Existence of shape-dependent thermodynamic limit in spin systems with short- and long-range interactions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Existence of shape-dependent thermodynamic limit in spinsystems with short- and long-range interactions
Takashi Mori
Department of Physics, Graduate School of Science,The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
August 16, 2018
Abstract
The existence of the thermodynamic limit in spin systems with short- and long-range interactions is established. We consider the infinite-volume limit with a fixedshape of the system. The variational expressions of the entropy density and the freeenergy density are obtained, which explicitly depend on the shape of the system. Thisshape dependence of thermodynamic functions implies the nonadditivity, which is oneof the most important characteristics of long-range interacting systems.
The aim of statistical mechanics is extracting thermodynamic properties from microscopicHamiltonian. Some thermodynamic properties and macroscopic phenomena can be well de-scribed by taking the thermodynamic limit [1, 2]. For example, thermodynamic quantitiesdo not fluctuate in thermodynamics, which is exactly true only in the thermodynamic limitfrom the microscopic point of view. A thermodynamic system sometimes exhibits a phasetransition, which is well characterized as a mathematical singularity only in the thermody-namic limit. Actually the system of interest is always finite, and thus the thermodynamiclimit should be regarded as a theoretical idealization to extract thermodynamic propertiesfrom a given Hamiltonian.From the statistical-mechanical point of view, it is a problem whether such a thermo-dynamic limit exists. In short-range interacting systems, existence of the thermodynamiclimit is well established, see [1, 2]. There, thermodynamic functions in the thermodynamiclimit are shown to have appropriate convexity or concavity consistent with thermodynam-ics. While, in long-range interacting systems, the existence of the thermodynamic limit hasnot been shown rigorously with sufficient generality. Since many works reveal the pecu-liarities of long-range interacting systems [3, 4] such as the ensemble inequivalence and thenegative specific heat, it is important to show the existence of the thermodynamic limitrigorously for general cases, e.g. the interaction potential is arbitrary under some naturalconditions and the shape of the system is arbitrary.In this paper, we shall establish the thermodynamic limit of classical spin systems withshort- and long-range pair interactions satisfying some natural conditions specified later forarbitrary spacial dimension d and arbitrary shape of the system specified by γ , see Sec. 2.1e also obtain the variational expression of the entropy density in the thermodynamic limit,which explicitly shows that the entropy density depends on γ even in the thermodynamiclimit. This dependence on the shape of the system implies the lack of additivity [7], whichis one of the most important characteristics of long-range interacting systems.This paper is organized as follows. In Sec. 2, the setup and the notation are explained.In Sec. 3, we mention the main result of this work, the existence of shape-dependent ther-modynamic limit and the variational expression of the entropy density. In Sec. 4 the proofis given. In Sec. 5, we discuss the result of the derived variational form of the entropydensity in the case of periodic boundary conditions. In Sec. 6, we conclude this work anddiscuss a future prospect. Let Γ ⊂ R d be a bounded domain with volume | Γ | on the d -dimensional space and ˆΓ = Γ ∩ Z d be the set of lattice points in Γ. The number of elements of ˆΓ is denoted by N Γ . We considera classical spin system put on ˆΓ. Each lattice point r ∈ ˆΓ has a spin variable σ ( r ), where σ ( r ) may be a scalar or a vector. The set of all the possible values of a spin variable isdenoted by S . Here we assume that S is identical for all r ∈ ˆΓ. The set of σ ( r ) for all r ∈ ˆΓ is denoted by σ Γ ∈ S N Γ .For simplicity, we consider the case in which σ ( r ) is a scalar variable, S ⊂ R , inthis paper, but the generalization to vector variables, S ⊂ R n , where n is the numberof components of a spin variable, is straightforward. Without loss of generality, we canassume 0 ∈ S . It is assumed that spin variables are bounded, | σ ( r ) | ≤ σ max , where σ max is independent of Γ. Furthermore, we assume that the “number of elements” of S is finite, P σ ∈S w < + ∞ . For a continuous spin, P σ ∈S should be interpreted as R S η ( σ ) dσ ,where η ( σ ) ≥ σ .We consider the system described by the following Hamiltonian, H Γ = H (0)Γ − X r , r ′ ∈ ˆΓ J ( r , r ′ ) σ ( r ) σ ( r ′ ) ≡ H (0)Γ + V Γ , (1)where H (0)Γ is the reference Hamiltonian , the condition on which will be specified later.The second term of Eq. (1) stands for the contribution of long-range interactions, and thecondition on the interaction potential J ( r , r ′ ) will be also mentioned later.In this paper, we mainly consider free boundary conditions, but the theorem presentedin Sec. 3 also holds for periodic boundary conditions as long as the distance | r − r ′ | isinterpreted by the minimum image convention (the distance between the two points appearsin the crucial conditions (13) and (14)).For convenience, we choose the zero point of energy so that, for any Γ ⊂ Γ ′ , H Γ = H Γ ′ if σ ( r ) = 0 for all r ∈ Γ ′ \ Γ. In other words, any spin in the “null state” σ ( r ) = 0 does notcontribute to the energy.The entropy S ( E, M, ∆ M ; Γ) is defined as S ( E, M, ∆ M ; Γ) = ln X σ Γ ∈S N Γ θ ( H Γ ≤ E ) θ X r ∈ ˆΓ σ ( r ) ∈ [ M, M + ∆ M ) . (2)2he function θ is defined as θ ( A ) = ( A is True,0 if A is False. (3)The magnetization is denoted by M , and the quantity ∆ M is some number which is largeenough to contain a large number of microscopic states with P r ∈ ˆΓ σ ( r ) ∈ [ M, M + ∆ M ),but macroscopically very small.Since the spin in the state σ ( r ) = 0 does not contribute to the energy and the magne-tization, for discrete spins we have S ( E, M, ∆ M, Γ) ≤ S ( E, M, ∆ M, Γ ′ ) for any Γ ⊂ Γ ′ . (4)This inequality is derived by restricting the spin configurations so that σ ( r ) = 0 for all r ∈ Γ ′ \ Γ. In other words, all the allowed spin configurations on Γ are included in thoseon Γ ′ , and hence Eq. (4) follows. For continuous spins, the inequality (4) does not hold asit is, but a slightly modified inequality can be derived if we assume the continuity of theenergy, | H Γ ′ − H Γ | ≤ ǫκN Γ ′ \ Γ with some constant κ > | σ ( r ) | ≤ ǫ for all r ∈ Γ ′ \ Γ. Theinequality in that case is given by S ( E + κǫN Γ ′ \ Γ , M, ∆ M − ǫN Γ ′ \ Γ ; Γ) + N Γ ′ \ Γ ln Z ǫ − ǫ η ( σ ) dσ ≤ S ( E, M, ∆ M ; Γ ′ ) . (5)The entropy density is given by s ( ε, m, δm ; Γ) = 1 | Γ | S ( | Γ | ε, | Γ | m, | Γ | δm ; Γ) . (6)We consider the thermodynamic limit. Now let us consider some fixed domain γ ⊂ R d of unit volume, | γ | = 1. We set Γ = Lγ , where the set kA with k ∈ R and A ⊂ R d isdefined as kA ≡ { x ∈ R d : x /k ∈ A } . Similarly, the set A + a with A ⊂ R d and a ∈ R d isdefined as A + a ≡ { x ∈ R d : x − a ∈ A } .By thermodynamic limit, we mean the limit of L → ∞ with fixed values of ε and m and with a fixed domain γ . It means that the system is made large with a fixed shape of thesystem. Later we will see that in long-range interacting systems the thermodynamic limitdepends on the shape of the system, γ . As is well known, it is not the case in short-rangeinteracting systems [1, 5]. Thermodynamic limit of the entropy density is, if it exists, givenby s γ ( ε, m ) = lim δm → lim L →∞ s ( ε, m, δm ; Lγ ) , (7)where δm = ∆ M/N Γ , see Eq. (2). The aim of this paper is proving the existence of Eq. (7)and finding its simple expression.Let us go back to our Hamiltonian, Eq. (1), on which we impose some conditions. Thecondition on H (0)Γ is as follows. Let Γ = Γ ∪ Γ with Γ ∩ Γ = ∅ and define H (0)Γ , Γ = H (0)Γ − H (0)Γ − H (0)Γ , (8)which expresses the interaction between subsystems Γ and Γ . Let us consider arbitrarytwo d -dimensional cubes of side l , Λ (1) l and Λ (2) l with Λ (1) l ∩ Λ (2) l = ∅ . Then we assume3hat there exist positive constants K > ν > d -dimensionalcubes, max σ Λ(1) l , σ Λ(2) l (cid:12)(cid:12)(cid:12)(cid:12) H (0)Λ (1) l , Λ (2) l (cid:12)(cid:12)(cid:12)(cid:12) ≤ Kl d R d + ν , (9)where R is the distance between the center of Λ (1) l and that of Λ (2) l , that is, R = 1 l d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Λ (1) l r d d r − Z Λ (2) l r d d r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Intuitively, the above condition means that the reference Hamiltonian H (0)Γ contains onlyshort-range interactions. We also assume that in the reference system the thermodynamiclimit of the entropy density s (0) ( ε, m ) = lim δm → lim L →∞ s (0) ( ε, m, δm ; Lγ ) (10)exists and is independent of γ . This has been rigorously proven for a wide class of short-range interacting systems, see Ref. [1].It helps us to give a few examples of the reference Hamiltonian. The Zeeman energyunder the magnetic field is represented by H (0)Γ = − h P r ∈ ˆΓ σ ( r ). The Hamiltonian H (0)Γ = − κ P r , r ′ ∈ ˆΓ θ ( | r − r ′ | = 1) σ ( r ) σ ( r ′ ) stands for nearest-neighbor exchange interactions.Next we mention the condition on V Γ . The potential J ( r , r ′ ) represents long-rangeinteractions between the spins at r and r ′ . By long-range interactions, we mean that J ( r , r ′ ) is written in the following form, J ( r , r ′ ) = 1 L d φ (cid:18) r L , r ′ L (cid:19) (11)for Γ = Lγ with | γ | = 1. The function φ is independent of Γ, symmetric φ ( x , y ) = φ ( y , x ),and integrable on γ × γ , Z γ d d x Z γ d d y φ ( x , y ) = N φ,γ < + ∞ . (12)The value of N φ,γ is not important, so we put N φ,γ = 1. Moreover, it is assumed that | φ ( x , y ) | ≤ J | x − y | α , (13) |∇ x φ ( x , y ) | ≤ J ′ | x − y | α +1 (14)with some J > J ′ >
0, and α ∈ [0 , d ).When we consider the translationally invariant interaction potential, φ ( x , y ) = φ ( x − y )and thus J ( r , r ′ ) = L − d φ (( r − r ′ ) /L ). It means that the interaction range and the size ofthe system are comparable. The sign of N φ,γ is important. By putting N φ,γ = 1, it is implicitly assumed that the interaction isferromagnetic as a whole. − X r , r ′ ∈ ˆΓ J ( r , r ′ ) σ ( r ) σ ( r ′ ) ∼ − L d Z γ d d x Z γ d d y L d φ ( x , y ) ∼ − L d , which is of the order of the volume of the system. For example, for the power-law in-teractions, φ ( x , y ) ∝ | x − y | − α with α ∈ [0 , d ), J ( r , r ′ ) has a scaling form of J ( r , r ′ ) ∝ L α − d | r − r ′ | − α . The factor L α − d makes the interaction energy per spin finite when theinteraction decays as 1 /r α .The thermodynamic limit is an idealization to describe a real finite but large system.The ideal limit should be taken in such a way that the thermodynamic properties of thesystem do not change by this limiting procedure. In order to do that, the energy should bemade extensive. The procedure to make the system extensive by introducing the system-sizedependence on V Γ as in Eq. (11) is referred