Giulio Pettini
email: pettini@fi.unifi.it

Matteo Gori
email: gori6matteo@gmail.com

Roberto Franzosi
email: franzosi@ino.it

Cecilia Clementi
email: cecilia@rice.edu

Marco Pettini
email: pettini@cpt.univ-mrs.fr

On the origin of Phase Transitions in the absence of Symmetry-Breaking

Keywords: numbers: 05.45.+b, 02.40.-k, 05.20.-y

In this paper we investigate the Hamiltonian dynamics of a lattice gauge model in three spatial dimensions. Our model Hamiltonian is defined on the basis of a continuum version of a duality transformation of a three dimensional Ising model. The system so obtained undergoes a thermodynamic phase transition in the absence of a global symmetry-breaking and thus in the absence of an order parameter. It is found that the first order phase transition undergone by this model fits into a microcanonical version of an Ehrenfest-like classification of phase transitions applied to the configurational entropy. It is discussed why the seemingly divergent behaviour of the third derivative of configurational entropy can be considered as the "shadow" of some suitable geometrical and topological transition of the equipotential submanifolds of configuration space.

I. INTRODUCTION

One of the main topics in Statistical Mechanics concerns phase transitions phenomena.

From the theoretical viewpoint, understanding their origin, and the way of classifying them, is of central interest.

Usually, phase transitions are associated with a spontaneous symmetry-breaking phenomenon: at low temperatures the accessible states of a system can lack some of the global symmetries of the Hamiltonian, so that the corresponding phase is the less symmetric one, whereas at higher temperatures the thermal fluctuations allow the access to a wider range of energy states having all the symmetries of the Hamiltonian. In the symmetry-breaking phenomena, the extra variable which characterizes the physical states of a system is the order parameter. The order parameter vanishes in the symmetric phase and is different from zero in the broken-symmetry phase. This is the essence of Landau's theory. If G 0 is the global symmetry group of the Hamiltonian, the order of a phase transition is determined by the index of the subgroup G ⊂ G 0 of the broken symmetry phase. The corresponding mechanism in quantum field theory is described by the Nambu-Goldstone's Theorem. However, this is not an all-encompassing theory. In fact, many systems do not fit in this scheme and undergo a phase transition in the absence of a symmetry-breaking. This is the case of liquid-gas transitions, Kosterlitz-Thouless transitions, coulombian/confined regime transition for gauge theories on lattice, transitions in glasses and supercooled liquids, in general, transitions in amorphous and disordered systems, folding transitions in homopolymers and proteins, to quote remarkable examples. All these physical systems lack an order parameter.

Moreover, classical theories, as those of Yang-Lee [START_REF] Yang | Statistical theory of equations of state and phase transitions I. Theory of condensation[END_REF] and of Dobrushin-Lanford-Ruelle [START_REF]A comprehensive account of the Dobrushin-Lanford-Ruelle theory and of its developments can be found[END_REF], require the N → ∞ limit (thermodynamic limit) to mathematically describe a phase transition, but the study of transitional phenomena in finite N systems is particularly relevant in many other contemporary problems [START_REF] Gross | Microcanonical Thermodynamics. Phase Transitions in "Small" Systems[END_REF], for instance related with polymers thermodynamics and biophysics [START_REF] Bachmann | Thermodynamics and Statistical Mechanics of Macromolecular Systems[END_REF], with Bose-Einstein condensation, Dicke's superradiance in microlasers, nuclear physics [START_REF] Ph | Caloric Curves and Energy Fluctuations in the Microcanonical Liquid-Gas Phase Transition[END_REF], superconductive transitions in small metallic objects. The topological theory of phase transitions provides a natural framework to get rid of the thermodynamic limit dogma because clear topological signatures of phase transitions are found already at finite and small N [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF][START_REF] Casetti | Phase transitions and topology changes in configuration space[END_REF].

Therefore, looking for generalisations of the existing theories is a well motivated and timely purpose. The present paper aims at giving a contribution in this direction along a line of thought initiated several years ago with the investigation of the Hamiltonian dynamical counterpart of phase transitions [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF][START_REF] Casetti | Dynamical and Statistical properties of Hamiltonian systems with many degrees of freedom[END_REF][START_REF] Casetti | Geometric approach to Hamiltonian dynamics and statistical mechanics[END_REF] which eventually led to formulate a topological hypothesis. In fact, Hamiltonian flows (H-flows) can be seen as geodesic flows on suitable Riemannian manifolds [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF][START_REF] Pettini | Geometrical hints for a nonperturbative approach to Hamiltonian dynamics[END_REF], and then the question naturally arises of whether and how these manifolds "encode" the fact that their geodesic flows/H-flows are associated or not with a thermodynamic phase transition (TDPT). It is by following this conceptual pathway that one is eventually led to hypothesize that suitable topological changes of certain submanifolds of phase space are the deep origin of TDPT. This hypothesis was corroborated by several studies on specific exactly solvable models [START_REF] Casetti | Topological origin of the phase transition in a mean-field model[END_REF][START_REF] Casetti | Exact result on topology and phase transitions at any finite N[END_REF][START_REF] Casetti | Phase transitions and topology changes in configuration space[END_REF][START_REF] Angelani | Topology and Phase Transitions: from an exactly solvable model to a relation between topology and thermodynamics[END_REF][START_REF] Santos | Topological approach to microcanonical thermodynamics and phase transition of interacting classical spins[END_REF] and by two theorems. These theorems state that the unbounded growth with N of relevant thermodynamic quantities, eventually leading to singularities in the N → ∞ limit -the hallmark of an equilibrium phase transition -is necessarily due to appropriate topological transitions in configuration space [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF][START_REF] Franzosi | Theorem on the origin of Phase Transitions[END_REF][START_REF] Franzosi | Topology and Phase Transitions I. Preliminary results[END_REF][START_REF] Franzosi | Topology and Phase Transitions II. Theorem on a necessary relation[END_REF][START_REF] Kastner | Phase Transitions Detached from Stationary Points of the Energy Landscape[END_REF].

Hence, and more precisely, the present paper aims at investigating whether also TDPT occurring in the absence of symmetry-breaking, and thus in the absence of an order parameter, can be ascribed to some major geometrical/topological change of the previously mentioned manifolds.

To this purpose, inspired by the dual Ising model, we define a continuous variables Hamiltonian in three spatial dimensions (3D) having the same local (gauge) symmetry of the dual Ising model (reported in Section II) and then proceed to its numerical investigation. The results are reported and discussed in Section III. Through a standard analysis of thermodynamic observables, it is found that this model undergoes a first order phase transition. It is also found that the larger the number of degrees of freedom the sharper the jump of the second derivative of configurational entropy, what naturally fits into a proposed microcanonical version of an Ehrenfest-like classification of phase transitions.

A crucial finding of the present work consists of the observation that this jump of the second derivative of configurational entropy coincides with a jump of the second derivative of a geometric quantity measuring the total dispersion of the principal curvatures of certain submanifolds (the potential level sets) of configuration space. This is a highly non trivial fact because the peculiar energy variation of the geometry of these submanifolds, entailing the jump of the second derivative of the total dispersion of their principal curvatures, is a primitive, a fundamental phenomenon: it is the cause and not the effect of the energy dependence of the entropy and its derivatives, that is, the phase transition is a consequence of a deeper phenomenon. In its turn, the peculiar energy-pattern of this geometric quantity appears to be rooted in the variations of topology of the potential level sets, thus the present results provide a further argument in favour of the topological theory of phase transitions, also in the absence of symmetry-breaking.

II. THE MODEL

Starting from the Ising Hamiltonian

-J ij ∈Λ σ i σ j (1)
with nearest-neighbor interactions ( ij ) on a 3D-lattice Λ, where the σ i are discrete dichotomic variables (σ i = ±1) defined on the lattice sites and J is real positive (ferromagnetic coupling), one defines the dual model [START_REF] Kogut | An introduction to lattice gauge theory and spin systems[END_REF] -

J U ij U jk U kl U li (2)
where the discrete variables U mm are defined on the links joining the sites m and m , and

U mm = ±1.
The summation is carried over all the minimal plaquettes (denoted by ) into which the lattice can be decomposed. The dual model in (2) has the local (gauge) symmetry

U ij → ε i ε j U ij (3)
with ε i , ε j = ±1, and i, j ∈ Λ. Such a gauge transformation leaves the model (2) unaltered, and after the Elitzur theorem [START_REF] Elitzur | Impossibility of Spontaneously Breaking Local Symmetries[END_REF] U ij does not qualify as a good order parameter to detect the occurrence of a phase transition because U ij = 0 always. In other words, no bifurcation of U ij can be observed at any phase transition point inherited by the model ( 2) from the Ising model [START_REF] Pettini | was on leave of absence from Osservatorio Astrofisico di Arcetri[END_REF].

In order to define a Hamiltonian flow with the same property of local symmetry -hindering the existence of a standard order parameter -we borrow the analytic form of (2) and replace the discrete dichotomic variables U ij with continuous ones U ij ∈ R. We remark that we donot want to investigate the dual-Ising model, rather we just heuristically refer to it in order to define a gauge model with the desired properties.

Moreover, we add to the continuous version of (2) a stabilizing term which is invariant under the same local gauge transformation (3); this reads

α ij U 2 ij -1 4 , (4)
where ij stands for nearest-neighbor interactions for link variables and α is a real positive coupling constant.

On a 3D-lattice Λ, and with I = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}, we thus define the following model Hamiltonian

H(π, U ) = i∈Λ µ∈I 1 2 π 2 iµ -J ∈Λ U ij U jk U kl U li + α i∈Λ µ∈I U 2 iµ -1 4 ( 5
)
whose flow is investigated through the numerical integration of the corresponding Hamilton equations of motion.

A more explicit form of ( 5) is given by

H(π, U ) = n i,j,k=1 3
ν=1 1 2 π 2 ijkν -J n i,j,k=1 [U ijk1 U i+1jk2 U ij+1k1 U ijk2 (6)
+ U ijk2 U ij+1k3 U ijk+12 U ijk3 + U ijk3 U ijk+11 U i+1jk3 U ijk1 ] + α n i,j,k=1 3
ν=1 U 2 ijkν -1 4 ,
where the summation is carried over trihedrals made of three orthogonal plaquettes. Here U ijk1 is the link variable joining the sites (i, j, k) and (i + 1, j, k), U ijk2 is the link variable joining the sites (i, j, k) and (i, j + 1, k), U ijk3 is the link variable joining the sites (i, j, k)

and (i, j, k + 1). Similarly, for example, U i+1jk2 joins the sites (i + 1, j, k) and (i + 1, j + 1, k), U ij+1k1 joins (i + 1, j + 1, k) and (i, j + 1, k), and so on. That is to say that the fourth index labels the direction, i.e which index is varied by one unit.

The Hamilton equations of motion are given by

Uijkν = π ijkν , πijkν = - ∂H ∂U ijkν , i, j, k = 1, . . . , n; ν = 1, 2, 3, (7)
periodic boundary conditions are always assumed.

The numerical integration of these equations is correctly performed only by means of symplectic integration schemes. These algorithms satisfy energy conservation (with zero mean fluctuations around a reference value of the energy) for arbitrarily long times as well as the conservation of all the Poincaré invariants, which include the phase space volume, so that also Liouville's theorem is satisfied by a symplectic integration. We adopted a thirdorder bilateral symplectic algorithm as described in [START_REF] Casetti | Efficient symplectic algorithms for numerical simulations of Hamiltonian flows[END_REF]. We used J = 1 and α = 1, the integration time step ∆t varied from 0.005 at low energy to 0.001 at high energy so as to keep the relative energy fluctuations ∆E/E close to 10 -6 .

III. DEFINITION OF THE OBSERVABLES AND NUMERICAL INVESTIGA-TION

Given any observable A = A(π, U ), one computes its time average as

A t = 1 t t 0 dτ A[π(τ ), U (τ )] (8)
along the numerically computed phase space trajectories. For sufficiently long integration times, and for generic nonlinear (chaotic) systems, these time averages are used as estimates of microcanonical ensemble averages in all the expressions given below.

