrum of the group of automorphisms ot (for more details see [20, 21]). By virtue of Proposition 4.2 we have Z ( A C ) = CI. Eq. (4.7) implies F(cr '°~o~))= Sp(a% ). We now consider the operator H ( V n ) =
~
~x,y. Let Sp(~/(V~)) denote the
(x,y)ELn
spectrum of ~/(V,). Setting trt~°z"" (x) = exp{itH(V~)}x e x p { - i t H ( V n ) } ,
x ~ A4,
we obtain Sp(a% .n) = Sp(H(Vn)) - Sp(I-/(Vn)) = {3. - / z I 3., IX e Sp(J?-/(Vn))}.
(4.8) 0.) n
It is clear that bii,k 6 Sp(H(V,)), ¥i, k 6 1, q. Eq. (4.8) then implies that Sp(tr% ' ) is generated by elements of the form bii,k -- bjj,1,
i, j, k, 1 ~ 1, q.
~'(rli'rlJ;J)-)~(rlm'rll;J) is a rational number, then there is a number 0 ~ (0, 1) Since ~.(rlk,rlp;j)_~.(rlu,rlv;j ) and integers mi.j.kj ~ Z (i, j, k, l ~ {1, 2 . . . . . q}) such that bii,k -- bjj,l ~- m i , j , k , l
log0.
(4.9)
12
F. MUKHAMEDOV
Hence we find that for some positive integer m the equality Sp(H(A)) = {nlogO}m=_m is valid. It follows that Sp(a '°(°z)) = {n log 0 },ez. Hence we get F ( a ~°~)) = {n log0}n~z. This means that A/[ is a factor of type III0. Otherwise F(a'°~~)) = II~, and this implies that .A4 is a factor of type III1. The theorem is proved. [] 5. 5.1.
Applications and examples Potts model We consider a Potts model on the Cayley tree F k whose Hamiltonian is regarded
as
H(a) = - Z
JScr(x)a(y),
(x,y)
where J 6 I~ is a coupling constant, as usual (x, y) stands for the nearest neighbour vertices and as before cr(x) ~ • = {r/i, r/2. . . . . r/q}. Here 8 is the Kronecker symbol. Eq. (2.1) implies that ~,(x)~(y) = q - 1 ~ ( x ) ~ ( y ) + q q-1
for all x, y ~ V. The Hamiltonian /-/(e) is therefore H(~r) = -
~
J'~r(x)cr(y),
(5.1)
i
(x,y)EL
where J ' -
q - 1 J. q Hence the ~.-model is a generalization of the Potts model, namely in this case the function ~. : • x • x / ~ --+ l~ is defined by JL(x, y; J') = - J ' ( x , y),
where x, y ~ ~q-1 and (x, y) stands for the scalar product in IRq-1. From (2.1) it is easy to see that J, if i = j , )~(r/i, r/j; J ) = J if i ~ j. (5.2) q-l' From (5.2) and (3.4) we can check that the assumption A (see Section 3) is true for the Potts model. So, there exists the unordered phase.
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
13
From (5.2) we can find that the fraction ~(rlk,~p; ~'(Tli'Oj;Jt)-~'(Om'Ol;Jt) Jt)-L(Ou,rlv; jr) takes values -4-1 and 0. So, by Theorem 4,3, the von Neumann algebra 34 is a Ilia-factor. From (4.9) we can obtain that 0 = exp /[ (q-1)T ~ }. Hence we obtain the following theorem. THEOREM 5.1. A v o n Neumann algebra 34 corresponding to the unordered phase of the Potts model (5.1) on a Cayley tree is a factor of type Ilia, where 0 = exp { ( q-J'q I -1)T/" REMARK 5.1. If q = 2, the considered Potts model reduces to the Ising model, for this model analogous results were obtained in [12]. For a class of inhomogeneous Potts model similar result were obtained in [13]. 5.2.
SOS model
In this subsection we consider the SOS (Solid-on-Solid) model on a Cayley tree. In the case of cubic lattice it was analyzed in [29]. The interest in these models is motivated by applications, in particular in the theory of communication networks [30]. The Hamiltonian of the SOS model is written as H(cr) = - J E let(x) - a(y)l, (x,y)
(5.3)
where J ~ IR and as before tr(x) ~ ~ = {rll . . . . . 0q}. It is clear that for this model the function ~. has the form Z(x, y; J) = - J I x - Y l , where x, y ~ ~q-1. From (2.1) we get 3.(0/,r/j; J) =
{0, ~/q--~-I _j ,
if i = j , if i 5~ j.
(5.4)
We note that from (5.4) and (3.4) it is easy to see that for the SOS model our assumption A is satisfied. Therefore there exists an unordered phase. From (5.4) we infer that the fraction ~.(rli, Oj; J) - )'07m, Or; J) ),(rlk, rlp : J) - ),(~u, 0~; J) can take values +1 and 0. Hence, according to Theorem 4.3, the von Neumann algebra 34 is a IIIa-factor. From the proof of Theorem 4.3 (to be precise from (4.9)), we can find the value of 0, i.e.