to as the “Kac prescription” [3, 6].In this paper, we only consider the microcanonical ensemble. We can do it withoutloss of generality because if we can show that the microcanonical entropy has its ther-modynamic limit, it is automatically shown that the free energies in the canonical andthe grandcanonical ensemble also have their thermodynamic limit. They are derived by theLegendre-Fenchel transformation from the microcanonical entropy. On the other hand, it ispointed out that the inverse transformation, i.e. transformation from the canonical ensem-ble to the microcanonical ensemble, is impossible as a result of the ensemble inequivalencein long-range interacting systems [3, 4].Before presenting the main result, we briefly explain the additivity and its consequences.The system is said to be additive if the following equality holds [7]: s γ ,γ ( ε, m , m ) = sup ε ,ε : λε +(1 − λ ) ε = ε [ λs γ ( ε , m ) + (1 − λ ) s γ ( ε , m )] , (15)where s γ ,γ ( ε, m , m ) is the entropy density of a state with the total energy density ε andthe magnetization densities m and m of the domains γ and γ , respectively.We can derive some important results from additivity, see Ref. [7] for the derivation.Firstly, when the system is additive, the entropy density is independent of γ , the shape ofthe system: s γ ( ε, m ) = s γ ′ ( ε, m ) ≡ s ( ε, m ) . (16)Secondly, we can show that the entropy density is a concave function of ε and m : s ( λε + (1 − λ ) ε , λm + (1 − λ ) m ) ≥ λs ( ε , m ) + (1 − λ ) s ( ε , m ) . (17)Concavity of the entropy ensures the ensemble equivalence, e.g., the microcanonicalensemble is equivalent to the canonical ensemble [9]. As we have seen above, such importantproperties immediately follow from our definition of additivity. Additivity in the sense ofEq. (15) is, therefore, considered to be a fundamental property of macroscopic systems.In short-range interacting systems with suitable conditions, it is rigorously shown thatthe system is additive [1]. While it is not necessarily the case in long-range interactingsystems [3, 4, 10]. As a result, in long-range interacting systems, the entropy density maydepend on γ and may not be concave. A nonconcave entropy implies the ensemble inequiv-alence. 5 Theorem on the thermodynamic limit
In this section we mention the theorem and discuss its consequence. The theorem we nowdiscuss is the following:
Theorem 1 (Thermodynamic limit of the entropy density) . Consider the system describedby Eqs. (1) and (11) with the conditions given by Eqs. (13) and (14). Then the thermody-namic limit of the entropy density exists and is given by the following variational formula : s γ ( ε, m ) = sup ε ( · ) ,m ( · ) ∈R γ (cid:20)Z γ d d x s (0) ( ε ( x ) , m ( x )) : Z γ d d x m ( x ) = m, − Z γ d d x Z γ d d y φ ( x , y ) m ( x ) m ( y ) + Z γ d d x ε ( x ) = ε (cid:21) , (18) where s (0) ( ε, m ) is the thermodynamic limit of the entropy density of the reference systemdescribed by H (0)Γ . The set of Riemann integrable functions on γ is denoted by R γ . Equation (18) means that thermodynamic properties can be described by the coarse-grained magnetization m ( x ) and the coarse-grained energy density ε ( x ). In the proof ofTheorem 1, we will divide the original system into a large number of cells of side l ≪ L . Wecan show that the entropy density is almost unchanged by averaging out the spin variableswithin each cell (this averaging procedure is called the coarse graining). This fact allowsus to express the entropy density in the variational form as Eq. (18).We can give the explicit expression of s (0) ( ε, m ) for some simple cases. When we considerthe case S = { , } , or σ ( r ) = 0 or 1, and there is no short-range interactions, H (0)Γ = 0,for example, we have s (0) ( ε, m ) = − m ln m − ( m + 1) ln( m + 1) for ε ≥ s (0) = 0 for ε < f γ ( β, m ) = lim δm → lim L →∞ − β ln X σ Lγ ∈S NLγ θ L d X r ∈ Lγ ∩ Z d σ ( r ) ∈ [ m, m + δm ) e − βH Lγ (20)is related to the entropy density via the Legendre-Fenchel transformation, f γ ( β, m ) = inf ε (cid:20) ε − β s γ ( ε, m ) (cid:21) . (21)By using Eq. (18), Eq. (21) becomes f γ ( β, m ) = inf m ( · ) ∈R γ (cid:20) − Z γ d d x Z γ d d y φ ( x , y ) m ( x ) m ( y ) + Z γ d d x f (0) ( β, m ( x )) (cid:21) , (22) The notation sup[ A : B ] means sup A under the condition B . γ (left) and ˜ γ ′ (right) for a given two-dimensional domain γ with a unit volume (the region inside of the thick lines). Eachsquare of side δ expresses Λ ( p ) l /L .where f (0) = inf ε [ ε − s (0) ( ε, m ) /β ] is the free energy density of the reference system. Equa-tion (22) is the variational expression of the free energy density of a short- and long-rangeinteracting spin system.We can see Eqs. (18) and (22) that the entropy density and the free energy densityexplicitly depend on γ , the shape of the system. We have seen that in any additive systemthe entropy density is independent of γ in the thermodynamic limit. This fact, therefore,implies that a system with long-range interactions is in general not additive as expected. In this section we give a proof of Theorem 1. In long-range interacting systems, it isexpected that short length-scale structure is not essential for thermodynamic properties.Hence the method of coarse graining is a powerful tool to examine macroscopic properties oflong-range interacting systems [23–25]. First we show that the procedure of coarse grainingis justified and then show that the entropy density calculated by the coarse graining has alimiting value predicted by Theorem 1 in the thermodynamic limit.