A. Thermodynamic observables

The basic macroscopic thermodynamic observable is temperature. The microcanonical definition of temperature depends on entropy -the basic thermodynamic potential in the microcanonical ensemble -according to the relation

1 T = ∂S ∂E V , ( 9
)
where V is the volume, E is the energy and the entropy S is given by

S(N, E, V) = k B log dπ 1 • • • dπ N dU 1 • • • dU N δ[E -H(π, U )] ( 10
)
where N is the total number of degrees of freedom, N = 3n 3 in the present context, and U k stands for any suitable labelling of them. By means of a Laplace transform technique [START_REF] Pearson | Laplace-transform technique for deriving thermodynamic equations from the classical microcanonical ensemble[END_REF],

from Eqs. ( 9) and [START_REF] Pettini | Geometrical hints for a nonperturbative approach to Hamiltonian dynamics[END_REF] one gets (setting k B = 1)

T = 2 (N -2) K -1 -1 . ( 11
)
where K -1 is the microcanonical ensemble average of the inverse of the kinetic energy

K = E -V (U )
, where V (U ) is the potential part of the Hamiltonian [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF].

In numerical simulations

K -1 = 1 t t 0 dτ n i,j,k=1 3
ν=1 1 2 π 2 ijkν (τ ) -1 . ( 12
)
For t sufficiently large K -1 attains a stable value (in general this is a rapidly converging quantity entailing a rapid convergence of T ).

Since the invariant measure for nonintegrable Hamiltonian dynamics is the microcanonical measure in phase space, the occurrence of equilibrium phase transitions can be investigated in the microcanonical statistical ensemble through Hamiltonian dynamics [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF][START_REF]For generic quasi-integrable systems[END_REF]. Standard numerical signatures of phase transitions (also found with canonical Monte Carlo random walks in phase space) are: the bifurcation of the order parameter at the transition point (somewhat smoothed at finite number of degrees of freedom), and sharp peaks -of increasing height with an increasing number of degrees of freedom -of the specific heat at the transition point. As already remarked above, our model [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF], because of the local (gauge) symmetry, lacks a standard definition of an order parameter as is usually given in the case of symmetrybreaking phase transitions. And in fact, in every numerical simulation of the dynamics we have computed the time average U ij t always finding U ij t 0 independently of the lattice size and of the energy value (the double average means averaging over the entire lattice first, and then averaging over time).

Thus, the presence of a phase transition is detected through the shape of the so-called caloric curve, that is, T = T (E). For the model in [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF] this has been computed by means of Eq. [START_REF] Casetti | Topological origin of the phase transition in a mean-field model[END_REF]. Then the microcanonical constant-volume specific heat follows according to the relation 1/C V = ∂T (E)/∂E. The numerical computation of specific heat can be independently performed, with respect to the caloric curve, as follows. Starting with the definition of the entropy, given in [START_REF] Pettini | Geometrical hints for a nonperturbative approach to Hamiltonian dynamics[END_REF], an analytic formula can be worked out [START_REF] Pearson | Laplace-transform technique for deriving thermodynamic equations from the classical microcanonical ensemble[END_REF], which is exact at any value of N . This formula reads

c V (E) = C V N = N (N -2) 4 (N -2) -(N -4) K -2 K -1 2 -1 , ( 13
)
and this is the natural expression to work out the microcanonical specific heat by means of Hamiltonian dynamical simulations.

In order to get the above defined specific heat, time averages of the kind

K α t = 1 t t 0 dτ n i,j,k=1 3 ν=1 1 2 π 2 ijkν (τ ) α are computed with α = -1, -2.
Then, for sufficiently large t, the microcanonical averages K α can be replaced by K α t . In Figure 1 the caloric curve is reported for different sizes of the lattice. A kink, typical of first order phase transitions, can be seen. This entails the presence of negative values of the specific heat, and, consequently, ensemble nonequivalence for the model under consideration.

And, in fact, in Figure 2, where we report the outcomes of the computations of the specific heat according to Eq.( 13), we can observe an energy interval where the specific heat C V is negative, and very high peaks are also found. Nevertheless, these peaks are not related with an analyticity loss of the entropy (see Section III C) but rather depend on the existence of two points of flat tangency to the caloric curve.

In Figure 3 The results so far reported provide us with a standard numerical evidence of the existence of a first order phase transition undergone by the model investigated. Besides standard thermodynamic observables, the study of phase transitions through Hamiltonian dynamics makes available a new observable, the largest Lyapunov exponent λ, which is of purely dy-namical kind, and which has usually displayed peculiar patterns in presence of a symmetrybreaking phase transition [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF][START_REF] Casetti | Geometric approach to Hamiltonian dynamics and statistical mechanics[END_REF][START_REF] Caiani | Geometry of dynamics, Lyapunov exponents and phase transitions[END_REF][START_REF] Caiani | Geometry of dynamics and phase transitions in classical lattice ϕ 4 theories[END_REF][START_REF] Caiani | Hamiltonian dynamics of the two-dimensional lattice ϕ 4 model[END_REF][START_REF] Firpo | Analytic estimation of the Lyapunov exponent in a mean-field model undergoing a phase transition[END_REF][START_REF] Barré | Lyapunov exponents as a dynamical indicator of a phase transition[END_REF]. Therefore in the following Section an attempt is made to characterise the phase transition undergone by our model also through the energy dependence of λ.

B. A Dynamic observable: the largest Lyapunov exponent

The largest Lyapounov exponent λ is the standard and most relevant indicator of the dynamical stability/instability (chaos) of phase space trajectories. Let us quickly recall that the numerical computation of λ proceeds by integrating the tangent dynamics equations, which, for Hamiltonian flows, read

ξi = ζ i , ζi = - N j=1 ∂ 2 V ∂q 1 ∂q j q(t) ξ j , i = 1, . . . , N (14)
together with the equations of motion of the Hamiltonian system under investigation. Then the largest Lyapunov exponent λ is defined by

λ = lim t→∞ 1 t log [ξ 2 1 (t) + • • • + ξ 2 N (t) + ζ 2 1 (t) + • • • + ζ 2 N (t)] 1/2 [ξ 2 1 (0) + • • • + ξ 2 N (0) + ζ 2 1 (0) + • • • + ζ 2 N (0)] 1/2 , (15)
In a numerical computation the discretized version of ( 15) is used, with ξ = (ξ 1 , . . . , ξ 2N )

and

ξ i+N = ξi λ = lim m→∞ 1 m m i=1 1 ∆t log ξ[(i + 1)∆t] ξ(i∆t) , (16)
where, after a given number of time steps ∆t, for practical numerical reasons it is convenient to renormalize the value of ξ to a fixed one. The numerical estimate of λ is obtained by retaining the time asymptotic value of λ(m∆t). This is obtained by checking the relaxation pattern of log λ(m∆t) versus log(m∆t).

Note that λ can be expressed as the time average of a suitable observable defined as follows. From the compact notation ξi = k J ik [q(t)]ξ k for the system [START_REF] Angelani | Topology and Phase Transitions: from an exactly solvable model to a relation between topology and thermodynamics[END_REF] and observing that 1 2 setting

d dt log(ξ T ξ) = ξ T ξ + ξT ξ 2ξ T ξ = ξ T J[q(t)]ξ + ξ T J T [q(t)]ξ 2ξ T ξ ,
J [q(t), ξ(t)] = {ξ T J[q(t)]ξ + ξ T J T [q(t)]ξ}/(2ξ T ξ), one gets λ = lim t→∞ 1 t log ξ(t) ξ(0) = lim t→∞ 1 t t 0 dτ J [q(τ ), ξ(τ )] , (17)
which formally gives λ as a time average as per Eq.( 8).

The numerical results summarised in Figure 4 qualitatively indicate a transition between two dynamical regimes of chaoticity: from a weakly chaotic dynamics at low energy density values, to a more chaotic dynamics at large energy density values. However the transition between these dynamical states is a mild one. At variance with those models where a phase transition stems from a symmetry-breaking, here there is no peculiar property of the shapes of λ = λ(E/N ) in correspondence of the phase transition. Therefore, in the following Section we directly tackle the numerical study of the differentiability class of the entropy.

C. Microcanonical definition of phase transitions

As is well known, according to the Ehrenfest classification, the order of a phase transition is given by the order of the discontinuous derivative with respect to temperature T of the Helmholtz free energy F (T ). However, a difficulty arises in presence of divergent specific heat C V associated with a second order phase transition because this implies a divergence of (∂ 2 F/∂T 2 ), and, in turn, a discontinuity of (∂F/∂T ) so that the distinction between first and second order transitions is lost. By resorting to the concept of symmetry-breaking, Landau theory circumvents this difficulty by classifying the order of a phase transition according to the index of the symmetry group of the broken-symmetry phase which is a subgroup of the group of the more-symmetric phase. As in the present work we are tackling a system undergoing a phase transition in the absence of symmetry-breaking, we have to get back to the origins as follows. According to the Ehrenfest theory, a phase transition is associated with a loss of analyticity of a thermodynamic potential (Helmholtz free energy, or, equivalently Gibbs free energy), and the order of the transition depends on the differentiability class of this thermodynamic potential. Later, on a mathematically rigorous ground, the identification of a phase transition with an analyticity loss of a thermodynamic potential (in the grancanonical ensemble) was confirmed by the Yang-Lee theorem. Now, let us consider the statistical ensemble which is the natural counterpart of microscopic Hamiltonian dynamics, that is, microcanonical ensemble. As already recalled in Section III.A, here the relevant thermodynamic potential is entropy, and considering the specific heat

C -1 V =
∂T (E) ∂E which, after Eq.( 9), reads

C V = - ∂S ∂E 2 ∂ 2 S ∂E 2 -1 , (18)
from the last expression we see that C V can diverge only as a consequence of the vanishing of (∂ 2 S/∂E 2 ) which has nothing to do with a loss of analyticity of S(E). This is why in Section III.A we have affirmed that the peaks of C V reported in Figure 2 stem from a rather trivial effect. For standard Hamiltonian systems (i.e. quadratic in the momenta) the relevant information is carried by the configurational microcanonical ensemble, where the configurational canonical free energy is

f N (β) ≡ f N (β; V N ) = 1 N log Z c (β, N ) with Z c (β, N ) = (Λ d ) ×n dq 1 . . . dq N exp[-βV N (q 1 , . . . , q N )]
and the configurational microcanonical entropy (in units s.t. k B = 1) is

S N (v) ≡ S N (v; V N ) = 1 N log Ω(N v, N ) ,
where v = v/N is the potential energy per degree of freedom, and

Ω(v, N ) = (Λ d ) ×n dq 1 • • • dq N δ[V N (q 1 , . . . , q N ) -v] , (19)
is the volume of the equipotential hypersurface Σ v (codimension-1 subset of configuration space), and δ[•] is the Dirac functional. Then S N (v) is related to the configurational canonical free energy, f N , for any N ∈ N, v ∈ R, and β ∈ R through the Legendre transform

-f N (β) = β • vN -S N (v N ) ( 20
)
where the inverse of the configurational temperature T (v) is given by

β N (v) = ∂S N ∂v (v) . (21)
Then consider the function

φ(v) = f N [β(v)], from φ (v) = -v [dβ N (v)/dv] we see that if β N (v) ∈ C k (R) then also φ(v) ∈ C k (R) which in turn means S N (v) ∈ C k+1 (R) while f N (β) ∈ C k (R).
Hence, if the functions {S N (v)} N ∈N are convex, thus ensuring the existence of the above Legendre transform, and if in the

N → ∞ limit it is f ∞ (β) ∈ C 0 (R) then S ∞ (v) ∈ C 1 (R), and if f ∞ (β) ∈ C 1 (R) then S ∞ (v) ∈ C 2 (R).
We are now ready for a classification of phase transitions à la Ehrenfest in the present microcanonical configurational context.

