We study the thermodynamics of quantum particles with long-range interactions at T = 0. Specifically, we generalize the Hamiltonian mean-field (HMF) model to the case of fermions. We consider the Thomas–Fermi approximation that becomes exact in a proper thermodynamic limit with a coupling constant k ~ N. The equilibrium configurations, described by the mean-field Fermi (or waterbag) distribution, are equivalent to polytropes of index n = 1/2. We show that the homogeneous phase, which is unstable in the classical regime, becomes stable in the quantum regime. The homogeneous phase is stabilized by the Pauli exclusion principle. This takes place through a first-order phase transition where the control parameter is the normalized Planck constant. The homogeneous phase is unstable for , metastable for and stable for . The inhomogeneous phase is stable for , metastable for and disappears for (for , there exists an unstable inhomogeneous phase with magnetization ). We point out analogies between the fermionic HMF model and the concept of fermion stars in astrophysics. Finally, as a by-product of our analysis, we obtain new results concerning the Vlasov dynamical stability of the waterbag distribution which is the ground state of the Lynden-Bell distribution in the theory of violent relaxation of the classical HMF model. We show that spatially homogeneous waterbag distributions are Vlasov-stable iff ≥ c = 1/3 and spatially inhomogeneous waterbag distributions are Vlasov-stable iff ≤ * = 0.379 and b ≥ b* = 0.37, where and b are the normalized energy and magnetization. The magnetization curve displays a first-order phase transition at t = 0.352 and the domain of metastability ranges from c to *.
We study the thermodynamics of quantum particles with long-range interactions at T = 0. Specifically, we generalize the Hamiltonian mean-field (HMF) model to the case of fermions. We consider the Thomas–Fermi approximation that becomes exact in a proper thermodynamic limit with a coupling constant k ~ N. The equilibrium configurations, described by the mean-field Fermi (or waterbag) distribution, are equivalent to polytropes of index n = 1/2. We show that the homogeneous phase, which is unstable in the classical regime, becomes stable in the quantum regime. The homogeneous phase is stabilized by the Pauli exclusion principle. This takes place through a first-order phase transition where the control parameter is the normalized Planck constant. The homogeneous phase is unstable for , metastable for and stable for . The inhomogeneous phase is stable for , metastable for and disappears for (for , there exists an unstable inhomogeneous phase with magnetization ). We point out analogies between the fermionic HMF model and the concept of fermion stars in astrophysics. Finally, as a by-product of our analysis, we obtain new results concerning the Vlasov dynamical stability of the waterbag distribution which is the ground state of the Lynden-Bell distribution in the theory of violent relaxation of the classical HMF model. We show that spatially homogeneous waterbag distributions are Vlasov-stable iff ≥ c = 1/3 and spatially inhomogeneous waterbag distributions are Vlasov-stable iff ≤ * = 0.379 and b ≥ b* = 0.37, where and b are the normalized energy and magnetization. The magnetization curve displays a first-order phase transition at t = 0.352 and the domain of metastability ranges from c to *.
We study the thermodynamics of quantum particles with long-range interactions at T=0. Specifically, we generalize the Hamiltonian Mean Field (HMF) model to the case of fermions and bosons. In the case of fermions, we consider the Thomas-Fermi approximation that becomes exact in a proper thermodynamic limit. The equilibrium configurations, described by the Fermi (or waterbag) distribution, are equivalent to polytropes with index n=1/2. In the case of bosons, we consider the Hartree approximation that becomes exact in a proper thermodynamic limit. The equilibrium configurations are solutions of the mean field Schrödinger equation with a cosine interaction. We show that the homogeneous phase, that is unstable in the classical regime, becomes stable in the quantum regime. This takes place through a first order phase transition for fermions and through a second order phase transition for bosons where the control parameter is the normalized Planck constant. In the case of fermions, the homogeneous phase is stabilized by the Pauli exclusion principle while for bosons the stabilization is due to the Heisenberg uncertainty principle. As a result, the thermodynamic limit is different for fermions and bosons. We point out analogies between the quantum HMF model and the concepts of fermion and boson stars in astrophysics. Finally, as a by-product of our analysis, we obtain new results concerning the Vlasov dynamical stability of the waterbag distribution.
We study the thermodynamics of quantum particles with long-range interactions at T = 0. Specifically, we generalize the Hamiltonian mean-field (HMF) model to the case of bosons. We consider the Hartree approximation that becomes exact in a proper thermodynamic limit with a coupling constant k ∼ 1/ N . The equilibrium configurations are solutions of the mean-field Schrödinger equation with a cosine interaction. We show that the homogeneous phase, which is unstable in the classical regime, becomes stable in the quantum regime. The homogeneous phase is stabilized by the Heisenberg uncertainty principle. This takes place through a second-order phase transition where the control parameter is the normalized Planck constant. The homogeneous phase is unstable for and stable for . The inhomogeneous phase is stable for and disappears for . We point out analogies between the bosonic HMF model and the concept of boson stars in astrophysics. We also discuss the differences between bosons and fermions for what concerns the thermodynamic limit, the order of the phase transition and the form of the density profiles.