We approximate Γ by an ensemble of d -dimensional cubes of side l , each of which is denotedby Λ ( p ) l , p = 1 , , . . . , with Λ ( p ) l ∩ Λ ( q ) l = ∅ . We consider the two ways of approximations,see Fig. 1. Firstly, we fill Γ with Λ ( p ) l so that ˜Γ = ∪ p Λ ( p ) l ⊂ Γ has the maximum volume.Secondly, we consider ˜Γ ′ = ∪ p Λ ( p ) l with the least volume satisfying ˜Γ ′ ⊃ Γ. We use the same It is noted that the nonadditivity does not imply the shape dependence of the entropy. In infinite-rangemodels, the spacial geometry is not important and the entropy density does not depend on γ , but they arenonadditive. The shape-dependent entropy density always implies nonadditivity. ( p ) l identifying ˜Γ and Λ ( p ) l identifying ˜Γ ′ may be different. The domains ˜ γ and˜ γ ′ are defined by ˜Γ = L ˜ γ and ˜Γ ′ = L ˜ γ ′ . Of course, ˜ γ ⊂ γ ⊂ ˜ γ ′ , where γ is defined by Γ = Lγ .The domains ˜ γ and ˜ γ ′ are ones that approximate γ by an ensemble of d -dimensional cubesof side δ = l/L . We assume that ˜ γ, ˜ γ ′ → γ in the limit of δ → +0.The coarse-grained Hamiltonian is obtained by averaging out σ ( r ) within each cell Λ ( p ) l : H ( δ,l )˜ γ = H (0)˜Γ − L d X p,q (Λ ( p ) l , Λ ( q ) l ⊂ ˜Γ) φ ( δ,l ) pq M p M q , (23)where φ ( δ,l ) pq = 1 l d X r ∈ ˆΛ ( p ) l l d X r ′ ∈ ˆΛ ( q ) l φ (cid:18) r L , r ′ L (cid:19) (24)and M p = l d m p = X r ∈ ˆΛ ( p ) l σ ( r ) . (25)Here, ˆΛ ( p ) l = Λ ( p ) l ∩ Z d . The coarse-grained Hamiltonian on ˜ γ ′ is obtained by replacing ˜ γ by ˜ γ ′ .Since L = l/δ , the difference between the exact Hamiltonian and the coarse-grainedone, 1 L d max σ ˜Γ ∈S N ˜Γ (cid:12)(cid:12)(cid:12) H ˜Γ − H ( δ,l )˜ γ (cid:12)(cid:12)(cid:12) ≡ ∆ ( δ,l )˜ γ (26)are determined by δ and l . If ∆ ( δ,l )˜ γ and ∆ ( δ,l )˜ γ ′ can be made vanishingly small in thethermodynamic limit, the procedure of coarse graining is justified. Indeed we can show thefollowing lemma, whose proof is given in Sec. 4.3, Lemma 1 (Justification of the coarse graining) . For any given γ , there exists ∆ δ > depending on δ such that lim δ → ∆ δ = 0 and ∆ ( δ,l )˜ γ , ∆ ( δ,l )˜ γ ′ ≤ ∆ δ for all l > . This lemma tells us that lim δ → L d max σ ˜Γ ∈S N ˜Γ (cid:12)(cid:12)(cid:12) H L ˜ γ − H ( δ,l )˜ γ (cid:12)(cid:12)(cid:12) = 0 , (27)where L = l/δ , and its convergence is uniform with respect to l .From Eq. (4), we have S ( E, M, ∆ M ; ˜Γ) ≤ S ( E, M, ∆ M ; Γ) ≤ S ( E, M, ∆ M ; ˜Γ ′ ) (28)for discrete spins. For continuous spins, the corresponding inequality is obtained by usingEq. (5), and it is slightly different from the above one. However, we can show the theoremby following the same line of the proof for discrete spins and finally taking the limit of ǫ → +0 (Remember that ǫ appears in Eq. (5)). Therefore, hereafter we focus on the caseof discrete spins for simplicity.By using Lemma 1, we obtain S ( δ,l ) ( E − L d ∆ δ , M, ∆ M ; ˜Γ) ≤ S ( E, M, ∆ M ; Γ) ≤ S ( δ,l ) ( E + L d ∆ δ , M, ∆ M ; ˜Γ ′ ) . (29)8igure 2: The set of Λ ( q ) l ∈ ∂ Λ ( p ) l in the case that the cells are tightly arranged on thetwo-dimensional space. The central cell is Λ ( p ) l and Λ ( p ) l itself is also the element of ∂ Λ ( p ) l .Here, S ( δ,l ) is the entropy calculated by H ( δ,l ) . The corresponding entropy density is denotedby s ( δ,l ) ( ε, m, δm ; ˜Γ). In terms of the entropy densities, the inequality (29) becomes s ( δ,l ) ( ε − ∆ δ , ˜ m, δ ˜ m ; L ˜ γ ) ≤ s ( ε, m, δm ; Lγ ) ≤ s ( δ,l ) ( ε + ∆ δ , ˜ m ′ , δ ˜ m ′ ; L ˜ γ ′ ) , (30)where ˜ m = M/ | ˜Γ | = m/ | ˜ γ | , δ ˜ m = δm/ | ˜ γ | , ˜ m ′ = m/ | ˜ γ ′ | , and δ ˜ m ′ = δm/ | ˜ γ ′ | .We take the limit of δ → l → ∞ is taken. In this limit, (˜ γ, ˜ γ ′ ) → γ , ∆ δ → m, ˜ m ′ ) → m , ( δ ˜ m, δ ˜ m ′ ) → δm . Thus if˜ s γ ( ε, m ) = lim δm → lim δ → lim l →∞ s ( δ,l ) ( ε, m, δm ; ˜Γ) (31)exists, the thermodynamic limit of the entropy density also exists and s γ ( ε, m ) = ˜ s γ ( ε, m )from the inequality (30). In Sec. 4.4 we show this fact summarized in the following lemma. Lemma 2 (Existence of the thermodynamic limit of the coarse-grained entropy density) . The limit of Eq. (31) exists and is expressed as ˜ s γ ( ε, m ) = sup ε ( · ) ,m ( · ) ∈R γ (cid:20)Z γ d d x s (0) ( ε ( x ) , m ( x )) : Z γ d d x m ( x ) = m, (32) Z γ d d x ε ( x ) − Z γ d d x Z γ d d y φ ( x , y ) m ( x ) m ( y ) = ε (cid:21) . (33)By