The original Ehrenfest definition associates a first or second order phase transition with a discontinuity in the first or second derivatives of

f ∞ (β), that is with f ∞ (β) ∈ C 0 (R) or f ∞ (β) ∈ C 1 (R)
, respectively. This premise heuristically suggests to associate a first order phase transition with a discontinuity of the second derivative of the entropy S ∞ (v), and to associate a second order phase transition with a discontinuity of the third derivative of the entropy S ∞ (v). Let us stress that this definition is proposed regardless the existence of the Legendre transform, which typically fails in presence of first order phase transitions which bring about a kink-shaped energy dependence of the entropy [START_REF] Gross | Microcanonical Thermodynamics. Phase Transitions in "Small" Systems[END_REF]. Thus, strictly speaking, the definition that we are putting forward does not mathematically and logically stem from the original Ehrenfest classification. The introduction of our entropy-based classification of phase transitions à la Ehrenfest is heuristically motivated, but to some extent arbitrary. This entropy-based classification no longer suffers the previously mentioned difficulty arising in the framework of canonical ensemble, including here both divergent specific heat in presence of a second order phase transition and ensemble non-equivalence. In the end the validity of the proposed classification has to be checked against practical examples. The gauge model, here under investigation, provides a first benchmarking in this direction.

It is worth mentioning that a thorough investigation of microcanonical thermodynamics, with special emphasis on phase transitions, can be found in Ref. [START_REF] Gross | Microcanonical Thermodynamics. Phase Transitions in "Small" Systems[END_REF]. However, our above proposed approach is rather different from (and complementary to) that discussed in [START_REF] Gross | Microcanonical Thermodynamics. Phase Transitions in "Small" Systems[END_REF].

In fact, in [START_REF] Gross | Microcanonical Thermodynamics. Phase Transitions in "Small" Systems[END_REF] the determinant and the eigenvalues of the curvature matrix associated with the two dimensional entropy surface S(E, N ) are the basic quantities to signal the presence, and define the order, of a phase transition; while no reference is made to the differentiability class of the entropy to characterise a phase transition. From the numerical results concerning the functions T (E) and u(E), reported in Figure 1 and Figure 3, respectively, we computed the first and second derivatives of the configurational entropy as The derivative (dE/du) entering Eq.( 22) is obtained after inversion of the function u = u(E)

∂S ∂u = ∂S ∂E dE du = 1 T (E) dE du , (22)
∂ 2 S ∂u 2 = ∂ ∂u 1 T (E) dE du . (23)
reported in Figure 3 and by means of a spline interpolation of its points. Whereas ∂ 2 u S(u) in Eq.( 23) is computed from the raw numerical data, and the derivatives with respect to u have been obtained by means of a standard central difference formula.

The four patterns of ∂ u S(u), computed for different sizes of the lattice and reported in Figure 5 and Figure 6, show that each ∂ u S(u) appears splitted into two monotonic branches, one decreasing and the other increasing as functions of u, respectively. Approximately out of the interval u ∈ (-1.6, -0.65) the four patterns are perfectly superposed, whereas within this interval -which contains the transition value u c -1.32 -we can observe that the transition from ∂ u S < 0 to ∂ u S > 0 gets sharper at increasing lattice dimension. This means that the second derivative of the entropy, ∂ 2 u S(u), tends to make a sharper jump at increasing N . And in fact, this is what is suggested by the four patterns of ∂ 2 u S(u) -computed for the same sizes of the lattice -reported in Figure 7. These are strongly suggestive to belong to a sequence of patterns converging to a step-like limit pattern. In this case the third order derivative (∂ 3 S/∂u 3 ) would asymptotically diverge entailing a loss of analyticity of the entropy which, in fact, would drop to S ∞ (u) ∈ C 1 . And this is in agreement with the above proposed classification à la Ehrenfest.

It is worth noting that we have found evidence of only one transition point, despite the presence of two peaks of the specific heat, thus confirming that in order to correctly characterize a phase transition in the microcanonical ensemble one has to look for the signals of analyticity loss of the entropy.

D. A geometric observable for the level sets Σ v in configuration space

We have seen in the preceding Section that -within the confidence limits of a numerical investigation -the first order phase transition of the gauge model under investigation seems to correspond to an asymptotic divergence of the third derivative (∂ 3 S/∂u 3 ) of the microcanonical configurational entropy. Under the main theorem in Ref. [START_REF] Franzosi | Topology and Phase Transitions II. Theorem on a necessary relation[END_REF] this should stem from a topological change of the potential level sets Σ u = V -1 (u). In order to get some information of topological kind about these level sets one has to resort to concepts and methods of differential topology. In fact, differential topology allows to catch some topological infor-mation on differentiable manifolds through suitable curvature integrals (the Gauss-Bonnet theorem being the first classical example of this type). Relevant geometric quantities can be computed through the extrinsic geometry of hypersurfaces of a Euclidean space. To do this one has to study the way in which an N -surface Σ curves around in R N +1 by measuring the way the normal direction changes as we move from point to point on the surface. The rate of change of the normal direction N at a point x ∈ Σ in direction v is described by the shape operator (sometimes also called Weingarten's map)

L x (v) = -∇ v N = -(v • ∇)N,
where v is a tangent vector at x and ∇ v is the directional derivative; gradients and vectors being represented in R N +1 .

For the level sets of a regular function, as is the case of the constant-energy hypersurfaces in the phase space of Hamiltonian systems or of the equipotential hypersurfaces in configuration space, thus generically defined through a regular real-valued function f as Σ a := f -1 (a), the normal vector is N = ∇f / ∇f . The eigenvalues κ 1 (x), . . . , κ N (x) of the shape operator are the principal curvatures at x ∈ Σ. For the potential level sets Σ v = V -1 (v) the trace of the shape operator at any given point is the mean curvature at that point and can be written as [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF][START_REF] Thorpe | Elementary Topics in Differential Geometry[END_REF]

M = - 1 N ∇ • ∇V ∇V = 1 N N i=1 κ i . (24)
We have numerically computed the second moment of M averaged along the Hamiltonian flow

σ M = N V ar(M ) t = N [ M 2 t -M 2 t ] 1 N N i=1 κ 2 i t - 1 N N i=1 κ i 2 t , (25)
where we have assumed that the correlation term N -2 i,j [ k i k j t -k i t k j t ] vanishes. In fact, on the one side there is no conserved ordering of the eigenvalues of the shape operator along a dynamical trajectory, and on the other side the averages are performed along chaotic trajectories (the largest Lyapounov exponent is always positive) so that k i and k j vary almost randomly from point to point and independently one from the other.

The numerical results are reported in Figure 8 and Figure 9, where an intriguing feature of the patterns of σ M (E) and σ M (u) is evident: below the transition point (marked with a vertical dashed line) the concavity of both σ M (E) and σ ( u) is oriented downward so that d 2 σ M /dE 2 and d 2 σ M /du 2 are negative, whereas just above the transition point both σ M (E) and σ M (u) are segments of a straight line, so that d 2 σ M /dE 2 and d 2 σ M /du 2 vanish. Thus both derivatives make a jump at the transition point. Again within the validity limits of numerical investigations, this means that the third order derivatives, and in particular d 3 σ M /du 3 , diverge. It is then natural to think of a connection with the asymptotic divergence of d 3 S/du 3 suggested by the results reported in the preceding Section.

Remark. There is a point of utmost importance to comment on. In presence of a phase transition (and of a finite size of a phase transition as is the case of numerical simulations) the typical variations of many observables at the transition point are the effects of the singular properties of the statistical measures and hence of the corresponding thermodynamic potentials (entropy, free energy, pressure). But this is not true for the geometric quantity σ M (u) which is independent of the properties of any statistical measure. Peculiar changes of the geometry -and possibly of the topology -of the potential level sets of configuration space (detected by σ M ) constitute the deep origin, the cause of phase transitions, not an effect. The singular pattern of σ M at the transition point is a primitive phenomenon. In other words, geometrical/topological variations of the spaces where the statistical measures are defined (phase space and configuration space) entail their singular properties [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF]. The vice versa is meaningless. Let us now see how the jump of the second derivative of σ M (u) and the jump of the second derivative of the configurational entropy S(u) can be both attributed to a deeper phenomenon: a suitable change of the topology of the {Σ u } u∈R . In what follows we resort to the best, non-trivial approximations at present available.

Consider the pointwise dispersion of the principal curvatures

s κ = 1 N N i=1 (κ i -κ) 2 (26)
where

κ = 1 N N i=1 κ i (27)
equation ( 26) is equivalently rewritten as [START_REF] Zhang | Some New Deformation Formulas about Variance and Covariance[END_REF]

s κ = 1 N 2 N i,j=1 (κ i -κ j ) 2 (28)
and the time average along the Hamiltonian flow of s κ is then equivalently written as

s κ t = 1 N N i=1 κ 2 i t -κ 2 t = 1 N 2 N i,j=1 (κ i -κ j ) 2 t . (29)
Now, from Eqs.( 25) and ( 29) we get

σ M -s κ t - 1 N N i=1 κ i 2 t + κ 2 t ( 30
)
so that, if we make a "mean field"-like approximation in the first term of the r.h.s. of [START_REF] Barré | Lyapunov exponents as a dynamical indicator of a phase transition[END_REF] by replacing the κ i with their average κ it follows σ M -s κ t -κ 2 t + κ 2 t , and as κ in Eq.( 27) is the same of M in Eq.( 24) one trivially gets

(1 - 1 N )σ M -s κ t 0 (31)
so that, under this "mean field"-like approximation, σ M in Eq.( 25) can be used to estimate s κ t . Then, as the ergodic invariant measure of chaotic Hamiltonian dynamics is the microcanonical one, the time averages • t provide the values of the surface averages • Σ E . Hence, and within the validity limits of the undertaken approximations, an interesting connection can be used between extrinsic curvature properties of an hypersurface of a Euclidean space R N +1 and its Betti numbers (the diffeomorphism-invariant dimensions of the cohomology groups of the hypersurface, thus topological invariants) [START_REF] Nakahara | Geometry, Topology and Physics[END_REF]. This connection is made by Pinkall's inequality given in the following.

Denoting by

σ(L x ) 2 = 1 N 2 i<j (κ i -κ j ) 2
the dispersion of the principal curvatures of the hypersurface, then after Pinkall's theorem

[34] 1 vol(S N ) Σ N v [σ(L x )] N dµ(x) ≥ N -1 i=1 i N -i N/2-i b i (Σ N v ) , (32)
where b i (Σ N v ) are the Betti numbers of the manifold Σ N v immersed in the Euclidean space R N +1 , S N is an N -dimensional sphere of unit radius, and µ(x) is the measure on Σ N v . With the help of the Hölder inequality for integrals we have

Σ N v [σ(L x )] 2 dµ(x) ≤ Σ N v {[σ(L x )] 2 } N/2 dµ(x) 2/N Σ N v dµ(x) 1/(1-2/N ) (33)
whence, at large N , [START_REF] Pinkall | Inequalities of Willmore Type for Submanifolds[END_REF] this inequality becomes an equality when |f | p / f p p = |g| q / g q q almost everywhere [START_REF] Reed | Functional Analysis, revised and enlarged edition[END_REF], where f p is the standard L p norm f p = S |f | p dµ(x)

Σ N v dµ(x) -1 Σ N v [σ(L x )] 2 dµ(x) ≤ Σ N v {[σ(L x )] 2 } N/2 dµ(x) 2/N
1/p , where S is a measurable space.

In the inequalities above g(x) = 1, thus the Hölder inequality becomes an equality when |f | p = f p p / S dµ(x), that is, when |σ(L x )| N equals its average value almost everywhere on Σ N v . Introducing a positive remainder function r(v), Eq.( 34) is rewritten as

Σ N v dµ(x) -1 Σ N v [σ(L x )] 2 dµ(x) = Σ N v {[σ(L x )] 2 } N/2 dµ(x) 2/N -r(v) . (35)
For the model under investigation, the pointwise dispersion of the principal curvatures of the potential level sets actually displays a limited variability from point to point. This follows from the observation that the numerically computed variance of the mean curvature in Eq.( 25) is very fastly convergent to its asymptotic value, independently of the initial condition [START_REF]The spread of the values of σ M = N V ar(M ) ∆t = N [ M 2 ∆t -M 2 ∆t ], numerically computed along short segments of time duration ∆t = 100[END_REF]. Hence, in the present case, the remainder r(v) appears to be a small correction and, consequently, the Hölder inequality is tight.

