Research Interests

Statistical Mechanics: Geometro-Topological Approach.

In a differential-geometrical treatment of Hamiltonian dynamics, the motions are seen as geodesics of suitable Riemannian manifolds endowed with an appropriate metric. Possible choices are the Jacobi and the Einsenhart metrics. Within this framework the instability properties of Hamiltonian dynamics, are caught by the Jacobi--Levi-Civita equation [see Phys. Rev. E 56 4872 (1997)].

Fig. 1

This interpretation allowed the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos in systems of arbitrarily large number of degrees of freedom , or arbitrarily far from quasi-integrability [see Phys. Rev. E 61 R3299 (2000), Chaos 15, 015106 (2005), Phys. Rev. E, 78, 046205 (2008)].

Fig. 2

Furthermore, a geometro-topological treatment of Hamiltonian systems, offers new tools to characterize the systems that - from the statistical mechanical point of view - undergo phase transitions. In Phys. Rev. E 60 R5009 (1999) it is reported a numerical investigation of the classicalmodels in one and two dimensions. The geometrical framework considered is the system configuration space and the topological tool is the Morse theory (1). In this, case the Morse function used to define the family submanifolds, is the potential energy, thus each is the set of points with potential energy lower than .and all its submanifoldsare endowed with a Riemannian metric. The topology of the familyis studied by computing2 the second variation3of the scalar curvaturei.e., the sum of all the possible products of two principal curvatures of the manifolds. This paper reports the results obtained with three different types of metrics, one conformally flat and the others nonconformal.

Peculiar behaviours ofare found only in the two-dimensional case, when a phase transition is present. Peaks ofappear at - the value of(obtained by both Monte Carlo averaging with the canonical configurational measure or Hamiltonian dynamics) at the phase transition point - in the two-dimensional case, whereas only monotonic patterns are found in the one-dimensional case, where no phase transition is present (see Fig. 2). Since these “singular” cuspy patterns ofare found independently from any possible statistical mechanical effects, and from the geometric structure given to the family , the observed phenomenology strongly hints that some major change in the topology of the configuration-space submanifoldsoccurs in correspondence of a second order phase transition.

Fig.3

Suitable topology changes of equipotential manifolds of configuration space can entail thermodynamic phase transitions. This is the result contained in Phys. Rev. Lett. 84, 2774 (2000), where is numerically computed the Euler characteristic(a topologic invariant) of the equipotential hypersurfacesof the configuration space of the two-dimensional lattice model. The pattern versus(potential energy) reveals that a major topology change in the familyis at the origin of the phase transition in the model considered. The direct evidence given in this work — of the relevance of topology for phase transitions — is obtained through a general method that can be applied to any other model. In fact, the Euler characteristic is a diffeomorphism invariant and expresses fundamental topological information4. In differential topology a standard definition ofiswhere also the numbers—the Betti numbers of —are diffeomorphism invariants. Anyhow, it would be hopeless to try to computefrom this definition in the case of nontrivial physical models at large dimension, nevertheless there is a possibility given by the Gauss-Bonnet-Hopf theorem, that relateswith the total Gauss-Kronecker curvature of the manifold:which is valid for even dimensional hypersurfaces of Euclidean spaces[]. Here is twice the inverse of the volume of an-dimensional sphere of unit radius,is Gauss-Kronecker curvature5 of the manifold, is the invariant volume measure of, andis the Riemannian metric induced from. Through the computation of thedependence of a topologic invariant, the hypothesis of a deep connection between topology changes of theand phase transitions has been given a direct confirmation. Moreover, it has been found that a sudden second order variation of the topology of these hypersurfaces is the “suitable” topology change that underlies the phase transition of second kind in the 2D latticemodel. Whereas no suitable topology change is present in the 1D lattice, where no phase transition takes place (see Figs. 3 4).

Fig. 4

These preliminary studies have paved the way for a general result definitely linking topology of configuration space and phase transitions. In fact, within the two papers Nuclear Physics B 782 (2007) 189-218 and Nuclear Physics B 782 (2007) 219-240 it has been given a proof of a general theorem about the necessity of a deep link between configuration space topology and the occurrence of thermodynamic phase transitions. These works enlighten of new light the problem of phase transitions, indeed the thermodynamic singularities that correspond to phase transformations, are found to necessarily stem from suitable changes in the topology of certain sub-manifolds of the configuration space. This is an independent and more fundamental mechanism with respect to the standard ones and it provides new tools and methods to tackle those transition phenomena (for which the standard methods fail) that are presently at the forefront of the research.

Classical and Quantum Dynamics.

Generalized coherent states can be profitably employed in the study of dynamics of quantum systems. When a quantum Hamiltonian is a linear function of the generators of a dynamical Lee group, the generalized coherent states of the corresponding dynamical algebra, will give rise to the exact quantum solutions. However, when the Hamiltonian of a system is a nonlinear function of the generators of the dynamical group, the exact solutions are hardly obtained also by coherent states techniques. In these cases, suitable coherent states can be used in combination with appropriate variational principles in order to derive a semiclassical description of quantum-system dynamics. The nonlinear character, inherent the dynamics of Bose-Einstein condensates in periodic (optical) potentials, is at the base of many interesting macroscopic phenomena I have studied. The quantum dynamics of interacting Bose-Einstein condensates, can be explored in terms of coherent states, whereas the corresponding semiclassical description can be derived by means a time dependent variational principle.