combining Lemma 1 and Lemma 2, we obtain Theorem 1.9 .3 Proof of Lemma 1 We evaluate the upper bound of ∆ ( δ,l )˜ γ , which is given by∆ ( δ,l )˜ γ = 1 L d max σ ˜Γ ∈S N ˜Γ (cid:12)(cid:12)(cid:12) H ˜Γ − H ( δ,l )˜ γ (cid:12)(cid:12)(cid:12) = 1 L d max σ ˜Γ ∈S N ˜Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L d X p,q (Λ ( p ) l , Λ ( q ) l ⊂ ˜Γ) X r ∈ ˆΛ ( p ) l r ∈ ˆΛ ( q ) l σ ( r ) σ ( r ) X r ∈ ˆΛ ( p ) l r ∈ ˆΛ ( q ) l l d h φ (cid:16) r L , r L (cid:17) − φ (cid:16) r L , r L (cid:17)i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ σ L d l d X p,q X r , r ∈ ˆΛ ( p ) l r , r ∈ ˆΛ ( q ) l (cid:12)(cid:12)(cid:12) φ (cid:16) r L , r L (cid:17) − φ (cid:16) r L , r L (cid:17)(cid:12)(cid:12)(cid:12) . (34)We divide the summation over q into that with Λ ( q ) l ∈ ∂ Λ ( p ) l and that with Λ ( q ) l / ∈ ∂ Λ ( p ) l ,where ∂ Λ ( p ) l is defined as Λ ( q ) l ∈ ∂ Λ ( p ) l ⇔ r pq < √ dl, (35)see Fig. 2 for visualizing the set { Λ ( q ) l ∈ ∂ Λ ( p ) l } in the case that the cells are tightly arrangedon the two-dimensional space. The distance between the central points of Λ ( p ) l and Λ ( q ) l hasbeen denoted by r pq . Then we have∆ ( δ,l )˜ γ ≤ σ L d l d X p,q Λ ( q ) l ∈ ∂ Λ ( p ) l X r , r ∈ ˆΛ ( p ) l r , r ∈ ˆΛ ( q ) l (cid:12)(cid:12)(cid:12) φ (cid:16) r L , r L (cid:17) − φ (cid:16) r L , r L (cid:17)(cid:12)(cid:12)(cid:12) + σ L d l d X p,q Λ ( q ) l / ∈ ∂ Λ ( p ) l X r , r ∈ ˆΛ ( p ) l r , r ∈ ˆΛ ( q ) l (cid:12)(cid:12)(cid:12) φ (cid:16) r L , r L (cid:17) − φ (cid:16) r L , r L (cid:17)(cid:12)(cid:12)(cid:12) ≡ ∆ + ∆ (36)Let us first evaluate ∆ . From Eq. (13), (cid:12)(cid:12)(cid:12) φ (cid:16) r L , r L (cid:17) − φ (cid:16) r L , r L (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) φ (cid:16) r L , r L (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) φ (cid:16) r L , r L (cid:17)(cid:12)(cid:12)(cid:12) ≤ J L α (cid:18) | r − r | α + 1 | r − r | α (cid:19) . (37)By substituting it into the expression of ∆ , we obtain∆ ≤ σ L d X p X q (Λ ( q ) l ∈ ∂ Λ ( p ) l ) X r ∈ ˆΛ ( p ) l X r ∈ ˆΛ ( q ) l ( r = r ) J L α | r − r | α . (38)10or r ∈ Λ ( p ) l and r ∈ Λ ( q ) l with Λ ( q ) l ∈ ∂ Λ ( p ) l , | r − r | ≤ √ dl + r pq < √ dl . Hence,∆ ≤ σ L d X r , r ∈ ˆ˜Γ( | r − r | < √ dl ) J L α | r − r | α ≤ σ JL d − α X r ∈ ˆ˜Γ Z √ dl drS d r d − α − = (3 √ d ) d − α σ J S d d − α N ˜Γ L d δ d − α ≈ (3 √ d ) d − α σ J S d d − α | ˜ γ | δ d − α , (39)where S d = 2 π ( d +1) / / Γ(( d + 1) /
2) is the surface area of the d -dimensional unit cube. Thisupper limit is independent of l and going to zero in the limit of δ → . By the mean-value theorem, there exists u ∈ [0 ,
1] suchthat φ (cid:18) r − r L (cid:19) − φ (cid:18) r − r L (cid:19) = [ ∇ x φ ( x , y )] · r − r L + [ ∇ y φ ( x , y )] · r − r L , (40)with x = [(1 − u ) r + u r ] /L and y = [(1 − u ) r + u r ] /L . Because r , r ∈ Λ ( p ) l and r , r ∈ Λ ( q ) l , L x ∈ Λ ( p ) l and L y ∈ Λ ( q ) l . Here, by the triangle inequality, L | x − y | ≥ r pq − √ dl .Since Λ ( q ) l / ∈ ∂ Λ ( p ) l , r pq ≥ √ dl and thus | x − y | ≥ r pq / L . Moreover, | r − r | ≤ √ dl and | r − r | ≤ √ dl . Due to the condition (14) and Eq. (40), (cid:12)(cid:12)(cid:12)(cid:12) φ (cid:18) r − r L (cid:19) − φ (cid:18) r − r L (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ J ′ | x − y | α +1 (cid:18) | r − r | L + | r − r | L (cid:19) ≤ α +2 √ dJ ′ (cid:18) Lr pq (cid:19) α +1 δ. (41)Because Λ ( q ) l / ∈ ∂ Λ ( p ) l , r pq ≥ √ dl , and hence r pq /L ≥ √ dδ . Thus we can evaluate ∆ as∆ ≤ α +1 √ dJ ′ σ δ d +1 X p X q (Λ ( q ) l / ∈ ∂ Λ ( p ) l ) (cid:18) Lr pq (cid:19) α +1 ≤ α +1 √ dJ ′ σ δ d +1 X p Z x max √ dδ dxS d x d − α − = A | ˜ γ | δ Z x max √ dδ dxx d − α − (42)where x max is defined as x max = max x , y ∈ ˜ γ | x − y | , which is assumed to be finite, and11 = 2 α +1 √ dJ ′ σ S d . By evaluating the integral, we obtain∆ ≤ A x d − α − d − α − | ˜ γ | δ ( α < d − A | ˜ γ | δ ln x max √ dδ ( α = d − A α − d + 1)(2 √ d ) α − d +1 | ˜ γ | δ d − α ( d − < α < d ), (43c)In any case, as long as α < d , ∆ → δ →
