Frothlike minimizers of a non local free energy functional with competing interactions
Abstract.
We investigate the ground and low energy states of a one dimensional non local free energy functional describing at a mean field level a spin system with both ferromagnetic and antiferromagnetic interactions. In particular, the antiferromagnetic interaction is assumed to have a range much larger than the ferromagnetic one. The competition between these two effects is expected to lead to the spontaneous emergence of a regular alternation of long intervals on which the spin profile is magnetized either up or down, with an oscillation scale intermediate between the range of the ferromagnetic and that of the antiferromagnetic interaction. In this sense, the optimal or quasioptimal profiles are “frothlike”: if seen on the scale of the antiferromagnetic potential they look neutral, but if seen at the microscope they actually consist of big bubbles of two different phases alternating among each other. In this paper we prove the validity of this picture, we compute the oscillation scale of the quasioptimal profiles and we quantify their distance in norm from a reference periodic profile. The proof consists of two main steps: we first coarse grain the system on a scale intermediate between the range of the ferromagnetic potential and the expected optimal oscillation scale; in this way we reduce the original functional to an effective “sharp interface” one. Next, we study the latter by reflection positivity methods, which require as a key ingredient the exact locality of the short range term. Our proof has the conceptual interest of combining coarse graining with reflection positivity methods, an idea that is presumably useful in much more general contexts than the one studied here.
1. Introduction and description of the model
The competition between shortrange attractive forces and longrange dipolar forces can give rise to the spontaneous formation of periodic patterns, such as stripes or bubbles, as observed in several quasi twodimensional (2D) systems, e.g., micromagnets and magnetic films, ferrofluids, quasi 2D electron gases and high temperature superconductors, liquid crystals, system of suspended lipidic molecules on the water surface, assemblies of diblock copolymers, martensitic phase transitions; see, e.g., [4, 5, 14, 21, 25, 35, 38, 39, 40, 42, 43, 47]. From a mathematical point of view, these systems are modelled by a microscopic or mesoscopic nonconvex energy functional, whose low energy states are expected to display the same pattern formation phenomenon. There are a number of rigorous indications of the emergence of regular structures, ranging from equipartition to rigorous upper and lower bounds on the minimizing energy [1, 6, 9, 15, 16, 17, 18, 19, 22, 30, 32, 33]. In a few cases, the existence of periodic ground states can be rigorously proved [2, 8, 13, 26, 27, 28, 29, 31, 36, 41, 44, 46]. Among these, a one dimensional (1D) Ising model with nearest neighbor ferromagnetic (FM) exchange and long range power law antiferromagnetic (AF) interaction, where periodicity of the ground states was proved by means of a generalized reflection positivity (RP) method [26]. Later, such proof of periodicity was extended to other systems, both in one and two dimensions, in the discrete or continuum setting [27, 28, 29, 30, 31]; in particular, we mention two continuous versions of the 1D spin model studied in [26], where the discrete spin Hamiltonian is replaced by an effective free energy functional and the configuration of discrete Ising spins is replaced by a magnetization profile with , either assuming all possible values between and (the “soft spin” case, see [28]), or assuming only values (the “Ising spin” case, see [29]). In both cases, a crucial technical assumption for the method of the proof to work is that the short range FM term appearing in the free energy functional is exactly local, i.e., it is modeled by a gradient term or by a local surface tension term, depending on whether one considers the soft or Ising spin case. Under this assumption, the minimizers are exactly periodic and consist of intervals of constant length (the optimal modulation length) in which the magnetization has constant sign, the sign oscillating from plus to minus or viceversa when one moves from a given interval to the following one. Moreover, the magnetization profiles with free energy close to the minimal one are very close to the periodic minimizers.
