Proof of Rounding by Quenched Disorder of First Order Transitions in Low-Dimensional Quantum Systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Proof of Rounding by Quenched Disorder of First OrderTransitions in Low-Dimensional Quantum Systems
Michael Aizenman ∗ Departments of Physics and Mathematics,Princeton University, Princeton NJ 08544-8019
Rafael L. Greenblatt † Dipartamento di Matematica, Universit`a degli Studi Roma Tre,Largo San Leonardo Murialdo 1, 00146 Roma, Italy
Joel L. Lebowitz ‡ Departments of Mathematics and Physics,Rutgers University, Piscataway NJ 08854-8019 (Dated: November 23, 2011)
Abstract
We prove that for quantum lattice systems in d ≤ T = 0. For systems with continuous symmetry the statement extendsup to d ≤ PACS numbers: 75.10.Jm,64.60.De ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION Quenched disorder is known to have a pronounced effect on phase transitions in lowdimensional systems. As Imry and Ma pointed out in 1975 [1], disorder can prevent theappearance of discontinuities, and of long range order, associated with first order transitions.Nonetheless, for a number of years there was considerable uncertainty about the mechanismsinvolved in such a “rounding effect”, and consequently also of the circumstances in whichthis effect appears [2]. Rigorous results in a series of works in the 1980s [3, 4] established theconditions for rounding in classical systems. The situation in quantum systems remaineduncertain, with recent suggestions [5] that for such the effect might not have the same generalcharacter as in the context of classical statistical mechanics.In this work we present a modified version of the argument of Ref. 4 which allows us(as previously announced in Ref. 6) to extend the classical results to quantum systems. Itwas not obvious that this should be possible, since the statistical mechanical descriptionof a d dimensional quantum systems often has the appearance of a ( d + 1) dimensionalclassical system, and in addition some of the tools used in the classical case have no clearquantum generalizations. The latter difficulty is resolved here by focusing on thermodynamicquantities, bypassing some of the subtle issues related to equilibrium states which played arole in Ref. 4.Before presenting the formalism in which the general result is expressed, we start withsome specific examples to which the results apply. II. EXAMPLES OF THE ROUNDING EFFECTA. Transverse Field Ising Model
The transverse-field Ising model in a random longitudinal field is defined by the Hamil-tonian H TFIM = − J X h x,y i σ ,x σ ,y − λ X σ ,x − X ( h + ǫη x ) σ ,x , (2.1)where σ j,x are Pauli matrices describing the j = 1 , , x , andthe first summation is over pairs of neighboring sites x, y ∈ Z d . The symbol η x representsa random field in the j = 3 direction, whose values at different sites are given by indepen-2ent and identically distributed (i.i.d.) random variables, satisfying certain mild regularityconditions (to be specified in Section IV B below).In d ≥ ǫ = 0) exhibits spon-taneous (or residual) magnetization at h = 0 if J/λ and
J β are large enough. In d = 1dimension this occurs only at zero temperature, T ≡ β − = 0, i.e. in the ground state. Moreexplicitly, existence of spontaneous magnetization means that M ( β ) := 12 [ M ( β ) − M − ( β )] > M ± ( β ) are the two limits of the Gibbs state expectation values in finite (rectangular)domains Γ ⊂ Z d , M ± ( β ) := lim h → ± lim Γ ր Z d | Γ | *X x ∈ Γ σ ,x + h,β Γ . (2.3)Since the h limit is taken last, the boundary conditions on the finite systems do not affectthe value of the limits in (2.3) (for the convenience of translation covariance our defaultchoice is periodic boundary conditions). In view of the symmetry of the model, M = − M − = M ( β ).The general result which is proven below (Theorem IV.1) implies: In d ≤ , the system described above, with the Hamiltonian (2.1) , at ǫ = 0 has M ( β ) = 0 at all β ≤ ∞ . In particular, even if there is more than one ground state, they all yield thesame bulk-average value for the mean magnetization. This can also be restated in thermodynamic terms: Letting F ( h, β ) denote the ther-modynamic limit of the specific free energy, which is defined below in (3.6), the differentmagnetizations can be expressed as directional derivatives [7] M ± ( β ) = ∂ F ∂h ± (cid:12)(cid:12)(cid:12)(cid:12) h =0 . (2.4)Non-vanishing spontaneous magnetization corresponds therefore to a discontinuity of a firstorder derivative of the free energy, here at h = 0, and thus to a first order phase transitionin the terminology of Ehrenfest [8].Note that the order parameter for the phase transition is the expectation of the volumeaverage of the local operator σ ,x which in the Hamiltonian (2.1) appears coupled to therandom field. Such conjugacy is the essential condition for the general result presented here.The theorem does not imply that the rounding of the magnetization in the ‘ z -direction’3ould occur when random terms are added only to the transverse field λ . Indeed, as haslong been known[9], when only such disorder is present spontaneous magnetization doespersist in two dimensions. Such transversal disorder can have other subtle effects on theferromagnetic-paramagnetic transition[10], but these are beyond the scope of the presentwork. B. Isotropic Heisenberg model
The isotropic, nearest-neighbor Heisenberg ferromagnet in a random magnetic field isdescribed by the Hamiltonian H Heis = − J X h x,y i ~σ x · ~σ y − X x ( ~h + ǫ~η x ) · ~σ x (2.5)where J is a positive real number, ~σ x is a vector spin operator associated with the site x ∈ Z d ,and ~η x is a magnetic field which varies randomly from site to site. We now assume that theserandom field variables are not only independently and identically distributed, but also thattheir distribution is rotation invariant. We also assume that ~η x is nonzero with probability 1.In the case of a uniform magnetic field ( ǫ = 0) it is universally believed that for d ≥ ~h = ~ J -dependent criticaltemperature (although a rigorous proof is lacking). Letting h·i ~h,β Γ now denote the Gibbsstate of this system on a finite domain Γ ⊂ Z d with periodic boundary conditions at inversetemperature β , spontaneous magnetization can be expressed, similarly to the previous cases,by the statement that ~M ( β ) := lim h ց lim Γ ր Z d | Γ | *X x ∈ Γ ~σ x + h ˆ e,β Γ = 0 (2.6)where ˆ e is any fixed unit vector (indicating the direction along which ~h is taken to ~ ~M ( β ) = lim h ց ddh F ( h ˆ e, β ) . (2.7)Note that as in the previous example the randomness is conjugate to the quantity whosedensity is the magnetization considered here.4heorem IV.2, below, has the following implication for this system at ǫ = 0: In d ≤ , the random-field Heisenberg ferromagnet described above has ~M ( β ) = 0 (andhence F ( ~h, β ) is differentiable in ~h at ~h = ~ ) for any β including β = ∞ . As will be seen in Section IV C, the vanishing of the spontaneous magnetization impliesalso the absence of ferromagnetic long range order.
