Out-of-equilibrium phase transitions in the Hamiltonian mean-field model: a closer look

Tricritical point in the quantum Hamiltonian mean-field model

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.

Violent relaxation in quantum fluids with long-range interactions

Violent relaxation is a process that occurs in systems with long-range interactions. It has the peculiar feature of dramatically amplifying small perturbations, and rather than driving the system to equilibrium, it instead leads to slowly evolving configurations known as quasistationary states that fall outside the standard paradigm of statistical mechanics. Violent relaxation was originally identified in gravity-driven stellar dynamics; here, we extend the theory into the quantum regime by developing a quantum version of the Hamiltonian mean field (HMF) model which exemplifies many of the generic properties of long-range interacting systems. The HMF model can either be viewed as describing particles interacting via a cosine potential, or equivalently as the kinetic XY model with infinite-range interactions, and its quantum fluid dynamics can be obtained from a generalized Gross-Pitaevskii equation. We show that singular caustics that form during violent relaxation are regulated by interference effects in a universal way described by Thom's catastrophe theory applied to waves and this leads to emergent length scales and timescales not present in the classical problem. In the deep quantum regime we find that violent relaxation is suppressed altogether by quantum zero-point motion. Our results are relevant to laboratory studies of self-organization in cold atomic gases with long-range interactions.

Spectral and formal stability criteria of spatially inhomogeneous stationary solutions to the Vlasov equation for the Hamiltonian mean-field model

Stability of spatially inhomogeneous solutions to the Vlasov equation is investigated for the Hamiltonian mean-field model to provide the spectral and formal stability criteria in the form of necessary and sufficient conditions. These criteria determine stability of spatially inhomogeneous solutions whose stability has not been decided correctly by using a less refined formal stability criterion. It is shown that some of such solutions can be found in a family of stationary solutions to the Vlasov equation, which is parametrized with macroscopic quantities and has a two-phase coexistence region in the parameter space.

Linear response theory in the Vlasov equation for homogeneous and for inhomogeneous quasistationary states

Response to small external forces is investigated in quasistationary states of Hamiltonian systems having long-range interactions. Quasistationary states are recognized as stable stationary solutions to the Vlasov equation, and, hence, the linear response theory to the Vlasov equation is proposed for spatially one-dimensional systems with periodic boundary condition. The proposed theory is applicable both to homogeneous and to inhomogeneous quasistationary states and is demonstrated in the Hamiltonian mean-field model. In the homogeneous case magnetic susceptibility is explicitly obtained, and the Curie-Weiss like law is suggested in a high-energy region. The linear response is also computed in the inhomogeneous case, and resonance absorption is investigated to extract nonequilibrium dynamics in the unforced system. Theoretical predictions are examined by direct numerical simulations of the Vlasov equation.

Nonequilibrium phase transitions and violent relaxation in the Hamiltonian mean-field model

We discuss the nature of nonequilibrium phase transitions in the Hamiltonian mean-field model using detailed numerical simulations of the Vlasov equation and molecular dynamics. Starting from fixed magnetization water bag initial distributions and varying the energy, the states obtained after a violent relaxation undergo a phase transition from magnetized to nonmagnetized states when going from lower to higher energies. The phase transitions are either first order or are composed of a cascade of phase reentrances. This result is at variance with most previous results in the literature mainly based on the Lynden-Bell theory of violent relaxation. The latter is a rough approximation and, consequently, is not suited for an accurate description of nonequilibrium phase transition in long-range interacting systems.

Precise determination of the nonequilibrium tricritical point based on Lynden-Bell theory in the Hamiltonian mean-field model

Existence of a nonequilibrium tricritical point has been revealed in the Hamiltonian mean-field model by a nonequilibrium statistical mechanics. This statistical mechanics gives a distribution function containing unknown parameters, and the parameters are determined by solving simultaneous equations depending on a given initial state. Due to difficulty in solving these equations, pointwise numerical detection of the tricritical point has been unavoidable on a plane characterizing a family of initial states. In order to look into the tricritical point, we expand the simultaneous equations with respect to the order parameter and reduce them to one algebraic equation. The tricritical point is precisely identified by analyzing coefficients of the reduced equation. Reentrance to an ordered phase in a high-energy region is revisited around the obtained tricritical point.