Then, using

σ M = N [ M 2 Σv -M 2 Σv ] ∼ Σ N v dµ(x) -1 Σv [σ(L x )] 2 dµ(x) (36)
together with Eq.( 35) and the Pinkall inequality, one finally gets

σ M ∼ Σ N v {[σ(L x )] 2 } N/2 dµ(x) 2/N -r(v) ∼ vol(S N ) N -1 i=1 i N -i N/2-i b i (Σ N v ) 2/N -r(v) , (37)
that is, the observable σ M (v) is explicitly related with the topology of the level sets Σ N v . This relation, even being an approximate one, is definitely non-trivial because there are very few possibilities of relating total curvature properties of a manifold with its topological invariants.

On the other hand, both Pinkall's inequality and the Hölder inequality are sufficiently tight to make Eq.( 37) meaningful. In fact, in addition to the already given arguments concerning the Hölder inequality in Eq.( 34), Pinkall's inequality stems from the Morse inequalities Therefore, the integral in the l.h.s. of Eq.( 37) necessarily follows the topological variations of the Σ N v described by the weighted sum of its Betti numbers. The consequence is that a suitable variation with v of the weighted sum of the Betti numbers of a Σ N v can be sufficient to entail a sudden change of the convexity of the function σ M (v), as reported in Figure 9, and thus entail a discontinuity of its second derivative [START_REF]The Betti numbers -as well as Morse indexes -are integers so that their sum, weighted or not, forms only staircase-like patterns which do not qualify as continuous and possibly differentiable functions. Actually the technical details of the reason why the corners of these staircase-like patterns are rounded can be found in Section 9[END_REF].

µ k (M ) ≥ b k (M )
On the other hand, the existence of a relationship between thermodynamics and configuration space topology is provided by the following exact formula [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF][START_REF] Franzosi | Topology and Phase Transitions II. Theorem on a necessary relation[END_REF] S (-)

N (v) = (k B /N ) log M N v d N q = k B N log   vol[M N v \ N (v) i=1 Γ(x (i) c )] + N i=0 w i µ i (M N v ) + R   , (38)
where S

(-)

N is the configurational entropy, and the µ i (M N v ) are the Morse indexes (in one-toone correspondence with topology changes) of the submanifolds

{M N v = V -1 N ((-∞, v]
)} v∈R of configuration space; in square brackets: the first term is the result of the excision of certain neighbourhoods of the critical points of the interaction potential from M N v ; the second term is a weighed sum of the Morse indexes, and the third term is a smooth function of N and v.

Again, sharp changes in the potential energy pattern of at least some of the µ i (M N v ) (thus of the way topology changes with v) affect S (-) N (v) and its derivatives. In other words, both the jump of the second derivative of the entropy and of the second derivative of σ M are possibly rooted in the same topological ground, where some adequate variation of the topology of the Σ N v -foliating the configuration space -takes place. Notice that even if in Eq.( 38) S (-) N (v) depends on the topology of the M N v through the Morse indexes µ i (M N v ), in the framework of Morse theory a topology change of a level set Σ N v is always associated with a topology change of the associated manifold M N v of which Σ N v is the boundary [START_REF] Milnor | Morse Theory[END_REF].

Summarizing, the topology changes indirectly detected by the function σ M (u) can affect the configurational entropy S N (v) and its tendency to develop an asymptotic discontinuity of ∂ 2 v S ∞ (v) (we use u and v interchangeably).

Finally, in Appendix we show that the non-trivial contribution to the homology groups of the energy level sets Σ N E comes from the homology groups of the configuration space submanifolds M N v ⊂ M N E and Σ N v ⊂ Σ N E . Therefore, the topology variations of the Σ N v imply topology variations of the Σ N E , and these necessarily affect also the functional dependence on E of the total entropy S N (E). In fact, the variation with v of the topology of the Σ N v is in one-to-one correspondence with some variation with v of the Betti numbers b i (Σ N v ) entering Eq. [START_REF]The Betti numbers -as well as Morse indexes -are integers so that their sum, weighted or not, forms only staircase-like patterns which do not qualify as continuous and possibly differentiable functions. Actually the technical details of the reason why the corners of these staircase-like patterns are rounded can be found in Section 9[END_REF], and this entails the variation with E of the Betti numbers b i (Σ N E ), so that, according to the following formula [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF] for the total entropy

S N (E) ≈ k B N log vol(S N -1 1 ) N i=0 b i (Σ N E ) + R 1 (E) + R 2 (E) , (39)
where R 1 (E), and R 2 (E) are smooth functions, we see that the variation with v of the topology of the Σ N v implies also the variation with E of the total entropy. After the Yang-Lee theory, phase transitions are commonly associated with a loss of analyticity of a thermodynamic potential entailing non-analytic patterns of thermodynamic observables (the pertinent potential depends on the statistical ensemble chosen). However, the caloric curve found for our gauge model is very regular, no bifurcating order parameter exists, and the peaks of specific heat are just due to horizontal tangencies to the caloric curve. In other words, apparently there is no evidence of the existence of genuinely nonanalytic energy pattern of some observable. However, by directly looking at the energy pattern of the entropy, we have identified a point of discontinuity of its second derivative (or at least a finite size version of such a discontinuity), hence an asymptotic divergence of the third derivative of the entropy. Then we have discussed how this fits into a classification scheme à la Ehrenfest, adapted to the framework of microcanonical ensemble, that allows to determine the order of a phase transition on the basis of the differentiability class of the entropy in the thermodynamic limit.

Remarkably, we have found a quantity, σ M (v), that -by measuring the total degree of inhomogeneity of the extrinsic curvature of the potential level sets Σ v = V -1 (v) in configuration space -identifies the phase transition point. This quantity is not a thermodynamic observable, has a purely geometric meaning, and displays a discontinuity of its second derivative in coincidence with the same kind of discontinuity displayed by the entropy. Rather than being a trivial consequence of the presence of the phase transition, the peculiar change of the geometry of the {Σ N v } v∈R so detected is the deep cause of the singularity of the entropy. In fact, the potential level sets are simply subsets of R N defined as Σ N v = {(q 1 , . . . , q N ) ∈ R N |V (q 1 , . . . , q N ) = v}, whose ensemble {Σ N v } v∈R foliates the configuration space; the volume Ω(v, N ) -of each leaf Σ N v -and the way it varies as a function of v is just a matter of geometrical/topological properties of the leaves of the foliation.

These properties entail the v-dependence of the entropy S N (v) = (1/N ) log Ω(v, N ), and, of course, its differentiability class. This is why the v-pattern of the quantity σ M (v) is not the consequence of the presence of a phase transition but, rather, the reason of its appearance. This is already a highly non trivial fact indicating that whether a physical system can undergo a phase transition is somehow already encoded in the interactions among its degrees of freedom described by the potential function V (q 1 , . . . , q N ), independently of the statistical ensemble chosen to describe its macroscopic observables. However, we can wonder if one can go deeper by looking for the origin of the peculiar changes with v of the geometry of the Σ N v . Actually, by resorting to a theorem in differential topology, and with some approximations, these geometrical changes appear to be due to changes of the topology of both the potential level sets in configuration space and the energy level sets in phase space.

Therefore, the results of the present work lend further support to the topological theory of phase transitions.

Moreover, since the practical computation of σ M (v), or of σ M (E), is rather straightforward, this can be used to complement the study of transitional phenomena in the absence of symmetry-breaking, as is the case of: liquid-gas change of state, Kosterlitz-Thouless transitions, glasses and supercooled liquids, amorphous and disordered systems, folding transitions in homopolymers and proteins, both classical and quantum transitions in small N systems.

With respect to the latter case, a remark about the topological theory is in order. In nature, phase transitions (that is major qualitative physical changes) occur also in very small systems with N much smaller than the Avogadro number, but their mathematical description through the loss of analyticity of thermodynamic observables requires the asymptotic limit N → ∞. To the contrary, within the topological framework a sharp difference between the presence or the absence of a phase transition can be made also at any finite and even very small N . At finite N , the microscopic states that significantly contribute to the statistical averages of thermodynamic observables are spread in regions of configuration space which get narrower as N increases, so that the statistical measures better concentrate on a specific potential level set thus better detecting its sudden and major topology changes, if any. Eventually, in the N → ∞ limit the extreme sharpening of the effective support of the measure leads to a topology-induced nonanalyticity of thermodynamic observables [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF].

Furthermore, even if somewhat abstract, the model studied in the present work has the basic properties of a lattice gauge model, that is, its potential depends on the circulations of the gauge field on the plaquettes, so that the geometrical/topological approach developed here could be also of some interest to the numerical investigation of phase transitions of Euclidean gauge theories on lattice. In fact, computing σ M (v), or σ M (E), is definitely easier than computing the Wilson loop, commonly adopted in place of an order parameter for gauge theories. Actually, a few decades ago, several papers on the microcanonical formulation of quantum field theories appeared [START_REF] Callaway | Lattice gauge theory in the microcanonical ensemble[END_REF][START_REF] Fukugita | Testing microcanonical simulation with SU(2) lattice gauge theory[END_REF], motivated by the fact that in statistical mechanics and in field theory there are systems for which the canonical description is pathological, but the microcanonical is not, also arguing, for instance and among other things, that a microcanonical formulation of quantum gravity may be less pathological than the usual canonical formulation [START_REF] Strominger | Microcanonical Quantum Field Theory[END_REF][START_REF] Iwazaki | Microcanonical formulation of Quantum field theories[END_REF][START_REF] Morikawa | Supercooled states and order of phase transitions in microcanonical simulations[END_REF][START_REF] Duane | Stochastic quantization versus the microcanonical ensemble: getting the best of both worlds[END_REF]. More recent works can also be found on these topics [START_REF] Cirilo-Lombardo | Quantum field propagator for extended-objects in the microcanonical ensemble and the S-matrix formulation[END_REF][START_REF] Casadio | Microcanonical Description of (Micro) Black Holes[END_REF][START_REF] Sinatra | Genuine phase diffusion of a Bose-Einstein condensate in the microcanonical ensemble: A classical field study[END_REF][START_REF] Strauss | Quantum field theory of classically unstable Hamiltonian dynamics[END_REF].

Finally, as a side issue, it is provided here an example of statistical ensemble nonequivalence in a system with short-range interactions. Ensemble non-equivalence is another topic which is being given much attention in recent literature [START_REF] Campa | Statistical mechanics and dynamics of solvable models with long-range interactions[END_REF].

S (-) (E) = k B 2N log E 0 dη d N p δ( i p 2 i /2 -η) Σ E-η dσ ∇V . ( 40
)
As N increases the microscopic configurations giving a relevant contribution to the entropy, and to any microcanonical average, concentrate closer and closer on the level set Σ E-η . A link among the topology of the energy level sets and the topology of configuration space can be established for systems described by a Hamiltonian of the form H N (p, q) = N i=1 p 2 i /2 + V N (q 1 , ..., q N ).

In fact, (using a cumbersome notation for the sake of clarity) the level sets Σ H N E of the energy function H N can be given by the disjoint union of a trivial unitary sphere bundle (representing the phase space region where the kinetic energy does not vanish) and the hypersurface in configuration space where the potential energy takes the total energy value (details are given in [? ])

Σ H N E homeomorphic to M V N E × S N -1 Σ V N E ( 41
)
where S n is the n-dimensional unitary sphere and

M f c = {x ∈ Dom(f )|f (x) < c} , Σ f c = {x ∈ Dom(f )|f (x) = c} . (42)
The idea that finite N topology, and "asymptotic topology" as well, of Σ H N E is affected by the topology of the accessible region of configuration space is suggested by the Künneth formula: if H k (X) is the k-th homological group of the topological space X on the field F then

H k (X × Y ; F) i+j=k H i (X; F) ⊗ H j (Y ; F) . (43)
Moreover, as H k N i=1 X i , F = N i H k (X i , F), it follows that:

H k Σ H N E , R i+j=k H i M V N E ; R ⊗ H j S N -1 ; R ⊕ H k Σ V N E ; R H k-(N -1) M V N E ; R ⊗ R ⊕ H k M V N E ; R ⊗ R ⊕ H k Σ V N E ; R (44)
the r.h.s. of Eq. [START_REF] Duane | Stochastic quantization versus the microcanonical ensemble: getting the best of both worlds[END_REF] shows that the topological changes of Σ H N E only stem from the topological changes in configuration space.

