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Topology and Phase Transitions II. Theorem on a necessary relation

In this second paper, we prove a necessity Theorem about the topological origin of phase transitions. We consider physical systems described by smooth microscopic interaction potentials V_N(q), among N degrees of freedom, and the associated family of configuration space submanifolds {M_v}_{v \in R}, with M_v={q \in R^N | V_N(q) \leq v}. On the basis of an analytic relationship between a suitably weighed sum of the Morse indexes of the manifolds {M_ v}_{v \in R} and thermodynamic entropy, the Theorem states that any possible unbound growth with N of one of the following derivatives of the configurational entropy S^{(-)}(v)=(1/N) \log \int_{M_v} d^Nq, that is of |\partial^k S^{(-)}(v)/\partial v^k|, for k=3,4, can be entailed only by the weighed sum of Morse indexes. Since the unbound growth with N of one of these derivatives corresponds to the occurrence of a first or of a second order phase transition, and since the variation of the Morse indexes of a manifold is in one-to-one correspondence with a change of its topology, the Main Theorem of the present paper states that a phase transition necessarily stems from a topological transition in configuration space. The proof of the Theorem given in the present paper cannot be done without Main Theorem of paper I.


INTRODUCTION
In Statistical Mechanics, a central task of the mathematical theory of phase transitions has been to prove the loss of differentiability of the pressure function -or of other thermodynamic functions -with respect to temperature, or volume, or an external field. The first rigorous results of this kind are the exact solution of 2d Ising model due to Onsager [1], and the Yang-Lee theorem [2] showing that, despite the smoothness of the canonical and grand canonical partition functions respectively, in the N → ∞ limit also piecewise differentiability of pressure or other thermodynamic functions becomes possible.
Another approach to the problem has considerably grown after the introduction of the concept of a Gibbs measure for infinite systems by Dobrushin, Lanford and Ruelle. In this framework, the phenomenon of phase transition is seen as the consequence of non-uniqueness of a Gibbs measure for a given type of interaction among the particles of a system [3,4].
Recently, it has been conjectured [5,6,7,8,9] that the origin of the phase transitions singularities could be attributed to suitable topology changes within the family of equipotential hypersurfaces {Σ v = V −1 N (v)} v∈R of configuration space. These level sets of V N naturally foliate the support of the statistical measures (canonical or microcanonical) so that the mentioned topology change would induce a change of the measure itself at the transition point. In a few particular cases, the truth of this topological hypothesis has been given strong evidence: i) through the numerical computation of the Euler characteristic for the {Σ v } v∈R of a two-dimensional lattice ϕ 4 model [7]; ii) through the exact analytic computation of the Euler characteristic of {M v = V −1 N ((−∞, v])} v∈R submanifolds of configuration space for two different models, the mean-field XY model [10] and the k-trigonometric model [11].
In the present paper we prove a necessity Theorem which implies that for a wide class of potentials (good Morse functions), a first or a second order phase transition can only be the consequence of a topology change of the submanifolds M v of configuration space, and this appears to be the truly primitive and deep mathematical origin of the phase transition phenomena, at least for the mentioned class of potentials.
The Theorem is enunciated as follows: Theorem Let V N (q 1 , . . . , q N ) : R N → R, be a smooth, non-singular, finite-  Thus the proof of the present Theorem crucially relies on Main Theorem of paper I which is a first step toward proving the necessity of topology changes of configuration space submanifolds (either the level sets Σ v of the potential function, or the manifolds M v bounded by them) for the occurrence of phase transitions [12,13]. Main Theorem of paper I is based on the assumption of the existence -at any number N of degrees of freedom -of an energy density interval [v 0 ,v 1 ] free of critical values. Under this assumption, in paper I, we proved the uniform convergence in the thermodynamic limit of configurational entropy in the class of three times differentiable functions. However, since in general it is a very hard task to locate all the critical points of a given potential function, it is also very hard to ascertain whether Main Theorem of paper I applies to it or not. To overcome this limitation, in the present paper we prove a new Theorem, based on a less restrictive assumption (allowing the existence of critical points), which we could not prove without the Main Theorem of paper I.
