On the uniqueness of the equilibrium state for plane rotators

Communications in Commun. math. Phys. 56, 281—296 (1977) Mathematical Physics © by Springer-Verlag 1977 On the Uniqueness of the Equilibrium State for Plane Rotators J. Bricmont and J. R. Fontaine Institut de Physique Theorique, U.C.L., Universite de Louvain, B-1348 Louvain-la-Neuve, Belgium L. J. Landau Department of Mathematics, Bedford College, London NW1 4NS, England Abstract. We study the classical statistical mechanics of the plane rotator, and show that there is a unique translation invariant equilibrium state in zero external field, if there is no spontaneous magnetization. Moreover, this state is then extremal in the equilibrium states. In particular there is a unique phase for the two dimensional rotator, and a unique phase for the three dimensional rotator above the critical temperature. It is also shown that in a sufficiently large external field the Lee-Yang theorem implies uniqueness of the equilibrium state. 1. Introduction Some new results have been obtained recently concerning the classical statistical mechanics of the plane rotator model, defined by the Hamiltonian H=-β^Jίjσrσj Jy£0 (1.1) where σt is a two-dimensional vector of unit length. For d ^ 3, it has been proven [6] that spontaneous magnetization occurs for large β. For d = 2,it is well-known that there is no spontaneous magnetization for sufficiently short range interactions [17] moreover Dobrushin and Shlosman have proven that for finite range interactions all the equilibrium states are invariant under the action of S0(2) on the configuration space [2]. One may ask: does the absence of spontaneous magnetization imply uniqueness of the equilibrium state, as for example in the Ising model [16,19]? We show in the present paper that it implies at least uniqueness of the translation invariant equilibrium state and that the latter cannot be decomposed into non-invariant equilibrium states. Compared with the similar problem in the Ising model [16,19], our method looks more complicated the reason is that for this model, we don't have correlation inequalities comparable to the F.K.G. inequalities [5] which are valid for all the different boundary conditions. Instead of considering boundary conditions, we characterize invariant equilibrium states as tangents to the pressure [9]. We introduce various perturbations to the pressure and control the derivatives of these perturbed pressures with correlation inequalities. 282 J. Bricmont et al. In a sufficiently large external field, the correlation inequalities are sufficient to control all boundary conditions, and together with the Lee- Yang theorem gives uniqueness of the equilibrium state. The extension to models where σt has more than two components seems difficult because of the lack of the necessary correlation inequalities. Acknowledgements. It is a pleasure to thank F. Dunlop and J. Slawny for helpful discussions, and A. Messager, S. Miracle, and Ch. E. Pfϊster for providing us with Lemma 3.2 [24] which leads to a simplification of our original proof of the extremality of the unique phase. 2. The Model At each point i of the lattice TLd is associated a spin variable σ^elR2. We use two parametrizations, given by : σt = (si9 tt) = (rt cos φi9 r . sin φt) where The a priori probability distribution for σt is assumed rotation invariant : and the spin is bounded: |σj^6 with probability 1. In addition the measure v 0 satisfies Condition A, which is important for the development of correlation inequalities : Condition A. J (s + sr(s - sTf(t + tj(t' Remark. 