Figure 1 .

 1 Figure 1. (Color online) Caloric curve. Temperature is computed according to Eq.(11). Lattice dimensions: n 3 = 6 × 6 × 6 (rhombs), n 3 = 8 × 8 × 8 (squares), n 3 = 10 × 10 × 10 (circles). The dashed lines identify the point of flat tangency at lower energy.

  the average potential energy per lattice site u = V /N is displayed as a function of the total energy density. Also in this case we observe a regular function which is stable with the number of degrees of freedom. The dashed lines identify the phase transition point which corresponds to E c /N -0.40 and u c /N -1.32.

Figure 2 .

 2 Figure 2. (Color online) Constant volume specific heat computed by means of Eq.(13). Lattice dimensions: n 3 = 6 × 6 × 6 (rhombs), n 3 = 8 × 8 × 8 (squares), n 3 = 10 × 10 × 10 (circles).

Figure 3 .

 3 Figure 3. (Color online) Internal potential energy density computed through Eq.(8) where the observable A is the potential function per degree of freedom of the system. Lattice dimensions: n 3 = 6 × 6 × 6 (rhombs), n 3 = 8 × 8 × 8 (squares), n 3 = 10 × 10 × 10 (circles).

Figure 4 .

 4 Figure 4. Largest Lyapunov exponent versus the energy per degree of freedom. Lattice dimensions: n 3 = 6 × 6 × 6 (rhombs), n 3 = 8 × 8 × 8 (squares), n 3 = 10 × 10 × 10 (circles). The dashed vertical l line indicates the phase transition point.

Figure 5 .

 5 Figure 5. (Color online) First derivative ∂S/∂u of the configurational entropy versus the average potential energy per degree of freedom u. Lattice dimensions: n 3 = 6 × 6 × 6 (red full circles), n 3 = 8 × 8 × 8 (green full circles), n 3 = 10 × 10 × 10 (blue full circles), n 3 = 14 × 14 × 14 (black full circles). The vertical dashed line locates the phase transition point.

Figure 6 .

 6 Figure 6. (Color online) Zoom of the first derivative ∂S/∂u of the configurational entropy versus the average potential energy per degree of freedom u. Lattice dimensions: n 3 = 6 × 6 × 6 (red full circles), n 3 = 8 × 8 × 8 (green full circles), n 3 = 10 × 10 × 10 (blue full circles), n 3 = 14 × 14 × 14 (black full circles). The vertical dashed line locates the phase transition point.

Figure 7 .

 7 Figure 7. (Color online) Second derivative ∂ 2 S/∂u 2 of the configurational entropy versus the average potential energy per degree of freedom u. Lattice dimensions: n 3 = 6 × 6 × 6 (thin solid line with small triangles), n 3 = 8 × 8 × 8 (dot-dashed line), n 3 = 10 × 10 × 10 (dashed line), n 3 = 14 × 14 × 14 (thick solid line).

Figure 8 .

 8 Figure 8. (Color online) Second moment of the total mean curvature of the potential level sets Σ u versus energy density E/N . n 3 = 6 × 6 × 6 (rhombs), n 3 = 8 × 8 × 8 (squares), n 3 = 10 × 10 × 10 (circles). The oblique dashed line is a guide to the eye. The vertical dashed line corresponds to the point where the second derivative d 2 σ M /dE 2 jumps from a negative value to zero.

Figure 9 .

 9 Figure 9. (Color online) Second moment of the total mean curvature of the potential level sets Σ u versus the average potential energy per degree of freedom u. n 3 = 6×6×6 (rhombs), n 3 = 8×8×8 (squares), n 3 = 10 × 10 × 10 (circles). The oblique dashed line is a guide to the eye. The vertical dashed line corresponds to the point where the second derivative d 2 σ M /du 2 jumps from a negative value to zero.

  which relate the Morse indexes µ k (M ) to the Betti numbers b k (M ) of a manifold M (Pinkall's inequality would be replaced by an equality if written with Morse indexes), and these Morse inequalities are very tight since the alternating sums of Morse indexes and of Betti numbers, respectively, give the same result (the Euler characteristic).

  IV. CONCLUDING REMARKSWe have tackled the problem of characterizing a phase transition in the absence of global symmetry breaking from the point of view of Hamiltonian dynamics and related geometrical and topological aspects. In this condition the Landau classification of phase transitions does not apply, because no order parameter -commonly associated with a global symmetry -exists. The system chosen is inspired by the dual of the Ising model, and the discrete variables are replaced by continuous ones. We stress that our work has nothing to do with the true dual Ising model, which has just suggested how to define a classical Hamiltonian system with a local (gauge) symmetry. Since the ergodic invariant measure for generically non-integrable Hamiltonian systems is the microcanonical measure in phase space, studying phase transitions through Hamiltonian dynamics is the same as studying them in the microcanonical ensemble.A standard analysis has been performed to locate the phase transition and to determine its order through the shape of the caloric curve, T = T (E), which appeared typical of a first order phase transition. The presence of energy intervals of negative specific heat are indicative of ensemble nonequivalence. At variance with what has been systematically observed for systems undergoing symmetry-breaking phase transitions, the energy pattern of the largest Lyapunov exponent does not allow to locate the transition point.

APPENDIXA.

  Relation between topological changes of the Σ v and of the Σ E Now, let us see why a topological change of the configuration space submanifolds Σ v = V -1 (v) (potential level sets) implies the same phenomenon for the Σ E . The potential level sets are the basic objects, foliating configuration space, that represent the nontrivial topological part of phase space. The link of these geometric objects with microcanonical entropy is given by


Kevin Falk

Examples of infinite direct sums of spectral triples

Keywords: noncommutative geometry, spectral triples, Toeplitz operators, self-similar sets

published or not. The documents may come

Examples of infinite direct sums of spectral triples

Introduction and motivation

The main idea of Connes's noncommutative geometry is to characterize the geometry of a space in the language of algebras [START_REF] Connes | Noncommutative geometry[END_REF]. We know for instance that a compact Hausdorff space can be equivalently seen as the commutative C * -algebra of continuous functions living on it. By analogy, a noncommutative algebra would correspond to a space of quantum nature: a noncommutative space. More precisely, the algebraic description of a Riemannian manifold is based on the notion of unital spectral triple, consisting of the data (A, H, D), where A is an involutive unital * -algebra A faithfully represented on a Hilbert space H via a representation π, and D is a selfadjoint operator acting on H with compact resolvent and such that for any a ∈ A, π(a) maps dom(D) into itself, and [ D, π(a) ] extends to a bounded operator on H. When A is not unital, replace the compactness of the resolvent by the compactness of π(a)(D -λ) -1 for any a ∈ A and λ / ∈ Spec(D): the induced triple is then called nonunital. Among the various geometric entities which are encoded in the spectrum of D, we are interested in the so-called spectral dimension, defined as the quantity

d := inf{s ∈ R , Tr |D| -s < +∞} .
As easily checked, the direct sum of a finite number of spectral triples is again a spectral triple. We are interested here in integrations of spectral triples which consist, roughly speaking, of the direct sum of an infinite number of spectral triples. Such constructions have already been encountered in [START_REF] Engliš | Spectral triples and Toeplitz operators[END_REF]: the spectral triple related to the Berezin-Toeplitz quantization over a smoothly bounded stricly pseudoconvex domain of C n can be viewed as the integration of an infinite family of spectral triples based on algebras generated by Toeplitz operators acting on weighted Bergman spaces.

The first idea is the following: given a countable family of spectral triples (A m , H m , D m ) m∈N (commutative or not), the corresponding infinite direct sum " m∈N (A m , H m , D m )", might not be necessarily a spectral triple again. Indeed, as m tends to infinity, the boundedness of the representations of A m , the boundedness of the commutator between A m and the operators D m , or the compactness of the resolvent of the direct sum of all operators D m is hard to control in general and the sum may fail to converge. In order to control the behaviour of the operators D m , we multiply them by some coefficients α m ∈ R\{0}.

Surprisingly, a strong link exists between direct summations of spectral triples and fractal sets, but before describing the second approach, let us recall some previous results on the topic. Since the works of A. Connes [START_REF] Connes | Noncommutative geometry[END_REF]Chapter 4,[START_REF] Christensen | Dirac operators and spectral triples for some fractal sets built on curves[END_REF].ε], we know that noncommutative geometry can detect the topology of fractal sets: it is shown that a commutative spectral triple involving the C * -algebra of continuous functions over the Cantor set can be used to recover its Hausdorff dimension and the Hausdorff measure. Later on, D. Guido and T. Isola proposed a commutative spectral triple, also based on a discrete approximation of the fractal, and extend Connes' result to more general self-similar sets in R n [10, Chapter 7], [START_REF] Guido | Dimensions and singular traces for spectral triples, with applications to fractals[END_REF] (the existence of such spectral triples was already conjectured in M. Lapidus' paper [START_REF] Lapidus | Analysis on fractals, Laplacians on self-similar sets, noncommutative geometry and spectral dimensions[END_REF]). See also [START_REF] Lapidus | Towards a noncommutative fractal geometry? Laplacians and volume measures on fractals[END_REF] for a review of open problems and questions about the links between analysis and spectral geometry on fractal sets.

In the latter works, each spectral triple is directly built over the fractal set. The approach we follow in the present paper is a constructive one: decompose the considered fractal set as the union of an infinite number of subdomains and associate to each of them a spectral triple. The spectral triple over the whole fractal set is obtained after the direct summation of all these spectral triples. This construction has already been used in [START_REF] Christensen | Dirac operators and spectral triples for some fractal sets built on curves[END_REF][START_REF] Cipriani | Spectral triples for the Sierpinski gasket[END_REF][START_REF] Lapidus | Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets[END_REF] to recover the Hausdorff dimension and the metric on p-summable infinite trees and the Sierpinski gasket, and also in [START_REF] Christensen | Spectral triples and the geometry of fractals[END_REF] to study the Hausdorff dimension of the Sierpinski gasket (and pyramid), its metric and describe its K-homology group.

For simplicity reasons, we restrict our study to self-similar sets E of the plane C which can be expressed as

E = E 0 ∪ N k=1 F k (E 0 ) ∪ N k,l=1 F k • F l (E 0 ) ∪ . . . , (1)
where the overline means taking the closure, E 0 is a polygonal Jordan curve in the complex plane or the unit disk, and (F k ) k=1,...,N is a finite family of contracting similarities.

The paper is organized as follows.

We present in Section 2 some sufficient conditions for the sum to be a spectral triple and we give an example of such integration using Toeplitz operators over the unit ball of C n . We show in Section 3 that is is possible to build a noncommutative spectral triple over such sets, involving algebras of Toeplitz operators, and whose spectral dimension corresponds to the Hausdorff dimension of E.

2 Abstract integration of spectral triples

(α m ) m∈N ∈ (R\{0}) N . Let D ⊕ := m∈N α m D m with domain dom(D ⊕ ) := N m=0 v m ∈ H ⊕ , N ∈ N , v m ∈ dom(D m )
.