The quantum dynamics of an ultracold dilute gas of bosonic atoms in an external trapping potential is described by the Hamiltonian operator

For periodic potentials, the “single atom” Hamiltonian eigenstates are Bloch functionsof quasi-momentum and the band index. Already for moderate optical potential depths, a good approximation for the lowest energy gap is given by, whereis the oscillation frequency of a particle trapped in the harmonic approximation of the potential close to a minimum. One can assume that only the fundamental energy band is involved into the dynamics and describe the system using Wannier functionslocalized at the minima of the optical potential. Accordingly, the boson-field operator can be expressed aswhere the boson operatordestroys (creates) an atom at the lattice site. By substituting the latter expression for the field operators into the Hamiltonian, and by keeping the lowest order in the overlap between the single-well wave-functions, one obtains the effective Bose-Hubbard Hamiltonian

In paper Int. Jour. of Mod. Phys. B Vol. 14, No. 9 (2000) 943-961 it is shown that, by combining the description for the Bose-Hubbard model quantum states as product of site Glauber coherent states and, a time dependent variational principle, the semiclassical dynamics of Bose-Hubbard model is well described by a discrete nonlinear Schroedinger equation (that is the discrete version of the Gross-Pitaevskii equation). Furthermore, it is therein analysed the classical dynamics of two interacting Bose-Einstein condensates (dimer). In spite of the integrable character of the system's dynamics, due to non-linearity, its equations of motion admit interesting and unexpected dynamical behaviours. In fact, this system displays peculiar behaviours like the population self-trapping.

In part of paper Int. Jour. of Mod. Phys. B Vol. 14, No. 9 (2000) 943-961 and, more deeply, in paper Phys. Rev. A 63, 043609-1 (2001), it is considered the dimer quantum dynamics and it is analysed the link between classical trajectories and quantum energy levels. The spectrum structure is studied within the framework of the Schwinger realization of the angular momentum. The latter allows to recognize the symmetry properties of the system Hamiltonian and, then, to use them for characterizing the energy eigenstates. The dimer spectrum is characterised by the presence of nondegenerate doublets. It is really the existence of these nondegenerate doublets in the spectrum that allows to recover, in the classical limit, the symmetry-breaking (self-trapping) effect that characterizes the system classically.

In Phys. Rev. A 65, 013601-1 (2001); Phys. Rev. E, 67, 046227 (2003) and Phys. Rev. Lett., 90, 050404 (2003), it has been shown that the apparently harmless addition of a further coupled condensate to the dimer system, is sufficient to make the dynamics of three coupled Bose-Einstein condensates (trimer) non-integrable. Indeed, in Phys. Rev. A 65, 013601-1 (2001); Phys. Rev. E, 67, 046227 (2003) and Phys. Rev. Lett., 90, 050404 (2003), is investigated the trimer classical dynamics in different inequivalent, and experimentally meaningful, configurations, showing that it displays instabilities in extended regions of the phase space. Besides the chaotic nature of the trimer dynamics, in the latter papers is investigated the collective modes associated with the system's equations and the non-linear self-trapping that emerges in the super-fluid regime.

The non-linear character of the classical equations of motion of Bose-Einstein condensates in arrays of arbitrary length, is at the base of the self-localization phenomenon that, in Phys. Rev. Lett. 97, 060401 (2006), I have predicted to take place in condensates loaded in dissipative optical lattices (see Fig. 5.). The dynamical configurations that in Phys. Rev. Lett. 97, 060401 (2006) has been shown to spontaneously appear, have the nature of (static or moving) “breathing dynamical solutions” (see Fig. 5) that are the very genuine non-linear discrete dynamical states. These macroscopic effects are, and have been, of primary importance in leading toward their experimental observation also in small (few interacting condensates) systems.

Fig. 4a

Fig. 5

In paper III.) I have considered a generic classical many particle system described by an autonomous Hamiltonian which, in addition, has a conserved quantity , so that the Poisson bracketvanishes. I have derived in detail the microcanonical expressions for entropy and temperature. I have shown that both of these quantities depend on multidimensional integrals over submanifolds given by the intersection of the constant energy hypersurfaces with those defined by . I have shown that temperature and higher order derivatives of entropy are microcanonical observable that, under the hypothesis of ergodicity, can be calculated as time averages of suitable functions. Finally I have derived the explicit expression of the function that gives the temperature.

In paper IV.) We have shown how negative temperature states can be obtained in the Nonlinear Discrete Schroedinger Equation. The definition of the microcanonical temperature given by me in paper III.), associated with the corresponding Hamiltonian, allows to obtain a consistent thermodynamic description for both positive and negative temperature states. We have also described how one can pass from positive to negative temperatures by applying energy dissipation to the Nonlinear Discrete Schroedinger Equation chain boundaries. We have found that the microscopic evolution in thermalized negative temperature states is characterized by the mechanism of focusing of particle density (and energy), characterized by the formation of localized breather states.