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Periodic minimizers in 1D local mean field theory

Using reflection positivity techniques we prove the existence of minimizers for a class of mesoscopic free-energies representing 1D systems with competing interactions. All minimizers are either periodic, with zero average, or of constant sign. If the local term in the free energy satisfies a convexity condition, then all minimizers are either periodic or constant. Examples of both phenomena are given. This extends our previous work where such results were proved for the ground states of lattice systems with ferromagnetic nearest neighbor interactions and dipolar type antiferromagnetic long range interactions.


INTRODUCTION
We consider the nature of the minimizers for a class of 1D free-energy functionals that model the continuum limit of microscopic systems with competing interactions on different length scales. An example is a spin system on a lattice with a nearest neighbor ferromagnetic interaction and a long range antiferromagnetic power law type interaction. In [17] we considered the ground states of such systems in one-dimension and in [18] we also investigated higher dimensional models with dipolar type interactions. In both cases we obtained periodic ground states whose period (which could be infinite) depended on both the strength of the short range interaction and the nature of the long range interaction. The technique used in those papers, reflection positivity, could not be extended to positive temperatures, for which only approximate methods and computer simulations are available now [1,19,24,31]. It turns out, however, that these reflection positivity methods are directly applicable to the Ginzburg-Landau type freeenergy functionals used to describe the continuum versions of such microscopic systems [26,28].
These include finite temperature effects, at least at a mean field level, via an inclusion of a local entropy term in the effective continuum description of the system. These free-energies functionals have been used extensively in both the physical and mathematical literatures as models for a great variety of systems, including micromagnets [7,10,14], diblock copolymers [20,23,27], anisotropic electron gases [29,30], polyelectrolytes [5], chargedensity waves in layered transition metals [25] and superconducting films [12]. Many of these systems are characterized by low temperature phases displaying spontaneous formation of periodic mesoscopic patterns, such as stripes or bubbles. It is, therefore, important to show that these free energy functionals have periodic minimizers. This has been proved rigorously in some cases [2,9] and argued for heuristically in others [6,7,12,14,16,20,23,26,29,30].
In this paper we use reflection positivity methods to prove the periodicity of minimizers for a certain class of such 1D free-energy functionals. These include cases that were not treated before, e.g., those with long-range power law type interactions. As noted before, reflection positivity methods have been succesfully applied to find periodic ground states for a class of microscopic 1D and 2D lattice spin models but has not, as far as we know, been used before for continuum systems. We begin in Section 2 by presenting the class of models under consideration and our results. These are proved in Section 3. In Section 4 we give an example of the transition from a state of finite periodicity to a uniform (infinite periodicity) state, as a parameter is varied.
We discuss the connection with related work in Section 5.

FORMULATION OF MODEL AND STATEMENT OF RESULTS
The formal infinite volume free energy functional to be minimized, in a sense to be made precise below, is constant. We shall also assume that F (t) is an even function of its argument, and that F (t) ≥ 0, with F (t) > 0 for |t| < 1 and F (t) = 0 for |t| = 1. Note that we do not need either that F is continuous or that it goes to +∞ as |t| → ∞. Some examples to keep in mind are (1), where, defining α = tanh β: (2. 2) The gradient term in (2.1) represents the cost of a transition between two phases, while the term F represents the local free energy density for a homogeneous system. We have chosen F (t) to have two symmetric minima corresponding to the case of a ferromagnetic Ising spin system (and its analogues) below T c . (Since we are not concerned here explicitly with the dependence on temperature we have used the scaling parameter α to set the value of the mean field spontaneous magnetization in a(t), at T < T c , equal to unity.) The third term on the right represents the long range antiferromagnetic type interaction. Note that v, which can include terms decaying as a power law, is reflection-positive, see [13], summable and completely monotone, i.e., (−1) n v (n) (x) ≥ 0 and ց 0 as x → ∞. The minimum value of this interaction term occurs when φ = 0. It competes, therefore, with the first two terms, which are minimized when φ(x) = 1 or φ(x) = −1 for all x. (Note, however, that in the absence of the gradient square term the minimizer would be an infinitely rapidly oscillating φ(x), between −1 and +1; see discussion of Kac potential in Section 5.) To state our results, let us first recall some standard notation. H 1 (R) is the space of functions whose first distributional derivative is in L 2 (R) and which → 0 as |x| → ∞.
is the space of functions whose derivatives are in L 2 ([a, b]) for all intervals −∞ < a < b < +∞ Then we define the infinite volume specific ground state free energy e 0 corresponding to the formal energy functional E(φ) to be We say that In the following we shall exploit the reflection-positivity of v in order to show existence of periodic minimizers for the functional (2.1). We need to introduce some more definitions and notation. and to be the function obtained by juxtaposing the functions f i on the real line, in such a way that, In order to visualize the meaning of Definition 3, we plot, in Figure 1, a function ϕ[F ](x) corresponding to M = N = 2.
x T + T (iv) The reflections of F − and F + are defined to be: Figure 2.
We are now ready to state our main results.
The limits in (2.5) exist, are equal and are given by where, . (2.8)

