Phase transition in ferromagnetic Ising model with a cellboard external field.
Abstract
We show the presence of a firstorder phase transition for a ferromagnetic Ising model on with a periodical external magnetic field. The external field takes two values and , where . The sites associated with positive and negative values of external field form a cellboard configuration with rectangular cells of sides sites, such that the total value of the external field is zero. The phase transition holds if is an interaction constant. We prove the firstorder phase transition using the reflection positivity (RP) method. We apply a key inequality which is usually referred to as the chessboard estimate. , where
Keywords: Ising model, periodic external field, Peierls condition, reflection positivity, phase transition.
1 Introduction
In many models of statistical physics the phase transition is a result of spontaneous breaking of the symmetry of a system. The best known model with phase transition is the ferromagnetic Ising model (system) in the absence of a magnetic field. Essentially, this fact has been shown by Peierls [24]. It has become a theorem by Griffiths [20] and Dobrushin [11] (see also [27] and [10]). Peierls ideas are referred to as Peierls arguments based on Peierls condition and Peierls transformation. Peierls condition means that energy required for a droplet formation of one of the phases surrounded by the sites occupied by another phase is proportional to the size of the droplet boundary. For a two dimensional model (on ) the boundary size is the length of the droplet’s boundary. The second component of Peierls arguments allows to perform Peierls transformation. It is based on a symmetry which a studied model has. By Peierls transformation, it is possible to remove a contour in the configuration such that only the energy of the contour is eliminated, and the energy of the rest part of the configuration is not changed. Peierls condition is unrelated to the model symmetry. Peierls condition is satisfied for the Ising model with a uniform external field, however there is no the symmetry in this case.
Peierls arguments show a type of “stability” of ground states. It means that at a low temperature the state is ensemble of small perturbations of the ground state which would result in a configuration “close” to the starting ground state.
Unlike Peierls argument (specifically Peierls transformation), the PirogovSinai theory of phase transitions allows one to find a lowtemperature phase diagram of models with no symmetry requirement. When there is no a symmetry, the lowtemperature phase diagram is shifted with respect to the ground state diagram.
In addition, there exist a several more approaches. One such approach, Reflection Positivity (RP), requires showing a type of reflection symmetry. Essentially, it is possible to prove a phase transition constructing a contour argument using the chessboard inequality obtained from the RP property.
An external field added to the Hamiltonian can change the whole phase diagram. In the case of the ferromagnetic Ising model, any nonzero uniform external field suppresses the phase transition. In some models where the magnetic field is not supposed to be uniform, it is possible to prove phase uniqueness, see for instance, [6], [7]. A random external field can also suppress the phase transition in a planar Ising model (see [1], [2]), even in the case when the total average of the external field is equal to 0.
In this paper we will address the problem of the existence of phase transitions in a planar Ising model where the external field is periodic, forming a cellboard configuration such that total value of the magnetic field is zero. The initial motivation is coming from image processing where Ising models with nonuniform external fields are used for analysing segmentation. The model in this study firstly were numerically studied by M. Sigelle in [26]. Reviews on the applications of Gibbs fields in image processing can be found in Descombes and Zhizhina [9] (see also [22] and the book [28]). Posteriorly, Darbon and Sigelle [8] proposed a grayscale fast and exact optimization method, it decomposing the target image in layers behaving as Ising models with cellboard external fields.
The models with staggered external fields can be useful in the theory of surfaces and domain theory of the solids.
In this work we consider the Ising model where the external field takes two values and , where . The lattice is split into the union of disjointed cells of the same size, and the signs of the external field are alternated similar to a chessboard. Specifically, the cell with one sign of the external field is surrounded by four neighbor cells with the opposite value of the external field. We propose the reflection positivity method for the studies of this model. We will use a specific term for the alternated external field, cellboard partition, to avoid a confusion with the chessboard estimate, to be used later in the paper.
In [23] (by F.R. Nardi, E. Olivieri, and M. Zahradník) the authors study the case when the cells have infinite horizontal length, while their height equals one. Except the phase coexistence at low temperature, a lot of effort in [23] are focused on the proof of the uniqueness in a parameter region where the ground states coexist.