As already remarked above, the necessity Theorem proved in the present paper applies to a very broad class of systems: those described by finite range potentials which are good Morse functions. In fact, checking whether a given potential is a good Morse function or not is not difficult and amounts to control whether the potential is smooth, bounded below and whether its Hessian is non-degenerate, that is free of vanishing eigenvalues; degeneracy typically occurs in presence of continuous symmetries and can be removed by arbitrarily small and standard perturbations [14] which do not alter neither the microscopic dynamics produced by the gradients of the potential nor the thermodynamics.

BASIC DEFINITIONS
For a physical system S of n particles confined in a bounded subset Λ d of R d , d = 1, 2, 3, and interacting through a real valued potential function V N defined on (Λ d ) ×n , with N = nd, the configurational microcanonical volume Ω(v, N) is defined for any value v of the potential V N as where dσ is a surface element of Now we can define the configurational thermodynamic functions to be used in this paper.
Henceforth, according to the need for explicit reference to the N-dependence of V , we shall use both V and V N to denote the potential.
Definition 1. Using the notationv = v/N for the value of the potential energy per particle, we introduce the following functions: -Configurational microcanonical entropy, relative to Σ v . For any N ∈ N andv ∈ R, log Ω(Nv, N) .
-Configurational microcanonical entropy, relative to the volume bounded by Σ v . For any N ∈ N andv ∈ R, with Θ[·] the Heaviside step function; M(v, N) is the codimension-0 subset of configuration space enclosed by the equipotential hypersurface Σ v . The representation of M(v, N) given in the r.h.s. stems from the already mentioned co-area formula in [15].
Definition 2 (First and second order phase transitions). We say that a physical system S undergoes a phase transition if there exists a thermodynamic function which -in the thermodynamic limit (N → ∞ and vol(Λ d )/N = const) -is only piecewise analytic. In particular, if the second-order derivative of the entropy S (−) ∞ (v) is discontinuous at some pointv c , then we say that a first-order phase transition occurs. If the third-order derivative of the entropy is discontinuous at some pointv c , then we say that a second-order phase transition occurs. These definitions stem from the standard definitions of first and second order phase transitions as due to a discontinuity of the first or second derivatives of the Helmoltz free energy, respectively, and from the existing relationship -through a Legendre transform -between the Helmoltz free energy and the entropy (see Definition 1 in paper I).
Definition 3 (Standard potential, fluid case). We say that an N degrees of freedom potential V N is a standard potential for a fluid if it is of the form where B N is a compact subset of R N , N = nd, Ψ is a real valued function of one variable such that additivity holds, and where U Λ is any smoothed potential barrier to confine the particles in a finite volume Λ, that is where Λ ′ ⊂ Λ and Λ ′ arbitrarily close to Λ ⊂ R N , closed and bounded. U Λ is a confining potential in a limited spatial volume with the additional property that given two limited d-dimensional regions of space, Λ 1 and Λ 2 , having in common By additivity we mean what follows. Consider two systems S 1 and S 2 , having N 1 = n 1 d and N 2 = n 2 d degrees of freedom, occuping volumes Λ d 1 and Λ d 2 , having potential energies v 1 and v 2 , for any (q 1 , . . . , q N 1 ) ∈ (Λ d 1 ) ×n 1 such that V N 1 (q 1 , . . . , q N 1 ) = v 1 , for = v be the potential energy v of the compound system S = S 1 + S 2 which occupies the volume where v ′ stands for the interaction energy between S 1 and S 2 , and if v ′ /v 1 → 0 and v ′ /v 2 → 0 for N → ∞ then V N is additive. Moreover, at short distances Ψ must be a repulsive potential so as to prevent the concentration of an arbitrary number of particles within small, finite volumes of any given size.
Definition 4 (Standard potential, lattice case). We say that an N degrees of freedom potential V N is a standard potential for a lattice if it is of the form where B N is a compact subset of d ≤ m, and 1 ≤ i d ≤ m, after gluing together the two systems through a common d − 1 dimensional boundary the new system has indexes i running over, for example, where v ′ stands for the interaction energy between the two systems and if v ′ /v 1 → 0 and v ′ /v 2 → 0 for N → ∞ then V N is additive.
Definition 5 (Short-range potential). In defining a short-range potential, a distinction has to be made between lattice systems and fluid systems. Given a standard potential V N on a lattice, we say that it is a short-range potential if the coefficients C ij are such that for any i, j ∈ I ⊂ N d , with c is definitively constant for N → ∞.