(Note that the limit in the r.h.s. of (2.8) exists, because ϕ[f ] is periodic and v is summable.)
Moreover e T is a continuous function of T and lim T →∞ e T exists and equals e 0 . It is given There is a function, φ T that is a minimizer for (2.7) and satisfies |φ T | ≤ 1. If F (t) is differentiable for t > 0, then φ T is twice differentiable and it satisfies the Euler-Lagrange equation If F (t) is convex for t > 0, then φ T is unique and the inf on the r.h.s. of (2.9) is a minimum, with a constant function as a minimizer.

Corollary 1. (Infinite volume ground states). (i) If there exists T 0 such that
Remark. Theorem 1 and its Corollary may be informally stated by saying that all the minimizers of E(φ) are either simply periodic, of finite period T , with zero average, or of constant sign (and are constant if F is convex on R + ). By "simply periodic" we mean that within a period the minimizer has only one positive and one negative region, with the negative part obtained by a reflection from the positive part.

PROOF OF THE MAIN RESULTS
As proved in [13], reflection-positivity of the long range potential v implies the following basic estimate.

11)
where In terms of the function ϕ[F ] in Figure 1, the statement of the Lemma is that the energy of this function is larger than the average of the energies of the two reflected configurations in Figure 2.
The key technical ingredient in the proof of Theorem 1 is the chessboard estimate, which is obtained from Lemma 1 by repeatedly reflecting around different nodes of the function. A chessboard estimate in the presence of periodic boundary conditions has appeared many times before in the literature, see for instance [13]. Here, however, we will need a generalization of it to the case of Dirichlet boundary conditions, and we proceed as proposed in the Appendix of [18].
Chessboard estimate with Dirichlet boundary conditions. Given a finite sequence of , we have: Proof of (3.12). We proceed by induction.
(i) If N = 1, let us first compare the energy of f 1 with that of {f 1 , ± θf 1 }. Using the fact that F (t) is even, the energy of {f 1 , ± θf 1 } can be rewritten as: At least one of the two interaction energies . By reflection positivity, i.e., by Lemma 1, we have in fact that Using (3.13) we find: (3.14) Iterating the same argument, we find: where, by definition, Taking the limit m → ∞ in (3.15) we get the desired estimate: (ii) Let us now assume by induction that the bound is valid for all 1 ≤ N ≤ n − 1, n ≥ 2, and let us prove it for N = n. There are two cases.
(a) n = 2p for some p ≥ 1. If we reflect once, by Lemma 1 we have: respectively, the two terms in the r.h.s. of (3.18) can be regarded as two terms with N = 2p − 1 and, by the induction assumption, they satisfy the bounds: where we used that e ∞ ((θf p+1 ) ⊗2 ) = e ∞ (f p+1 ) and e ∞ (f ⊗2 p ) = e ∞ (f p ). Therefore, the desired bound is proved.
(b) n = 2p + 1 for some p ≥ 1. If we reflect once we get: The first term in the r.h.s. corresponds to N = 2p − 1 so by the induction hypothesis it is bounded below by 2p+1 i=p+2 T i e ∞ (f i ). As regards the second term, using reflection positivity again, we can bound it from below by By the induction hypothesis, the first term is bounded below by (1/2) p i=1 T i e ∞ (f i ), and the second can be bounded using reflection positivity again. Iterating we find: Note that the last term is equal to T p+1 e ∞ (f p+1 ), so (3.22) is the desired bound. This concludes the proof of (3.12).
Proof of Theorem 1.
(1) Let us first prove that e 0 = inf T inf f ∈CT e ∞ (f ). First of all, let us because the interaction v is absolutely summable. Moreover, it follows by the variational We then just need to prove that lim given L > 0, let φ be any function in H 1 0 ([0, L]) and let us denote by , it will be understood that φ has two nodes between a and b, the first located at x = a the second at x = b). We define: By construction any f i is either nonnegative or nonpositive and φ = ϕ[{f 1 , . . . , f N }]. Using the chessboard estimate we find: All the e ∞ (f i ) in the r.h.s. can be bounded below by inf T inf f ∈CT e ∞ (f ), and the proof of this first claim is concluded.
(2) Next, let us show that for any fixed T > 0 there exists a function φ T ∈ C T such that |φ T | ≤ 1 and e ∞ (φ T ) = inf f ∈CT e ∞ (f ) ≡ e T . Note that, for any f ∈ C T , e ∞ (f ) can be rewritten as: Since F ≥ 0, the first two terms on the r.h.s. of (3.24) are clearly nonnegative. Since v is absolutely summable, the potential v T (x, y) can be rewritten as and, using the fact that v ′′ ≥ 0, it is straightforward to check that each term in the sum on the r.h.s. is pointwise positive, for all 0 ≤ x, y ≤ T . This implies that the third term in the r.h.s. of (3.24) is nonnegative as well. We will denote the kinetic energy, i.e., the first term in the r.h.s. of (3.24), by K f , and the second and third terms by V f and W f , respectively. Now let f j be a minimizing sequence, i.e., e ∞ (f j ) → e T as j → ∞ and f j ∈ C T . First we note that K f j is bounded by a constant independent of j, because K f j ≤ T e ∞ (f j ) and e ∞ (f j ) is uniformly bounded from above by some constant C. Moreover, we can assume without loss of generality that |f j | ≤ 1: in fact, since F (t) has an absolute minimum at t = 1 and the potential v(x, y) is pointwise positive, we see that the energy in are weakly sequentially compact (see [21], Section 7.18), we can therefore find a function φ T in H 1 0 ([0, T ]) and a subsequence (which we continue to denote by f j ) such that f j → φ T weakly in H 1 0 ([0, T ]). By Corollary 8.7 in [21] ("weak convergence implies a.e. convergence"), we can assume without loss of generality that f j (x) → φ T (x) for almost every x ∈ [0, T ]. This function φ T satisfies |φ T | ≤ 1 and will be our minimizer: in fact, since the kinetic energy K f is weakly lower semicontinuous (see [21], Section 8.2), and since V f j → V φT and W f j → W φT as j → ∞, by the dominated convergence theorem, we have that and this shows that φ T is the minimizer.
(3) Now, let us show that e T is continuous in T . We shall do this by deriving bounds from above and below on e T +ε , tending to e T as ε → 0. Let us take ε > 0. In order to get the bound from above on e T +ε , let us consider a variational function f T +ε ∈ H 1 0 ([0, T + ε]), coinciding with φ T , i.e., the minimizer of e T , for x ∈ [0, T ], and equal to 0 in x ∈ [T, T + ε]. , y) , (3.28) and clearly e ∞ (f T +ε ) → e ∞ (φ T ) as ε → 0. In order to get a lower bound, let us use the variational estimate e T ≤ e ∞ (g T ), where g T = φ T +ε x(1 + ε/T ) . Using (3.24), we get

