Topology of configuration space of mean-field model by Morse theory
In this paper we present the study of the topology of the equipotential hypersurfaces of configuration space of the mean-field model with a symmetry. Our purpose is discovering, if any, the relation between the second-order -symmetry breaking phase transition and the geometrical entities mentioned above. The mean-filed interaction allows us to solve analytically either the thermodynamic in the canonical ensemble or the topology by means of Morse theory. We have analyzed the results at the light of two theorems on necessary or sufficient topological-geometrical conditions for phase transitions and symmetry breaking recently proven. This study makes part of a research line based on the general framework of geometric-topological approach to Hamiltonian chaos and critical phenomena.
pacs:75.10.Hk, 02.40.-k, 05.70.Fh, 64.60.Cn
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 ising () by Onsager onsager (), these singularities have been found in many other models, and later developments like the renormalization group theory goldenfeld () 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 hereafter). 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.
Consider an degrees of freedom system with Hamiltonian given by
Let be the configuration space. The partition function is by definition
where (in unit ), is the kinetic part of , and is the configurational part. In order to develop what follows we assume the potential to be lower bounded. can be written as follows
where is the potential per degree of freedom, and the ’s are the -level sets defined as
The ’s are the boundaries of the ’s () defined as
The set of the ’s is a foliation of configuration space while varying between and . The ’s are very important submanifolds of because as the canonical statistic measure shrinks around , where is the average potential per degree of freedom. Thus, becomes the most probably accessible -level set by the representative point of the system.
This fact may have significant consequences on the symmetries of the system and on the analyticity of because of the very complex topology in general of the which changes while varying .
We can make the same considerations for , but the related submanifolds , where is the kinetic energy per degree of freedom, are all trivially homeomorphic to an -sphere, thus they cannot affect the symmetry properties of the system by topological reasons. Furthermore, is analytic at any in the thermodynamic limit, so that it cannot entail any loss of analyticity in .
In Sec. II we present a detailed analytical study of the canonical thermodynamic and of the topological changes which occur in the manifolds of the mean-field model, which allows a complete and constructive analytical characterization of the topology of the ’s and also a computation of their Euler characteristic. In Sec. III we present the same study for a case where no SBPT is present, i.e. the model without interaction, in order to compare the two cases and obtain hints towards a general understanding of the general relation between topology changes and SBPTs. In both cases we use Morse theory as a mathematical tool that allows one to study the topology of a manifold in terms of the analytical properties of suitable functions (called Morse functions) . The connection between this technique and physics is made by choosing the potential per degree of freedom as our Morse function. In Sec. IV we try to discover any possible relation between the topology and geometry and the -SBPT of the mean-field model at the light of the results obtained in bc (); ccp (); ccp1 (); ccp2 (); gfp () where some necessary and sufficient geometric-topological condition for SBPTs are showed.
Ii Mean-field model
The lattice models are a class of models with an symmetry for . We have restricted our study to the model with an symmetry (known even as symmetry) and with mean-field (m-f hereafter) interactions, i.e. every degree of freedom interacts with every other. The Hamiltonian is as follows
The ’s are the canonically conjugated momenta of the coordinates ’s, is the coupling constant, and is the number of degrees of freedom.
ii.1 Canonical thermodynamic
In what follows we will disregard the kinetic terms for the reasons already exposed in the previous section. The configurational partition function is
is the local potential. The order parameter, i.e. the magnetization in our case, is
which, introduced in , gives
Now, for the sake of completeness, we briefly recall the solution of the thermodynamic by means of m-f theory, but it is available in literature, for example in dl (). M-f interactions imply that the potential is a function of , so that we can analytically solve by the Hubbard-Stratonovich transformation goldenfeld () based on the equality
which, inserted in (10), yields
and the variable changing , we get
is the configurational free energy per degree of freedom.