Since our work is concentrated to the cases where Peierls condition is fulfilled then PirogovSinai theory can be applied in this case. However, we used the reflection positivity method based on the periodicity of the cellboard external field. In this sense, as we will state in Corollary 2.4, a particular case of cellboard models when the size of the cells is , is trivially related with antiferromagnetic Ising model with uniform external field (see [13]). For that model, in [16] the RP method has been used to prove the phase transition showed by Dobrushin [12]. Also RP property has been used to prove phase transition in planar rotor models with staggered external field (van Enter and Ruszel [14]). In addition, Frohlich et al. [15] claimed that RP methods would produce better bounds of the critical temperature than PirogovSinai approach can propose.
The paper is organized as follows: In sect. 2 we define the model we study, and present our main result (Theorem 2.2). Sect. 3 contains a brief description of main ideas of the proof. The reflection positivity technique, which is the main tool we use for the proofs, will be discussed in sect. 4.1, where we will also describe the chessboard estimates. The proof of the main result using the RP technique follows the standard scheme, (see for example [4], Chapters 5, 6) and are given in sect. 4.2. The chessboard estimates as constructed in sect. 4.1 does not encompass the external field in [23], therefore in sect. 5 we study using RP a generalization of that model, again in the region with Peierls condition.
2 Definitions and results
We study the ferromagnetic Ising model on with a periodic external field introduced in [26] (see also [22]). Represent the lattice as the union of rectangular cells of the size , : for each pair of integers we define
(2.1) 
That is . Let us define subsets and of :
(2.2) 
A site of is colored white if it is from and black otherwise. Thus, the whole lattice is like a chessboard (see Figure 2, where and ).
Further we use a term cellboard since the term chessboard is used by reflection positivity technics which we apply.
Let be the set of all configurations on . The formal Hamiltonian is defined by
(2.3) 
for any , where is a spin value of configuration at the site , is an interaction constant, a symbol denotes unordered pairs of nearest neighbors , that is the Euclidean distance between the sites is one, , and an external field is given by
(2.4) 
Further, for any subset and any configuration , we will use the notation for the configuration of restricted to the set of sites .
We recall the standard definitions of a Gibbs field on the infinite lattice and related notations. Let be a finite subset from , and let be the set of all configurations on : . The Gibbs probability of the configuration with boundary conditions , is given by
(2.5) 
where is a positive constant usually interpreted as the inverse temperature, and is a normalizing constant, called a partition function.
Let be a set of all Gibbs states on obtained by the thermodynamic limit.
A configuration is a local perturbation of a configuration if there exists a finite set such that
(2.6) 
A configuration is called a ground state for the Hamiltonian , if for any local perturbation of the configuration the inequality
is valid. Following [27] we say that the Peierls condition holds true, if there exists a positive constant such that for any local perturbation (as in (2.6)) of a ground state the inequality
(2.7) 
holds, where is the boundary of the set . The constant is called the Peierls constant.
The following theorem provides the known results from [22] about the ground states and the Peierls condition for our model.
Theorem 2.1.
If
(2.8) 
then there exist two periodical ground states, namely the constant configurations and . In addition, the Peierls condition holds, and the Peierls constant is equal to . If (2.8) does not hold and
(2.9) 
then the configuration
(2.10) 
is the unique periodic ground state.
2.1 Main result. Phase transition for cellboard model
The next theorem provides the presence of a firstorder phase transition in the cellboard model.
Theorem 2.2.
Let the condition (2.8) hold true, then there exists some , such that for any , there exist two distinct measures and , which satisfy
(2.11) 
That means . Moreover
(2.12) 
where are defined in (4.13) and is a combinatorial constant related to the number of contours of a given size.
Remark 2.3.
Theorem 2.2 is the main result in the paper. The proof is in sect. 4.2. It is based on the reflection positivity machinery. We explain the reflection positivity (RP) technique in a way adapted to our model in the section 4.