Using (3.24) we get
and clearly e ∞ (g T ) − e ∞ (φ T +ε ) → 0 as ε → 0. A similar proof applies to the case ε < 0, therefore continuity of e T is proved.
(4) Let us prove that lim T →∞ e T exists and is equal to: Using the fact that 0 ≤ φ T ≤ 1, as proved in part (2), and the fact that v ∈ L 1 (R) ∩ L ∞ (R) is completely monotone, we find that the last two terms in (3.31) tend to 0 as T → ∞. Therefore: Repeating the proof in part (2), we find that the inf in the r.h.s. is a minimum, and we denote by f T the corresponding minimizer. Note that the On the other hand, using summability and complete monotonicity of v, we find that for any T where ϕ([f T ]) ∈ C ∞ is the function obtained by periodically repeating the sequence {f T , −θf T } infinitely many times and o(1) is a remainder that goes to 0 as T → ∞. Clearly, the first term in the r.h.s. can be bounded from below by inf f ∈C∞ lim inf L→∞ E F −L,L (f )/(2L) and this concludes the proof of the claim.
(5) Finally, it is straightforward to check that if the distributional derivative of F is a function, then the minimizer φ T satisfies the Euler-Lagrange equation in the sense of distributions. If F (t) is differentiable for t > 0, by the smoothness of v, it follows by a standard "bootstrap" argument (see Theorem 11.7 in [21]), that φ T ∈ C 2 .
If F (t) is convex for t > 0, then the functional e ∞ (f T ) is strictly convex (because (f ′ ) 2 is strictly convex) for f T ∈ C T and the minimizer φ T is unique. Similarly, for any L > 0, the functional E F 0,L (f )/L is strictly convex for f ∈ C * L and so is E per 0,L (f )/L, where E per 0,L (f ) is the analogue of E F 0,L (f ) with periodic boundary conditions: with f ∈ C per L ≡ {g ∈ C * L : g(0) = g(L)}.
By the summability of v, . By convexity, the latter quantity is bounded below by E per 0, . This shows that the limit as T → ∞ of e T is min t∈R + {F (t) + t 2 +∞ −∞ dx v(x)} and concludes the proof of Theorem 1.
Corollary 1 is a simple consequence of Theorem 1 and of its proof.
Proof of Corollary 1. (i). Let T 0 be such that e 0 = e T0 and let us assume by contradiction that there exists an interval [a, b] and a function f ∈ H 1 loc (R), coinciding with for some k ∈ N, and f (a ′ ) = f (b ′ ) = 0. We denote by f 1 the restriction of f to [a ′ , b ′ ] and we write: T 0 e 0 and, by the chessboard inequality, so that we find e ∞ (f 1 ) − e 0 < 0, which is a contradiction.
(ii) Let e 0 = lim T →∞ e T and F (t) convex for t > 0. As proved in Theorem 1, and a repetition of the proof in part (i) shows that f (x) ≡ t 0 is an infinite volume ground state.

