The origin phase transitions enlightened by an elementary classical spin model
We introduce a classical spin model with long-range interaction undergoing a first-order -symmetry breaking phase transition (SBPT) which is in our knowledge one of the simplest models showing such a phenomenon. Its aim is to enlighten the generating-mechanism of a SBPT in general, at least for long-range systems, even though it may be give hints useful also for short-range systems. Further, we present a general rule to model the shape of the long-range potential energy of a Hamiltonian system for a -SBPT to occur. The main feature is a double-well potential which competes with the concavity of the entropy in shaping the free energy, accordingly to Landau mean-field theory of SBPTs. Finally, we revisit the Ising model and the spherical model (Berlin-Kak) in mean-field version at the light of the results obtained here. The model introduced here may be suitable also for didactic purposes and for numerical investigation of the dynamic near the transition point.
pacs:75.10.Hk, 02.40.-k, 05.70.Fh, 64.60.Cn
Keywords: Phase transitions; potential energy landscape; configuration space; symmetry breaking
Phase transitions are sudden changes of the macroscopic behavior of a physical system composed by many interacting parts occurring while an external parameter is smoothly varied, generally the temperature, but e.g. in a quantum phase transition it is the external magnetic field. From a mathematical viewpoint, a phase transition is a non-analytic point in the partition function emerging as the thermodynamic limit has been performed. The successful description of phase transitions starting from the properties of the microscopic interactions among the components of the system is one of the major achievements of equilibrium statistical mechanics.
From a statistical-mechanical point of view, in the canonical ensemble, a phase transition occurs at special values of the temperature called transition points, where thermodynamic quantities such as pressure, magnetization, or heat capacity, are non-analytic functions of . These points are the boundaries between different phases of the system. Starting from the exact solution of the -dimensional Ising model  by Onsager , these singularities have been found in many other models, and later developments like the renormalization group theory  have considerably deepened our knowledge of the properties of the transition points. Typically, but non necessarily, these singularities are associated with spontaneous symmetry breaking phenomenon, giving rise to symmetry breaking phase transitions (SBPT). In this paper we consider this case only. But in spite of the success of equilibrium statistical mechanics, the issue of the deep origin of SBPTs remains open, and this motivates further studies of SBPTs.
In this paper we introduce a -symmetric classical spin model undergoing a first-order SBPT, that, in our knowledge, is one of the most elementary models showing such a phenomenon in the canonical treatment. We think that the general mechanism of entailing SBPT acts in our model, and that it becomes manifest because the dramatic simplicity of the model. In more physically realistic models the mechanism may be hidden because the complexity of the models, also in that cases in which the canonical thermodynamic is analytically solvable. Indeed, even if the analytic solution is at disposal, we can find out whether a system undergoes SBPTs by analyzing the solution ’a posteriori’, but we do not know any general criterion capable to predict SBPTs founded only on the properties of the potential energy landscape.
2 General picture of symmetry breaking phase transitions in long-range systems
The leading idea that has inspired the building of our model is the following. Our aim is finding out a general criterion to define a potential shape capable to entail a SBPT. It has been natural to start from the simplest symmetry group in our knowledge, i.e. 111 is the group formed by reflection of coordinates and the identity identity transform which is isomorphic to the ring integers modulo , from which the name. is also known as . Anyway, we hope that the considerations exposed here may be generalized to other symmetry groups, e.g. for and for , .
Let us consider a Hamiltonian system with degrees of freedom and standard kinetic energy, described by the Hamiltonian
where () is the potential energy and the ’s and the ’s () are, respectively, the canonical conjugate momenta, and are continuous variables. , where is the -dimensional configuration space manifold, and is bounded from below. In what follows we will disregard the kinetic term because it does not affect the SBPTs. Indeed, the partition function can be written as
where the first integral in the right hand side is trivial and gives rise to a specific kinetic energy which does not affect the analytic properties of the thermodynamic functions and the symmetry properties. Hereafter, by we will refer to the configurational partition function alone, i.d. the second integral.
In  two straightforward theorems (theorem 1, theorem 2) on a sufficient topological condition for -SBPTs has been proven. In order to show how the theorems work, define the equipotential hypersurfaces
We will simplify a bit the hypotheses of the theorems, in the sense that we do not assume the most general scenario, but maybe the most common one occurring in the models, which is enough for our purposes.