We conclude this section with a well known fact about connection between a particular case of our model and the antiferromagnetic Ising model with a constant external field. The formal Hamiltonian for the antiferromagnetic model is
(2.13) 
where the interaction constant is negative, , that creates the antiferromagnetic interactions between the nearest spins, the external field is a real constant. The external field, , of the cellboard model, when , should be equivalent to the antiferromagnetic model with the constant external field . This fact has been discussed by Frohlich et al. [16] in the context of RP applications (see also [13]). In our settings this result is the consequence of Theorem 2.2.
Corollary 2.4.
Let . If and for some , then the antiferromagnetic Ising model (2.13) with and has two phases.
Proof.
Consider the cellboard model with , as defined in (2.3) and (2.4). Now, define the transformation of the configuration space , ,
is an onetoone transformation of . Note that if in (2.13) we choose , where , and , then the transformation does not change energy of the configurations and provides the direct equivalence of the models. ∎
3 Plan of the proof of Theorem 2.2
Our model has a set of reflection symmetries that allow us to apply the reflection positivity technique (see subsection 4.1). The proof of the RP property is Proposition 4.3. The reflections are with respect to lines parallel to the coordinate axes. Depending on the parity of sides and , the reflecting lines can either go through the sites of or bisect edges of . In our model, not all such lines are reflecting. As a result there are blocks of the sites in which do not have the reflection property, those blocks entirely reflected with respect to the reflecting lines. Definitions of the blocks see in (4.11) and (4.12).
We take a torus as a main scene of our considerations. The thermodynamical limit is corresponding to the growth of the torus size.
Proving the main Theorem 2.2 we estimate the probability to have different spin values +1 and 1 at remote sites on the torus (see Proposition 4.5). The goal is to show that this probability is small. It is clear that the event
when , should generate Peierls contour which is a set of the edges having the different values on the edge ends. We use the contour arguments for the proof, however we have to use thick contours (block contours) consisted of the blocks. The block contour is composed of the blocks in which Peierls contour is passing. Any configuration on each such block takes the different spin values (a bad block). There is an exclusion which should be treated separately (see about doubleblocks in section 4.2). A small probability of the configurations on the bad block follows from the chessboard estimate (Theorem 4.2) and from the Peierls condition (Lemma 4.7, see also Proposition 4.4). The chessboard estimate is applied to find an upper bound of the bad block probability, see (4.35). The Peierls condition is applied to make this upper bound small at small temperature.
4 A detailed plan of the proof. Constructions
Together with the lattice we often consider a graph
(4.1) 
where is a set of edges between the neighbouring sites. Along with the discrete spaces and sets we consider “continuous” spaces (manifolds) as and tori.
Now, we place the spin system on a twodimensional torus. Let
(4.2) 
be a toric manifold. A map , is such that for every rectangle
the restricted map , is a bijection. Let be an image of by
The coordinate system of is naturally induced by from . An ambiguity because of multivaluedness of the map will not lead to confusions hereinafter.
We assume that is even. Then is composed by cells of and the same amount of cells of type .
Let be the set of all configurations on the torus . We consider Hamiltonian with so called periodical boundary conditions: for any
(4.3) 
The Gibbs measure is
(4.4) 
where is the corresponding partition function:
(4.5) 
4.1 Reflection Positivity and chessboard estimate
In this section we define the Reflection Positivity (RP) technique that we use. The main consequence of RP is the chessboard estimate, which is used to prove phase coexistence in the models with RP property. This technique was developed in the works of Frohlich et al. [15, 16, 17, 19]. Surveys about this method can be found in Georgii [18] and Shlosman [25].
We include some detailed explanations of the RP method, because in our case there exists the dependence of chessboard estimates on the size of the cells of the external field (2.4). We will mainly use the notation and definitions of Biskup and Kotecký [5] and Biskup [4].
4.1.1 Reflection positivity.
We define reflection symmetries with respect to lines orthogonal to one of the lattice directions. Assuming the lattice embedded in , we denote by the group of all transformations of generated by reflections of with respect to lines orthogonal to one of the lattice directions such that is invariant for any : . Let denote the reflection with respect to the line . The group is composed by two distinct subgroups , generated by reflections for which the corresponding lines are
(4.6) 
for or , integer and or . Reflections from we will call the reflections through sites: the corresponding reflection lines pass through the sites of . Reflections from the set we will call reflections through bonds: the corresponding reflection lines bisect bonds of , (4.1).