0 =°xp{ Thus we can formulate the following result. THEOREM 5.2. The yon Neumann algebra 34 corresponding to the unordered phase of SOS model (5.3) on a Cayley tree is a factor of type IIIa, where 0 =
14
E MUKHAMEDOV
5.3.
An example
Consideration of the above models gives the rise to the question: will the von Neumann algebra corresponding to the unordered phase of every ~.-model be a factor of type III0 for some 0 e (0, 1)? The next example shows that the answer to this question is negative. EXAMPLE 5.1. We define a function ~.(x, y; J ) as follows: ~,(01, 01) = I n
J,
)~(01, 02) = •(02, 01) = l n ( J V ~ ) , )~(01, 03) = ~-(03, 01) = ln(1.5Jv/-2),
(5.5) )~(02, 03) = ln)~(03, 02) = ln((1 - 0 . 5 V ~ ) J ) , )~(02, 02) = l n ( 2 J v ~ ) , ~-(03, 03) = ln(1.5J~¢/2), where J > 0. For the function ~.(x, y; J ) thus defined we can consider the corresponding )~-model on the Cayley tree with spin values in ~ . From (5.5) one easily verifies that the assumption A is satisfied. Hence, we can consider the unordered phase. From (5.5) it is not hard to check that the fraction in Theorem 4.3 takes an irrational value, hence, again by this Theorem 4.3, the von Neumann algebra A4 is a factor of type III1.
5.4.
Markov random fields
In this subsection we consider a case when )~(x, y) is not a symmetric function and the corresponding Gibbs measure is a Markov random field. Recall that matrix (Pij)2i,j=l is called stochastic if Pij >_ 0 and Pil • PiE = 1, i = 1, 2. Now let P = (Pij)2,j=l be a stochastic matrix such that Pij > 0 for all i, j e 1, 2. Define a function )~(x, y) as follows:
{ ~-(01,01) ~----- logpal,
~-(01, 0"]2) = --
log P12,
logp21,
~,(02, 02) = --
log P22-
)~(02, 01) = --
From now on we will consider the case /~ = 1 and q = 2. In this case Eq. (3.4) can be written as
Zx =
-I
P l l Z v + P12
: , yeS(x) P21Zy -q- P22
(5.6)
with Zx = exp{2hx}. It is clear that the function Zx -- 1 is a solution of (5.6),
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
15
which means that Assumption A is satisfied. By /z we denote the corresponding unordered phase of the ~.-model. Observe that if k = 1 then the measure /z is a Markov measure, associated with the stochastic matrix P (see [18]). By w , one denotes the diagonal state corresponding to the measure /~ on the C*-algebra A = ®r, M2(C). Repeating the reasoninig of Section 4 we find a quantized Hamiltonian corresponding to the state w,:
H(Vn)=
~)x,y, ~)x,y=
~
(o11 o o ) log P12 O 0 logp21
(x v~L,
O
O
.
(5.7)
log P22
As before let A4 = zro~(A)'. Let aft be the modular group of A4 associated with w , and .A4°u be the centralizer of co,. LEMMA 5.3. If there exist integers m j, j = 1, 2, 3 and the smallest number ~. e (0, 1) such that Pll = ~ml, P12
P2__~l= ~m2, P22
Pll = ~m3, P21
(5.8)
then for the modular group ~rt~ and the number to = -2zr/log~., the equality tri0u = Id is valid, here and below Id stands for the identity mapping. Proof: From (5.8) we have ~ml
Pll
-
-
1
/~ml + 1'
P12 -- ~ml + 1
~ml--m3
~ml--m2--m3
P21 - - -
~ml- + 1'
P22 - -
(5.9)
~.ml + 1
It follows from (5.7) that
[
,,~
N
exptitWx,y)
=
(~ml
1 .~- 1)it
( )~itml 0
0
0
0
0
0
1
0 0 ~.it(ml-m3) 0
0
0
)
0 ~it (m I --m2--m3)
" = Id. This completes Then the last equality together with (4.5) implies that trto the proof. [] Let us prove the following useful result•
16
F. MUKHAMEDOV
PROPOSITION 5.4. Let A/" be a factor, ~o be a faithful normal state on A/" and let trf be its modular group. If the equality tr~ = Id is valid for the number to = - 2 z r / l o g ) ~ then Af cannot be a factor of type III1.