1) The following measures (suitably normalized) are known to satisfy Condition A [1,3,12,18,22]: i) δ(r-b)drdφ b>Q ii) χ(r e [0, b^rdrdφ 4 2 (fixed length) b>0 (uniform distribution) iii) exp( - αr + cr )χ(re [0, bj)rdrdφ a, b > 0 . The interaction between spins is given by a translation invariant, ferromagnetic pair interaction where (2.1) We consider the case of finite range interactions : J. = 0 if |ΐ-;| However our results are not sensitive to this restriction and carry over to long range interactions with sufficiently rapid fall-off. Uniqueness of the Equilibrium State 283 If A is a finite region, an exterior configuration ω is a specification of the spins in ΛC9 the complement of A, and a boundary condition is a probability measure µbc(ω) on the exterior configurations. A particularly useful boundary condition is denoted s — b.c. for which all spins in Λc are in the s-direction with maximum value : for all σi =(st= max r, ί . = 0\ . iεAc \ suppA / The boundary condition ί-b.c. is obtained by the interchange of s and ί variables. An exterior configuration ω induces an effective Hamiltonian for the region A given by H J σ σ Λ,O=- Σ ij i' j~ i, jeA Σ Jifii ieΛ JεΛc σ3{ω) where σ7 (ω) takes the value specified by ω. The joint probability distribution of the spins in the region A is given by the Boltzmann factor (-#Λ,ω) Π ^θK> ieΛ where ieΛ Expectations with respect to µΛ>ω will be denoted < >^ ω. For a general boundary condition The pressure P^ f ω = MI" and for a general boundary condition The pressure P= and is independent of the boundary condition [20]. An equilibrium state is an infinite volume limit of finite region states < yA >bc with some boundary condition, or equivalently a probability measure on the infinite volume configuration space satisfying the DLR equations [9, 13]. We will generally use the symbol < > for a finite region state and the symbol ρ for an infinite volume equilibrium state. The set of equilibrium states for the Hamiltonian H is denoted Δ(H). A phase is an equilibrium state invariant under the lattice translations, and the set of phases is denoted A ///). In an external field hn the Hamiltonian (2.1) is modified to = -Σ Ji^j + ¥j) - Σ (*A + V, ) - (2.2) 284 J. Bricmont et al. Definition. The spontaneous magnetization M is defined by M=limρ h (n σί) fc\0 where Remark. By translation invariance of the phase ρh, M is independent of/ by rotation covariance, M is independent of the direction n. Using the fact that the pressure is a convex function of h and the relation between phases and tangents to the graph of the pressure [9] it is seen that M is the right-derivative of the pressure with respect to h at ft = 0 and is independent of the choice of phase ρheAj(Hh). Our basic results can now be formulated : 2.1. Theorem. If the spontaneous magnetization M = 0 then there is a unique phase in zero external field. Moreover, this phase is then extremal in the equilibrium states. This phase is also the unique quasi-periodic state. 2.2. Theorem. Let dλ(r) = δ(r — b)dr. Then for a large enough external field the equilibrium state is unique. 2.3. Corollary. If the lattice dimension d = 2, there is a unique phase In zero external field. If d — 3, there is a unique phase for T> Tc where Tc is the critical temperature for spontaneous magnetization. We point out that for simplicity of presentation we have not given each theorem or lemma in its most general form. We discuss extensions in Section 6. 