III. A GENERAL FORMULATIONA. The system and its Hamiltonian
We consider here systems on homogeneous d dimensional lattices, which for simplicitywe take to be Z d . Associated with each lattice site x ∈ Z d is a quantum system whose statespace is isomorphic to a common finite dimensional Hilbert space H . The local systemsare coupled through a Hamiltonian which for a finite region Γ ⊂ Z d , with free boundaryconditions, takes the form H h,ǫ,η Γ , = X A ⊂ Γ Q A + X x ∈ Γ ( h + ǫη x ) κ x . (3.1)Here Q A is an operator which acts on the quantum degrees of freedom in A (i.e. it isdescribed by an operator acting in the space ⊗ x ∈ A H x ). The interaction is assumed tobe translation invariant ( Q T x A = U † x Q A U x , with T x denoting translations on the latticeand U x the corresponding unitary operators). Unless stated otherwise it is assumed herethat the interactions are of finite range (i.e., Q A = 0 for all sets with diam A > R , atsome finite R ). However the stated results apply also to a a broader class of long rangeinteractions with a sufficiently fast power law decay, which is described in Appendix A. Inparticular, Theorem IV.1 applies to two-body interactions satisfying k Q x,y k ≤ C/ | x − y | d/ ,and Theorem IV.2 applies if k Q x,y k ≤ C/ | x − y | α with some α > d − η x . Their variance is fixedat 1, so that the strength of the disorder is controlled by the parameter ǫ . The terms η x multiply are assumed to be of the form κ x ≡ U † x κ U x , with κ an operator which acts on afinite cluster of sites, whose size may be greater than 1. It should be noted that { η x } appearin (3.1) as additions to a uniform parameter h . This parameter plays an essential role forthe results presented here. The symbol η without the subscript indicates the collection of5andom fields at all sites, and η Γ denotes the random fields in some finite region Γ ⊂ Z d . Insystems with several families of independent random terms our results hold for the observableassociated with each family considered separately, but we will not discuss this point here(see [11]).Hamiltonians with other boundary conditions, H h,ǫ,η Γ ,B , are defined analogously, so that theterms within Γ are the same as for free boundary conditions, while at a finite distance fromthe boundary there can be additional terms which are uniformly bounded in norm and inthe size of the clusters of the directly affected sites. Among the allowed options are periodicboundary conditions, denoted by B = per, which will be the default choice if no other isexplicitly specified.As usual, the finite volume partition function is denoted Z Γ ,B ( h, β, ǫ ; η ) := Tr exp( − βH h,ǫη Γ ,B ) , (3.2)and the corresponding Gibbs state is h ... i h,β,ǫ Γ ,B ( η ) := Tr (cid:16) ... e − βH h,η Γ ,B (cid:17) Z Γ ,B ( h, β, ǫ ; η ) . (3.3)When the values of the superscripts is deemed obvious from the context, they will sometimesbe omitted. B. The quenched free energy and its infinite volume limit
The free energy for a finite system is denoted here by F Γ ,B ( h, β, ǫ ; η ) := − β log Z Γ ( h, β, ǫ ; η ) , (3.4)We will mostly be concerned with square domains Γ K = [ − K, K ] d ∩ Z d with the periodicboundary conditions, and so we introduce the abbreviation F hK ( η ) ≡ F Γ K , per ( h, β, ǫ ; η ) . (3.5)The free energy density, per unit volume, is F Γ ,B ( h, β, ǫ ; η ) := F Γ ,B ( h, β, ǫ ; η ) / | Γ | . (3.6)where | Γ | is the number of sites in Γ. As the notation suggests, for finite systems the freeenergy depends on the choice of boundary conditions and on the disorder variables. In the6hermodynamic limit, however, the dependence of its density on the boundary conditionsdisappears and so does the dependence of its typical value on the disorder variables. Thefollowing generally known result (we refer here to such statements as Propositions) appliesto all systems of the type described above. Proposition III.1.
If the random fields are of finite variance and form a translation in-variant and ergodic process, then for any β ∈ [0 , ∞ ] there is a full measure set N of fieldconfigurations for which the infinite volume limit F ( h, β, ǫ ) := lim L →∞ F Γ L ,B ( h, β, ǫ ; η ) (3.7) exists for all h and its value is independent of η and the boundary conditions B . Since much of the early discussion of this result [12] was limited to the classical case letus add that the argument used in that context extends directly also to quantum statisticalmechanics. In essence: the free energy density can be approximated up to an arbitrarilysmall correction, of order O ( L − ) (see Inequality (6.1)) by that of a system obtained bypartitioning the space into large blocks (of length L ) among which the couplings wereremoved. For the approximants the density converges by either the law of large numbers, incase of independent { η x } , or by the ergodic theorem in the more general case stated above.That implies Proposition III.1.The following remarks are of relevance: i. Since convexity is automatically inherited by the limit, F ( h, β, ǫ ) is concave as afunction of h . ii. The case β = ∞ corresponds to the ground state energy: for almost every η the limit F ( h, ∞ , ǫ ) := lim β →∞ F ( h, β, ǫ ) (3.8)exists and is equal to the limit of the energy densities of the finite volume ground states ofthe random Hamiltonian (whose value is independent of the boundary conditions). In otherwords, for the free energy the limits β → ∞ and L → ∞ are interchangeable. iii. The uniqueness of the free energy density does not extend to uniqueness of the Gibbsstates, or ground states in case β = ∞ . The question which our results address is whetherthe different Gibbs states (or ground states) of the given Hamiltonian can differ in theirmean magnetization, that is in the volume averages of h κ x i .7 v. The following condition will allow us to weaken other assumptions. We will say thatour system satisfies the weak FKG condition with respect to κ if h κ x i h Γ ( η ) ≥ h κ x i h ′ Γ ( η ′ ) (3.9)whenever x ∈ Γ, h ≥ h ′ , and η y ≥ η ′ y for all y ∈ Γ. This condition is known to hold in thetransverse field Ising model described in Section II A [9].