Proof: Let us assume that ./V be a factor of type III1 and denote a = o't0 ~. According to Lemma 2.9 [31] the crossed product .IV"x,~ Z is a factor of type III4. On the other hand, according to Section 22.6 [21], every element XA = ~ zr~(a(n))Un, n
A = {a(n)} C A/',
of the crossed product A/" x~ Z belongs to the centre of this algebra if and only if the following relations hold: (i) a(n)otn(x) = xa(n) for all x ~ A/', n e Z, (ii) a n ( a ( m ) ) = a(m) for all m, n ~ Z. Here zr~ and Un stand for the generators of the crossed product algebra A/" x~ Z. Now define a sequence AD = {ao(n)} as follows
ao(n)=
1, 0,
if I n l < D , if Inl > D ,
where D is an arbitrary fixed positive integer. Then it is easy to see that the conditions (i) and (ii) for the element XAD are satisfied. So the element Xao belongs to the centre. But Xao q[ CI, and thus we have a contradiction with the fact that A r x~ Z is a factor. The proposition is proved. [] Since ~o, is invariant with respect to the group of translations Gk+l then according to Corollary 5.3 [32] the factor .At has either type III1 or 1118, 8 e (0, 1). Let us assume that the condition (5.8) is satisfied, then according to Lemma 5.3 and Proposition 5.4 we conclude that the factor .M has type II18 for some 8 e (0, 1). This means that the Connes' spectrum lV(tr°~,) of tr'%, i.e. the set {nlog8 I n ~ Z} is discrete. Then using Proposition 16.4 [21] we find that the centralizer .M*v is a factor. Now we are going to show that 8 = ~. To do this we compute the Connes' spectrum. We have just proved that .M *~ is a factor, this means Z ( M °) = C1. Then Eq. (4.8) implies F(tr °~) = Sp(tr'%). Now repeating the argument of the proof of Theorem 4.3 we can prove that the factor M has type 1114. Otherwise M is a factor of type 1111. So we can assert the following. THEOREM 5.5. Let P = (Pij)ij=l 2 be a stochastic matrix such that Pij > 0 for all i, j = 1, 2 and at least one element of this matrix is different from 1/2 and o9~ be the corresponding Markov state. If the condition (5.8) is satisfied then zr~ (A)" is a factor of type 1114, otherwise it is a factor of type III1.
17
ON A FACTOR ASSOCIATED WITH THE UNORDERED PHASE
REMARK 5.2. If all elements of the stochastic matrix P are equal to 1/2 then the corresponding Markov state wu is a trace and consequently ~r,o,(A)" is a factor of type lib REMARK 5.3. Theorem 5.5 can be proved by using an analogous proposition as Proposition 4.2. On the other hand while proving Theorem 5.5 we have used another Proposition 5.4. Conversely, Theorem 4.3 can also be proved by using Proposition 5.4. Let us give some examples of stochastic matrices which satisfy and do not satisfy (5.8). EXAMPLE 5.2. Let p ~ (0, 1). Consider a stochastic matrix defined by 1 l+p
P=
(1
p) 1 p
Then one easily sees that the condition (5.8) is satisfied with the parameters ). = p, ml = - 1 , me = - 1 , m3 = 0. Hence, according to Theorem 5.5, we conclude that the corresponding Markov state generates a factor of type Hip. Observe that in the considered case the Markov state is a product state, and the last result agrees with the Powers' result (see [51). EXa_MPLE 5.3. Consider the equation x 5 - x 3 - x + 1 = 0. It is clear that this equation is equivalent to ( x - 1)(x3(x + 1 ) - 1 ) = 0. Suppose that 8 is a solution of the equation x3(x + 1) - 1 = 0 belonging to the interval (0, 1). Now define a stochastic matrix as follows 1
1
Pll -- 1 + 85,
P21 - - 1 q- 8 2 ,
P12
1
PI1,
P22
1
P21.
Then (5.8) is satisfied with the parameters ~. = ~, ml = - 5 , mE ----- 2, m 3 ---- --1. Now let us show that 8 is the smallest number which satisfies (5.8). Assume that there exist a number v e (0, 1) and integers rhj e Z, j = 1, 3, such that v < / z and Pll
--
- , ])/'r/I
P12
P21 ~
] ) m- 2 ,
P22
Pll
~
] ) m- 3
P21
hence we obtain ])rhl = 8 - 5 ,
])th2 = 82,
Vrh3 = 8 -1.
(5.10)
It is clear that rhj < 0, j = 1, 3. Denoting n3 = --rn3 we get 8 = vn3. This contradicts our assumption. Hence, according to Theorem 5.5, we find that the Markov state corresponding to the matrix P generates a factor of type III~. EXAMPLE 5.4. Define a stochastic matrix as follows Pll = 1/3, P12 = 2/3, P21 = 1/4, P22 = 3/4, then it is easy to see that (5.8) is not satisfied. So, the von Neumann algebra zro,~(A)" is a factor of type 1Ill.
18
E MUKHAMEDOV Acknowledgements
The final part of this work was done within the scheme of Mathematical Fellowship at the Abdus Salam International Center for Theoretical Physics (ICTP). The author thanks ICTP and IMU/CDE-program for providing financial support and all facilities. The author is also grateful to the referees for useful comments and observations. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [153 [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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