From a physical point of view, the locality of the surface tension term is a phenomenological (often unjustified) assumption and it should be essentially irrelevant as far as the results are concerned. In other words, if we replace a local surface tension term by a short but finite ranged one, with range much smaller than the range of the AF interaction, the magnetization profiles minimizing the free energy functional, or with free energy sufficiently close to the minimum, should still consist of a regular alternation of intervals where the magnetization is positive or negative. The exact periodicity of the minimizers may be a special feature of models with a local surface tension, but approximate periodicity should be a robust property. Therefore, it is important to understand whether the results of [28, 29] can be extended to cases where the generalized RP method breaks down, due to the non locality of the short range interaction. The extension has, on the one hand, a specific interest for the class of 1D magnetic models we are considering: in fact, we are not aware of examples of free energy functionals with strictly local penalization of gradients which can be directly derived as continuum limit of microscopic particle models. On the other hand, it has a more general conceptual importance: it is of great interest to develop methods allowing one to extend the validity of results based on RP to cases where RP does not hold exactly, possibly by combining it with coarse graining or averaging methods.
In this paper we attempt a first extension in this direction, by focusing on a free energy functional that arises naturally in 1D Ising models with competing long range interactions at positive temperature in a specific mean field limit, known as Kac limit. To be more precise, let us consider the 1D spin system described by the Hamiltonian
(1.1) 
where and should be thought of as small parameters, the sums over and run over the set , is an even, nonnegative, smooth monotone function with support equal to and is smooth and reflection positive, that is: can be written as the Laplace transform of a positive measure, , with a probability measure on such that a positive constant. We remark that this implies is a function on , with derivatives of all orders extendable up to . To simplify some technical points in the following, we shall further assume that has compact support, well separated from (that is, is a superposition of exponentials, with range comparable with ). Given and , on a coarse grained scale of the order , the typical configurations with respect to the Gibbs measure with Hamiltonian Eq.(1.1) and inverse temperature are described by a nonlocal large deviation functional . Roughly speaking, this means that the probability of the spin configurations compatible with the coarse grained profile is approximately given by as ; for a more precise statement, proved in the case that and are both of finite range, see [12] . The nonlocal large deviation functional corresponding to Eq.(1.1) has the form, ,
(1.2) 
where , and
(1.3) 
with . If , the local potential has a double well shape, with two degenerate minima located in , where is the positive solution to the selfconsistency equation . Therefore, for , the minimizers are the homogeneous profiles or . For there is a competition between the short ranged part , which favors the “FM phase” or , and the long ranged part
2. Main results and strategy of the proof
Our goal is to characterize the shape of the “quasiminimizers” of in the case that , that is when the function in Eq.(1.2) has a double well shape. Loosely speaking, a quasiminimizer is a magnetization profile with energy “sufficiently close” to the minimum; we shall clarify and quantify what we mean by that below. However, before doing this, we find convenient to introduce the sharp interface counterpart of the functional of interest, which was studied in [29] by RP, and to briefly review the key bounds that characterize its quasiminimizers: these will justify and motivate the statement of the corresponding results in the non local functional studied in this paper.
Let be the surface energy associated to the shortranged part of the functional, namely
(2.1) 
where
and
(2.2) 
We recall that the variational problem (2.1) has a minimizer which is unique up to translations [23, 24, 37]. More precisely, any minimizer has the form , , where is a strictly monotone antisymmetric function, solution to the local mean field equation , and converging exponentially fast to as . In particular, . Sometimes the profile is pictorially called the “instanton”.