Then D ⊕ is essentially selfadjoint, with selfadjoint extension D ⊕ . Proof. Let v ⊕ := m∈N v m ∈ H ⊕ .
For any m ∈ N, the operator D m is densely defined so there is a sequence (v mj ) j∈N of elements in dom(D m ) converging to v m as j → ∞. Thus for any fixed (m, j) ∈ N 2 , there is

M mj ∈ N such that v m -v m,M mj +k 2
Hm < 2 -j for any k ∈ N. Define for any j ∈ N the vector w ⊕ j := j m=0 v m,M mj ∈ dom(D ⊕ ). For any j ∈ N, w ⊕ j ∈ dom(D ⊕ ) and

v ⊕ -w ⊕ j 2 H ⊕ = j m=0 v m -v m,M mj +k 2 Hm + m>j v m 2 Hm < j2 -j + m>j v m 2 Hm -→ j→+∞ 0 .
Thus for any ε > 0, there exists N ∈ N such that v ⊕ -w ⊕ N H ⊕ < ε, which shows that D ⊕ is densely defined. Using the same reasoning and the fact that for any m ∈ N, Ran(α m D m ±i) = H m (since α m D m is selfadjoint), it can be shown that for any v ⊕ ∈ H ⊕ and ε > 0, there is N ∈ N and w ⊕ N ∈ dom(D ⊕ ) defined as above and such that v

⊕ -(D ⊕ ± i)w ⊕ N H ⊕ < ε, thus Ran(D ⊕ ± i) is dense in H ⊕ . The operator D ⊕ is also symmetric since for any v ⊕ := N m=0 v m and v ⊕ := N m=0 v m in dom(D ⊕ ), D ⊕ v ⊕ , v ⊕ H ⊕ = min(N,N ) m=0 α m D m v m , v m Hm = min(N,N ) m=0 v m , α m D m v m Hm = v ⊕ , D ⊕ v ⊕ H ⊕ ,
which shows that D ⊕ is essentially selfadjoint (see [17, Chapter VIII.2, Corollary p.257]).

The following result establishes sufficient conditions on an infinite family of spectral triples together with a family of weights (α m ) m∈N ∈ (R\{0}) N so that the corresponding weighted direct sum is a spectral triple.

Proposition 2.2. Let (A m , H m , D m ) m∈N be a family of (not necessarily unital) spectral triples, with corresponding representations (π m ) m∈N , and denote . m the norm on H m . Let (α m ) m∈N be a sequence of non-zero real numbers such that

(1 + α 2 m D 2 m ) -1/2 m -→ m→+∞ 0 . (2)
Define the following objects:

• H ⊕ := m∈N H m ,
• D ⊕ := m∈N α m D m and D ⊕ as above, both acting on H ⊕ ,

• A ⊕ := (a m ) m∈N ∈ m∈N A m : sup m∈N π m (a m ) m < +∞ , and sup m∈N [ α m D m , π m (a m ) ] m < +∞ , • π ⊕ (a ⊕ ) := m∈N π m (a m ), for a ⊕ ∈ A ⊕ .
Then (A ⊕ , H ⊕ , D ⊕ ) is a (not necessarily unital) spectral triple.

Proof. For two elements a ⊕ = (a m ) m∈N and b ⊕ = (b m ) m∈N in A ⊕ , we have:

sup m∈N π m (a m b m ) m ≤ sup m∈N π m (a m ) m sup m∈N π m (b m ) m < +∞ , and
sup m∈N [ α m D m , π m (a m b m ) ] m ≤ sup m∈N π m (a m ) m sup m∈N [ α m D m , π m (b m ) ] m + sup m∈N [ α m D m , π m (a m ) ] m sup m∈N π m (b m ) m < +∞ , hence A ⊕ is an algebra with involution * : a ⊕ = (a m ) m∈N → (a ⊕ ) * := (a * m ) m∈N . For a ⊕ ∈ A ⊕ , we have π ⊕ (a ⊕ ) 1 + (D ⊕ ) 2 -1/2 = m∈N π m (a m ) (1 + α 2 m D 2 m ) -1/2 .
For any m ∈ N, the summand

π m (a m ) (1 + α 2 m D 2 m ) -1/2 is compact. From (2) and the fact that π ⊕ is a bounded representation, π m (a m ) (1 + α 2 m D 2 m ) -1/2 m tends to 0 as m → +∞. As a consequence, π ⊕ (a ⊕ ) 1 + (D ⊕ ) 2 -1/2 is compact. From Lemma 2.1, D ⊕ is essentially selfadjoint with selfadjoint extension D ⊕ . For a ⊕ ∈ A ⊕ and v ⊕ N := N m=0 v N,m ∈ dom(D ⊕ ), for some N ∈ N, we have π ⊕ (a ⊕ )v ⊕ N = N m=0 π m (a m )v N,m
and each summand on the right-hand side belongs to dom(D m ) since

(A m , H m , D m ) is a spectral triple for any m ∈ N. Thus π ⊕ (a ⊕ ) maps dom(D ⊕ ) into itself for any a ⊕ ∈ A ⊕ . Moreover, for any a ⊕ ∈ A ⊕ and v ⊕ N := N m=0 v N,m ∈ dom(D ⊕ ) of norm 1, we have [ D ⊕ , π ⊕ (a ⊕ ) ]v ⊕ N = sup m=0,...,N [ α m D m , π m (a m ) ]v N,m ≤ sup m∈N [ α m D m , π m (a m ) ] m < +∞ , so [ D ⊕ , π ⊕ (a ⊕ ) ] is bounded on dom(D ⊕ ). Moreover, since D ⊕ | dom(D ⊕ ) = D ⊕ | dom(D ⊕ ) = D ⊕ , then dom(D ⊕ ) is a core for D ⊕ . Using [16, Proposition A.1], we conclude that for any a ⊕ ∈ A ⊕ , π ⊕ (a ⊕ ) dom(D ⊕ ) ⊂ dom(D ⊕ )
and [ D ⊕ , π ⊕ (a ⊕ ) ] extends to a bounded operator on H ⊕ .

Definition 2.3. The spectral triple (A ⊕ , H ⊕ , D ⊕ ) as above is called the integration of the fivetuple

(A m , H m , D m , π m , α m ) m∈N , where (A m , H m , D m
) is a spectral triple for any m ∈ N, with corresponding representations π m and weights α m in R\{0}.

As a consequence of (2), the sequence (D m ) m∈N is such that

m∈N dim (Ker D m ) < ∞.
In particular, if we take the same D m = D 0 at each level m ∈ N, the latter must be invertible.

The two conditions in the definition of A ⊕ correspond to the boundedness of both the representation π ⊕ and the commutator [ D ⊕ , π ⊕ (A ⊕ ) ] for the norm . ⊕ := sup m∈N . m on π ⊕ (A ⊕ ). The parameter (α m ) m∈N has been introduced in order to control the behaviour of the sequence (D m ) m∈N as m tends to infinity. This can be avoided by putting some constraints directly on the operators D m , but this restricts the set of summable families of spectral triples. For instance, when D ⊕ := m∈N D 0 , with D 0 invertible, then the resolvent of D ⊕ is not compact.

We make use of the following notations for the rest of the document. For a multiindex α ∈ N n and z ∈ C n , denote

z α := z α 1 1 z α 2 2 . . . z αn n and |α| := α 1 +• • •+|α n |.
For any set X ⊂ C n , let Pol(X) be the set of polynomial functions in z and z over X. We denote

B n := {z ∈ C n , |z| < 1} the unit open ball of C n , B n its

An example of integration over the unit ball

We choose for B n the following defining function (i.e. a smooth function r over B n such that r| B n < 0, r| ∂B n = 0 and dr| ∂B n = 0) and a weight on B n :

r(z) := |z| 2 -1 , and w m (z) := (-r(z)) m , z ∈ B n , m ∈ (-1, +∞) . (3)
The weighted Bergman space over B n with weight w m is

A 2 m (B n ) := {φ ∈ L 2 (B n , w m dµ) , φ holomorphic in B n } ,
where dµ is the usual normalized Lebesgue measure over B n . Denote Π m the orthogonal projection from

L 2 (B n ) onto A 2 m (B n ). The Toeplitz operator T (m) f : A 2 m (B n ) → A 2 m (B n ) associated to the function f ∈ C ∞ (B n ) is defined as T (m) f : φ → Π m (f φ) .
In particular, Toeplitz operators enjoy the following properties:

f → T (m) f is linear , T (m) f ≤ f ∞ , and (T (m) f ) * = T (m) f . (4)
Since in general the product of two Toeplitz operators is not a Toeplitz operator anymore, we will consider the * -algebra generated by the Toeplitz operators, the involution being the Hilbert space adjoint operation (4).

The following result is a corollary of [8, Proposition 5.4]:

Proposition 2.4. For any real number m > -1, let A m be the algebra generated by the Toeplitz operators

T (m) f , f ∈ C ∞ (B n ), with the identity representation on H m := A 2 m (B n ),
and also

D m := (T (m) -r ) -1 . Then (A m , H m , D m ) is a spectral triple of spectral dimension n = dim C B n .
In order to get a family of spectral triples, we make m vary in the set of integers, and for the rest of this section m ∈ N. Let us present a preliminary result which establishes the dependence on m of the commutator between a Toeplitz operator with polynomial symbol and the previous operator

D m := (T (m) -r ) -1 . Denote the operators R := n j=1 R j and R := n j=1 R j with R j := z j ∂ z j and R j := zj ∂ zj , acting on C ∞ (B n ).
p αβ z α zβ ∈ Pol(B n ), we have [ (T (m) -r ) -1 , T (m) p ] = 1 m+1 T (m) (R-R) p , on A 2 m (B n ) .
Proof. We denote briefly

T p = T (m) p . An orthonormal basis of A 2 m (B n ) is given by (see [19, (2.9)]) u m,α (z) := (|α|+m+n)! (m+n)! α! 1/2 z α . ( 5
)
Using the shift operators S j : u m,α → u m,α+1 j , with α + 1 j := (α 1 , . . . , α j + 1, . . . , α n ) and j = 1, . . . , n, we have the relations

T z j = S j ( R j +1 R+m+n+1 ) 1/2 , [ R j , S k ] = δ(j = k) S j , S * j S j = 1 , for j = 1, . . . , n, and (6)
T -1 -r = (1 - n j=1 T |z j | 2 ) -1 = (1 - n j=1 (T z j ) * T z j ) -1 = (1 - n j=1 R j +1 R+m+n+1 ) -1 = 1 m+1 (R + m + n + 1) .
Hence we get

[ T -1 -r , T z j ] = 1 m+1 (R + m + n + 1) S j ( R j +1 R+m+n+1 ) 1/2 -S j ( R j +1 R+m+n+1 ) 1/2 (R + m + n + 1) = 1 m+1 S j ( R j +1 R+m+n+1 ) 1/2 R + m + n + 2 -(R + m + n + 1) = 1 m+1 T z j .
From this last equality and the fact that [ T z j , T z k ] = 0, for any j, k = 1, . . . , n, we get by iteration of the formula

[ A, BC ] = B[ A, C ] + [ A, B ]C [ T -1 -r , n j=1 T α j z j ] = |α| m+1 n j=1 T α j z j and [ T -1 -r , n j=1 (T * z j ) β j ] = -|β| m+1 n j=1 (T * z j ) β j , α, β ∈ N n .
Hence, the relation

T z α zβ = n j=1 (T * z j ) β j n j=1 T α j z j yields to [ T -1 -r , T p ] = |α|≤d,|β|≤d [ T -1 -r , T z α zβ ] = |α|≤d,|β|≤d [ T -1 -r , n j=1 (T * z j ) β j n j=1 T α j z j ] = |α|≤d,|β|≤d p αβ n j=1 (T * z j ) β j [ T -1 -r , n j=1 T α j z j ] + [ T -1 -r , n j=1 (T * z j ) β j ] n j=1 T α j z j = 1 m+1 |α|≤d,|β|≤d p αβ (|α| -|β|) n j=1 (T * z j ) β j n j=1 T α j z j = 1 m+1 |α|≤d,|β|≤d p αβ (|α| -|β|) T z α zβ = 1 m+1 T (R-R) p .
An example of previous integration of noncommutative spectral triples is given here for the unit ball B n : Proposition 2.6. For m ∈ N, let

• H m := A 2 m (B n ), • D m := (T (m) -r ) -1 ,
• A m be the * -algebra generated by Toeplitz operators T If we let H ⊕ , D ⊕ , π ⊕ as in Proposition 2.2 and A ⊕ be the algebra generated by elements of the form (T (m) p ) m∈N , with p ∈ Pol(B n ) (i.e. keeping the same polynomial at all levels m ∈ N), then the previous quintuple is integrable and