AN EXAMPLE
One expects that when v(x)x is summable, i.e., ∞ 0 ν(dα)α −2 < +∞, the minimizer is periodic when λ = v(0) is large, while small λ produces a function with constant sign, say positive. If v(x)x is not summable one expects that the minimizer is always periodic. We shall not prove this last statement, but see [17] for a similar discussion in the discrete microscopic case.
Here we give an illustrative example that will make this small λ, large λ dichotomy clear.
This example is generic, in the convex case, at least, it is only a question of estimating orders of magnitude in the two regimes of λ.

DISCUSSION AND CONNECTION WITH RELATED WORK
We investigated a class of 1D free-energy functionals, characterized by a competition between a local term, prefering a constant minimizer (equal to 1 or −1, the positions of the minima of a "double-well" even function F ), and a long range positive interaction, which is assumed to be reflection positive and summable. We showed by reflection positivity that, for any strength of the long range interaction, the ground state is either periodic (with mean zero) or of constant sign. If the local term F (φ), besides being even in φ, is assumed to be convex on R + , then the ground state is either periodic or constant. The proof is simple and does not depend on the details of the function F or v (as long as v is positive and reflection positive). Note, however, that the assumption that F is even is crucial: this means that we cannot include a chemical potential different from zero.
Our results extend or complement some known results first proved by Alberti and Müller [2] and by Chen and Oshita [9] on periodicity of the minimizers of certain 1D free-energy functionals. They consider models similar to (2.1), with non-zero chemical potential. They require however a smallness condition on the gradient term and there are certain classes of potentials, e.g., power law type interactions, which are not included in their proofs. Their analysis is based on detailed apriori asymptotic estimates that are not needed in our approach.
Let us conclude by mentioning the connection of our results with the so-called "froth problem" put forward by Lebowitz and Penrose in [22]. They consider d-dimensional systems of particles with density ρ (or spin systems with magnetization m) interacting both with a short range interaction and with a long range Kac potential of the form γ d v(γr), with R d v(r)dr = α.
When γ → 0, the exact free energy per unit volume, a(ρ), for the case where v is positive definite (which includes the cases considered here) is given by a(ρ) = a s (ρ) + 1 2 αρ 2 ; a s (ρ) is the free energy due to the short range potential. In cases where the short range interaction induces a phase separation, as indicated by a linear segment in a s (ρ), the long range positive definite Kac potential, with α > 0, will lead to a strictly convex a(ρ). This means that the global phase segregation, due to the short range interaction, is destroyed by the long range positive definite Kac potential in the limit γ → 0. The interpretation given in [22] was that there is no phase separation on the scale γ −1 , but non trivial structures may appear on an intermediate scale 1 ≪ γ −δ ≪ γ −1 . In this sense the system for finite, but small, γ is expected to be a sort of froth, with structures invisible on large scales, but observable on intermediate scales (and these structures may form periodic patterns, as discussed in the introduction). While the scale γ −δ is unknown in general, our results on microscopic models show that, at least in 1D lattice models, the correct scale to look at, at zero temperature, is γ −2/3 , see [17] and the discussion in Section VIII of [18].
The problem of understanding these mesoscopic structures can be related to the minimization problem studied in this paper, thanks to a result by Gates and Penrose [15], who proved that the free energy a(ρ) can also be obtained from a minimization of a free energy functional similar to (2.1), but without a gradient term. Such functional has the interpretation of large deviation functional for observing a mesoscopic density [3,4,8,11]. As already noted, the absence of a gradient term in this functional means that its minimizers (the "typical mesoscopic configurations") will oscillate on a scale small compared to γ −1 , e.g., the γ −2/3 found in [18].
It would be nice to understand the correspondence between these oscillations for γ ≪ 1 and the ones found here and in [2].