Let be -symmetric. Theorem 1 states that if, for , the ’s are made by two disjoint connected components, which are one the imagine of the other under , then the symmetry is broken for such that and 222We are assuming be a monotonically increasing function of because a negative configurational heat capacity is forbidden in the canonical ensemble.. This occurs because in the thermodynamic limit the canonical statistical measure shrinks around and, as a consequence, the representative point (RP) of the system is forced to choose between one of the two disjoint components, breaking the symmetry and giving rise to a non vanishing spontaneous magnetization. Furthermore (theorem 2), if there exists such that for every is made by a single connected component, then at such that the symmetry is unbroken, so that the presence of at least a non-analytic point is a necessary consequence. is a critical temperature, i.e. the boundary between the broken phase and the unbroken one.
Hereafter, we will focus on the following question: which shape of the potential may make the ’s fulfilling the hypotheses of the theorems in ? The most natural answer is a double-well potential with a minimum gap between the wells proportional to . To this condition we add the request that the two absolute minima have to correspond to a finite magnetization, which will be the spontaneous magnetization at . Indeed, at the RP is frozen in one of the absolute minima. The fact that the symmetry is broken at is not enough to be considered properly a SBPT, e.g. consider the case of -dimensional Ising model. Call the minimum gap between the wells of the potential. Since , we can set , where is a fixed constant. For every , where is the absolute minimum of , is made by two disjoint connected components, so that the hypotheses of theorem 1 are fulfilled. At converse, as is made by a single connected component, so that the symmetry is unbroken because theorem 2 333Here we are assuming that increasing enough , can be raised. This is a reasonable hypothesis, but in general it may not be verified.. Rigorously speaking, the topology of the ’s can be much more complicated than that just described, but it does not affect the substance of the line of reasoning.
To show how in general a double-well potential can be built, we resort to the blazoned ferromagnetic classical Ising model, described by the potential
where means that the sum is extended over the neighbor lattices in a certain range, e.g. the nearest neighbor lattices, and , . The coupling constants has been set to . Now, to make more clear the explanation, we replace the spins by continuous coordinates: , and, in order to confine the RP, we identify the configuration space with the -cube of side centered in the center of coordinates, so that its vertexes coincide with the Ising model lattice sites.
Define the line in configuration space passing between the points and . This line can be parametrized as with . is nothing but the magnetization, defined as , corresponding to the vector . describes a -symmetric concave function of which takes the maximum at and the minimum at the extremes and . The gap is proportional to , as wanted. But to fulfill the hypotheses of theorem 1 in  is requested. It can be shown, without enters the detail of the proof, that at large . A way to fix this difficulty is considering the mean-field case. This case is the easiest to treat because is given by
where the factor has been introduced to maintain the potential intensive. results to be a function of , i.e. is constant on the whole submanifold at constant magnetization defined as follows
Now we will make the link to Landau mean-field theory of second-order -SBPTs. The free energy is assumed an even function of the magnetization (more generally the order parameter)
where are real constants and is the temperature. is a convex function at , while it is a double-well function at . Since the stable configuration of the system corresponds to the minimum of , if the symmetry is unbroken, while if the symmetry is broken.
where , is the average value of the specific potential on the and
is the density of states at magnetization . For the potential (5) we get
where is the entropy at magnetization . After performing the thermodynamic limit , consider the Taylor power series expansion , where 444We are making the reasonable assumption that is concave, e.g. it is true for configuration space of the models considered in this paper., finally getting
which is of the same form of (7). At this point is clear that the phase transition arises from the competition between the double-well potential and the concavity of the entropy modulated by the temperature. At the double-wellness of the potential transfers to the free energy, breaking the symmetry, while at the concavity of the entropy makes the free energy convex, so that the symmetry is unbroken.
What exposed above is a sort of heuristic ’recipe’ for building models undergoing a -SBPT, even though not extensible to the short-range case. Indeed, the mean-field assumption is crucial. Summarizing, for the sake of mathematical formalism, we can condense part of the results in a theorem.
Theorem. Let us consider a system described by a Hamiltonian (1) with degrees of freedom and a double-well potential energy with a symmetry. Let be the coordinates of one of the two absolute minima of . Let be the minimum gap between the wells. If is non vanishing as , and if , or equivalently there exists a constant such that , then in the thermodynamic limit the is broken for where is such that the average potential density .