Given a standard potential V N for a fluid system, we say that it is a shortrange potential if there exist R 0 > 0 and ǫ > 0 such that for q > R 0 it is Definition 6 (Stable potential). We say that a potential V N is stable [16] if there exists B ≥ 0 such that for any N > 0 and (q 1 , . . . , q N ) ∈ (Λ d ) ×n , or for q i ∈ R D , i ∈ I ⊂ N d , Definition 7 (Confining potential). With the above definitions of standard potentials V N , in the fluid case the potential is said to be confining in the sense that it contains U Λ which constrains the particles in a finite spatial volume, and in the lattice case the potential V N contains an on-site potential such that -at finite energy -q i is constrained in compact set of values.
Remark 1 (Compactness of equipotential hypersurfaces). From the previous definition it follows that, for a confining potential, the equipotential hypersurfaces Σ v are compact (because they are closed by definition and bounded in view of particle confinement).
In view of formulating and proving the Main Theorem of the present paper, we have to define some neighborhoods, that we call "pseudo-cylindrical", of critical points of a potential function V N . Before defining these pseudo-cylindrical neighborhoods of critical points, let us remember the following basic result in Morse theory.
Theorem. Let f be a smooth real valued function on a compact finite dimensional manifold M. Let a < b and suppose that the set is compact and contains no critical points of f , that is ∇f ≥ C > 0 with C a constant. Let y ∈ (a, b). Then there exists a diffeomorphism Corollary. The manifolds f −1 (y), a < y < b, are all diffeomorphic.
This result is based on the existence of a one-parameter group of diffeomor- associated with the vector field Details can be found in standard references as [17,18,19].
Applied to the configuration space M, if the function f is identified with the potential V N , then in the absence of critical points of V in the interval Definition 8 (Pseudo-cylindrical neighborhoods). Let Σ vc be a critical level set of V N , that is a level set containing at least one critical point of V N .
Around any critical point q is the distance measured through the metric induced on Σ vc by the euclidean metric of the immersion space, and ρ is such that ρ < 1 c ), i, j label all the critical points on the given critical level set. Moreover, set the thickness of all the pseudo-cylinders equal to ε 0 = min j∈N (v j+1 c − v j c ). After Sard Theorem, both ρ and ε 0 are finite because, at finite dimension, there is a finite number of isolated critical points and, consequently, a finite number of critical values. We define a pseudo-cylindrical neighborhood Γ(q c as the open subset of M bounded by the following set of points. By mapping γ(q , and from Σ vc to Σ (vc−ε 0 ) , through the flow generated by the vector field X = ∇V N (q)/ ∇V N (q) 2 , we obtain the walls of c , ε 0 ), which are transverse to the Σ v , and then we close the neighborhood with the pieces of Σ (vc+ε 0 ) and Σ (vc−ε 0 ) bounded by the images γ(q Lemma 1 (Generalization of Corollary 1, paper I). Let V N be a standard, smooth, confining, short-range potential bounded from below (Definitions 3-7) with V N given by Definition 3 (fluid case), or by Definition 4 (lattice case).
The following two statements hold: Proof. For what concerns point (a), we note that the flow associated with the C ∞ vector field X = ∇V N (q)/ ∇V N (q) 2 is well defined at any point Then, after the "non-critical neck theorem" [17], for where Ω(Nv, N) = vol(Σv N )], having proved the statement (a), we can apply Lemma 4 of paper I which entails that Whence, after Lemma 3 of paper I, it follows The next step is to prove that also S because, after Lemmas 3 and 4 of paper I, this entails the truth of statement (b). Let us begin by considering the microcanonical configurational inverse is well known to be intensive and well defined also in the thermodynamic limit, at least for extensive potential energy functions. Then we work out a representation of β N (v) in the form of a microcanonical average of a suitable function. To this purpose we derive Ω(Nv, N) with respect to v by means of Federer's derivation formula [15,20], and then we integrate it. Federer's derivation formula (see Lemma 5 of paper where A stands for the operator Then we can write where dµ = d N q, so that we finally obtain which holds forv ∈ [0,v 0 ). An important remark is in order. We have used Federer's derivation formula apparently ignoring that it applies in the absence of critical points of the potential function. However, if the potential V is a good Morse function (not a very restrictive condition at all) we know, after Sard theorem [17], that the ensemble of critical values, here of the potential, for Ω(Nv, N), that is where Γ ⋆ stands for the union of all the pseudo-cylindrical neighborhoods of the critical points of V in the interval [v 0 ,v 1 ]. Then we consider the restriction we get By comparing Eq. (14) with Eq. (17), we see that also β N (v) has to be intensive up to the N → ∞ limit, like β N (v). In fact the excision of the set Γ ⋆ out of M v , no matter how the measure of Γ ⋆ depends on N, cannot change the intensive is given by the logarithm of both sides of (16) so that, resorting to Lemma 3 of paper I, the truth of statement (b) follows.