0. The convergence is uniformwith respect to l because the derived upper bound is independent of l .By collecting the results for ∆ and ∆ , we complete the proof of Lemma 1. We prove Lemma 2 by evaluating the upper bound and the lower bound of s ( δ,l ) ( ε, m, δm ; ˜Γ)and showing that these bounds become indistinguishable in a suitable limit.We decompose the reference Hamiltonian as H (0)˜Γ = X p (Λ ( p ) l ⊂ ˜Γ) H (0) pp ( σ Λ ( p ) l ) + X p ν > | H (0) pq | ≤ Kl d r d + νpq ≡ E (0) pq (45)with r pq = 1 l d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Λ ( p ) l r d d r − Z Λ ( q ) l r d d r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (46)The coarse-grained entropy density is explicitly given by s ( δ,l ) ( ε, m, δm ; ˜Γ) = 1 | ˜Γ | ln X σ ˜Γ ∈S N ˜Γ θ (cid:16) H ( δ,l )˜ γ ≤ | ˜Γ | ε (cid:17) θ | ˜Γ | X r ∈ ˜Γ σ ( r ) ∈ [ m, m + δm ) . (47)We have θ X p H (0) pp + V ( δ,l )˜ γ ≤ | ˜Γ | ε − X p
−∞ as the possible minimum value of H (0) pp /l d . Decompose the possiblevalues of energy and magnetization as1 l d H (0) pp − ε (0) g ∈ [ n p δε, ( n p + 1) ε ) (49)12nd 1 l d X r ∈ Λ ( p ) l σ ( r ) = m p ∈ (cid:20) k p δm , ( k p + 1) δm (cid:19) (50)with integers { n p } and { k p } . An arbitrary positive constant δε has been introduced.Because | σ ( r ) | ≤ σ max or | m p | ≤ σ max , we can restrict the summation over k p to k min ≤ k p ≤ k max . Here, k min is the maximum integer satisfying k min ≤ − σ max /δm and k max is the minimum integer satisfying k max > σ max /δm − n p . Since H (0) pp − l d ε (0) g ≥
0, we have n p ≥ X p H (0) pp + V ( δ,l )˜ γ ≤ | ˜Γ | ε − X p 0, we obtain X { n p } θ X p H (0) pp + V ( δ,l )˜ γ ≤ | ˜Γ | ε + X p 1. Hence there is some positive constant D = O (ln( δε ) /δε ) independent of δ and l with ln X { n p } θ X p H (0) pp + V ( δ,l )˜ γ ≤ | ˜Γ | ε + X p 16n the limit of l → ∞ and then δm → δε → 0, by the assumption that the entropydensity of the reference system has a thermodynamic limit, Eq. (10), we obtainlim inf δm → l →∞ s ( δ,l ) ( ε, m, δm ; ˜Γ) ≥ max { ε p ,m p } " δ d | ˜ γ | X p s (0) ( ε p , m p ) : δ d X p ε p + v ( δ )˜ γ ≤ | ˜ γ | ε, δ d X p m p = | ˜ γ | m , (67)where v ( δ )˜ γ = lim l →∞ V ( δ,l )˜ γ /L d .Furthermore, we shall take the limit of δ → 0. Now we define the domain λ ( p ) δ =Λ ( p ) l /L = δ (Λ ( p ) l /l ). The domain λ ( p ) δ is independent of l and | λ ( p ) δ | = δ d . The centralposition of λ ( p ) δ is denoted by x p . By introducing the functions ε ′ ( x ) and m ′ ( x ) so that ε ′ ( x ) = ε p and m ′ ( x ) = m p for any x ∈ λ ( p ) δ , we can write δ d X p s (0) ( ε p , m p ) = Z ˜ γ d d x s (0) ( ε ′ ( x ) , m ′ ( x )) ,δ d X p ε p = Z ˜ γ d d x ε ′ ( x ) ,δ d X p m p = Z ˜ γ d d x m ′ ( x ) , ˜ v ( δ )˜ γ = − Z ˜ γ d d x Z ˜ γ d d y φ ( x , y ) m ′ ( x ) m ′ ( y ) . (68)In the limit of δ → 0, ˜ γ → γ and any Riemann integrable functions ε ( x ) and m ( x ) can beapproximated by step functions ε ′ ( x ) and m ′ ( x ) if { ε p } and { m p } are suitably chosen. Wetherefore obtain lim inf δ → lim inf δm → l →∞ s ( δ,l ) ( ε, m, δm ; ˜Γ) ≥ sup ε ( · ) ,m ( · ) ∈R γ (cid:20)Z γ d d x s (0) ( ε ( x ) , m ( x )) : Z γ d d x m ( x ) = m, Z γ d d x ε ( x ) − Z γ d d x Z γ d d y φ ( x , y ) m ( x ) m ( y ) ≤ ε (cid:21) (69) Evaluation of an upper bound By using the right part of Eq. (48), we have θ (cid:16) H ( δ,l )˜ γ ≤ | ˜Γ | ε (cid:17) ≤ X { n p } θ | ˜Γ | ε (0) g + l d δε X p n p + V ( δ,l )˜ γ ≤ | ˜Γ | ε + X p 0, we obtainlim sup δm → l →∞ s ( δ,l ) ( ε, m, δm ; ˜Γ) ≤ max { ε p } , { m p } " δ d | ˜ γ | X p s (0) ( ε p , m p ) : δ d X p m p = | ˜ γ | m,δ d X p ε p + v δ ˜ γ ≤ | ˜ γ | ε . (77)Here we again introduce the step functions ε ′ ( x ) and m ′ ( x ). Since ε ′ ( x ) and m ′ ( x ) areRiemann integrable, we have, by using Eq. (68),lim sup δm → l →∞ s ( δ,l ) ( ε, m, δm ; ˜Γ) ≤ max { ε p } , { m p } (cid:20) | ˜ γ | Z ˜ γ d d x s (0) ( ε ′ ( x ) , m ′ ( x )) : Z ˜ γ d d x m ′ ( x ) = | ˜ γ | m, Z ˜ γ d d x ε ′ ( x ) − Z ˜ γ Z ˜ γ φ ( x , y ) m ′ ( x ) m ′ ( y ) ≤ | ˜ γε (cid:21) ≤ sup ε ( · ) ,m ( · ) ∈R γ (cid:20) | ˜ γ | Z ˜ γ d d x s (0) ( ε ( x ) , m ( x )) : Z ˜ γ d d x m ( x ) = | ˜ γ | m, Z ˜ γ d d x ε ( x ) − Z ˜ γ Z ˜ γ φ ( x , y ) m ( x ) m ( y ) ≤ | ˜ γ | ε (cid:21) . (78)In the limit of δ → 0, ˜ γ → γ and thuslim sup δ → lim sup δm → l →∞ s ( δ,l ) ( ε, m, δm ; ˜Γ) ≤ sup ε ( · ) ,m ( · ) ∈R γ (cid:20)Z γ d d x s (0) ( ε ( x ) , m ( x )) : Z γ d d x m ( x ) = m, Z γ d d x ε ( x ) − Z γ Z γ φ ( x , y ) m ( x ) m ( y ) ≤ ε (cid:21) . (79)This is identical to the derived lower bound. We therefore have finished to prove Lemma 2.19 Application of