3. Inequalities Let A be a finite subset of ΊLά with multiplicities, and define ίeA For example sfSj = sA where A = {i,iJ}. \A\ denotes the (finite) cardinality of A Similarly we define tA and rA. If AT is a ά function from TL to Z which is equal to zero except at a finite number of points, we define Note that if \A\ is even, sA±tA may be expressed in terms of cosines. (3.1) M and similarly for SA — tA: We decompose sA±tA into a sum of products of rfj (cosφicosφj±sinφisinφj) = rirjcos(φί + φj) and then use the identity cosNφcosMφ = j[c ' Uniqueness of the Equilibrium State 285 We denote by supp^ the set A without multiplicities: su Also We define the algebra 9ί as the set of finite linear combinations of the functions sAt& and the even and odd subspaces 2ίe? 9I0 as linear combinations of {rAcosNφ}, respectively {rAsmNφ}. Equivalently these subspaces are defined as linear combinations of {sAtB} with \B\ even, respectively odd. The subspaces may equally well be defined according to transformation properties under the transformation ί -> — ί Vz (Φi-+ — φiVϊ). Consider the interaction (2.2) with the external field in the positive s-direction. The Hamiltonian has the form H =~Σ Jifirj cos (Φt - ΦJ ~hΣrί cos Φi - With s-b.c. the effective interaction in the region A has the form H Λ,*=- Σ i,jeA Jijrirjcos(φi-φJ)-Σθίίricosφi ίeΛ where α^O. Then Ginibre's inequalities hold [Appendix, Theorems Al, A4] (rAcosNφrBcosMφys>A^(rAcosNφys>A(rBcosMφyS}Λ. (3.2) Now consider the interaction (2.2) with the external field such that hs, ht^0. Then the generalized Griffiths' inequalities hold [Theorems A2, A3] In addition there are comparison inequalities relating different states [Theorems A3, A4]. We derive here some basic results which will be useful in the proofs of Theorems 2.1 and 2.2. The discussion will be given for the interaction (2.1). 3. 1. Theorem. ρs= lim < >s A exists by monotonicίty and is an extremal equilibrium state. Remark. We note a useful property of the state ρs. By monotonicity in both h and A it follows that M = ρs(si). The proof of Theorem 3.1 is based on ί. 3.2. Lemma. Let ω be any exterior configuration for the region A. Then <rA cosMφys>A ^ <rA c Proof. We must show J (r^ cos Nφ - r'A cos Nφ')dµs({rίf φ 1 Lemma 3.2 is due to A. Messager, S. Miracle, and Ch. E.Pfister. It simplifies our original proof of Theorem 3.1 286 J. Bricmont et al. This is proved in the same way as Ginibre's inequalities [7] once the effective Hamiltonians are combined as J r c co - HA>S -HA>ω = Σ ij^i j ™(Φi ~ Φj) + W *(Φi ~ Φ'ji\ + £ α f r. cos φt + αjr; cos( where This can be written as a sum with positive coefficients of products of (rf + rj), . ίφ' -φ + w\ cos —-—-^—— cos —-—-^—— and similar terms with sine instead of cosine. r (rAcosMφ — r'AcosMφ ) can be written similarly. Then one expands the exponential in series and, due to the above decompositions, each term is a product of integrals over r ί5 rj and over φ , φ't. The integral r over φ., φ . factorizes into a product of two integrals of two identical functions, one in the variables ($ + φ^ and the other in (φ't - φt). Being a square, this expression is positive. The integral over the variables r , r( reduces to a product of integrals of the type: Kη+r r^r.