MAIN THEOREM
In this Section we prove the following Theorem 1 (Entropy and Topology).
where ν(v) = max{j|v j c ≤ v}, and an unbound growth with N of one of the derivatives |∂ k S (−) (v)/∂v k |, for k = 3, 4, and thus the occurrence of a first or of a second order phase transition respectively, can be entailed only by the The proof of formula (18) with where we have set equal to zero the minimum of the potential. Let {Σ v j c } be the family of all the critical level sets (in general not manifolds) of the potential, that is the constant potential energy hypersurfaces that contain at least one critical point q c , ε 0 ); ε 0 is the thicknessin potential energy -of the neighborhood. The assumption that the number of critical valuesv j c is at most linearly growing with N entails, together with Sard theorem, that ε 0 is finite. (21) where N (v) is the number of critical points of V N (q) up to the level v. We can equivalently write

Let us now split the integration on
where N cl (v) is the number of critical levels Σ v j c such that v j c < v, and N j cp is the number of critical points on the critical hypersurface Σ j vc and where we have changed the notation of the pseudo-cylindrical neighborhoods to Γ j (q (m) c , ε 0 ) labelling with j the level set to which it belongs and numbering with m the critical points on the j−th level set. Notice that N (v) = N cl (v) j=1 N j cp . Then we use the co-area formula in the r.h.s. of Eq.(20) to rewrite Eq.(22); a distinction is necessary between two cases for Σ v = ∂M v : its label v is closer than ε 0 to a critical level or not; thus we obtain when v Near to any critical point, a second order power series expansion of V (q)

is a good Morse function, then a coordinate transformation exists to the so-called
Morse chart [18] such that where k is the Morse index of q c . Using Morse chart we have where J is the Jacobian of the coordinate transformation.
Using Morse coordinates inside the pseudo-cylinder Γ j (q where the Morse index of q (m) c is denoted by k m , so that | X | 2 = km l=1 x 2 m and | Y | 2 = N l=km+1 x 2 l . Thus we rewrite the r.h.s. of Eq.(25) as where dΩ r is the solid angle element in r dimension, whose integration yields the surface C r of the r-dimensional hypersphere of unit radius. Putting z = |X| 2 and integrating on the angular coordinates we get where α(ξ, r) = ( ξ 2 + 4r 2 − ξ)/2 and β(ξ, r) = ( ξ 2 + 4r 2 + ξ)/2. These expressions stem from the definition of Γ j (q c , ε 0 ) whose boundaries have to be orthogonal to the potential level sets described by Eq. (26). These boundaries are given by the equation |X||Y | = r. Putting y = |Y |, from Eq.(28) when ξ > 0 we obtain and when ξ < 0 we obtain where C N−km−1 and C km are surfaces of hyperspheres of unit radii, that is C n = 2π n/2 /(n/2 − 1)! (for n even) and C n = 2 (n+1)/2 π (n−1)/2 /(n − 2)!! (for we can now write where, since the integrals in Eqs.(33), (34) are independent of the index j, we have defined a set of positive coefficients A(N, i, ε 0 ) as We remark that the term (35), being a function of the Morse indexes µ i (M v ), has topological meaning. In order to make clearer this topological meaning, we rewrite (35) in the equivalent form of a weighed sum of Morse indexes as follows. From the set of positive numbers J j(i,k)m(i,k) we define and rewrite the second term of the r.h.s. of Eq.(33) as Equation (35) is here recast in a form which makes somewhat clearer its dependence on Morse indexes. Moreover, we introduce also the coefficients where k(i) stems from j(i, k) =j, such that for For the purposes of the present proof, we are not concerned about the complication of the coefficients A(N, i, ε 0 ) and what we need, in order to make the link between configuration space topology and thermodynamics, is that the second term in the volume splitting in Eq. (21) can be written in the form (38). In fact, now we can write the entropy per degree of freedom as The equation above links thermodynamic entropy with the Morse indexes of the configuration space submanifolds M v , that is with their topology. In fact, according to Bott's "critical-neck theorem" [21], any change with v of any index µ i (M v ), i = 0, . . . , N, which can only be due to the crossing of a critical level, is associated with a topology change of the M v .