Finally, in order to apply the saddle point approximation to calculate , we minimize with respect to at fixed obtaining the spontaneous magnetization . From the latter we get the free energy, the average potential, and the specific heat
respectively. They are plotted in Fig. 1 as functions of . The picture is the well known one of a second-order -SBPT with classical critical exponents.
ii.2 Topology of the submanifolds ’s by Morse theory
Morse theory allows us to characterize the topology of the submanifolds of configuration space defined in (5) by a Morse function . The last is a function whose critical points are non-degenerate, i.e. such that the Hessian matrix of has rank at any critical point. For some introductive details of Morse theory we refer to App. A.
The potential is our Morse function. We cannot show in advance that is a Morse function, but we have verified that the output of our analysis is a set of isolated critical points. The fact that a critical points is isolated does not imply that it is also non-degenerate, but anyway if the set of critical points are isolated Morse theory con be successfully applied the same, even though the potential cannot be properly considered a Morse function. For more information about this delicate question we refer to mhk ().
The fact that all the critical points are isolated is not surprising, because the set of the Morse functions, or non-properly Morse functions in the sense above-specified, is dense in the set of the smooth functions. Furthermore, the discreteness of the symmetry does not create the problem created by continuous symmetries which entail sets of critical points describing submanifolds of configuration space.
Since in the thermodynamic limit the canonical statistical measure shrinks around the corresponding to the average potential density , we are interested in the topology of the ’s rather than the ’s. Anyway, the topology of the ’s are strictly related to that of the ’s, in particular if the ’s are diffeomorphic in an interval , then the same holds also for the ’s.
The critical points are the stationary points of , i.e. the solutions of , which for the potential of the Hamiltonian (6) takes the form
where is the magnetization defined in (9). This is a system of coupled non linear equations of degree , thus, if we aspect at most solutions. Since the equations of the system (19) are all equal, let us omit the index , and consider the equation
Consider the following cases.
(i); . The equation (20) has one real solution.
(ii); . The equation (20) has three real solutions, two of them are coinciding in the limiting case ’’.
Case (i) is easier to treat because the system (19) has an unique solution with components , , where is solution of
therefore, the solutions, with the respective potential values, are
Case (ii). The solutions of the system (19) are given by
with all the permutations of , , and , whose number is the multinomial coefficient
for every choice of such that
Furthermore, , , and have to satisfy the constraint
There are independent equations of the form (28).
To summarize, for a given choice of we obtain, if there exist, some solutions of which yield solutions of the form (24) with multiplicity and with critical value . In the following sections we will see how to calculate the index of every critical point. We have limited to show the results up to because some numerical problems makes the results for not entirely reliable.
It is easy to prove analytically that all the critical levels of the potential are bounded from above by zero. Start by observing that can be written as
if is a solution of , the conclusion is immediately reached. In km () it has been shown the same results for the model with nearest-neighbors interaction.
ii.2.1 Index of the critical points
In Morse theory the index of a critical point is the number of negative eigenvalues of the Hessian matrix , which for the potential (6) takes the form
can be written as , with
where is the matrix whose elements equal . The eigenvalues of read directly on its diagonal, while has an unique non-zero eigenvalue of value because its rank is . Because of the form of , we can apply a results based on the Wilkinson theorem which guarantees that, to get the number of negative eigenvalues of , if we take the number of the negative eigenvalues of we make at most an error of . For more details we refer to ccp1 (); ccp2 ().
ii.2.2 Euler Characteristic
The Euler characteristic is a topological invariant, i.e. a function of a manifold which does not change value if the manifold is deformed without varying its topology. is defined by the Betti numbers pettini (), Morse theory allows us to calculate it for the ’s by the relation
where the Morse number is the number of critical points of that have index . In Fig. 4 we have plotted because it approximately does not depend on . This is due to the fact that the number of critical points grows exponentially with . The modulus appears because is in general an oscillatory function of above and below zero. At we have found . This is coherent with the fact all the critical levels are below zero, because as a consequence for is homeomorphic to an -ball which has .