-r ^AWAW). This integral obviously vanishes if bt is odd and is positive if bt is even. D Definition. We say the state ρ is clustering if for all /, 0e2l 0(/τ«0)-0(/)β(τα0)->0 as |α|->oo where τ fl denotes a translation by the amount a. Proof of Theorem 3.1. From Lemma 3.2 it follows that 2 <r^cosMφ>s>yl (^0) decreases as Λ. / Zd. Thus the limit exists. Since (rA sin MφySfA = 0 the state ρs is welldefined. Again from Lemma 3.2 it follows that ρ(rA cos Nφ) ^ ρs(r^ cos AΓφ) for all ρ 6 A (H) , which shows that ρs is extremal as a state on the even subspace 9Ie. The extremality of ρs on the even subspace 9ϊe implies that ρs is clustering on 3ϊe. Indeed, since A (H) is a metrizable Choquet simplex [9], there is a unique probability measure ωs carried by the extremal elements of A(H) such that ρs(f) = $σ(f)dωs(σ), V/6ΪI. By Lemma 3.2 it follows that ρs|"2ϊe = σ|"2ϊeωs-a.e. Since σ is clustering, ρs is clustering on 9Ie. By correlation inequalities it follows that ρs is clustering on the full algebra 91: (Theorem A5 or [3]) is even and ίδ °dd ' 2 Lemma 3.2 holds for all exterior configurations and hence for any boundary condition b. Take b as the boundary condition induced by s-b.c. in Λ'c where Λ'^Λ Uniqueness of the Equilibrium State 287 By construction, the restriction of ρs to the odd subspace 910 is identically zero. Thus the extremality of ρs follows from the following lemma. 3.3. Lemma, (a) Let ρ, σ be two states on 21 satisfying (i) ρ, σ are clustering (ii) (iii) then Um (b) Let ω be a translation invariant probability measure on a set E of states σ, satisfying lim σ(τ{(/)) = 0 ω-a.e. for some /e2ϊ. - (c) Let ρ = J σdω(σ) be a decomposition of the invariant state ρ into a set of states E E, with the measure ω translation invariant. Suppose ρ and ω-a.e. σ satisfy the hypothesis in (a). Then ρ = σ ω-a.e. Proof, (a) By (i) and (iii) µmρ(/τite)) = 0 ι->oo V and by (ii) limσ(/τ^)) = 0. ϊ->oo Therefore by (i) σ(f) lim σ(τi(g)) = 0 V/, 0e2l 0 and the conclusion follows. i->oo (b) Since ω is translation invariant, J \σ(f)\dω(σ) = ^\σ(τί(f))\dω(σ), £ by dominated convergence, (^(τ^/))!^ ||/||) VieZ d and £ ι->oo So σ(/) = 0 ω-a.e. (c) By points (a) and (b) σ(f) = 0 ω-a.e. for each /e 3Ϊ0. Since 9ί0 is separable, we conclude σ/2I0 = 0 ω-a.e. and σ = ρ ω-a.e. D We now begin the argument leading to the proof of Theorem 2.1. There is a useful "bootstrap" principle which allows one to conclude that certain higher order correlation functions are zero if certain lower order ones are zero. In particular we shall conclude from M = 0 and Lemma 3.2 that the equilibrium states have certain symmetry properties. 3.4. Theorem. Let ρ be any weak* limit point of the finite volume states < >^ as A / Zd, where the effective Hamίltonian for the region A has the form -HΛ= iJeΛ Σ ^M+^+ΣβA+M ίeΛ with (3-4) 288 J. Bricmont et al. Then if ρ(si-ti) = Q for all z, it follows that ρ(sA-tA) = Q for all A. Proof. Define the random variables xi = 2~^(si + tί\ yi = 2~^(si — tί). The effective Hamiltonian in terms of the variables x, y is Σ Jtfaxj + yjj) + 2 - * £ (α< + J8ί)xj + (α, - ft)?, . Also the a priori measure for (x, y) is the same as for (s, ί) by rotation invariance. Therefore the generalized Griffiths' inequalities are valid (Theorem A2) and these extend to the limit point ρ : (3.5) 0^ρ(xByc)^ρ(xB)ρ(yc). We first show : Given n(odd), if ρ(yA) = 0 for all |^4|(odd) < n then ρ(sA) = ρ(tA) for all \A\<n. 3 Indeed 1 X ρ(sA-tA) = 2 -^ Q(xByc) = 0 if \A\<n BuC = A |C|odd by (3.5) and the hypothesis ρ(j;c) = 0. We next show : ρ(yA) = 0 for all \A\ odd. Indeed, by hypothesis ρ(si) = ρ(ίf) for all i and so ρ(;y.) = 0 for all i. The proof proceeds by induction. Let τt(odd) be given and suppose ρ(y^) = 0 for all |^4|(odd)<n. To show ρ(^) = 0 if |^4| = n we write Q(sBtc)-ρ(sctB) \B\odd \<\A\ But = ρ(sB)ρ(sc)-ρ(tB)ρ(tc) by a comparison inequality (A6) where ρ' denotes the state obtained from ρ by the interchange of s and t variables4. Since Q(sB) = ρ(tB), ρ(sc) = ρ(tc) by the induction hypothesis, we have \C\odd which implies ρ(yA) = Q. Π 3 In the summation over subsets of A we distinguish different occurences of the same lattice point (multiplicities). Otherwise combinatorial factors should be included in the expression 4 Note that for ρ' the effective Hamiltonian for the region A has the form ~H'Λ= Σ ΛM + ^ + Σ α ώ + flίί IJeΛ ieΛ where αJ = /?