C. Quenched disorder with continuous symmetry
For systems with a continuous symmetry, the Hamiltonian (with the free boundary con-ditions) is of the form H ~h,,ǫ,~η Γ , = X A ⊂ Γ Q A + X x ∈ Γ ( ~h + ǫ~η x ) · ~κ x , (3.10)where the Q interaction is not only translation invariant but also rotation invariant in thefollowing sense. We assume that the group of rotations SO ( N ) has a representation by localunitary operators whereby for each R ∈ SO ( N ) there is a corresponding b R x acting in H x .The interaction Q then is rotation invariant if and only if Q A = Y x ∈ A b R − x ! Q A Y x ∈ A b R x ! (3.11)for all finite A ⊂ Z d and all R ∈ SO ( N ). At each site x ∈ Z d instead of a single operator κ x there is now a collection of operators ~κ x = ( κ x, , ..., κ x,N ) which under the above actionof SO ( N ) transform as a vector. That is: for each R ∈ SO ( N ) and ~v ∈ R N b R − x ( ~v · ~κ x ) b R x = ( R~v ) · ~κ x . (3.12)Hamiltonians with other boundary conditions, Gibbs states, etc. are defined in analogywith the previous section. (In the classical case [4] it was also necessary to assume thatthe interactions transformed smoothly under nonuniform rotations, however for a finite-dimensional local state space this is automatically true.)Likewise, the random fields are now given by random vectors ~η x in R N . Clearly rotationinvariance is lost in the presence of such terms in the Hamiltonian. However, we will assumethat the symmetry is retained in the distributional sense, i.e., that for any rotation R therandom variables ~η x and R~η x have the same distribution.8 V. STATEMENT OF THE MAIN RESULTSA. Two perspectives on st order phase transitions As was done in the context of the examples of Section II, the general statements will bepresented in two equivalent ways: in their thermodynamic formulations, which is the levelat which the results are derived here, and then in the language of statistical mechanics,i.e. properties of Gibbs states. Since the results concern (the absence of) first order phasetransitions let us recall the latter’s dual manifestations.1. In thermodynamic terms, a 1 st order phase transition is associated with the disconti-nuity of the first derivative of the free energy with respect to one its parameters. Bydefault this parameter will be denoted here by h .2. In the terms of statistical mechanics a 1 st order phase transition is expressed in the non-uniqueness, among the infinite volume Gibbs equilibrium states, of the bulk density ofsome extensive quantity. Our discussion concerns the case when this is the quantitywhose coupling parameter in H is the field h which is randomized by the disorder.One occasionally finds that the equivalence of these two statements is not fully trusted inthe present context (of disordered systems)[13]. We shall therefore briefly recall below, inSection IV C, some pertinent known results. B. Thermodynamic formulation
Our first result concerning systems of the type described in Section III A, with η x givenby independent and identically distributed (i.i.d.) random variables, is: Theorem IV.1.
In dimensions d ≤ , assuming the variables η x are i.i.d. with absolutelycontinuous distribution and with more than two finite moments, the quenched free energydensity F is differentiable in h at all values of h , ǫ = 0 , and β ≤ ∞ .Furthermore, for β < ∞ , the assumption of absolute continuity can be relaxed, requiringinstead that the distribution of η has no isolated point masses, or alternatively that the systemsatisfies the weak FKG property with respect to κ . q is said to be absolutely continu-ous if it is of the form ρ ( q ) dq , with a density function ρ ( q ). In particular, the correspondingmeasure on R has neither ‘delta function’ terms (point masses), nor any component whichis supported on a Cantor type fractal set. Point masses are isolated if they are separatedfrom the continuous part of the distribution. The statement that the random variables η x have more than two finite moments means thatAv | η x | δ < ∞ (4.1)for some δ >
0, where Av denotes the average over the random fields.As in the classical case, in the presence of continuous symmetry, as understood above,the ‘rounding effect’ extends to higher dimensions:
Theorem IV.2.
In dimensions d ≤ , any isotropic system of the type described in Sec-tion III C, with the random terms being independent with an identical, rotation invariantdistribution with more than two finite moments and Prob( ~η = ~
0) = 0 , the quenched freeenergy density F is differentiable in ~h at ~h = ~ , for any ǫ = 0 and β ≤ ∞ .For β < ∞ , the conclusion still holds if there is a nonzero probability of a random termbeing zero. We note that an apparently weaker condition on the distribution of ~η compared to The-orem IV.1 is adequate because what will ultimately be important is the distribution of thecomponent in an arbitrary direction. With the assumption of an isotropic distribution forthe vector, the components can easily be seen to satisfy the stronger conditions used inTheorem IV.1 or Proposition VII.2. C. Statistical mechanical implications (no long range order)
In this section, we establish the relationship between phase transitions (understood interms of non-differentiability of free energy) and long range order, culminating in corollariesof Theorems IV.1 and IV.2 which reframe the results in statistical mechanics terms.We say that a system exhibits long range order with order parameter κ (for some set of pa-rameters and disorder variables) if the mean value of the bulk averages ¯ κ Λ = | Λ | − P x ∈ Λ κ x ,for Λ → Z d , depends on the boundary conditions B with which the infinite volume state is10onstructed. That is: if there are two sequences of cubic domains Λ and Γ increasing to Z d ,and two sets of boundary conditions which yield different values forlim Λ ր Z d lim Γ ր Z d h ¯ κ Λ i Γ ,B . (4.2)When the sensitivity to the boundary conditions is found for the full bulk averages, i.e.if different values occur for lim Γ ր Z d h ¯ κ Γ i Γ ,B , (4.3)we say that there is long - long range order. Remark:
If for a given system there are multiple limits of the expectation values in (4.2),then the system has multiple infinite volume KMS states [14]. Convex combinations of thedifferent states (which will also form KMS states) will exhibit non-decaying correlations of κ : h κ x κ y i − h κ x ih κ y i 6→ | x − y | → ∞ . The latter condition provides another aspect of(“short”-) long range order, which is often used as its definition.The basis of the connection to the thermodynamic quantities discussed in the previoussection is the relation ∂ F h Γ ,B ( η ) ∂h = 1 | Γ | X x ∈ Γ h κ x i h Γ ,B ( η ) . (4.4)A useful implication of convexity is that if a sequence of differentiable convex functions,such as F of Theorem III.1, converges pointwise (in h ) then also their derivatives convergeto the derivative of the limiting function, wherever that function is differentiable. Withoutassuming differentiability of the limiting function one may still conclude that all the deriva-tives’ accumulation points lie in the interval spanned by the left and right derivative of thelimiting function. (An elementary proof of that can be obtained by considering the relationsamong quotients of the form [ F ( h ) − F ( h )] / [ h − h ] for suitably chosen collections ofintervals [ h , h ].) This has the following relevant implication. Proposition IV.3.
Under the assumptions of Proposition III.1, for any set of the parame-ters ( β, h, ε ) at which F is differentiable in h lim L →∞ | Γ L | X x ∈ Γ L h κ x i h Γ L ,B ( η ) = ∂ F ∂h (4.5) for almost every realization of the disorder η , and any choice of the boundary conditions B .Furthermore, also: lim L →∞ lim K →∞ | Γ L | X x ∈ Γ L h κ x i h Γ K ,B ( η ) = ∂ F ∂h . (4.6)11 ithout assuming differentiability, one may still conclude that LIM
L,K →∞ ; K ≥ L | Γ L | X x ∈ Γ L h κ x i h Γ L ,B ( η ) ∈ (cid:20) ∂ F ∂h − , ∂ F ∂h + (cid:21) , (4.7) with LIM denoting the collection of accumulation points, for different boundary conditions,and possibly different sequences of volumes, and ∂∂h ± denoting one-sided derivatives withrespect to h . Of particular relevance for us is the conclusion that when F is differentiable there isonly one possible value for the limit (4.2), which by Proposition III.1 is independent of theboundary conditions. For classical systems more can be said: if F is differentiable not onlydo the mean values of the observable ¯ κ Γ converge, but the distribution of this quantity withrespect to the Gibbs state collapses onto a point. Such a stronger statement concerningquantum fluctuations is not known to be true. Nevertheless, Equation (4.5) holds also forthe quantum systems. Proof.