Proof. As already explained in this Section, a system under these hypotheses fulfills that ones of theorem 1 in , so that the -SB is guaranteed. These hypotheses are not sufficient to generate also a PS, other ones are needed, but here there is no space to deal with this question. Anyway, if there exists a critical temperature , it is such that the critical average potential , which comprehends also the limiting case .
As the reader will have understood, there is a great freedom in creating a double-well potential generalized to dimensions with minimum gap proportional to . For example, we have constrained the degrees of freedom by an -cube, but in the spherical model (Berlin-Kak) the constraint is an -sphere. Another example is the on lattice model, where the constraint is made by a local potential of the form added for every degree of freedom.
3 The new model
In  a simple model with first-order -SBPT has been introduced. It has been called hypercubic model because its double-well potential is built with -cubes. Further, the minimum gap between the wells is assumed proportional to . As we have shown in previous Section, a double-well potential with is a sufficiency condition to entail the -SBPT.
From the hypercubic model we derive a classical spin model by replacing the real coordinate with the classical spin for . In order to reproduce the double-well shape of the potential, we define it in the following way
where is a configuration of the system. This way, the potential has the gap between the wells proportional to . The constant plays the role of the strength of the interaction, and it modulates the critical temperature of the SBPT.
From a physical viewpoint such a potential may be regarded as the tendency of the spins to lie in the sites and by a completely delocalized interaction proportional to their number . This is similar to what happens in some quantum systems, where the non-locality of the wave function can entail particular kinds of interaction depending only on the number of particles lying in some given eigenstates, independently on the distances among the particles.
The thermodynamic can be solved in one shot by using the decomposition formula
where is the density of states, thus
In the limit of large , only one of the two exponential addends in the right hand side of the last equation survives, thus can be approximated as
where is the critical temperature. is the boundary between two different analytic forms of .
In the thermodynamic limit, the free energy, the average potential and the specific heat are, respectively
We can also give the relative expressions for finite
Now consider the magnetization per degree of freedom . As , the average potential is , so that the representative point can access only the sites and , where and , respectively. Since in the thermodynamic limit the probability of overturning simultaneously all the spins is vanishing, has to take only one of the two possible values. As , the average potential is , so that the representative point can freely go around the whole lattice expect the sites and . is vanishing because in the thermodynamic limit the most probable configurations are the ones with half spins taking the value , and half spins taking the value . Thus
Summarizing, we are in front of the complete picture of a first-order -SBPT, despite the dramatic simplicity of the model. As already noted above, the potential (12) can be considered as the effect of a completely delocalized interaction among the spins, thus it belongs to the class of long-range potentials.
3.1 Mapping in the class of the hypercubic models
We recall briefly how configuration space of the hypercubic model introduced in  is defined. We start with an -cube centered in the origin of coordinates where the potential takes the value , except in two smaller -cubes , , where the potential takes the value . In the remaining part of configuration space the potential is assumed to be . and are -symmetric. The hypercubic model undergoes a SBPT with critical temperature , where is the side of and is the side of and . Thus, the class of the hypercubic models depends on two real parameters: and , with and .
The class of the models introduced in this paper depends only on the real parameter . The mapping in the class of the hypercubic model is made by identifying the lattice sites and with the centers of mass of the -cubes and . Since the statistical weight of a single lattice site is and the weight of the whole lattice is , the same proportion has to hold between the volume of (or ) and , thus has to be assumed. Finally, in order to complete the mapping, we make the identification . Thus, and the thermodynamic functions are the same given in previous Section. Obviously, the mapping is injective, but not surjective.
3.2 Free energy as a function of and
In Sec. 3 we have calculated the partition function by decomposing it in a sum over all the possible values of the potential , i.e. and , but we can also choose to decompose in a sum over the possible values of the magnetization , that becomes an integral in the limit . This way, we can find out the -shape of the free energy as a function of , and moreover we can evaluate the effect of an external magnetic field applied to the system.
Let be a positive integer which counts the number of spins taking the value in a given configuration of the system, thus . labels the subsets of configuration space at constant magnetization defined as
Configuration space is restored by the disjoint union of all the ’s: . The relation which links to the corresponding magnetization is , so that .