As explained above, if is very small, we expect that the profiles with minimal energy, or close to the minimal energy, consist of jumps from the negative to the positive phase separated by a distance that typically is much larger than the range of and much smaller than the range of . The transition from negative to positive or viceversa is performed so to make the short range part of the functional happy: therefore, we expect the quasiminimizers to have a shape essentially equal to the instanton in the vicinity of the transition point and we expect the energy cost of the transition to be essentially . The soliton tends to exponentially fast with the distance from the jump; if seen from “far away”, i.e., on scale much larger than 1, the soliton is seen as a sudden jump from to . Therefore, a natural effective functional that should describe well the energy cost of the quasiminimizers, if seen on a scale intermediate between 1 and , is the following “sharp interface” functional, which was studied in [29],
(2.3) 
where , it has finite bounded variation and is the number of jumps from to or viceversa. Given such a , let be the partition of consisting of the maximal intervals on which is constant, which are separated among each other by the jump points; moreover, we define to be the lengths of these intervals. Due to the exact locality of the surface tension term, the functional Eq.(2.3) can be studied by RP methods, which imply the remarkable estimate [29],
(2.4) 
where is the energy per unit length in the thermodynamic limit of the periodic configuration consisting of intervals of constant length equal to and . More precisely,
(2.5) 
and an explicit computation shows that
(2.6) 
and
(2.7) 
Combining the lower bound Eq.(2.4) with the upper bound
gives,
(2.8) 
Moreover, the correction term in Eq.(2.4) provides an explicit bound on the energy cost for picking a magnetization profile different from or from one of its translates. In particular, it characterizes the quasiminimizers of , in the sense that, if , for some , then Eq.(2.4) easily implies that, for all ,
(2.9) 
Since the RP methods yielding Eq.(2.4) break down in the presence of a non local short ranged interaction, the idea is to study the quasiminimizers of Eq.(1.2) by first coarse graining the system on a scale intermediate between 1 and , which is the expected oscillation scale of the quasiminimizers, and to correspondingly reduce the study of to that of a sharp interface functional, analogous to ; next, the effective sharp interface functional will be studied by the same RP methods of [29]. The combination of these two ingredients yields detailed informations and estimates on the shape of the quasiminimizers; we are not able to prove the exact periodicity of the minimizers, because the coarse graining procedure produces error terms (which we explicitly bound in the following) that causes the optimal or quasioptimal profiles to be close to a periodic profile but not necessarily periodic (at least, this is the most we can say).
The main results on the non local functional Eq.(1.2) are summarized in the following theorems.
Theorem 2.1.
For any , there exists such that, if is small enough,
(2.10) 
where the infimum in the lefthand side runs over measurable functions such that for any .
In order to state our results on the shape of the quasiminimizers we need a few more definitions. Let be the ground state energy associated to the functional at finite and finite . Moreover, given , let be the partition of into intervals of length , where ; in the following, we shall refer to these intervals of length as “blocks”. We define to be the coarse version of on , that is, is the function, measurable with respect to , whose value at is equal to , where and . Given , we shall say that the block is of type (resp. ) if (resp. ); we shall say that it is of type if . The coarse version of any function with energy sufficiently close to the minimum consists of long sequences of blocks of type , of total length , followed by long sequences of blocks of type , of total length , except for an infinitesimal fraction of , on which looks “wrong” (i.e., has long sequences of blocks of type or sequences of or of of length different from ). This statement is made more precise by the following theorem.
Theorem 2.2.
Given and , there exists a positive constant such that, if and , then:
For any for which
(2.11) 
there exists a set , measurable with respect to , such that , all of length larger than contained in can be grouped into maximal connected sequences of blocks of constant type, either or , to be called , . On each , the intervals have alternating sign and are separated among each other by at most one block of type . Moreover, . The blocks , which is a disjoint union of intervals
(2.12) 
where indicates the subset of obtained from by depriving it of its first and last block in the sequence it consists of.
Remark 2.1.
The condition is not sharp and it can be easily weakened to , with , at the price of adding some extra conditions on and . For such long intervals, the specific choice of boundary conditions that we made (“open boundary conditions”) is irrelevant as far as the validity of Theorem 2.2 is concerned, that is, the errors in energy that we make in changing from open to “ boundary conditions” (which are of order 1) can be absorbed into the error of order appearing in the statement of the theorem. Here, given any “boundary condition” , i.e., any arbitrarily prefixed function , the functional with boundary conditions on is defined as
The cases in which is equal to , or , or to the periodic extension of to , or to the Neumann extension of to (i.e., the function obtained from by repeated reflections about the endpoints of ), are special and are refereed to as boundary conditions (b.c.), or b.c., or periodic b.c., or Neumann b.c., respectively. Let us note that these four special boundary conditions “are better than others”, in particular they are better than open b.c.: by this we mean that in the presence of such boundary conditions there are no error terms entering the estimates due to the boundary conditions. This makes possible to study the limiting behavior of the functional with (say) periodic boundary conditions as on intervals of length of the order , that is the same scale as the optimal oscillation length . This is an interesting case by itself, which is discussed in Subsection 3.1.
A useful corollary of Theorem 2.2, and in particular of Eq.(2.12), is the following: define the sets
where . Note that the sets and can be thought of as the intervals where “things go wrong”, either because is substantially different from , or because is substantially different from the expected optimal length . and
Corollary 2.1.