(A ⊕ , H ⊕ , D ⊕ ) is a spectral triple of spectral dimension n + 1.
Proof. First, we know from Proposition 2.4 that for any m ∈ N, (A m , H m , D m ) defines a spectral triple of dimension n. Moreover,

(1 + α 2 m D 2 m ) -1/2 m = (1 + α 2 m (T -2 -r ) (m) ) -1/2 m ≤ |α m | -1 T (m) -r m ≤ |α m | -1 r ∞ -→ m→+∞ 0 . Let us show that A ⊕ is a subalgebra of A ⊕ of Proposition 2.2: if (a m ) m∈N = (T (m) p ) m∈N of A ⊕ , with p ∈ Pol(B n ),
(m) (R-R)p ≤ (R -R) p ∞ < +∞ .
These inequalities remain valid for a general element of A ⊕ , which is composed, at each level m ∈ N, by the same finite sum of finite products of Toeplitz operators acting on A 2 m (B n ). Since A ⊕ form a * -algebra, we conclude that it is a * -subalgebra of A ⊕ and from Proposition 2.2, (A ⊕ , H ⊕ , D ⊕ ) is a spectral triple. We now compute its spectral dimension. For s ∈ R, we have

Tr |D ⊕ | -s = m∈N α -s m Tr (T (m) -r ) s = m∈N ( αm m+1 ) -s Tr (R + m + n + 1) -s = m∈N k∈N k+n-1 n-1 (k + m + n + 1) -s .
For any k ∈ N and s > 1, we have

k+m+n+1 k+m+n x -s dx < (k + m + n + 1) -s < k+m+n+2 k+m+n+1
x -s dx , so summing over m ∈ N leads to

1 s-1 (k + n) 1-s < m∈N (k + m + n + 1) -s < 1 s-1 (k + n + 1) 1-s . Since k+n-1 n-1 ∼ k→+∞ k n-1
(n-1)! , the operator |D ⊕ | -s if and only if

1 (1-s)(n-1)! k∈N (k + 1) n-1 (k + n + 1) 1-s < +∞ ⇔ m∈N k n-s < +∞ ,
i.e. for s > n + 1.

Remark 2.7. The previous result is restricted to the case of polynomial symbols. Indeed, we cannot apply the Stone-Weierstrass theorem in order to extend the result for general smooth functions over

B n since f → [ (T (m) -r ) -1 , T f ] is not continuous on A 2 m (B n ) for the norm . ∞ .
A possible extension of Proposition 2.6, in which any (a m ) m∈N ∈ A ⊕ is defined as the copy of the same element on each level m ∈ N, consists of replacing a finite number of a m by arbitrary elements of A m . Thus the representation of an element a ⊕ of this new algebra A ⊕ is of the form

π ⊕ (a ⊕ ) = m≤N π m (a m ) ⊕ m>N p i=1 q i j=1 T (m) p ij ,
for some integer N , some arbitrary a m ∈ A m , m ≤ N , and fixed family of polynomials p ij in Pol(B n ), i = 1, . . . , p, j = 1, . . . , q i .

We can also consider a more general sequence (α m ) m∈N such that α m ∼ m δ , as m tends to infinity, for 0 < δ ≤ 1 (the upper bound comes from the boundedness of the commutator between the representation of an element of the algebra and D ⊕ ). Then, the conclusions of Proposition 2.2 remain valid but the spectral dimension lies in [n + 1, +∞).

The "n + 1 phenomenon" also appears in the spectral dimension of the spectral triple built from the Berezin-Toeplitz quantization [8, Section 6]. For short, the latter spectral triple can be expressed as a summation of spectral triples on a smoothly bounded strictly pseudoconvex domain like the ones in Proposition 2.4. This can be equivalently seen as a spectral triple over the boundary of a disk bundle over the domain, whose spectral dimension is exactly n + 1, and which brings a geometric explanation for the "extra dimension".

Spectral triples with arbitrary real positive spectral dimension have already be encountered in C. Ivan and E. Christensen's paper [START_REF] Christensen | Spectral triples on AF C * -algebras and metrics on the Cantor set[END_REF]; the construction uses algebras of continuous functions over the Cantor set.

3 Integration along decomposable self-similar sets in the plane

F k (z) = a k z + b k , z ∈ C , with |a k | = c , ii) there is a non-empty open bounded set V ⊂ C such that N k=1 F k (V ) ⊂ V (open set condi- tion),
iii) there is a polygonal Jordan curve E 0 defined by the points (p j ) j=1,...,M , M > 2, such that the attractor E of (F k ) k=1,...,N can be decomposed as

E = m∈N ω∈{1,...,N } m F ω 1 • • • • • F ωm (E 0 ) . ( 7
)
The set E 0 is called the generator.

The set E is a non-empty closed bounded set in the metric space R 2 [12, 3.1.(3)(i)]. Moreover, since E is defined from similarities of same ratios and verifies the open set condition, its Hausdorff dimension dim H is given by (see [START_REF] Falconer | Fractal Geometry, Mathematical Foundations and Applications[END_REF]Theorem 9.3])

dim H (E) = log(N ) log(1/c) . ( 8
)
For the rest of this section, (F k ) k=1,...,N denotes an element of S with fixed ratio c ∈ (0, 1), E 0 a generator and E the corresponding attractor. Let (L j ) j=1,...,M -1 be the family of closed line segments between the points p j and p j+1 , and L M between p M and p 1 . If |L j | is the length of the segment L j , we assume that the perimeter M j=1 |L j | of E 0 is 2π and we denote θ j := j-1 =1 |L | for any j = 2, . . . , M , and θ 1 = 0. For j = 1, . . . , M -1, let A j be the closed arc of the unit circle ∂B between e iθ j and e iθ j+1 , and A M the one between the points e iθ M and 1. For any m ∈ N, ω ∈ {1, . . . , N } m and j = 1, . . . , M , we use the following notations:

B m := {z ∈ C , |z| < c m } , C m := ∂B m , F ω := F ω 1 • • • • • F ωm , E ω := F ω (E 0 ) , p ωj := F ω (p j ) , L ωj := F ω (L j ) , A ωj := c m A j . (9)
(note that A ωj are the closed arcs whose union over j is C m .)

Example 3.2. The Sierpinski gasket E SG [START_REF] Sierpinski | Sur une courbe dont tout point est un point de ramification[END_REF] is the attractor of (F 1 , F 2 , F 3 ) ∈ S, where F k , k = 1, 2, 3, is the homothety of center the kth vertex p k of an equilateral triangle E 0 , and of ratio c = 1/2. It can be expressed as the union between E 0 and all of its images by F ω , for any ω ∈ {1, . . . , N } m and m ∈ N, and its Hausdorff dimension is dim H (E SG ) = log(3) log(2) . On Figure 1, the sets F k (E 0 ) and F k • F l (E 0 ), k, l = 1, 2, 3, are denoted E k and E kl respectively.

p 1 p 2 p 3 E 0
Step 0

E 1 E 2 E 3 p 1 p 2 p 3
Step 1

E11 E13 E23 E21 E22 E23 E31 E21 E33 p 1 p 2 p 3
Step 2

E p 1 p 2 p 3
Step 2

Figure 1

In order to define Toeplitz operators on the polygonal Jordan curves E ω , we define a sufficiently nice homeomorphism from the circle C m into E ω based on Möbius transforms that send holomorphically each arc A ωj into the line segments [p ωj , p ωj+1 ].

Lemma 3.3. For any m ∈ N, ω ∈ {1, . . . , N } m and j = 1, . . . , M , let the maps κ ωj (z) := (p ωj -i/(δ ωj τ ωj ))z + c m e iθ j (p ωj +i/(δ ωj τ ωj ))

z + c m e iθ j
, from C m into E ω , where

δ ωj := (p ωj+1 -p ωj ) -1 , τ ωj := tan((θ j+1 -θ j )/2) , if j = 1, . . . , M -1 , and
δ ωM := (p ω1 -p ωM ) -1 , τ ωM := tan((θ 1 -θ M )/2) .
Then the map κ ω defined as

κ ω | A ωj := κ ωj is an homeomorphism from C m into E ω .
Proof. Since M > 2, |L j | = θ j+1 -θ j < π so τ ωj < +∞ for any j = 1, . . . , M . Expressing any point z in A ωj as

z = z(t) = c m e i(θ j +t|L j |) , for t ∈ [0, 1] , (10)
we have for any t ∈ [0, 1]

κ ωj (z(t)) = (p ωj -i/(δ ωj τ ωj ))c m e i(θ j +t|L j |) + c m e iθ j (p ωj +i/(δ ωj τ ωj )) c m e i(θ j +t|L j |) + c m e iθ j = (p ωj -i/(δ ωj τ ωj ))e it|L j | +p+i/(δ ωj τ ωj ) e it|L j | +1 = p ωj + i δ ωj τ ωj 1-e it|L j | 1+e it|L j | = p ωj + (p ωj+1 -p ωj ) tan(t |L j |/2) tan(|L j |/2) . (11)

The inner product in

H m between φ(z) = k∈N φ k z k and ψ(z) = k∈N ψ k z k , z ∈ B m , is given by φ , ψ Hm := 2π 0 φ(c m e it ) ψ(c m e it ) dt 2π = k∈N φ k ψ k c 2mk .
An orthonormal basis for H m is given by the vectors v mj (z) := c -m z j and the reproducing kernel

S (m) z (w) = k∈N c -2mk zk w k verifies φ(z) = φ , S (m)
T (ω) p•κω : H m φ → Π m (p • κ ω )φ ∈ H m . (12)
The integral representation of the action of such operators is

T (ω) p•κω φ (z) = (p • κ ω )φ , S (m) z Hω = M j=1 θ j+1 θ j dt 2π (p • κ ωj )(c m e it ) φ(c m e it ) S (m) z (c m e it ) , z ∈ B m . (13)
As in previous section, define the operators R := z∂ z and R := z∂ z acting on H ω . Since Spec(R) = N consists of the positive part of the usual Dirac operator on the circle C m , we choose for D ω the following expression

D ω := α m R + β m , (14)
where

(
p•κω| Cω ] is bounded and

[ D ω , T (ω) p•κω| Cω ] ≤ α m K p ,
for some constant K p > 0 independent on m.

Proof. We fix m ∈ N, ω ∈ {1, . . . , N } m , and denote T u := T

as operators acting on H ω , where π ωab defined by

π ωab | Åωj = (R -R)κ a ωj κ ωj b , for any j = 1, . . . , M ,
where Åωj denotes the interior of the closed arc A ωj , is extendible to a piecewise continuous, hence bounded, function on C m . For any j = 1, . . . , M , the map κ a ωj is holomorphic in a neighborhood of A ωj , and we write κ a ωj (z) = k∈N κ ωjak z k , z ∈ A ωj . For any z ∈ B m , n ∈ N, and using (13), we have on one hand e iθ j+1 (k-l+n-s) -e iθ j (k-l+n-s) i(k-l+n-s) e iθ j+1 (k-l+n-s) -e iθ j (k-l+n-s) i(k-l+n-s)

δ(k + n = l + s) + (θ j+1 -θ j ) δ(k + n = l + s) .
δ(k + n = l + s) + (θ j+1 -θ j ) δ(k + n = l + s) .
Thus, setting B := T π ωab -[ R, T κ a ω κω b ], we obtain

Bz n = s∈N c m(n-s) z s M j=1 k,l∈N κ ωjak κ ωjbl c m(k+l) (k -l + n -s)
e iθ j+1 (k-l+n-s) -e iθ j (k-l+n-s) i(k-l+n-s)

δ(k + n = l + s) = -i s∈N c m(n-s) z s M j=1 k,l∈N κ ωjak κ ωjbl c m(k+l) (e iθ j+1 (k-l+n-s) -e iθ j (k-l+n-s) ) = -i s∈N c m(n-s) z s e i M j=1 κ a ωj (p ωj+1 )κ ωj b (p ωj+1 )e iθ j+1 (n-s) -κ a ωj (p ωj )κ ωj b (p ωj )e iθ j (n-s) .
Since κ ω is continuous on C m , κ ωj (p ωj+1 ) = κ ωj+1 (p ωj+1 ) for any j = 1, . . . , M -1, and κ ωM (p ω1 ) = κ ω1 (p ω1 ), so the summation over j on the right-hand side vanishes and we proved [START_REF] Lapidus | Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets[END_REF]. By linearity, we have [ D ω , T p•κω| Cm ] = α m a,b p ab T π ωab , which is a bounded operator on H ω with

[ D ω , T p•κω| Cm ] ≤ α m a,b |p ab | π ωab ∞ .
We have for any z ∈ Åωj Rκ ωj (z) = z∂ z (p ωj -i/(δ ωj τ ωj ))z + c m e iθ j (p ωj +i/(δ ωj τ ωj )) z + c m e iθ j = -2i/(δ ωj τ ωj )c m e iθ j z (z+c m e iθ j ) 2

, so using [START_REF] Guido | Fractals in noncommutative geometry[END_REF] we get Proof. The circle C ω is the boundary of a strictly pseudoconvex domain with complex dimension 1 and A ω is a subalgebra of the algebra of generalized Toeplitz operators of order 0 (see [START_REF] Boutet De Monvel | The spectral theory of Toeplitz operators[END_REF]), and the operator D ω is a selfadjoint elliptic generalized Toeplitz operator of order 1 on H ω , so the proof is similar to the one of [8, Proposition 5.2], except that here [ D ω , A ω ] / ∈ A ω . The boundedness of the commutator is nonetheless proved by Lemma 3.4.