The potential can by written as a function of
and thus the partition function can be decomposed as a sum over all the ’s
where and is the density of states at magnetization . In our model equals the number of lattice sites belonging to indicated as 555( stands for volume, even though this word generally refers to continuous sets, but with a small abuse of language we use it also for discrete sets. A more appropriate word may be cardinality).. is linked to the entropy by the relation . It turns out that
where are the binomial coefficients. Thus
In order to perform the thermodynamic limit, we use the Stirling formula666 for ., getting
Remembering that , we can express in terms of , and then make the substitution , where is a continuous real variable belonging to the interval . This substitution is possible because, as , the values of become dense and equally spaced in . Thus
The potential and the entropy too remain defined as functions of , respectively, as follows
The last quantities and the free energy are plotted in Fig. 3. is a limiting case of a square double-well where the width of the wells have been set to zero.
The central minimum of , i.e. , competes with the other two extreme minima in order to determine the absolute minima of , thus giving rise to the critical temperature . The magnetization is already given in (22).
Remark. The fact that the potential (12) is a function of the magnetization lets us separate the contribution of the Boltzmann factor from the entropy factor in the calculation of the partition function and therefore of the free energy. Since the density of states increases as because the -dimensionality of configuration space, setting the gap between the wells of the potential proportional to is a necessary and sufficient condition to entail the -SBPT, in accordance with the results found out in previous Section. Indeed, if the gap increased faster than the potential would not be intensive, while if the gap increased more slowly than the -SBPT would disappear.
3.3 Fisher zeros
To give a complete analysis of the thermodynamic of the model introduced here, we will locate the Fisher zeroes of the partition function . Fisher zeros  are the zeros of in the complex-temperature-plane. Because the analyticity of , the Fisher zeroes have not to lie on the real axis, but if becomes non-analytic for , they converge to the real axis at the critical temperature. We recall that Fisher zeros are the analogs of the complex-fugacity-plane zeros of the grand canonical partition function, introduced by Yang and Lee in .
We solve the equation
. The solutions are
As expected, the limit of each solution is the inverse critical temperature
3.4 Effect of an external magnetic field
In this section we determine the effect of an external magnetic field applied to the system. The new potential has to take into account the magnetic interaction by the term
thus, the new potential is
where is given in (30). Therefore, the free energy is
In order to find out the magnetization, we have to minimize with respect to . We start by solving the following equation
whose solution is
Then, we have to confront with if , or with if . This can be easily made by introducing the quantity
if and if , respectively. The root shows a sort of pseudo-critical temperature separating two different analytic forms of
in the limit , thus takes the values or for any value of .
We observe that the presence of an external magnetic field breaks the symmetry of the system, but does not remove the phase transition regarded as a non-analytic point in the thermodynamic function, as we would expect if the -SBPT were continuous. This is not surprising, because the -SBPT is of the first order.
4 Mean-field Ising model
The solution of the classical mean-field Ising model is by means of the mean-field theory and it is a paradigmatic example in literature, e.g. [15, 13]. Nevertheless, we revisit it enlightening how the double-wellness of potential competes with the convexity of the entropy in shaping the free energy as a function of the magnetization, entailing the second-order SBPT with classical critical exponents occurring in this model.
The potential is as follows
where for , , and the factor is introduced to guarantee the intensive property of the potential per degree of freedom.
Following the same method used in Sec. 3, we introduce the positive integer which labels the subsets , defined in (23), of configuration space at constant magnetization , i.e the density of states. can be easily written as a function of because the interaction among the spins is mean-field
Since is linked to the magnetization by the relation , the potential can be expressed as a function of
Then, as , we can make the substitution , where is a continuous real variable, finally getting
It is plotted in Fig.5.
In order to find out the spontaneous magnetization, we have to differentiate with respect to , and set to zero. The resulting equation gives in the implicit form
The last equation has two symmetric solutions with respect to the -axes as , and the solution as , thus the critical temperature is , as well known.
We do not add nothing about the effect of an external magnetic field because it is well known in literature, e.g. .
5 Mean-field spherical model (Berlin-Kac)
In this section we add another example of mean-field model solvable in the same way applied to the model introduced in this paper and to the Ising model. The model was introduced by Berlin and Kac [3, 19] (also known as spherical model). It approximates the picture of the Ising model by substituting the discrete classical spins variables with the continuous real ones for and constraining them on an -sphere777With -sphere we refer to a hypersphere embedded in , some authors call it -sphere. of radius , where is the number of degrees of freedom, centered in the origin of coordinates. Hence, the -sphere is the configuration space and contains the lattice sites of the Ising model as for .