the variational formula to the case of peri-odic boundary conditions We have considered spin systems with free boundary conditions, but Theorem 1 also holdsfor periodic boundary conditions as long as | x − y | in the conditions (13) and (14) isinterpreted by the minimum image convention. In this section, we briefly mention someconsequences from the variatonal expression of the entropy density. In this section, we assume periodic boundary conditions and fully ferromagnetic and trans-lational invariant couplings φ ( x , y ) = φ ( x − y ) ≥ x and y , we can obtain someresults mentioned below.In periodic boundary conditions, we set γ to be the d -dimensional unit cube, γ =[0 , d ≡ Λ . In general, we define Λ dl ≡ [0 , l ) d , that is, Λ dl is the d -dimensional cube ofside l . The point x ± e k is identified with x , where e k is the unit vector along k -direction( k = 1 , , . . . , d ). Of course, the translational invariant potential satisfies φ ( x ) = φ ( x ± e k )for k = 1 , , . . . , d .In the above setting, it is shown that the interaction potential φ ( x − y ) can be replacedby the mean-field (MF) coupling, φ ( x − y ) → x , y ∈ Λ in a wide region ofthe parameter space, ( ε, m ) or ( β, m ) depending on the ensemble, called the “MF region”without changing the value of the entropy density [11] or the free energy density [12, 13].On the other hand, if the density of an extensive quantity such as ε or m is held fixed,it is also shown that there is a parameter region called the non-MF region, in which thevalue of the entropy density or the free energy density crucially depends on the details of φ ( x − y ), and hence replacing φ by 1 is not allowed. The fact that replacing φ by 1 isallowed is called the “exactness of the MF theory” [15–18], because it is well known thatthe spin model with the all-to-all couplings are thermodynamically equivalent to the spinmodel with the MF approximation [14].In earlier works , the exactness of the MF theory and its violation has been investi-gated for the case with the homogeneous magnetic field but without any short-range inter-actions, H (0)Γ = − h P r ∈ ˆΓ σ ( r ). As we will show below, the results of the earlier works arestraightforwardly extended to the case with short-range interactions by using the variatonalexpression of the entropy density (18).We shall derive the exactness of the MF theory and its violation for the canonicalensemble. In periodic boundary conditions, the translationally symmetric potential energycan be diagonalized by the Fourier expansion, Z Λ d d x Z Λ d d y φ ( x − y ) m ( x ) m ( y ) = X n ∈ Z d φ n | m n | , (80)where φ n = Z Λ d d x φ ( x ) e − πi n · x = Z Λ d d x φ ( x ) cos(2 π n · x ) (81) Exactness of the MF theory and its violation has been also discussed for quantum spin systems [19],but the derivation of the microcanonical entropy in quantum systems has not been fully rigorous as pointedout by Olivier and Kastner [20]. However, the results discussed in classical spins are also true in quantumspin systems at least for the canonical ensemble. m n = Z Λ d d x m ( x ) e πi n · x . (82)From Eq. (81), as long as φ ( x ) ≥ x ∈ Λ , φ n ≤ Z Λ d d x φ ( x ) = φ = 1 . (83)Remember the normalization of N φ,γ = 1 in Eq. (12).We define the second largest Fourier component of φ ( x ) as φ max , φ max = max n ∈ Z d \ φ n . (84)The interaction term is bounded as [21] X n ∈ Z d φ n | m n | ≤ m + φ max Z Λ d d x m ( x ) − φ max m . (85)By using this inequality, we find that the free energy density satisfies f ( β, m ) ≥ inf m ( · ) ∈R Λ1 (cid:26) − m + f (0) ( β, m ) + (cid:20)Z Λ d d x (cid:18) − φ max m ( x ) + f (0) ( β, m ( x )) (cid:19) − (cid:18) − φ max m + f (0) ( β, m ) (cid:19)(cid:21)(cid:27) . (86)We define the free energy of the reference system with the MF couplings as f MF ( β, m ; J ) = − J m + f (0) ( β, m ) . (87)Then the lower bound of the free energy is written as f ( β, m ) ≥ f MF ( β, m ; 1) − [ f MF ( β, m ; φ max ) − f ∗∗ MF ( β, m ; φ max )] , (88)where f ∗∗ MF ( β, m ; φ max ) is the convex envelope of f MF ( β, m ; φ max ) with respect to m . Inother words, f ∗∗ MF ( β, m ; φ max ) is the maximum convex function of m satisfying f ∗∗ MF ( β, m ) ≤ f MF ( β, m ; φ max ). We have used the relation f ∗∗ MF ( β, m ; φ max ) = inf m ( · ) ∈R Λ1 (cid:20)Z Λ d d x f MF ( β, m ( x ); φ max ) : Z Λ d d x m ( x ) = m (cid:21) . (89)The upper bound of the free energy density is easily obtained by putting m ( x ) = m inEq. (22), f ( β, m ) ≤ − m + f (0) ( β, m ) = f MF ( β, m ; 1) . (90)Thus we have obtained the following inequality: f MF ( β, m ; 1) − [ f MF ( β, m ; φ max ) − f ∗∗ MF ( β, m ; φ max )] ≤ f ( β, m ) ≤ f MF ( β, m ; 1) . (91)21his inequality is an extension of the inequality derived in the previous work [12, 13], inwhich only the case of H (0)Γ = − h P r ∈ ˆΓ σ ( r ), i.e., without short-range interactions, wasconsidered . From the inequality (91), if f MF ( β, m ; φ max ) is convex with respect to m ,the lower bound coincides with the upper bound, and thus f ( β, m ) = f MF ( β, m ; 1). Inparticular, at the minimum point of f MF ( β, m ; 1) with respect to m , which correspondsto an equilibrium state when the value of the magnetization is not fixed, the convexity of f MF ( β, m ; φ max ) is always satisfied and thus min m f ( β, m ) = min m f MF ( β, m ; 1). This isnothing but the statement of the exactness of the MF theory.We shall consider the case where m is held fixed at some value, not necessarily theminimum of f ( β, m ). From the stability analysis around the uniform solution m ( x ) = m , itis found that in the region of the parameter space ( β, m ) with ∂ f MF ( β, m ; φ max ) /∂m < F ( β, { m ( x ) } ) = − Z Λ d d x Z Λ d d y φ ( x − y ) m ( x ) m ( y ) + Z Λ d d x f (0) ( β, m ( x )) , (92)which means that the uniform solution is unstable. We therefore have f ( β, m ) < f MF ( β, m ; 1) (93)for ( β, m ) satisfying ∂ f MF ( β, m ; φ max ) /∂m < 0. This is a part of the non-MF region. Inthe non-MF region, macroscopic heterogeneity emerges, see Ref. [13, 21] in more detail. In short-range interacting systems, large clusters with the same spin state appear near thecritical point, which implies the divergence of the correlation length [14]. The universalityclass depends on the type of symmetry breaking, spacial dimension, and so on. On theother hand, in long-range interacting systems, the system tends to be homogeneous evenat the critical point because all the spins interact with each other and spacial geometrybecomes less important. Indeed, in the model only with the all-to-all interactions (the MFmodel), the spin configuration is always uniform and the universality class of the criticalphenomena belong to the MF universality class independently of the spacial dimension.When the system possesses both the short- and long-range interactions, it has beenargued that critical phenomena always belong to the MF universality class even if thestrength of long-range couplings is infinitesimal [22]. We can see it by using the exactnessof the MF theory. When the temperature is above the critical temperature, the free energyis convex with respect to m and f ( β, m ) = f MF ( β, m ; 1) = − (1 / m + f (0) ( β, m ). Weassume that m = 0 is the minimum point of f ( β, m ). If there were no long-range inter-action, the macroscopic ordering due to short-range interactions would occur at β (0) c with ∂ f (0) ( β (0) c , m ) /∂m | m =0 = 0. With the presence of long-range interactions, at the criticalinverse temperature β c , ∂ f ( β c , m ) /∂m | m =0 = − ∂ f (0) ( β c , m ) /∂m | m =0 = 0, whichimplies that ∂ f (0) ( β c , m ) /∂m | m =0 = 1 > 0. From this observation, we can say β c < β (0) c and phase transitions in a system with short- and long-range interactions are always drivenby long-range interactions before growing large clusters due to short-range interactions. For a reference Hamiltonian of the form H (0)Γ = − h P r ∈ ˆΓ σ ( r ), f MF ( β, m ; φ max ) = f MF ( βφ max , m ; 1). Ifwe write f MF ( β, m ) ≡ f MF ( β, m ; 1), the inequality (91) is reduced to the inequality obtained in Ref. [12,13]. 22s a result, critical phenomena are governed by long-range interactions and the criticalphenomena belong to the MF universality class. The crossover between short-range Isingmodel and the long-range Ising model is investigated in Ref. [22]. We have proven the existence of the thermodynamic limit in spin systems where bothshort-range interactions and long-range ones are present. We have obtained the variationalexpression of the entropy density explicitly depending on the shape of the system γ . 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