Iii model without interaction
iii.1 Canonical thermodynamic
In order to make a confront with a model without SBPT, we have studied the model without interaction. The potential is nothing but that of the Hamiltonian (6) when the interacting terms have been deleted, i.e. as is set. The configurational partition function is
where is the local potential (8). The analytic solution can be obtained by the following integral
where are parabolic cylinder functions. After some algebraic manipulation, we get
No SBPT can occur because the thermodynamic function do not depend on , so that the thermodynamic limit cannot generate any emergent behavior.
iii.2 Topology of the submanifolds ’s
The potential of the Hamiltonian (6) can be written as , so that the potential of our model is
As already made for the m-f case, we have to solve , which is the following system
It is immediate to see that the solutions are of the form
The Hessian matrix takes the form
so that the index of the critical points is simply . For the Euler characteristic, all proceeds as for the m-f model.
Iv Discussion of the results
For convenience, in what follows we will consider the ’s instead of the ’s, but they are perfectly equivalent for our purpose. From a topological viewpoint, it is convenient to divide the range of the accessible ’s of the m-f model in three regions:
(i) ; the ’s are equivalent to the disjoint union of two -spheres. , while is a functions of of which some points are plotted in Fig. 11.
(ii) ; the topology of the ’s are dramatically intricate. Simplifying the situation, the interval plays the role of a critical level which divides region (i) from region (iii).
(iii) ; the ’s are equivalent to an -sphere.
is bounded from above by . This is shown in Fig. 11 at least up to , furthermore it can be analytically proven in the following way. Among the solutions (39) of , where is defined in (37), consider the ’s for , and . The corresponding magnetization is vanishing because it is given by . Since
we see that the ’s are solutions even for . The corresponding value of the potential is . Now distinguish two cases: even and odd. If is even, for , so that the latter is an upper bound of . If is odd, for , where is the integer part, , so that is an upper bound of even this case.
We conclude that, neither the growing of , nor the growing of can restrict the critical region (ii). In a further paper we will see how it is possible to reduce it to an unique critical level with an unique critical point of index , i.e. a saddle point.
Since as increases the total amount of critical points tends to reduce from a maximum of for (see Fig. 2, 10), we are led to conjecture that there exists , eventually dependent on , such that for the critical points are the only ones with vanishing magnetization, i.e. with . In other words, the degeneracy on would be removed by values of large enough. Our conjecture is reinforced by a similar result that has been found out in km (); mhk () for the model with nearest-neighbors interaction via numerical analysis.
In what follows we will analyze these results at the light of two theorems recently proven.
iv.1 Theorem on a sufficient topological condition for -Sbpt
In bc () two straightforward theorems on a sufficient geometric-topological condition for -SBPTs have been shown. For the sake of clarity, in the following considerations we will simplify a little bit the picture.
Consider an degrees of freedom Hamiltonian system with a -symmetric, and bonded from below potential. The sufficient condition is as follows: if the ’s are made by two disjoint connected components , for such that is the image of under the symmetry for any , then the symmetry is broken for , where is the average potential (density) selected by the temperature . Furthermore, if there exists a critical potential at which the symmetry breaks, then has to hold.
Again, the second theorem states that, if the ’s are topologically equivalent to an -sphere and ergodic for with , then the symmetry is intact for , so that a phase transition, meant as a non-analytic point in the magnetization, has to occur as a consequence.
Now will see that the m-f model satisfies the hypotheses of the first theorems, at least for the values of considered here, anyway, we have no reasons to doubt that this can hold for every . Let us see why. If a double-well potential has the minimum gap between the wells proportional to , then it satisfies the hypotheses of the first theorem. Indeed, for the values of the potential comprised between the absolute minimum and the minimum gap between the wells the ’s are topological equivalent at least to two disjoint connected components which are non-symmetric under singularly considered. Fig. 11 shows that is proportional to , at least for the ’s considered here. The minimum gap is bounded from below by and from above by , so that it is proportional to . We can so conclude that the m-f model satisfied the hypotheses of the first theorem.