Conversely, any topology change, in the sense of a loss of diffeomorphicity, occurring to the M v when v is varied, is signaled by one or more changes of the Morse indexes µ i (M v ) because, after the "non-critical neck theorem" [17], this has to be the consequence of the crossing of a critical level.
We remark that the two terms Noting that d k B(N, i, v, ε 0 )/dv k = d (k−1) F (ξ, k m , N)/dξ (k−1) , we focus on the smoothness of F ± (ξ, k m , N). Let us consider F + (ξ, k m , N). There are two cases: k m is even so that (k m − 2)/2 is an integer; k m is odd so that (k m − 2)/2 = n + 1/2 with n ∈ {−1} ∪ N. In the first case, by iteratively applying the derivation formula d dα ψ(α) c , ε 0 ) of thickness ε 0 and we take ε < ε 0 arbitrarily close to ε 0 . From Morse theory, we know that passing a critical value v c entails that for each critical point of index k i a k i -handle H N,k i is attached to M v<vc so that the following diffeomorphism holds where a k i -handle in N dimensions (0 ≤ k i ≤ N) is the product of two disks, , and where φ i stands for the attachment of H N,k i to an hypersphere; details can be found in [17,18,19]).
The excision of the pseudo-cylindrical neighborhoods Γ(q i c , ε 0 ) of all the critical points q (i) c ∈ Σ Nvc implies that all the manifolds c , ε 0 ) with Nv c − ε 0 < v < Nv c + ε 0 are free of critical points and, consequently, are diffeomorphic. In fact, for any v, v ′ ∈ R such that the flow associated with the vector field X = −∇V N / ∇V N 2 [17,18].
By applying the Main Theorem of paper I and its Corollary 1 to the first term in the r.h.s. of the equation above, we know that 1 N |∂ k log M(v, N) /∂v k | for k = 1, . . . , 4, are uniformly bounded in N, and thus no phase transition can be attributed to this term.
Then, let us consider the second term of the r.h.s. of the equation above.
By computing its first derivative we obtain d dv Then we compute the second derivative  (N, i(n), v) is shaped as a "staircase" with "rounded corners".
Remark 3. About the applicability domain of the Main Theorem proved in the present paper, note that V is required to be a finite range potential and a good Morse function. The former is a physical assumption, the latter a mathematical property of V . Finite range potentials are typical in condensed matter systems, where even Coulomb interactions are effective only at a finite distance because of the Debye shielding. The mathematical property of being a good Morse function is absolutely generic, in fact it requires that a potential function is bounded from below and that the Hessian of the potential is nondegenerate (i.e. its eigenvalues never vanish). Moreover, given a real-valued function f of class C 2 defined on an arbitrary open subset X of R N , the mapping x f (x) − (a 1 x 1 + · · · + a N x N ) : X R is nondegenerate for almost all (a 1 , . . . , a N ) ∈ R N (see Chapter 6 of Ref. [14]). This means that nondegeneracy is generic whereas degeneracy is exceptional. Continuous symmetries are the only physically relevant source of degeneracy, however this kind of degeneracy can be removed by adding a generic term (a 1 x 1 + · · · + a N x N ) to the potential with an arbitrarily small vector (a 1 , . . . , a N ). This removal of degeneracy is a rephrasing, within the framework of Morse theory, of a standard procedure undertaken in statistical mechanics to explicitly break a continuous symmetry, that is the addition of an external field whose limit to zero is taken after the limit N → ∞. where critical levels are separated by a finite minimum energy amount, as is the case of "spin flips" in the 1d-XY model [10], mean-field XY model [10], p-trig model [11], or of elementary configurational changes that, in lattices and fluids, correspond to the appearance of a new critical value of V at an energetic cost which is independent of N.