Theorem 2.2 and its corollary characterize the quasiminimizers of for small, asymptotically as . In particular, the two estimates in Eq.(2.13) are the analogues of Eq.(2.9). The proofs of these claims are based on a coarse graining procedure that maps every measurable function into a piecewise constant function such that . An essential condition on the coarse grain procedure is that it induces a small change in the long range contributions to the energy. This is realized by conserving the averages on suitably long blocks , where we require . As a consequence we are forced to solve constrained variational problems (in the “canonical ensemble”) on such blocks and to face delicate finite size effects. Among them the arising of a critical droplet size which is treated by means of arguments similar to those employed in [11] in higher dimension. As a result of this construction, the original functional is bounded from below in terms of a new functional , acting on the space of the ’s, which is simpler than the original one, because it has a local surface tension term and, therefore, can be studied by the reflection positivity methods of [26, 28, 29].
The rest of the paper is organized as follows. In Section 3 we describe the coarse graining procedure that maps every profile into a piecewise constant function and, correspondingly, bounds the original functional from below in terms of a simplified functional for , which can be studied by RP methods; the main results of this section is summarized in Proposition 3.1. In Section 4 we prove Theorems 2.1 and 2.2, by using Proposition 3.1. In Section 5 we prove Proposition 3.1. Finally, in Section 6 we draw the conclusions and discuss some open problems. Some technical aspects of the proofs are deferred to the appendices.
3. The coarse graining procedure
In this section, we describe the coarse graining procedure that maps every profile , , into a piecewise constant function such that , where is a suitable positive constant, to be fixed below. We denote by the space of such functions. Given , we denote by the maximal intervals on which has constant sign and by their lengths, with . We shall say that an interval is of type + or , depending on whether is positive or negative on it; in this sense, induces a partition of consisting of intervals of alternating type, on which is correspondingly positive or negative. The relevance of the map relies on the fact that the energy of can be bounded from below in terms of the energy of , which is computed by using a modified functional ; moreover, the minimizers of the latter can be estimated by using the methods of [26, 28, 29]. The result is summarized in the following proposition, which is proved in Section 5.
Proposition 3.1.
Let and . There exists such that, if and , then for any measurable function there exists a piecewise constant function , such that
(3.1) 
where
(3.2) 
Moreover, for any and any ,
(3.3) 
Remark 3.1.
Of course, in order to make the statement of Proposition 3.1 more explicit, we need to explain how the reference profile is defined, which is done in Subsection 3.2 below. However, before doing that, let us add a few more remarks about the connection between Proposition 3.1 and the notion of convergence.
3.1. On the relation with convergence
Proposition 3.1, which is the key technical result behind the proofs of our main results announced in Section 2, is in many respects stronger than Theorems 2.1 and 2.2. In fact, Proposition 3.1 provides us with detailed informations about the “excited states”, rather than just the minimizers or the quasiminimizers, of our variational problem. Consider the space of functions on that assume only values and note that if , the functional is the same as the functional discussed in Section 2, for which the energy of any (not necessarily minimal) configuration can be efficiently estimated by RP methods. If is close to a profile (here “close” means that both and are small as ), then Proposition 3.1 tells us that is bounded from below by plus small error terms as . An inequality in the opposite direction is valid, too: if is “reasonable” (i.e., if the distance between its jump points is larger than ) then one can find a smooth profile (which is obtained from by replacing the sharp interfaces by cutoffed instantonic profiles) such that is bounded from below by , up to small errors as . In other words, our functional of interest is bounded from above and below by up to small error terms as : in this sense Proposition 3.1 can be thought of as a quantitative version of De Giorgi’s convergence [7, 20] in the “thermodynamic limit” (i.e., with error terms scaling proportionally to the length of the interval ).
We recall that the sequence of functionals on a metric function space is said to converge to as if

(liminf). For each and each sequence converging to in , it holds that .

(limsup). For each there exists a sequence converging to in such that .
Moreover, see [7, Theorem 1.21], if the sequence is equicoercive (i.e., if any sequence such that is precompact) then convergence implies the convergence of the minimizers.