Now we will show that holds for every , as requested by first theorem. Fig. 11 shows that the critical average potential for . To demonstrate that this holds for every we resort to the result obtained in hk () which provides that
Now we will discuss about the second theorem. If the ergodicity hypothesis were satisfied for all the ’s belonging to the region (iii), then the SBPT would be located in the region (ii) for any . But since this is not true, the ergodicity cannot be resort to explain the SB for , despite the fact that the ’s are topologically equivalent to an -sphere. This means that, besides to the topological one, another SB-mechanism has to be at work. In the next Section we will deep this question.
iv.2 Theorem on a necessary topological condition for phase transitions
In fp (); fp1 () a theorem on a necessary topological condition for phase transitions (PT) has been proven. PTs are regarded as a non-analytic point in the thermodynamic functions that arises in the thermodynamic limit. PTs are not necessarily related to symmetry breaking. Simplifying a little bit the picture, the theorem states that if a Hamiltonian system with a potential which is confining, bounded from below, and short-range undergoes a PT, then a topological change has to occur in correspondence of the critical average potential of the PT.
Our results show that may belong either to the region (ii) or to the region (iii). If we are in the second case, then cannot be in correspondence of any critical because they are all topologically equivalent to an -sphere. Anyway, this dose not affect the theorem because the m-f model is a long-range system which does not satisfies a hypothesis of the theorem.
In km (); mhk () the critical points of the model with nearest neighbor interaction have been studied. It has been shown that all the critical points are below zero, like in m-f case. This affects the theorem because the critical potential is unbounded above, so that as it is greater than zero, it cannot be in correspondence to any .
In gfp () the authors of the theorem have answered with a generalized version of it where a topological change is only a particular case of a more general necessary condition on the . In particular, the authors extend the concept of diffeomorphicity among manifolds to the concept of asymptotic diffeomorphicity, according to which two manifolds can be topologically equivalent at any but not asymptotic diffeomorphic as . This concept has been successfully applied to the model. We suggest that the same results may be obtained also for the m-f version, and further numerical and analytic studies in this direction may be a natural extension of this line of research. We suggest that, if this were true, the hypotheses of the theorem may be enlarged to include also long-range potentials.
iv.3 Euler characteristic
In ccp (); ccp1 (); ccp2 () the model in m-f version and without interaction has been studied following the same procedure of this paper. We recall that the m-f model undergoes a second-order -SBPT with classical critical exponent, as the m-f interaction requires. The thermodynamic critical average potential is in correspondence of a topological critical level at which a topological change occurs with critical points of all possible indexes, and the graph of the modulus of the Euler characteristic shows a jump. Obviously, has a jump at any critical level, but provided that is large enough it can be approximated by a continuous function. In the model without interaction no one of these behaviors shows. At converse, has a continuous shape everywhere.
These results have suggested a strong relation between SBPTs and topological changes, but this scenario does not seem confirmed by the results of the m-f model founded out here. shows a jump at followed by a very intricate shape and by an angular point at . never corresponds to the thermodynamic critical average potential, while corresponds only for a particular choice of . Worst, this shape is showed even by the model without interaction where no SBPT occurs, even though the graph is more regular. We conclude that the scenario depicted by the m-f model is a particular case without any particular significance from a viewpoint of the relation between SBPT and topology of configuration space.
V Concluding remarks
In this paper we have analytically characterized the topology of the ’s of the m-f model and the same model without interaction by means of Morse theory. Then, we have tried to discover any possible link with the -SBPT occurring in the m-f model. From a topological viewpoint we have not spotted any qualitative difference between the two models. The critical levels remain confined in an interval of the potential which is for the model without interaction and for the m-f model, where as increases.
Despite the huge number of critical points growing as , but decreasing at the increasing of , in our opinion all that critical points do not have any particular significance except that to separate the region (i), where the ’s are homeomorphic to two disjoint -balls, from the region (iii), where ’s are homeomorphic to an -ball alone. This is true only in the m-f model, where the potential is double-well with the two absolute minima separated by a minimum gap proportional to . This is the effect of the m-f interaction and it is the very peculiar difference between the two models. Indeed, the absolute minima of the model without interaction are and remain located at .
We wonder weather the scenario of the m-f model may be transferred to any short-range version. In our opinion the answer is negative, because a double-well potential with minimum gap proportional to implies a non-concave graph of the microcanonical entropy , while in a short-range system it must be concave or non-strictly concave in the presence of a phase transition. Furthermore, it can be shown that for a short-range -dimensional Ising model the minimum gap is proportional to in the limit of large .