The groups naturally generate the reflections of the tori and . Thus and . The reflecting line in becomes two antipodal lines in the torus which splits the torus into two symmetric components, say and , the left and the right halfs. We denote those lines with the same symbol as well as the reflection between the left and right halfs such that and vice versa (see Figure 1). Note that for the reflections through the sites () and are disjoint for the reflections through the bonds ().
Let () be a algebra on generated by all functions . As in [5] we introduce a reflection operator : for any spatial reflection . The operator obeys the following properties:

is an involution, ;

is a reflection in the sense that if depends only on configurations on , then depends only on configurations on .
Definition 4.1.
(Reflection Positivity [15, 16], and see Definition 2.2 of [5]). Let be a probability measure on , denote the corresponding expectation, and let be a reflecting line. We say that is a reflection positive measure with respect to if for any two bounded measurable functions and
(4.7) 
and
(4.8) 
where is the measurable function .
A consequence of RP is an inequality like the CauchySchwarz inequality
(4.9) 
4.1.2 Chessboard estimates.
In this section, we recall the chessboard estimate in a form fitted to our case. The symmetries of which are used for the applications, are related to the symmetries of the external field. Since the external field is periodical any symmetry transformation should save block periods. The symmetry transformation of are reflections of with respect to lines in . Let be a set of those lines being the union of the lines
(4.10) 
where and .
Note that if is odd then the corresponding reflection , and if is even then the corresponding reflection . Any such line cuts in half corresponding cells (see (2.1)). The set of lines provides the decomposition of into rectangular blocks (see Figure 2). In each block the total value of external field is equal to zero.
Let be the minimal block, obtained with divisions by , which contains the origin, that is
(4.11) 
A corresponding block on is
(4.12) 
Note that the block contains sites, where
(4.13) 
The torus can be covered by translations of ,
(4.14) 
where is a qoutient subgroup of . Correspondently the torus can be covered by sets :
The neighboring translations of can have a side in common. Let be the set of all configurations defined on and let be a algebra of events on . We call event the events .
Next we introduce some notions. For each , the map is the translation by defined as . We consider the lines
(4.15) 
which bisect the block . The reflections and are out of . In particular, these reflections do not shift and if , the corresponding operators do not preserve energy
A propagation operator on is defined with the help of two operators
(4.16) 
, as following
(4.17) 
Here . The symbol above means the identical operator . The configuration is shifted such that its values on is moved to , with the possible reflections and , depending on the parities of and . The event is a cylindrical set of configurations from .
We remark that a propagator is based on reflection through the sides between two neighbors blocks. Let mean the set of all propagations corresponding the torus . Any propagation is a bijection
Thus we can use the inverse map .
As we will see in the proof of Proposition 4.5, depending on parity of and , we work with four similar propagators , , related to doubleblocks or blocks. Particularly, when (resp. ) is even, we work with horizontal (resp. vertical) blocks, defined by
(4.18) 
The associated quotient subgroups of are, respectively, given by
(4.19) 
Further we use a notion events for sets configurations defined on each of the blocks in (4.18). Moreover, for brevity we denote
(4.20) 
Now, we state the chessboard estimates.
Theorem 4.2.
(Chessboard estimate [15, 16, 4, 25]) Let a measure on which is RP with respect to all reflections between the neighboring blocks . Then for any events and any distinct sites ,
(4.21) 
Moreover, for any events and any distinct sites ,
(4.22) 
The following quantities play the main role in the proof of the phase transition
(4.23) 
where is event. The function is not additive. However, given the additivity of and using the chessboard estimate, it is easy to prove that it is subadditive (see [4], Lemma 5.9). That is, for any collection of events such that , the inequality
(4.24) 
holds. The limiting version of this quantity will be of particular interest for us. Thus, we define
(4.25) 
The existence of the limit follows from the subadditivity. Furthermore, we define for event , where , the similar quantities
(4.26) 
and
4.2 Phase coexistence
The basis of the proof of Theorem 2.2 are Propositions 4.3, 4.4 and 4.5 which we shall prove in the subsection 4.3.