The relations (4.5) and (4.7) follow directly by the above stated property of convexfunctions (i.e., the implication for the derivatives of convex functions’ pointwise conver-gence).To prove (4.6) and the corresponding extension of (4.7), let F h,δ, ΛΓ ,B denote the free energywith the fixed field within Λ changed by δ , so that1 | Γ L | X x ∈ Γ L h κ x i h Γ K ,B ( η ) = 1 | Γ L | ∂F h,δ, ΛΓ K ,B ∂δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ =0 (4.8)A standard estimate, which is presented in Proposition VI.1 below, implies that F h,δ, Γ L Γ K ,B − F h Γ K ,B = F h + δ Γ L , − F h Γ L , + O (cid:0) k V η Γ L k (cid:1) (4.9)uniformly in K , where V ηL denotes the terms in the Hamiltonian crossing the boundary ofΓ L , i.e. V η Γ := X A : A ∩ Γ / ∈{∅ ,A } Q A + X x ∈ ∂ Γ ( h + ǫη x ) κ x . (4.10)Then for any η in the full measure set N on which the limits in Theorem III.1 are defined,this implies that lim L,K →∞ ; K ≥ L (cid:16) F h,δ, Γ L Γ K ,B − F h Γ K ,B (cid:17) / | Γ L | = F ( h + δ, β ) − F ( h, β ) . (4.11)12ach of these differences is a concave function of δ , so the above mentioned convexity argu-ment gives LIM L,K →∞ ; K ≥ L ∂∂δ F h,δ, Γ L Γ K ,B / | Γ L | ∈ (cid:20) ∂ F ∂h − , ∂ F ∂h + (cid:21) (4.12)and the desired results follow immediately from Equation (4.8).In particular, the results which are formulated in Theorems IV.1 and IV.2 as statementsof differentiability of the free energy directly imply the following. Theorem IV.4.
In dimensions d ≤ , under the assumptions of Theorem IV.1 the systemalmost certainly does not exhibit (short or long) long range order.Furthermore, for systems with continuous symmetry which meet the assumptions of The-orem IV.2, in dimensions d ≤ almost certainly there is no (short or long) long range orderat ~h = ~ . V. THE FREE-ENERGY-DIFFERENCE FUNCTIONAL
The above discussion reduces our main results to Theorems IV.1 and IV.2. A key tool forthe proof of these theorems is a quantity which describes the differences in the finite volumecontent of free energies between two of the system’s equilibrium states, which are constructedto have the two extremal mean magnetizations, ∂ F ∂h + and ∂ F ∂h − , assuming they are not thesame. The difference, which is denoted below as G L , will be shown to satisfy contradictorybounds: i. an absolute upper bound on | G L | which is due to the observation that the twofree energies cannot differ by more than the magnitude of the interaction across the finitevolume boundary, ii. a lower bound on the fluctuations of this quantity which reflects theidea that with a systematic difference in the magnetization the states’ free energies willrespond differently to the fluctuating random fields.More specifically, the free energy differences (properly defined) will be shown to satisfy: | G L | ≤ CL d − + DL d/ , (5.1)with D = 0 for finite range interactions. In the presence of continuous symmetry for finiterange interactions the bound is improved to: | G L | ≤ CL d − . (5.2)13or the opposite bound it will be shown that if there is a first order transition then G L /L d/ converges to a normally distributed random variable with variance b > G L L d/ → N (0 , b ) . (5.3)The construction of a quantity with the above characteristics is the main subject of thissection. The contradiction between the two bounds yields the main result.Estimates similar to the above form the basis of the argument of Imry and Ma[1], andthe more precise argument outlined above is similar to the one which was employed forthe analysis of in the classical case [4]. However the technique introduced there for theconstruction of the auxiliary quantity G L employed probability measures over states, definingwhat has since been called metastates [15]. Unfortunately, although metastates can bedefined for quantum systems [16], not all the steps taken in Ref. 4 for the construction of G L have such an extension. The progress made in this work is enabled by the observationthat one can define a suitable quantity G L in terms of just free energy differences, avoidingmore delicate issues of quantum states. This makes it possible to formulate a proof parallelto the classical case (and indeed the nonrigorous arguments which inspired it).As a simple illustration of the concept one may first consider the zero temperature case ofa system such as the random field Ising model with two distinct states, labeled by + and − ,of different mean magnetizations and with only the ‘canonical dependence’ on η (cf. [4]).In that case, for G L one may take the difference in the two states’ energy contents in finitevolumes: E L ( η ) := E + ,L ( η ) − E − ,L ( η ) = ǫ * X x ∈ Γ L η x κ x + + − ǫ * X x ∈ Γ L η x κ x + − ≈ ǫ X x ∈ Γ L η x ! M → ǫM L d/ N (0 , , (5.4)where the last step, in which the central limit theorem is invoked, is valid only to the extentthat one may ignore the adaptation of the system to the disorder. (A more precise statementis possible taking into account the dependence of the ± states on η [4]). The above providesonly a suggestive example in the context of a special (and classical) system at T = 0. Weturn now to a more general definition which would be suitable for our purpose.Our choice of G L ( η ) for the general case is based on the following free energy ‘second14ifference’ (in the field h , and disorder η Γ L ): b G δL,K ( η L ) := 12 Av (cid:2) F h + δK ( η ) − F h + δK ( r L ( η )) − F h − δK ( η ) + F h − δK ( r L ( η )) (cid:12)(cid:12) η L (cid:3) , (5.5)where r L η is the field obtained by setting η x = 0 for all x ∈ Γ L , η L is the restriction of η to Γ L and Av [ ·| η L ] is the conditional average over the random terms outside Γ L , i.e. with η Γ L held constant. It should be noted that F K (which is defined in (3.5)) refers to the freeenergy with the periodic boundary conditions. This choice assures translation covariance,which is mentioned below (condition 4. in Lemma (V.1)) and used in the argument.When the double limit lim δ ց lim K →∞ b G δL,K ( η L ), exists (for all η L ), the quantity definedby it has properties we desire of G L ( η L ). (The order of the limits is important here: taking δ ց − states.) Inconveniently, the limits are not generally known (or expected in full generality)to exist. However compactness arguments can be applied to prove sufficient convergencealong subsequences. The essential statement is the following. We use here the ℓ -Lipschitzseminorm |||·||| (in lieu of a uniform bound on the derivative) which is defined as ||| f ||| := sup η,η ′ ∈E < k η − η ′ k < ∞ | f ( η ) − f ( η ′ ) |k η − η ′ k , (5.6)with k η k := P x ∈ Z d | η x | . Lemma V.1.