The potential is the same of the Ising model
where the sum is extended over all the couples of variables. Since we consider only the ferromagnetic mean-field case, we also set . The factor has been introduced in order to guarantee the intensive property of the potential per degree of freedom. The potential density can be written as a function of the magnetization
thus, the free energy can be derived as a function of and .
In order to make this, define the subset of at constant magnetization
which is the intersection of the hyperplane at constant magnetization with the -sphere of radius centered in the origin. Hence, is an -sphere, and its volume is
By the Pythagorean theorem, results
where . The sketch in Fig. 6 can help the reader.
The entropy results
and by applying the Stirling formula we get, in the limit ,
Finally, the free energy results
By minimizing with respect to we find out
where is the critical temperature. shows the well known classical second-order -SBPT with a non-analytic point between the ferromagnetic phase and the paramagnetic one. By inserting in (55), we get the free energy
that shows a discontinuity in the second derivative at , as expected.
The picture of the -SBPT is identical to the mean-field Ising model one, apart non-substantial differences in the -shape of the entropy.
6 Concluding remarks
In  two straightforward theorems on sufficient topological conditions for -SBPT has been shown. The conditions are given on the equipotential hypersurfaces (). Loosely speaking, if the ’s are made by two, or more, disconnected components -non-symmetric in a finite interval of the average specific potential , then a -symmetry breaking occurs (theorem 1). Furthermore, if above a certain value of the ’s are made by a single connected component on which the ergodicity is assumed to hold also in the thermodynamic limit, then also a phase transition occurs (theorem 2), in the sense of a loss of analyticity at least in the spontaneous magnetization.
In this paper we have shown how it is possible to shape the potential in order to fulfill the hypotheses of the theorems. The potential is double-well with a minimum gap between the wells proportional to the number of degrees of freedom . The last conditions can be satisfied assuming the potential to be mean-field, i.e. a function of the magnetization. We wonder if this picture may be transferred also in short-range system. Some results found in  for the Ising model with nearest-neighbors interactions seem to give an affirmative answer. Anyway, further studies are needed to clarify the situation. The crucial difference with respect to the long-range case is that the free energy at fixed cannot be a non-convex function, neither in the broken phase. This is clear thinking of a ferromagnetic material modeled by the short-range -D Ising model which can exhibit a vanishing magnetization also in the broken phase.
The classical spin model introduced in Sec. 3 shows in the most elementary way the mechanism of entailing a first-order -SBPT at work. There are no substantial differences with respect to the second-order ones, as highlighted by the Ising model and the spherical model in mean-field version revisited in the Sec. 4 and 5. The critical temperature arises from the competition between the double-wellness of the potential and the density of states at fixed magnetization: the former tends to give the free energy a double-well shape entailing the -symmetry breaking, while the latter tends to make the free energy convex leaving the symmetry intact. This is nothing but the Landau picture of SBPT [15, 13] that has been here linked with the potential shape in configuration space.
For reasons of space, we do not include in this paper the results about the topology of the ’s of the model studied here, which show that a topological change occurs exactly in correspondence with the thermodynamic critical potential. This interesting topic  will be treated in a further paper.
In this paper we have considered only the symmetry. What can we say about other symmetry groups? Since , the most natural symmetry group to consider in order to extend the consideration developed here is , . The concept of a double-well potential with , that we have defined in configuration space of a symmetry system, can be transferred to the reduces configurational space of the magnetization (in general the order parameter). For fixing the ideas, consider . The magnetization is , so that the reduced configurational space is . Since we have limited to mean-field potentials, i.e. the potential is a function of , where , the double-well potential of is replaced by a sombrero-shaped potential, and is replaced by . The potential takes the value over the circle , where is the spontaneous magnetization at . This condition can be immediately generalized to every , for which . At this level this is only a conjecture, even though supported by some results in [4, 5, 31]. e.g. the mean-field model with an symmetry. This may be another line for further investigations.
Finally, the model introduced in this paper may be suitable for didactic purposes, as it is the case of the hypercubic model introduced in , and for numerical simulations of the dynamic near the transition point.
I would like to thank Lapo Casetti for helpful discussions on the subjects of this paper.
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