£, $ = 0^. Since α^lβj the hypotheses of comparison Theorem A3 are satisfied Uniqueness of the Equilibrium State 289 4. Uniqueness of the Phase We consider the plane rotator in zero external field and suppose there is no spontaneous magnetization. We will show that there is a unique phase. We use the equivalence between phases and tangents to the graph of the pressure [9]. For fe 9ί we consider the perturbation λf to the Hamίltonian H. That is, we consider where we sum over the lattice translates of/. (Equilibrium states for Hλ are defined as in Section 2, via finite volume states with some effective Hamiltonian Hλ Λ, or directly by the DLR equations appropriate for Hλ.) If we can show that the pressure pλ is differentiable at λ = 0 it follows that all invariant equilibrium states take the same value on /. This may be formulated as 4.1. Lemma. Let there exist a sequence of positive numbers (Aπ)ne]N and another one of negative numbers (λ'n)neN, both converging to zero and invariant equilibrium states ρ^, ρλn of Hλ,n, Hλn such that lim ρλίι(/) = lim ρλn(f). Then all the ρeA^H) take the same value on f 1= lim ρλn(f) Proof. The equivalence between invariant equilibrium states and tangents to the graph of the pressure [9], together with the hypothesis and the convexity of Pλ shows that Pλ is differentiable at λ = Q and this is equivalent to the conclusion [9,13]. D We shall apply Lemma 4.1 to various perturbations and first to {SA, tA}. Once we have shown uniqueness on these functions (Lemma 4.2) we use this result to get uniqueness on the even subspace 2Ie. 4.2. Lemma. All phases take the same value on SA and on tA. Proof. The proof proceeds in three steps. a) Qs(sA) = Qs(tA) for all A, and ρs(sA) = 0 if |4|odd: Since M = 0 Lemma 4.1 implies ρs(sf) = 0. Since ρs(ί.) = 0 by construction, we have ρs(5f —ί f ) = 0. By the "bootstrap" Theorem 3.4, ρs(sA — tA) = 0 for all A. Since ρs(ί^) = 0 if \A\ odd by construction, we may also conclude ρs(s^) = 0 if \A\ odd. b) For all ρeA(H) ρ(sA) = ρ(tA) for all A and ρ(sA) = Q if \A\odd: By Lemma 3.2 ρ(sA — tA)^ρs(sA — tA) = 0 if \A\ even [expanding in terms of cosines—Eq. (3.1)]. Thus by the symmetry of interchanging s and t variables, we conclude ρ(sA — tA) = 0 for all ρe A (H), \A\ even . If \A\ odd, ρ(sA)^ρs(sA) = Q. By the symmetry s-»— s, we conclude ρ(sA) = 0 for all ρeA(H), \A\ odd. By the symmetry of interchanging s and t variables we now conclude ρ(tA) = Q for all ρεA(H), \A\ odd. 290 J. Bricmont et al. c) All phases take the same value on SA and on tA : Consider the perturbation + λsA, λ > 0. d Let ρ+λ be the (translation invariant) limit (Theorem 3. 1) as Λ / TL of < > Sj + λfΛ and ρ+ = lim ρ+λ (Griffiths' inequalities). Let ρ_λ be any weak* limit point as d Λ/TL of < >s _λ Λ and let ρ_λ be an average over translations of ρ_λ. Let ρ_ be a weak* limit point ofρ.