For any β ≤ ∞ , there are sequences K j → ∞ and δ i → such that the limits G L ( η L ) := lim i →∞ lim j →∞ (cid:16) b G δ i L,K j − Av h b G δ i L,K j i(cid:17) (5.7) exist for all L and η , and have the following properties:1. Av G L ( η L ) = 0 G L ( η L ) depends on the values of η x only for x ∈ Γ L ||| G L ||| ≤ ǫ Av [ G L | η Λ ] = G L ′ ( T x η ) whenever T − x Γ L ′ = Λ ⊂ Γ L G (that is, G L with L = 1 , which is a function of one variable) has a distributionalderivative G ′ satisfying (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂G ∂η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ βǫ and . Av G ′ ( η ) = ǫ (cid:0) ∂ F ∂h + − ∂F∂h − (cid:1) Furthermore, if the system satisfies the weak FKG condition with respect to κ , then G ismonotone. For β = ∞ the same results hold assuming that the distribution of η is absolutelycontinuous. The proof is postponed to section VIII, as it is based on compactness arguments par-alleling those in Ref. 4. Before turning to it, in the next two sections we will show thatany such G L ( η L ) must obey the upper bound (5.1), and where appropriate (5.2), and alsosatisfies (5.3). VI. UPPER BOUNDS ON THE FREE ENERGY DIFFERENCE
We will frequently use the following estimate to control the effects of changes in theHamiltonian on the free energy.
Proposition VI.1 ([7]) . For any Hermitian matrices
C, D of the same finite size, (cid:12)(cid:12) log Tr e C − log Tr e D (cid:12)(cid:12) ≤ k C − D k (6.1) where k · k is the operator norm. As a simple application, this means that the change in free energy for the system definedby the Hamiltonian (3.1) when boundary conditions are added is bounded in terms of thetotal of the norms of all terms in the interaction which cross the boundary.
A. A general surface bound
The following lemma shows generally that the free energy difference due to random fieldfluctuations in a finite region can be bounded by the norm of the interaction terms crossingthe surface of that region, and that for short range interactions this is proportional to thearea of that surface.
Lemma VI.2.
Assuming the interactions are either of finite range or satisfy (A.1) (ofAppendix A), for any pair of sequences δ i → , K j → ∞ for which the following sequenceof functions converges for all η L G L ( η L ) = lim i →∞ lim j →∞ (cid:16) b G δ i L,K j − Av h b G δ i L,K j i(cid:17) (6.2)16 he limiting function satisfies (with some C, D < ∞ ): | G L | ≤ CL d − + DL d/ . (6.3) Proof.
We will show that | b G δL,K | ≤ cL d − + bL d/ + O ( δL d ) (6.4)with c , b independent of L , K , and δ . The desired result holds then with C = 2 c , D = 2 b .To see this, we let Λ L be the smallest cube in Z d so that κ x acts within Λ L for all x ∈ Γ L ,and then split the Hamiltonian into contributions from Λ L and Γ K \ Λ L and terms connectingthe two. We consider the quantity F hK | L ( η ) := − β log Tr exp (cid:16) − β h H h,η Λ L , + H h,η Γ K \ Λ L, ∗ i(cid:17) , (6.5)where the subscript 0 refers to free boundary conditions, and the subscript ∗ refers to periodicboundary conditions on the edge of Γ K and free boundary conditions on the edge of Λ L . TheHamiltonian in Equation (6.5) differs from H h,η Γ K only in the absence of terms connecting Λ L to the rest of Γ K , which as in Equation (4.10) we denote by V η Λ L . Then by Proposition VI.1 (cid:12)(cid:12) F hK | L ( η ) − F hK ( η ) (cid:12)(cid:12) ≤ (cid:13)(cid:13) V η Λ L (cid:13)(cid:13) , (6.6)and therefore (cid:12)(cid:12) F h + δK ( η ) − F h + δK ( r L ( η )) − F h − δK ( η ) + F h − δK ( r L ( η )) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) F h + δK | L ( η ) − F h + δK | L ( r L ( η )) − F h − δK | L ( η ) + F h − δK | L ( r L ( η )) (cid:12)(cid:12)(cid:12) + 4 (cid:13)(cid:13) V η Λ L (cid:13)(cid:13) . (6.7)Each F K | L is a sum of a part from Λ L and a part from the rest of Γ L ; the latter canceleach other exactly, and the former can be arranged in two pairs which differ only in theconstant field term which has norm of order δL d . Taking the average conditioned on η Γ L and noting that the definition of Λ L makes V η Λ L independent of η Γ L , (cid:12)(cid:12)(cid:12) b G δL,K ( η L ) (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13) V η Λ L (cid:13)(cid:13) + O ( δL d ) (6.8)for all η , and since the δ term is uniform in K , | G L ( η L ) | ≤ (cid:13)(cid:13) V η Λ L (cid:13)(cid:13) . (6.9)This reduces Inequality (6.3) to a comparable bound on V η Λ L , i.e.Av (cid:13)(cid:13) V η Λ L (cid:13)(cid:13) ≤ cL d − + bL d/ . (6.10)For finite range interactions it is obvious that Av (cid:13)(cid:13) V η Λ L (cid:13)(cid:13) ≤ cL d − , and Appendix Adiscusses more general conditions under which Inequality (6.10) holds.17 . An improved bound for systems with continuous symmetry For systems with continuous symmetry the stronger bound of Inequality (5.2) can beobtained through the analysis of soft deformations. The argument is similar to the onewhich was carried out in the classical context in Ref. 4. Although the estimate may at firstsight appear to be simply a calculation of the Bloch spin wave energy, that alone would notyield the desired result since the first order term is potentially much larger than the requiredbound. However, by comparing the distortion energies of opposite deformations one findsthat in any situation for one of the distortions the first order term is of the desired sign.Therefore the analysis can be limited to the second order term - and that yields the claimedbound.We need to modify the definition of G L slightly to accommodate random vector fields.The argument will be based on the effect of the fluctuations of a single component, describedby expressions where the others have been averaged out. We let b G δ ˆ eL,K ( η L ) = 12 Av (cid:2) F δ ˆ eK ( ~η ) − F δ ˆ eK ( r L ( ~η )) − F − δ ˆ eK ( ~η ) + F − δ ˆ eK ( r L ( ~η )) (cid:12)(cid:12) ˆ e · ~η L (cid:3) (6.11) G L ( η L ) = lim i →∞ lim j →∞ (cid:16) b G δ i ˆ eL,K j ( η ) − Av h b G δ i ˆ eL,K j (ˆ e · ~η ) i(cid:17) (6.12)where ˆ e is an arbitrary unit vector; note that the rotation symmetry of the system and ofthe distribution of ~η mean that the right hand sight of the last expression is independent ofˆ e . Lemma VI.3.