If we insist in taking rather than keeping it finite with explicit error terms, we can translate part of the results of Proposition 3.1 into a convergence result, for instance in the case that is chosen proportionally to (which is a possible choice for if, say, periodic boundary conditions are chosen, see Remark 2.1). Let us explain this in some more detail: we choose periodic boundary conditions and , with fixed , and we adopt the rescaled variable . The energy, as a function of the periodic profiles , is easily seen to be given by the following functional,
where
with
Then, it is easy to show that the sequence on is equicoercive and converges to
where is the inverse of the Laplacian on with periodic boundary condition. The minimizers of this functional have been studied in [44] where it is proved that they are periodic. We do not belabor the details of this statement, which is a simple consequence of an analogous statement for the short ranged part of the functional, see e.g. [37, Section 7.1.7], and a straightforward bound on the difference between and .
Of course, as in any convergence result, the form of the limit depends on the rescaling chosen both for the lengths and the energies. The special rescaling chosen above is interesting and natural, because in the limit both the short and the long range interaction terms survive and compete on equal footings, as in the original finite functional of interest. Still, it may be interesting to investigate in a more detailed way the possibility of defining more precisely a notion of convergence in infinite (or at least larger than ) volume. We hope to come back to this issue in a future publication.
Let us now come back to the description of the map .
3.2. Partitioning the big interval.
The replacement of into is a local procedure, defined in each single element of a suitable partition of and depending on the average of on each such element. Therefore, the first thing that we need to explain is how to define the partition , which depends on the shape of itself. We start from a regular partition and then modify some of the intervals, adapting them to the shape of . Let us fix and consider the partition of into intervals of constant length , which was defined after Theorem 2.1. Given on , let us label each block in by the value of its internal energy . We start by selecting the blocks whose internal energy is not bigger than . Not surprisingly, if has energy that does not exceed this cutoff value, then it is possible to find a long segment in where stays close either to or to ; moreover, this segment can be chosen to stay sufficiently far from the boundary of . For a precise statement, see the following lemma (and see Appendix B for its proof).
Lemma 3.1.
Given a block , let us partition it into a sequence of small blocks of size , with a small number, independent of . Given a small block , let us denote by the average of over . If , then for any there exists such that it is possible to find a sequence of contiguous small blocks with the following properties:
1) if , then ;
2) the total length of is larger than , for a suitable constant (possibly depending on );
3) the distance of each block in from the boundary of is larger than .
Now, we modify the original partition of in the following way. For each block with internal energy not bigger than , we remove its boundary lines and draw a new “boundary line” in the middle of the segment defined in Lemma 3.1, see Fig. 1. We end up with a sequence of segments of delimited by two boundary lines, each of which “well in the middle” of a region where is essentially constant. Each has the property of being partitioned into new blocks of size (i.e., of size consits of a single block , to be called a “good block” (note that by construction each good block – with the possible exception of a “boundary” good block, i.e., a good block that is adjacent to the boundary of – has boundary conditions, where is the sign of at the left boundary and is the sign of at the right boundary), or each of its blocks has an internal energy (in which case we will say that consists of a collection of bad blocks). These new blocks form the desired partition of . We will further denote by the set of good blocks in and by the set of bad blocks. ) and, either
3.3. Replacing in each block of the partition.
We now need to explain how to replace by within each element of . Given , we shall assume without loss of generality that , with , and let . We also introduce a tolerance , with a suitable constant to be fixed below, which will be used to distinguish the blocks where the average of is larger than in absolute value, from those where it is smaller. The replacement procedure within depends on whether is good or bad, as explained in the following. For an example, see Fig. 2.

In the case that , the replacement procedure is very simple: if , then on . If , then on and on , with fixed in such a way that . We remark that the choice to have positive to the left and negative to the right is arbitrary and not necessary.

In the case that , the replacement procedure is more elaborated and depends on the specific boundary conditions associated with .

Suppose that is not a boundary block and the boundary conditions are . If , then on and on , with fixed in such a way that . In the opposite case, let us assume (the occurrence can be treated analogously). We decompose as the union of three intervals, , with and [resp. ] on the side where the boundary condition is positive [resp. negative], and set
(3.4) Note that . Moreover, it will be proved below that for , as it should.

Suppose that is not a boundary block and the boundary conditions are . Without loss of generality, let us assume that (of course, the case of boundary conditions is treated similarly). If , with , then