Nevertheless, in mhk () the authors have found the potential of the nearest-neighbor version to have only three critical points, i.e. a saddle at the -level set and two absolute minima at a level set proportional to , for values of the coupling constant large enough. Let be the minimum value of for which the critical points are only three at fixed . It turns out that allegedly goes as , so that, starting from and , new critical points, whose critical levels belong to , necessarily arise while increasing . This is not a proof that the minimum gap cannot be proportional to , but if this were not true, then the minimum gap should be necessarily proportional to even in the nearest-neighbor version.
Another remarkable fact is that the SBPT occurs even in correspondence of ’s which are topological equivalent to an -sphere. This means that a further SB mechanism must exist besides the topological one working for . In other words, the ergodicity of a cannot be guaranteed by the only assumption that it is homeomorphic to an -sphere, as is already arisen from the model. Further studies about the asymptotic diffeomorphicity gfp () of the in correspondence to the critical point may be a natural line of future research. To conclude, we hope that this work may give useful hints to deepen our understanding of the role of geometry and topology of configuration space in phase transitions phenomenon.
Appendix A A glance at Morse theory
Morse theory links the topology of a given manifold with the properties of the critical points of smooth functions defined on it. Two manifold and are topologically equivalent if they can be smoothly deformed one into the other, i.e. if there exists a diffeomorphism that maps into . Here we consider only compact, finite-dimensional manifolds, but most of the results can be extended to non-compact manifolds. The key ingredient of Morse theory is to consider the manifold as decomposed into the level sets of a function . We recall that the -level set of is the set
M being compact, any function has a minimum and a maximum , so that one can be build starting from and then adding continuously to it all other level surfaces up to . Further, we define the ’part of M below ’ as
As is varied between and , describes the whole manifold .
For our purposes, we need to restrict the class of functions into the class of Morse functions, which are defined as follows. A point is called a critical point of if , while the value is called a critical value. is called a Morse function on if its critical points are non-degenerate, i.e., if the Hessian matrix of at , whose elements in local coordinates are
has rank , where is the dimension of . As a consequence, one can prove that the critical points of a Morse function, and also its critical values, are isolated. It can be proved also that the set of the Morse functions are dense in the space of the smooth functions from to . A level set is called critical level if there is at least a critical point belonging to it.
If the interval contains no critical values of , then the topology of does not change for any . This result is sometimes called the noncritical neck theorem.
If the interval contains critical values, the topology of changes in correspondence with the critical values themselves, in a way that is completely determined by the properties of at the critical points. The number of negative eigenvalues of , , is the index of the critical point. The change undergone by the submanifolds as a critical level is passed is described using the concept of ’attaching handles’. Suppose that the critical level contains a critical point of index . We define a -handle in dimensions () as a product of two disks, one -dimensional and the other -dimensional:
Having defined handles, we can state the main result of Morse theory. Let a smooth embedding ( is a -sphere). Then one can build the topological space , i.e. with a -handle attached by . This procedure admits a generalization to the simultaneous attachment of -dimensional handles of indexes .
Appendix B Mean-field model
Equation (28) is equivalent to a th degree equation in the magnetization . Only numerical solutions are available for this equation. In order to check the results obtained here, we have introduced a slight modification in the m-f model to obtain an equation (28) solvable analytically, i.e. of a degree less or equal to the th. The goal has been get by lowering the degree of the local potential by one unit, i.e. by substituting the quartic term by a cubic term where the modulus has been added to conserve the symmetry, so that the local potential takes the form
where the sign has to be taken if , while the if . The solutions are
where . By inserting in (28) we get
where the coefficients , , , and have been introduced. The last equation is equivalent to
which is a th degree equation in , q.e.d..
The results do not show any qualitative difference with respect to the m-f model, neither in the canonical thermodynamic, nor in the topology of the -level sets. All the considerations made in Sec. II for the m-f model can be perfectly transferred to the m-f model.
Acknowledgements.The most part of the work in this paper is contained in my Master’s thesis baroni (), hence I would like to warmly thank the supervisor L. Casetti. Furthermore, I would like to thank also M. Pettini for priceless discussion, suggestions, and for having created and continuously supported this line of research.
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