For any system with continuous symmetry and isotropic disorder as describedin Section III, of interactions which are either of finite range or satisfy (A.7) , at ~h = 0 : | G L ( η L ) | ≤ CL d − , (6.13) with C < ∞ , uniformly in L and η L .Proof. We shall derive (6.13) through uniform bounds on g δ ˆ eL,M ( ~η ) := Av (cid:2) F δ ˆ eK ( ~η ) − F − δ ˆ eK ( ~η ) (cid:12)(cid:12) ˆ e · ~η L (cid:3) , (6.14)for which we shall show that Assumption (A.7) implies | g δ ˆ eL,K ( ~η ) | = O ( L d − ) . (6.15)18he two quantities are related through G L ( η L ) = 12 lim i →∞ lim j →∞ (cid:16) g δ i ˆ eL,K j ( η ) − Av g δ i ˆ eL,K j (ˆ e · ~η ) (cid:17) , (6.16)and hence (6.15) implies the desired statement.Let ρ be the generator (in so ( N )) of a rotation in a plane containing ˆ e , and for each x ∈ Z d let ρ x be the generator of the corresponding rotation in the single-site algebra A x (Unlike in Ref. 6, we will use the “mathematician’s” convention that rotations are given by e θρ , so that ρ is an antihermitian matrix and ρ x is an antihermitian operator). We introducethe slowly varying angles θ x := , x ∈ Γ L k x k − LL π, < d L ( x ) < Lπ, d L ( x ) ≥ L , (6.17)where d L ( x ) is the distance from x to Γ L in the largest-component metric, i.e. d L ( x ) := min y ∈ Γ L k x − y k ∞ . (6.18)We also introduce the associated rotations on fields and on A Γ defined by b R x := e θ x ρ (6.19)( R θ ( ~η )) x ≡ R x ~η x (6.20) b R θ = O x ∈ Γ e θ x ρ x . (6.21)where Γ is the relevant finite subset of Z d which can be inferred from the context. b R θ is unitary, and so we can rewrite the free energy F − δ b eK ( ~η ) appearing in (6.14) as F − δ b eK ( ~η ) = − β log Tr exp (cid:16) − β b R − θ H − δ b e,~η Γ K b R θ (cid:17) . (6.22)At this stage we could carry out a spin-wave analysis, writingAv (cid:2) F − δ b eK ( ~η ) (cid:12)(cid:12) ˆ e · ~η L (cid:3) = − β Av h log Tr exp (cid:16) − β [ H δ b e,~η Γ K + ∆ H θ ] (cid:17)(cid:12)(cid:12)(cid:12) ˆ e · ~η L i , (6.23)which by Lemma VI.1 implies | g δ b eL,K ( ~η ) | ≤ k ∆ H θ k . (6.24)However this will not lead to the desired result, since k ∆ H θ k includes ‘first order’ terms(in ∇ θ ) which scale as L d − as well as the ‘second order’ terms (and higher) which scale as L d − . To cancel the former we rewrite F − δ b eK ( ~η ) using an opposite rotation and average thetwo expressions, to obtain 19v (cid:2) F − δ b eK ( ~η ) (cid:12)(cid:12) ˆ e · ~η L (cid:3) = − β Av h log Tr exp (cid:16) − β [ H δ b e,~η Γ K + ∆ H θ ] (cid:17) + log Tr exp (cid:16) − β [ H δ b e,~η Γ K + ∆ H − θ ] (cid:17)(cid:12)(cid:12)(cid:12) ˆ e · ~η L i . (6.25)A combination of the Cauchy-Schwarz inequality, the Golden-Thompson inequality [17]and Lemma VI.1 yields the general relationlog Tr e A −
12 log Tr e B −
12 log Tr e C = log (cid:18) Tr e A (Tr e B ) / (Tr e C ) / (cid:19) ≤ log (cid:18) Tr e A Tr e B/ e C/ (cid:19) ≤ log Tr e A − ( B + C ) / ≤ (cid:13)(cid:13)(cid:13)(cid:13) A − B + C (cid:13)(cid:13)(cid:13)(cid:13) (6.26)for arbitrary Hermitian matrices A, B, C . Applying this to Equation (6.25) gives g δ b eL,K ( ~η ) ≤ k ∆ H θ + ∆ H − θ k . (6.27)We now write out the terms in the rotated Hamiltonian in Equation (6.22) as b R − θ H h,~η Γ b R θ = b R − θ X A P Γ ( Q A ) + X x ∈ Γ ( − δ b e + ǫ~η x ) · P Γ ( ~κ x ) ! b R θ , (6.28)where P Γ ( Q A ) and P Γ ( ~κ x denote the terms appearing in the Hamiltonian with periodicboundary conditions, obtained by mapping each site outside Γ to the corresponding sitewithin Γ.Since ~κ are vector operators (recall Equation (3.12)), ~η x · (cid:16) b R − θ ~κ x b R θ (cid:17) = ~η x · R θ ( ~κ ) x = (cid:2) R − θ ( ~η ) x (cid:3) · ~κ x ; (6.29)inside Γ L there is no rotation, and outside we are performing an average with respect to anisotropic distribution, so this term makes no contribution to ∆ H .As for the fixed field terms, we have b e · (cid:16) b R − θ ~κ x b R θ (cid:17) = (cid:0) R − x b e (cid:1) · ~κ x . (6.30)The choices of ρ and θ were intended precisely to make R x b e = − b e for d L ( x ) > L ; and forthe remaining (3 L ) d sites we have (cid:13)(cid:13)(cid:13)b e · (cid:16) b R − θ ~κ x b R θ (cid:17) + b e · ~κ x (cid:13)(cid:13)(cid:13) ≤
2, so these terms make acontribution to ∆ H which is uniformly bounded in norm by 2(3 L ) d δ .20e are left with the terms arising from the transformation of the nonrandom interaction.For legibility we let Γ = Γ K for the rest of this section. For any A and any (arbitrarilychosen) x ∈ A ∩ Γ, b R − θ P Γ ( Q A ) b R θ = O y ∈ A ∩ Γ e − ( θ y − θ x ) ρ y e − θ x ρ y ! P Γ ( Q A ) O z ∈ A ∩ Γ e − θ x ρ z e ( θ z − θ x ) ρ z ! = O y ∈ A ∩ Γ e − ( θ y − θ x ) ρ y ! P Γ ( Q A ) O z ∈ A ∩ Γ e ( θ z − θ x ) ρ z ! , (6.31)(using the rotation invariance of Q A ). Expanding the exponentials, we obtain b R − θ P Γ ( Q A ) b R θ = P Γ ( Q A ) + X y ∈ A ∩ Γ ( θ x − θ y ) ( ρ y P Γ ( Q A ) − P Γ ( Q A ) ρ y )+ O (cid:18) (diam A ) | A | L k Q A k (cid:19) , (6.32)where the estimate of the higher order terms uses | θ x − θ y | ≤ π k x − y k ∞ L ≤ π diam AL (6.33)and the observation that the n th order term in the expansion is potentially a sum of | A | n terms, as well as k P Γ ( Q A ) k ≤ k Q A k . The first order terms are odd in θ , and will cancel in∆ H θ + ∆ H − θ , with the leading term being second order. What appears there is X A ∩ Γ = ∅ (cid:16) b R − θ P Γ ( Q A ) b R θ − P Γ ( Q A ) (cid:17) = O L d X A ∋ | A | (diam A ) | A | L k Q A k ! = O ( L d − ) , (6.34)where for non-finite-range interactions the last equality requires the assumption (A.7), whichis presented in Appendix A.Then the right hand side of Inequality (6.27) is k ∆ H θ + ∆ H − θ k = O ( L d − ) + O ( δL d ) . (6.35)This provides only an upper bound on g δ b eL,M ( ~η ), rather than a bound on its absolutevalue. However it is obvious from the definition (6.14) of g that g δ b eL,M ( ~η ) = − g − δ b eL,M ( ~η ), so theneeded lower bound follows automatically. This proves (6.14), and thus, thorough (6.16),the claimed bound (6.13). 21 II. STOCHASTIC LOWER BOUNDS ON THE LOCAL FREE ENERGY DIF-FERENCE
To prove the Imry and Ma[1] phenomenon, Aizenman and Wehr[4] employed a somewhatgeneralized form of the central limit theorem which is suitable for the families of randomvariables presently under consideration [4, 18]. We will use the following reformulation ofthat statement. Since the proof is in the literature it will be omitted here; one incorporatinga correction due to Bovier[19] can be found in Ref. 11.Recalling that T x is the operation of translation by x ∈ Z d and Av [ ·| η Λ ] is the conditionalaverage conditioned on the values of { η x } x ∈ Λ =: η Λ , we present: Proposition VII.1.