^asλ-^O. Since by a comparison inequality [Eq. (A4)] \SB/s, λ,Λ = \SB/s, - λ,Λ it follows that Similarly Since ρ+eA^H), it follows by point (b) that ρ + (sB-tB) = 0 and so ρ + (sA) = ρ_(sA). By Lemma 4.1 all phases agree on SA. By the symmetry of interchanging 5 and t variables all phases agree on tA. D 4.3. Lemma. All phases take the same value on the even subspace 2Ie. Proof, a) Consider the perturbation ±λrAcosMφ. From a comparison inequality (Theorem A4) we have (rBcosNφys>λ9Λ^(rBcosNφySί_λ}Λ for λ>0 . Taking weak* limit points as in Lemma 4.2 we obtain states ρ+ which satisfy (4.1) ρ+ (rBcosNφ)^ρ_(rBcosNφ). b) If \B\ is even, inequality (4.1) implies and Q-(sAtB)=-Q-(sA(SBS S S S ^-Q + ( A( B ~ **)) + Q - ( A B) Since ρ + (sAsB) = ρ_(sAsB) by Lemma 4.2 the above inequalities imply and Thus Q-(sAtB)==Q + (sAtβ) ^n particular ρ+ (r^c Therefore all phases agree on the even subspace 2Ie by Lemma 4.1. Π Proof of Theorem 2.1. From Lemma 4.3 ρ(f) = ρs(f) for all /e2ϊe, and by construction ρs(f) = 0 for all / in the odd sub-space 9I0. If there were a phase ρ such that, ρ(/)Φθ for some /e9I0, then defining ρ' from ρ by the transformation Uniqueness of the Equilibrium State 291 Φi-^ — φiVi we would have a nontrivial decomposition of £ s = 2 (# + £?') which contradicts the extremality of ρs (Theorem 3.1). Thus ρ(/) = 0 for all ρeA^H) if /e9ϊ0. Thus ρ5 is the unique phase, and by Theorem 3.1 this phase is an extremal equilibrium state. That ρs is also the unique quasi-periodic state follows from the extremality of the unique phase, as in [21, Appendix C]. D Proof of Corollary 2.3. If the lattice dimension d = 29 we know by the theorem of Mermin [17] that lim <<τ w> h periodic = 0 with periodic boundary conditions. This h \0 implies that M = 0 and the corollary follows from Theorem 2.1. If d = 3 we define Tc as the lowest temperature such that M = 0 for all T> Tc. (Tc is finite by the high temperature cluster expansion and T C ΦO by [6].) Theorem 2.1 implies the uniqueness of the phase for T> Tc. D 5. Uniqueness of the Equilibrium State It will be shown that for sufficiently large external field, the plane rotator with fixed length (dλ = δ(r — b)dr) has a unique equilibrium state. We define two phases ρM and ρm which suitably bound all equilibrium states. Proving that ρM = ρm then gives the uniqueness of the equilibrium state. We take the external field interaction for the case hs = ht = h. (By rotation covariance the result is true for the external field in an arbitrary direction.) The effective Hamiltonian for the region A has the form (3.4) where α ί? βt depend on the exterior configuration. If h^b^J^ then α f , β.^0 for all exterior conj figurations. This allows the use of correlation inequalities for all equilibrium states. Let Ai = max αf = max βi coeΛc (oeΛc Bt = min a. = min βt ωeΛc ωeΛc and define the states < >M>yl (resp. < > J by taking a;=4, βt=Bt (resp. at = Bt, A = 4). Note that < >m<A is obtained from < >M>/1 by the interchange of s and t variables. Then from comparison inequalities (A4, A 5) for any exterior configuration ω , ί B M , ί B ω , ί ΰ m ^ and l<MB>M f Λ ~ < S ^ ί B>ω f ^l ^ <SA>MtA<tB>ωtΛ ~ <SA>(o,Λ<tB>M,Λ from (A6). 5.1. Lemma. The following infinite volume limits exist. (52) 292 J. Bricmont et al. Proof. It follows from Equations (5.1) that 5 <s^>Mjyl decreases as A increases , (ΪAΪM Λ increases as A increases. Thus the limits as A / TLd exist. From (5.2) it now follows that 5 <s^£β>M A converges as A/7Ld. By the interchange of s and t variables the same holds for < > m>yl . D Proof of Theorem 2.2. The Lee- Yang theorem for this model has been deduced by Dunlop and Newman [4] from a theorem of Suzuki and Fisher [23]. Although it is not stated in this form in [4], one can deduce from [23] analyticity of the pressure in hs with ht fixed, real, and non-zero and analyticity in ht with hs fixed, real and nonzero. Here hs (resp. ht) is the field in the s-direction (resp. the ί-direction). (In our case we are interested in a neighbourhood of hs = ht = h.) Therefore for all h Φ 0, ρ(si) and ρfa) are independent of ρeAj(H). In particular QM(si) = QM(ti)- The "bootstrap" Theorem 3.5 then implies QM(sA) = ρM(tA) for all A. By the symmetry of interchanging s and t variables we conclude s = s that QM( A) Qm( A) Now by Equations (5.1) we conclude that all equilibrium states take the same value on SA similarly for tA. By Equation (5.2) all equilibrium states take the same value on sAtB. D 6. Generalizations In this section we discuss extensions of results derived in preceding sections. Using a modification of Condition A, Messager et al. [24] have obtained uniqueness of the phase in any nonzero external field with the Lee- Yang theorem (e.g. fixed length rotator). Theorems 2.1, 2.2, and Corollary 2.3 extend to long-range interactions with sufficiently rapid fall-off. For d = 2, Corollary 2.3 is valid with Jtj such that £ ^/[/I 2<00 i α e f°r Λ/ jeZ 2 decreasing like \i—j\ with α>4. Kunz and Pfister [11] have shown that if Jtj decreases like \i— j\~a with 2<α<4, spontaneous magnetization occurs at low temperatures. They remark that in the borderline case, α = 4, there is no spontaneous magnetization. The conclusion of Corollary 2.3 should hold in this case too. Theorem 2.2 is valid for any a priori measure satisfying Condition A and such that the pressure is differentiable in hs and ht (the s- and ί-components of the external field). In particular a weak version of the Lee- Yang property would be sufficient. We will denote by "generalized interaction" a Hamiltonian of the form where J(A)^Q and \A\ is even. 5 Inequalities (5.1) and (5.2) hold for any exterior configuration and hence for any boundary condition b. Take b induced by < > M < y l ' where Λ ' D Λ L Uniqueness of the Equilibrium State 293 Lemma 3.2 and Theorem 3.1 depend only on Ginibre's inequalities [Theorem Al]. Lemma 3.3 is a general result for states on a separable algebra. Lemma 4.1 is a general statistical mechanical result. Given that Qs(sA) = Qs(tA)VA and ρs(sA) = 0 for \A\ odd, then Lemma 4.2b) follows for any generalized interaction, c) follows for a translation invariant generalized interaction, as does Lemma 4.3. Given the correlation inequalities of Dunlop [3], one can extend all the above results to get results on the Heisenberg model and on 4-component models. The correlation inequalities are not sufficient to get uniqueness of the translation invariant equilibrium state on all the correlation functions but only on a subset of those. It has already been noticed that Griffiths' inequality for rotators (Theorem A2—A3) has some analogy with Lebowitz' inequality [15] for Ising models [3, 1, 22]. In support of this, one may note that using only Lebowitz' inequality (in a similar way to the use of Theorem A3 in this paper), and assuming that the equilibrium states are invariant under the spin-flip symmetry, at zero external field, one can show that the equilibrium state is unique. Of course, in this case, the use of FKG inequalities [5] gives much stronger results [16,19]. Appendix: Correlation Inequalities We first recall Ginibre's and (generalized) Griffiths' inequalities : Theorem Al. (Ginibre [7], Dunlop-Newman [4].) Let dµ = ZΛ1 exp £ suppAcΛ suppMcA (J(A, M)rA cos Mφ) Π dλί(ri)dφί izA λt(r •) any measure on R+ of compact support. Then <JA cosMφrB cosNφy ^ (rA cosMφ) (rB cosNφy . (Al) Theorem A2. (Generalized Griffiths' inequality [1, 3, 12, 18, 22].) Let AcA I ίeΛ ί/v;(s;ί;) satisfy Condition A and are of compact support. Then <S^B>^<S^><SB> (A2) 0^<s Λ ί Jl >S<s^><tB> (A3) Proof. Stated in this form, the proof is in [1, 