Engineering long-range interactions in experimental platforms has been achieved with great success in a large variety of quantum systems in recent years. Inspired by this progress, we propose a generalization of the classical Hamiltonian mean-field model to fermionic particles. We study the phase diagram and thermodynamic properties of the model in the canonical ensemble for ferromagnetic interactions as a function of temperature and hopping. At zero temperature, small charge fluctuations drive the many-body system through a first-order quantum phase transition from an ordered to a disordered phase. At higher temperatures, the fluctuation-induced phase transition remains first order initially and switches to second-order only at a tricritical point. Our results offer an intriguing example of tricriticality in a quantum system with long-range couplings, which bears direct experimental relevance. The analysis is performed by exact diagonalization and mean-field theory.
We consider a gas of indistinguishable bosons, confined to a ring of radius R, and interacting via a pair-wise cosine potential. This may be thought of as the quantized Hamiltonian Mean Field (HMF) model for bosons originally introduced by Chavanis as a generalization of Antoni and Ruffo’s classical model. This thesis contains three parts: In part one, the dynamics of a Bose-condensate are considered by studying a generalized Gross-Pitaevskii equation (GGPE). Quantum effects due to the quantum pressure are found to substantially alter the system’s dynamics, and can serve to inhibit a pathological instability for repulsive interactions. The non-commutativity of the large-N , long-time, and classical limits is discussed. In part two, we consider the GGPE studied above and seek static solutions. Exact solutions are identified by solving a non-linear eigenvalue problem which is closely related to the Mathieu equation. Stationary solutions are identified as solitary waves (or solitons) due to their small spatial extent and the system’s underlying Galilean invariance. Asymptotic series are developed to give an analytic solution to the non- linear eigenvalue problem, and these are then used to study the stability of the solitary wave mentioned above. In part three, the exact solutions outlined above are used to study quantum fluctuations of gapless excitations in the HMF model’s symmetry broken phase. It is found that this phase is destroyed at zero temperature by large quantum fluctuations. This demonstrates that mean-field theory is not exact, and can in fact be qualitatively wrong, for long-range interacting quantum systems, in contrast to conventional wisdom.
A comprehensive theory of the quantum phase transition in clean itinerant Heisenberg ferromagnets is presented. It is shown that the standard mean-field description of the transition is invalid in spatial dimensions d<3 due to the existence of soft particle-hole excitations that couple to the order parameter fluctuations and lead to an upper critical dimension dc+=3. A generalized mean-field theory that takes these additional modes into account predicts a fluctuation-induced first-order transition. In a certain parameter regime, this first-order transition in turn is unstable with respect to a fluctuation-induced second-order transition. The quantum ferromagnetic transition may thus be either of first or of second order, in agreement with experimental observations. A detailed discussion is given of the stability of the first-order transition and of the critical behavior at the fluctuation-induced second-order transition. In d=3, the latter is mean field like with logarithmic corrections to scaling, and in d<3 it can be controlled by means of a $3\ensuremath{-}\ensuremath{\epsilon}$ expansion.
In this Thesis we describe the theoretical-computational study performed on the behavior of isolated systems, far from thermodynamic equilibrium. Analyzing models well-known in literature we follow a path bringing to the classification of different behaviors in function of the interaction range of the systems' particles. In the case of systems with long-range interaction we studied the "Quasi-Stationary states" (QSSs) which emerge at short times when the system evolves with Hamiltonian dynamics. Their interest is in the fact that in many physical systems, such as self-gravitating systems, plasmas and systems characterized by wave-particle interaction, QSSs are the only experimentally accessible regime. QSS are defined as stable solutions of the Vlasov equation and, as their duration diverges with the system size, for large systems' size they can be seen as the true equilibria. They do not follow the Boltzmann statistics, and it does not exists a general theory which describes them. Anyway it is possible to give an approximate description using Lynden-Bell theory. One part of the thesis is devoted to shed light on the characteristics of the phase diagram of the "Hamiltonian mean field" model (HMF), during the QSS, calculated with the Lynden-Bell theory. The results of our work allowed to confirm numerically the presence of a phase re-entrance. In the Thesis is present also a detailed description on the system's caloric curves and on the metastability. Still in this context we show an analysis of the equivalence of the statistical ensembles, confirmed in almost the totality of the phase diagram (except for a small region), although the presence of negative specific heat in the microcanonical ensemble, which in Boltzmannian systems implies the non-equivalence of statistical ensembles. This result allowed us to arrive to a surprising conclusion: the presence of negative specific heat in the canonical ensemble. Still in the context of long-range interacting systems we analyze the linear stability of the non-homogeneous QSSs with respect to the Vlasov equation. Since the study of QSS find an application in the Free-electron laser (FEL) and other light sources, which are characterized by wave-particle interaction, we analyze, in the last chapter, the experimental perspectives of our work in this context. The other class of systems we studied are short-range interacting systems. Here the behavior of the components of the system is strongly influenced by the neighbors, and if one takes a system in a disordered state (a zero magnetization state for magnetic systems), which relaxes towards an ordered equilibrium state, one sees that the ordering process first develops locally and then extends to the whole system forming domains of opposed magnetization which grow in size. This process is called "coarsening". Our work in this field consisted in investigating numerically the laws of scale, and in the Thesis we characterize the temporal dependence of the domain sizes for different interaction ranges and we show a comparison between Hamiltonian and Langevin dynamics. This work inserts in the open debate on the equivalence of different dynamics where we found that, at least for times not too large, the two dynamics give different scaling laws.