Let η x be a collection of i.i.d. random variables (indexed by x ∈ Z d )with Av | η x | δ < ∞ for some δ > , and let G L be a family of real functions indexed by L ∈ N , each with the following properties:1. Av G L ( η ) = 0 ||| G L ||| ≤ ǫ (with |||·||| the Lipschitz seminorm defined by (5.6) )3. G L ( η ) depends on the values of η x only for x ∈ Γ L Av [ G L ( η ) | η Λ ] = G L ′ ( T x η Λ ) whenever T − x Γ L ′ = Λ ⊂ Γ L Then G L ( η ) /L d/ −→ N (0 , b ) (7.1) in distribution as L → ∞ , for some b satisfying Av G ≤ b ≤ ǫ (Av | η | ) . (7.2)Assumption 4 in essence states that the different functions involved are essentially thesame quantity at different scales, and that it is in some sense translation invariant. Moreprecisely: the family forms a stationary Martingale, in a multidimensional sense of thiscondition.In order to use this result, we will need to establish some control over the conditionsunder which Av G >
0. To do this, we employ the following criterion (proven in AppendixIII of Ref. 4): 22 roposition VII.2.
Let ν be a Borel probability measure on R , and let g be a continuousfunction with ||| g ||| = 1 and Z g ′ ( t ) ν ( dt ) > . (7.3) Then any of the following is a sufficient condition for R g ( t ) ν ( dt ) > :1. ν is absolutely continuous2. g is differentiable with ||| g ′ ||| finite, and ν has no isolated point masses.3. g is differentiable with ||| g ′ ||| finite, g ′ ≥ , and ν is not concentrated at a single point. To apply this we note that G is a function of one variable ( η ) which satisfies theconditions given above. In particular, it will be shown that the criteria 2. and sometimes 3.are satisfied at finite temperatures, in which case (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂G ∂η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is uniformly bounded. In addition,we will see that the weak FKG property implies monotonicity of G . VIII. CONCLUSION OF THE PROOFS OF THE MAIN RESULTSA. Existence of an infinite-system limit for free energy fluctuations
Having established the properties we require of the infinite-system free energy fluctuation G L , we now proceed to show that such a suitable quantity exists. This will be done in twooverlapping cases. We will have occasion to use the following equicontinuity arguments: Lemma VIII.1.
Let f ij : R N → R be a family of functions labeled by i, j ∈ N , eachsatisfying ||| f ij ||| ≤ and f ij (0) = 0 . Then there are subsequences i k , j l such that f ( z ) = lim k →∞ lim l →∞ f i k j l ( z ) (8.1) exists for all z ∈ R N . Furthermore the convergence is uniform on any compact domain Ξ ⊂ R N . Lemma VIII.2.
Let f ij : R N → R be as in Lemma VIII.1 above, without assuming f ij (0) =0 . Then the same result holds assuming instead that there is some c such that | f ij ( z ) | ≤ c < ∞ for all z ∈ R N , i, j ∈ N .
1. Finite temperature
We now begin with the proof of Theorem V.1 in the case β < ∞ . Firstly, note that it isobvious that if the specified limits exist, then they will have the properties labeled 1 and 2in that theorem. To define the sequence used we proceed as follows. For x ∈ Γ L , let φ xL,K,δ ( η ) := ∂ b G δL,K ∂η x = ǫ h h κ x i h + δK ( η ) − h κ x i h − δK ( η ) (cid:12)(cid:12)(cid:12) η L i ; (8.2)then evidently | φ xL,K,δ ( η ) | ≤ ǫ (8.3)and (cid:12)(cid:12)(cid:12)(cid:12) ∂φ xL,K,δ ∂η y (cid:12)(cid:12)(cid:12)(cid:12) = βǫ (cid:12)(cid:12)(cid:12) h κ x κ y i h + δK ( η ) − h κ x κ y i h − δK ( η ) − h κ x i h + δK ( η ) h κ y i h + δK ( η ) + h κ x i h − δK ( η ) h κ y i h − δK ( η ) (cid:12)(cid:12)(cid:12) ≤ βǫ . (8.4)For β < ∞ , this means that for each L , φ xL,K,δ is an equicontinuous family of functionsof the L d variables η Γ L . We can apply Lemma VIII.2 to find a decreasing sequence δ i → K j → ∞ (by applying the diagonal subsequence trick, we canchoose them to be independent of x and L ) so that ψ xL ( η ) := lim i →∞ lim j →∞ φ xL,K j ,δ i ( η ) (8.5)exists, and by uniformity of convergence G L ( η L ) := lim i →∞ lim j →∞ (cid:16) b G δ i L,K j ( η L ) − Av b G δ i L,K j (cid:17) (8.6)also exists with ∂G L ∂η x = ψ xL ( η ) (8.7)for all L , η , x ∈ Γ L . Property 3 and the relation of FKG to monotonicity then follow fromEquation (8.2), and property 5 follows from Equation (8.4).24o show property 6, we use Equations (8.7) and (8.5) to writeAv G ′ = Av ψ = Av lim i →∞ lim j →∞ φ ,K j ,δ i . (8.8)Recalling the definition of φ and using dominated convergence to exchange limits and aver-ages we have Av G ′ = lim i →∞ lim K →∞ ǫ h h κ i h + δ i K − h κ i h − δ i K i (8.9)and since the random fields are i.i.d. we can apply the Birkhoff ergodic theorem [20] toreplace the disorder average with a volume average, which is related to the derivative of thefree energy by Proposition IV.3, givingAv G ′ = ǫ (cid:18) ∂ F ∂h + − ∂ F ∂h − (cid:19) . (8.10)Note that Equation (8.2) implies that, for any L ′ < L ≤ K , φ xL ′ ,K,δ = Av (cid:2) φ xL,K,δ (cid:12)(cid:12) η Γ L ′ (cid:3) , (8.11)and we apply the conditional form of the dominated convergence theorem to the limits usedto define ψ to obtain ψ xK = Av [ ψ xL | η K ] . (8.12)Since b G δ i L,K j was defined in terms of periodic boundary conditions and the limits used toobtain φ are independent of x , we also have φ xL,K,δ ( T y η ) = φ x − yL,K,δ ( η ) , (8.13)and taking the limit (8.5) we have ψ xL ( T y η ) = ψ x − yL ( η ) . (8.14)Together with Equations (8.12), this lets us obtainAv [ ψ yL | η Λ ] = Av (cid:2) ψ y + xL ◦ T x (cid:12)(cid:12) η L ′ (cid:3) (8.15)or in other words (in light of (8.7))Av (cid:20) ∂G L ∂η y (cid:12)(cid:12)(cid:12)(cid:12) η Λ (cid:21) = ∂∂η y G L ′ ( T x η ) (8.16)The uniform bounds proven above allow us to take the partial derivative with respect to η y outside the conditional average [21], and the resulting expression can then be integrated toprove Property 4. 25 . Absolutely continuous random fields With β = ∞ , it is still obvious from Equation 5.5 that b G δL,K (0) = 0; Inequality (8.3)implies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b G δL,K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ , so we can apply Lemma VIII.1 to obtain (8.6), with ||| G L ||| ≤ ǫ .Av G L = 0 is obvious. We can apply the diagonal subsequence trick to obtain sequencesindependent of L , which implies that the consistency condition 4 of Theorem VII.1 is satis-fied.The difference arises in proving that Av G ′ = ǫM . Without equicontinuity of the deriva-tives, there is no reason to expect that functions constructed like φ of the previous sectionwill converge uniformly, or that the pointwise limit will be differentiable. This problem canbe resolved applying the following convergence criteria: Lemma VIII.3.