3]. D 294 J. Bricmont et al. We want to extend these inequalities to some cases where J x or J2 are not necessarily positive. This is an extension of a remark of Griffiths [8] and of a recent inequality of Lebowitz [14]. Theorem A3. Let dµ = Z-ι exp / £ J1(A)sA + J2(A)t \suppAcA dµ' = Z'- ' exp Σ W)sA + J'2(A)tA ieΛ dv{ satisfy Condition A and are of compact support. \J2(A)\ZJ'2(A). Then (i) \<sAy\^sA-> (A4) (ii) KOI^X (A5) (iii) |<S^ίB> - <s^B>'| ^ <s^> <ίB>' - < V<tB> . (A6) Proof. We use the method of duplicate variables [7]. We consider the integral J (SA - slS'A) (t'B - e2gdµ({siίί})dµ'({sίί/i}) (*) where ε f = ±1. Now J1(A)sA + J2(Λ)tA + J\(A)s'A + J'2(A)tfA = i K^i^) + W)) (^ + ^) + (JiW - J'lW) (SA - s'A) + (J'2(A) + J2(A)) (tA + t'A) + (J'2(A) - J2(A)) (t'A - ίJ] . By hypothesis all the coefficients JiW + J'^A), J,(A)-J\(A) J2(A) + J2(A) , J'2(A) - J2(A) are positive . We expand the exponential in series and develop in each term the factors SA + s'A, SA — s'A, etc. with the iterated formula : We get a series with positive coefficients of products of integrals which are all positive by Condition A. Thus the integral (*) is ^0, which gives This gives where ε= +1. Uniqueness of the Equilibrium State 295 (i) follows taking B = φ, ε= ±1 (ii) follows taking A = φ, ε= ±1 (iii) follows taking ε = — 1. D Combining this method with Ginibre's method of proving his inequalities, we have: Theorem A4. Let dµ = Z-ι exp/ £ J(A, M)rA cosMφ\ IsuppMcΛ \suppAcA dµ' = Z- l exp/ Σ I / J(A, M)'rA cosMφ\ Π dλjrjdφ, . 1 suppM CΛ \suppAcA I / If for all M, A J(M9A)^\J(M,A)'\ then (rB cosNφy ^ (TB cosNφy . Proof. J (rA cosMφ - r'A cosMφ^dµ^ φ^dµ^φ',}) ^ 0 by the same method as above. The inequalities used in the proof of Theorem 3.1 are derived as follows. Theorem A5. (Generalized Dunlop-Newman inequalities [3].) Let ρ be a state satisfying Ginibre's inequalities (Theorem Al) and such that ρ(sAtB) = Q if\B\ is odd. Then i) \Q(sAtBsctD)-ρ(sAtB)ρ(sctD)\ ^ρ(sAsBscsD)-ρ(sAsB)ρ(sctD) 2 if \B\ is even; ii) ρ(sAtBsctD) ^ ρ(sAsBscsD) - ρ(sAsB)2ρ(scsD)2 if 2 \B\ is odd. Proof. We assume that |D| is even in i) and odd in ii) otherwise the l.h.s. vanishes. i) Let lt = ± 1. Using Ginibre's inequalities and Equation (3.1), we have Q(SA(SB + Ί ^)SC(SD + '2^) ^ Q(SA(SB +lι tB))ρ(sc(sD + l2tD)) which gives ^ Q(SASBSCSD) ~~ Q(SASB)Q(SC - ρ(sAtB)ρ(scsD)) + l2(ρ(sAsBsctD) - ρ(sAsB)ρ(sctD)) . Take the case / 1 = 1, 12 = 1, add it to the case l± = — 1, 12 = — 1 and divide by 2 : - (ρ(sAtBsctD) - ρ(sAtB)ρ(sctD)) ^ Q(SASBSCSD) - ρ(sAsB)ρ(scsD) . 296 J. Bricmont et al. With /! = +1, / 2 = — 1 added to / 1 = — 1,12 = +1, we get the same result with the sign instead of — in front of the l.h.s. So i) is proved, ii) We use duplicate variables and apply i). Then 2 2 2 \ρ(sAtBsctD) -ρ(sAtB) ρ(sctD) \ 2 ^ ρ(sAsBscsD) 2 - ρ(sAsB) ρ(scsD) 2 but Q(sAtB) = ρ(sctD) = 0 since \B\9 \D\ are odd. D References 1. Bricmont, J.: Correlation inequalities for two-component fields. Ann. Soc. Sc. Brux. 90, 245—252 (1976) 2. Dobrushin, R. L., Shlosman, S. B.: Absence of breakdown of continuous symmetry in twodimensional models of statistical physics. Commun. math. Phys. 42, 31—40 (1975) 3. Dunlop,F.: Correlation inequalities for multicomponent rotators. Commun. math. Phys. 49, 247— 256 (1976) 4. Dunlop,F., Newman, C.M.: Multicomponent field theories and classical rotators. Commun. math. Phys. 44, 223—235 (1974) 5. Fortuin,C.M, Ginibre,J., Kastelyn,P.W.: Correlation inequalities on some partially ordered sets. Commun. math. Phys. 22, 89—103 (1971) 6. 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