It would be impossible to prove MT-II without resorting to MT-I, though MT-II is much stronger than MT-I. Actually

FINAL REMARKS
Let us conclude with a few general comments. Earlier attempts at introducing topological concepts in statistical mechanics concentrated on macroscopic low-dimensional parameter spaces. Actually this happened after Thom's remark that the critical point shown by the van der Waals equation corresponds to the Riemann-Hugoniot catastrophe [22]. Hence some applications of the theory of singularities of differentiable maps to the study of phase transitions followed [23]. An elegant formulation of phase transitions as due to a topological change of some abstract manifold of macroscopic variables was obtained by using the Atiyah-Singer index theorem [24,25] and deserves special attention because it applies to the 2d Ising model, whose phase transition is associated with a jump of the Atiyah index of some suitable vector bundle. This shows that also for discrete variables systems, like spin systems, topological concepts can be useful in the study of phase transitions, provided that the relevant manifolds are identified.
The Main Theorem, that we have proved above, makes a new kind of link between the study of phase transitions and differential topology. In fact, in the present work we deal with the high-dimensional microscopic configuration space of a physical system. The level sets of the microscopic interaction potential among the particles -or the regions of configuration space bounded by them -are the configuration space submanifolds that necessarily have to change their topology in correspondence with a phase transition point. The topology changes implied here are those described within the framework of Morse theory through attachment of handles [18].
We have explicitly investigated these topology changes in some particular models. The results so far obtained, already reported in the literature, are: i) numerical results on the lattice ϕ 4 model [7], ii) exact analytical results on the two models considered in Refs. [10,11], iii) analytical results on the coupled rotators model. Although the models in item (ii) do not fulfil the condition of short-range interactions assumed by MT-I and MT-II in their present formulations, we already get a coherent scenario illustrating in practice how topology changes in presence and in absence of phase transitions. In fact, we observe that when thev-pattern of the Euler characteristic χ(v)which is the probe that we use to detect the variations of topology withvapproaches a smooth curve as N increases, then phase transitions are absent; this is the case of χ(Σ v ) for the one-dimensional ϕ 4 model and of χ(M v ) for the one-dimensional chain of coupled rotators. At variance, sharp jumps or "cuspy"v-patterns of χ(Σ v ) or of χ(M v ) are associated with phase transitions, in qualitative agreement with what is expected after MT-II; in Ref. [11] it is also shown that first and second order phase transitions are signaled by markedly differentv-patterns of χ(M v ).
Notice that in our approach the role of the potential V is twofold: it determines the relevant submanifolds of configuration space and it is a good Morse function on the same space. However, for example, in the case of entropy driven phase transitions occurring in hard sphere gases, the fact that the (sin-gular) interaction potential cannot play any longer the role of Morse function does not mean that the connection between topology and phase transitions is lost, it rather means that other Morse functions are to be used. Just to give an idea of what a good Morse function could be in this case, let us think of the sum of all the pairwise euclidean distances between the hard spheres of a system: it is real valued, it has a minimum when the density is maximum, that is for close packing, meaning that this function is bounded below. The discussion of non-degeneracy is more involved and here would be out of place, let us simply remark that Morse functions are dense and degeneracy is easily removed when necessary.
The topology of configuration space submanifolds makes also a subtle link between dynamics and thermodynamics because it affects both of them, the former because it can be seen as the geodesic flow of a suitable Riemannian metric endowing configuration space [9], the latter because an analytic (though approximate) relation between thermodynamic entropy and Morse indexes of the critical points of configuration space submanifolds can be worked out [10].
Moreover, there are "exotic" kinds of transitional phenomena in statistical physics, like the glassy transition of amorphous systems to a supercooled liquid regime, or the folding transitions in polymers and proteins, which are qualitatively unified through the so-called landscape paradigm [26,27] which is based on the idea that the relevant physics of these systems can be understood through the study of the properties of the potential energy hypersurfaces and, in particular, of their stationary points, usually called "saddles". That this landscape paradigm naturally goes toward a link with Morse theory and topology has been hitherto overlooked. However, though at present our Main Theorem only applies to first and second order phase transitions, the topological approach seems to have the potentiality of unifying the mathematical description of very different kinds of phase transitions.