Let g be a measurable function and g n a sequence of measurable functionssuch that k g n k ∞ ≤ , k g k ∞ ≤ , and lim n →∞ Z ba g n ( t ) dt = Z ba g ( t ) dt (8.17) for all a, b ∈ R . Then for any finite measure ν which is absolutely continuous with respectto the Lebesgue measure, lim n →∞ Z g n dν = Z g dν. (8.18)This can be proven by elementary measure theory techniques[11].Thanks to Rademacher’s theorem [22], Lipschitz continuity guarantees that f n have dis-tributional derivatives, i.e. functions g n satisfying Z ba g n ( t ) dt = f n ( b ) − f n ( a ) (8.19)for any a, b ∈ R ; and furthermore that k g n k ∞ = ||| f n ||| ≤
1. We therefore have
Corollary VIII.4.
Let f n be a sequence of functions R → R such that f n → f pointwise,and ||| f n ||| ≤ . Then f has a distributional derivative f ′ satisfying lim n →∞ Z g n ( t ) dν ( t ) = Z f ′ ( t ) dν ( t ) . (8.20)Applying Corollary VIII.4 twice to b G δ i ,K j ( η L ) − Av b G δ i ,K j and G , we obtainAv G ′ = lim i →∞ lim j →∞ Av (cid:16) b G δ i ,K j (cid:17) ′ ( η ) (8.21)26nd applying Proposition IV.3 and the Birkhoff ergodic theorem to the right hand side asin the finite temperature case we obtainAv G ′ = ǫ (cid:18) ∂ F ∂h + − ∂ F ∂h − (cid:19) (8.22)and the proof of Theorem V.1 is complete. B. Concluding the proofs of Theorems IV.1 and IV.2
In summary, we have established that under the hypotheses of Theorems IV.1 and IV.2 thefree energy fluctuations G L ( η ) are uniformly bounded above by CL d − , and correspondingly CL d/ , and that if there is a 1 st order phase transition (i.e. a discontinuity in ∂ F ∂h ) then G L ( η )has asymptotically normal fluctuations on a scale of L d/ . That is is a contradiction evenin the marginal dimensions (where the powers match) as a normally distributed randomvariable will take with positive probability values which are arbitrarily large on any givenscale. An alternative way to present this case is to note that assuming the existence of a 1 st order phase transition, by Proposition VII.1 we getlim L →∞ Av exp (cid:0) tG L /L d/ (cid:1) = exp( t b / . (8.23)At the same time, if k G L k ∞ ≤ AL d/ then for all positive t :Av e tG L /L d/ ≤ e tA . (8.24)Clearly if b = 0, the two relations are incompatible at sufficiently large t . Finally Propo-sition VII.2 allows to conclude that in any of the cases under consideration, b = 0 implies M = 0, and the proof is complete. Appendix A: Decay conditions for long range interactions
In this Appendix we turn to the general class of interactions under which the resultsderived in this work apply.The estimate which is relevant for Theorem IV.1 is valid if: X A ∋ A ≤ L diam A | ∂A || A | k Q A k ≤ c ′ L (2 − d ) / (A.1)27or some constant c ′ . In case of two body interaction, that is when Q A is non zero only when A is a two point set, this condition is met if k Q { x,y } k ≤ Const. / | x − y | d/ . (A.2)As noted in the proof, the essential estimate which is required for Lemma VI.2, andtherefore Theorem IV.1, is that Av (cid:13)(cid:13) V η Λ L (cid:13)(cid:13) ≤ cL d − + bL d/ , (A.3)with some finite constants b and c . The contribution in V η Λ L of terms involving the randomfields η consists of a collection of O ( L d − ) independent summands with finite average norm.Thus the assumption (A.3) requires only a bound of the same form on the nonrandom partof the interaction. The following simple calculation [4] shows that (A.1) provides a sufficientcondition for that.By the triangle inequality k V , L k ≤ X A : A ∩ Γ L / ∈{∅ ,A } k Q A k . (A.4)In this sum the terms with diameter L or less contribute at most X A ∋ A ≤ L dL d − diam A | A | ≤ dc ′ L d/ , (A.5)and the remaining portion is bounded by X A ∋ A ≤ L L d | ∂A || A | k Q A k≤ L d − X A ∋ A ≤ L diam A | ∂A || A | k Q A k ≤ c ′ L d/ . (A.6)Putting the two parts back together we have Inequality (A.3) with C ′ = (2 d + 1) c ′ .The recent result of Cassandro, Orlandi and Picco [23] allows to conclude that at d = 1the condition (A.2) does indeed provide the threshold decay rate for the validity of Theo-rem IV.1. They prove that the phase transition is stable under weak disorder in a family ofone dimensional Ising spin systems with long range interactions with decay rates arbitrarilyclose to 3 /
2, more explicitly with k Q { x,y } k ≈ Const. / | x − y | / − γ at any γ ∈ (0 , . γ , the lower end of the interval coincides with (A.2).) The questionwhether the above criterion is optimal also for d = 2 is of interest, as there are some notablesystems with inverse cube interactions in two dimensions [24].For the stronger conclusion which is derived here for interactions with continuous sym-metries the relevant assumption is not (A.3) but X A ∋ (diam A ) | A |k Q A k < ∞ , (A.7)which is used in (6.34), in the proof of the free energy estimate of Lemma VI.3. For pairinteractions this reduces to the statement X x ∈ Z d k Q { ,x } k | x | < ∞ , (A.8)found (in slightly different notation) in Ref. 6. ACKNOWLEDGMENTS
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