Rκ ωj (t) = -2i δ ωj τ ωj e it |L j | (1+e it |L j | ) 2 , hence |Rκ ωj |(t) ≤ 2 τ M |p j+1 -p j |k j , ∀t ∈ (0, 1) , with k j := inf t∈(0,1) (1 + e it |L j | ) 2 > 0. Moreover, since E is a compact set in C, there is a constant K > 0 such that for any m ∈ N and ω ∈ {1, . . . , N } m , κ ω ∞ ≤ K. Thus π ωab ∞ ≤ sup j=1,...,M a(Rκ ωj )κ a-1 ωj κ ωj b | Åωj ∞ + bκ a ωj (Rκ ωj )κ ωj b-1 | Åωj ∞ ≤ sup j=1,...,M 2
We assumed that the attractor E is the union of all the components E ω , so we sum the spectral triples obtained in Proposition 3.5 in order to obtain a noncommutative spectral triple over the whole set E. It is still possible to adjust the coefficients (α m ) m∈N and (β m ) m∈N so that the spectral dimension of the integrated spectral triple corresponds to dim H (E). Theorem 3.6. Let (F k ) k=1,...,N be an element of S with ratio c ∈ (0, 1) such that 1 < cN . Let Then D ⊕ is essentially selfadjoint and one can choose the sequences (α m ) m∈N and (β m ) m∈N so that (A ⊕ , H ⊕ , D ⊕ ) is a spectral triple of spectral dimension dim H (E).

• H ⊕ := m∈N ω∈{1,...,N } m H ω , with H ω := H 2 (C m ), • A ⊕
Proof. The attractor E is compact, so for any p ∈ Pol(E), the norm 1) , and

T ⊕ p ≤ p ∞ is finite. Let ∈ ( log(N ) log(cN ) , +∞) , α m := c -m N -m( -
β m := c -m , ∀m ∈ N . (16)
For any m ∈ N and ω ∈ {1, . . . , N } m , Spec(D ω ) = {α m j +β m , j ∈ N} ⊂ R + \{0}, so the operator D -1 ω is compact and Because of the condition 1 < cN , the previous operator D ⊕ encodes the Hausdorff dimension of E when the latter is strictly greater than 1.

D -1

Spectral triple on fractal sets generated by the unit disk

The integration of Section 2.2 is obtained from a family of spectral triples over a fixed domain which is the unit disk B, and the dimension is recovered after adjusting the family of weights and the sequence (α m ) m∈N . Here, the considered domain is a union of disks of different sizes, forming a self-similar set. This approach of integration seems more natural in the sense that the geometrical structure of the fractal keeps us to put by hand the coefficients α m on the operators D ω , and the only degree of freedom remains the choice of the weights on each disk.

In this section, we consider a family of similarities (F k ) k=1,...,N on C which verify i) and ii) from Definition 3.1, whose attractor E can be expressed as

E = m∈N ω∈{1,...,N } m F ω (B)
(we keep the same notations as in ( 9)).

Again, the attractor E is a self-similar set and its Hausdorff dimension is given by [START_REF] Engliš | Spectral triples and Toeplitz operators[END_REF]. The spectral triple over E is obtained in a similar way as in Section 3.1, except that the algebras we consider here are the algebras generated by Toeplitz operators on Bergman spaces over the disks.

For any m ∈ N, we consider for the disk B m the following defining function and weight

  closure, and simply B := B 1 the unit open disk in C. The corresponding boundaries are denoted respectively ∂B n and ∂B.

Proposition 2 . 5 .

 25 For any polynomial function p(z) = |α|≤d,|β|≤d

  on H m , with p ∈ Pol(B n ), • π m be the identity representation on H m , • . m be the usual norm of operators, • α m := m + 1.

  is a generator, the conditions are satisfied since sup m∈N π m (a m ) m ≤ p ∞ < +∞ and from Proposition 2.5, sup m∈N [ α m D m , π m (a m ) ] m = sup m∈N m+1 m+1 T

3. 1

 1 Spectral triple on fractal sets generated by a polygonal Jordan curve Definition 3.1. Let S be the set of families (F k ) k=1,...,N of similarities on C such that i) for any k = 1, . . . , N , F k have the same ratio c ∈ (0, 1):

Thus κ ωj maps

  continuously the closed arc A ωj into the line segment [p ωj , p ωj+1 ] for any j = 1, . . . , M -1, so does κ M from A M into the line segment [p ωM , p ω1 ]. Let m ∈ N and ω ∈ {1, . . . , N } m . The Hardy space over the circle C m , denoted H 2 (C m ), H ω or H m , is the space of functions φ that are holomorphic on the corresponding open disk B m and such that it )| 2 dt 2π < +∞ .

z

  Hm for any φ ∈ H m and z ∈ B m . The orthogonal projection Π m : L 2 (C m ) → H m is called the Szegö projector. Any bounded function u on the circle C m gives rise to the Toeplitz operator T u : φ → Π m (uφ), φ ∈ H m . Thus for any polynomial function p ∈ Pol(E ω ), the function p • κ ω is bounded on C m , and we can consider Toeplitz operators of the form

  α m ) m∈N and (β m ) m∈N are two sequences of strictly positive real numbers: these sequences depend only on m since each D ω acts on a Hardy space over the same circle C m of radius c m . Lemma 3.4. Let m ∈ N, ω ∈ {1, . . . , N } m and p ∈ Pol(E w ). The operator [ D ω , T

u

  , u ∈ L ∞ (C ω ), for clarity reasons. If p(z) = a,b p ab z k zl is a polynomial on the set E ω , we have [ R, T p•κω ] = a,b p ab [ R, T κ a ω κω b ]. We now show that for any a, b ∈ N, [ R, T κ a ω κω b ] = T π ωab ,

T κ a ω κω b z n = M j=1 θ j+1 θ j dt 2π k∈N κ ωjak c mk e itk l∈N κ ωjbl c ml e itl c mn e itn s∈N c -2ms z s c ms e -κκ

  ωjak κ ωjbl c m(k+l) e it(k-l+n-s) ωjak κ ωjbl c m(k+l)

  On the other hand, since on eachA ωj we have (R -R)κ a ωj κ ωj b = (Rκ a ωj ) κ ωj b -κ a ωj Rκ ωj b , we get for any z ∈ B m and n ∈ N T π ωab z n = l) κ ωjak κ ωjbl c m(k+l) c mn e itn s∈N c -2ms z s c ms e -its = s∈N c m(n-s) z s M j=1 k,l∈N κ ωjak κ ωjbl c m(k+l) (k -l)

Proposition 3 . 5 .

 35 (a+b)K a+b-1 τ ωj |p ωj+1 -p ωj |k j =: K ab .Finally, we take K p := a,b |p ab |K ab and the proof is complete. Let ω ∈ {1, . . . , N } m , m ∈ N. Let A ω be the algebra generated by Toeplitz operators of the form (12), H ω := H 2 (C m ) and D ω as in[START_REF] Lapidus | Towards a noncommutative fractal geometry? Laplacians and volume measures on fractals[END_REF]. Then for any sequences (α m ) m∈N , (β m ) m∈N of stricly positive real numbers, (A ω , H ω , D ω ) is a spectral triple of spectral dimension 1.

  be the algebra generated by Toeplitz operators of the formT ⊕ p := m∈N ω∈{1,...,N } m T (ω) p•κω , with p ∈ Pol(E), • D ⊕ := m∈N ω∈{1,...,N } m D ω , where D ω := α m R ω + β m ,for some sequences (α m ) m∈N , (β m ) m∈N of strictly positive real numbers and with domain dom(D ⊕ ) := {v ⊕ := N m=0 ω∈{1,...,N } m v ω , N ∈ N , v ω ∈ dom(D ω )} .

1 s- 1

 11 j + β m ) -1 = β -1 m = c m -→ m→+∞ 0 ,hence D ⊕ has compact resolvent. Using Lemma 3.4, for any p ∈ Pol(E), we have[ D ⊕ , T ⊕ p ] = sup m∈N sup ω∈{1,...,N } m [ D ω , T (ω) p•κω ] ≤ K p sup m∈N α m ≤ K p(indeed α 0 = 1 and log(N ) log(cN ) < ⇔ α m < 1 for any m = 0). From Proposition 3.5, the spectral dimension of (A ω , H ω , D ω ) is 1 for any ω ∈ {1, . . . , N } m , m ∈ N, so we study Tr(|D ⊕ | -s ) for s > 1:Tr(|D ⊕ | -s ) = m∈N N m j∈N (α m j + β m ) -s = m∈N N m α -s m j∈N (j + βm αm ) -s .A similar calculation as in the proof of Proposition 2.6 shows that j∈N (j+ βm αm ) -s ∼ m→+∞ ( βm αm ) 1-s , so Tr(|D ⊕ | -s ) is finite if and only if m∈N N m α -s m ( βm αm ) 1-s = m∈N (c s N ) m < +∞ ⇔ c s N < 1 ,i.e. for s > log(N ) log(1/c) = dim H (E).

r

  m (z) := |z| 2 -c 2m , and w m (z) := (-r m (z)) N m , z ∈ B m , and we denote the corresponding weighted Bergman spaces A 2 m (B m ). For any m ∈ N and ω ∈ {1, . . . , N } m , we consider the translation ι ω : z → z + q ω from B m into F ω (B), where q ω := F ω (0) is the center of the open disk F ω (B). Of course Proposition 2.4 remains valid when H m is replaced by A 2 m (B m ), and we get

  Lemma 2.1. Let (H m ) m∈N be a family of Hilbert spaces, (D m ) m∈N be a family of unbounded selfadjoint operators with corresponding dense domains (dom(D m ) ⊂ H m ) m∈N , and

2.1 Conditions of integrability

Acknowledgements:

The author expresses his gratitude to M. Engliš, B. Iochum and O. Gabriel for useful advices.

Proof. For any m ∈ N, an orthonormal basis of A 2 m (B m ) is given by

and similarly as in ( 6), we have for any ω ∈ {1, . . . , N } m

. These sequences are equivalent, as m → ∞, to (α m ) m∈N and (β m ) m∈N of ( 16) when = 2. Since we assumed 1 < c 2 N , the compactness of the resolvent of D ⊕ and the computation of the spectral dimension are shown similarly as in the proof of Theorem 3.6. Moreover, as in Proposition 2.5, we have

for any m ∈ N and ω ∈ {1, . . . , N } m , so [ D ⊕ , T ⊕ p ] ≤ sup m∈N α m p ∞ < +∞.