The main applied technics is the reflection positivity technique. The proof essentially consists on two steps. First, the easiest step, Proposition 4.3, we apply a known criterion for establish RP property for our model. Second, we construct two measures and and prove that the probabilities and can be made less than for large . This will prove the phase coexistence. In order to provide this we use the chessboard estimate (4.21) for a sort of Peierls arguments evaluating the contour probabilities. It is implemented in the proof of Proposition 4.5.
The contour technique we work with are based on usage of thick contours partly assembled of a block set , where and partly assembled of the doubleblocks . We shall describe later all details.
Proposition 4.3.
In order to apply the chessboard inequality we introduce bad block events we deal with. Let and be the constant configurations on with all spins plus and all spins minus, respectively.
For each configuration on we define the event
(4.27) 
Let be the set of all bad configurations, . Remember that the size of the block is equal to sites, and as defined in (4.13). This implies that .
Let denote the event that the block is bad for , that is,
(4.28) 
The event is called bad event. Which represents all the torus configurations that are not constant on block.
The proof of the main theorem about the phase coexistence is based on the contour technique. Both Peierls contours and thick contours are applied. The thick contours are consisted of the blocks and blocks. The block is included to the thick contour if Peierls contour touches this block, the block appears in the thick contour when Peierls contour at least partly passes between neighbouring blocks. It happens when the size in the direction of the block localisations is even.
If then the neighbouring blocks and are not intersected, that is when () is even.
As in (4.28) we define the badblock events,
(4.29) 
where and are the constant configurations on each of the blocks, .
In the next proposition we show that the bad and bad events have small probability independently on , when is large.
Proposition 4.4.
If the condition (2.8) holds true, then for any even
(4.30) 
Moreover, for any , and any multiple of 4,
(4.31) 
Now, we can state the main proposition.
Proposition 4.5.
Let the condition (2.8) holds true. There exists a constant such that for any , the following inequality holds
(4.32) 
for any
(4.33) 
The constant appears from the combinatorial argument related to the number of contours builded from  and blocks.
4.2.1 Proof of Theorem 2.2.
First of all, we use the following symmetry of the torus measure. Let be the block containing the site , then
(4.34) 
for any . In order to check this equality we apply the following two transformations for any configuration , such that . First, we apply on the reflection operator , defined as in section 4.1, where is given by (4.15). That is, takes the value , for all . Second, we obtain , flipping all the spin values. In other words, , for all . Clearly, and given , the Hamiltonians are equal . It proves (4.34).
Since the symmetry property of the model the following equalities hold
for any . Therefore we omit sometimes the index at .
Using the chessboard estimate (4.21), we obtain the inequality
(4.35) 
Then, from (4.34)
(4.36) 
Let such that , and define
(4.37) 
By (4.36), (4.30), and Proposition 4.5 we have
(4.38) 
and
(4.39) 
When we extract from the sequences of the measures and , two converging subsequences. Let and be corresponding limits. Those measures are infinitevolume Gibbs measures corresponding to the Hamiltonian ((2.3) and (2.4)). It follows from DLRequation that those measures are Gibbsian (see [4]).
This inequality means that the phase transition holds for all
(4.41) 
∎
4.3 Remaining proofs
4.3.1 Proof of Proposition 4.3.
The proof is the application of the known criteria for a measure to be reflection positive. Fix a line of the reflections and let be the corresponding reflection operator. The criteria applied to our case claims that the measure is reflection positive, if its Hamiltonian can be represented in the form
(4.42) 
where are measurable functions. Then for all the torus Gibbs measure , is RP with respect to (see Definition 4.1). The criteria can be found in Theorem 2.1 of Shlosman [25] or Corollary 5.4 of Biskup [4].
In our case there are two possibilities for : passes trough sites of or not. In the case of passing through the sites of choose