Violent relaxation in quantum fluids with long-range interactions

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.

Quantum fluctuations inhibit symmetry breaking in the Hamiltonian mean-field model

It is widely believed that mean-field theory is exact for a wide range of classical long-range interacting systems. Is this also true once quantum fluctuations have been accounted for? As a test case we study the Hamiltonian mean-field (HMF) model for a system of bosons which is predicted (according to mean-field theory) to undergo a second-order quantum phase transition at zero temperature. The ordered phase is characterized by a spontaneously broken O(2) symmetry, which, despite occurring in a one-dimensional model, is not ruled out by the Mermin-Wagner theorem due to the presence of long-range interactions. Nevertheless, a spontaneously broken symmetry implies gapless Goldstone modes whose large fluctuations can restore broken symmetries. In this work we study the influence of quantum fluctuations by projecting the Hamiltonian onto the continuous subspace of symmetry-breaking mean-field states. We find that the energetic cost of gradients in the center-of-mass wave function inhibits the breaking of the O(2) symmetry, but that the energetic cost is very small, scaling as O(1/N^{2}). Nevertheless, for any finite N, no matter how large, this implies that the ground state has a restored O(2) symmetry. Implications for the finite-temperature phases, as well as the classical limit, of the HMF model are discussed.

Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model

The Hamiltonian mean-field (HMF) model describes particles on a ring interacting via a cosine interaction, or equivalently, rotors coupled by infinite-range XY interactions. Conceived as a generic statistical mechanical model for long-range interactions such as gravity (of which the cosine is the first Fourier component), it has recently been used to account for self-organization in experiments on cold atoms with long-range optically mediated interactions. The significance of the HMF model lies in its ability to capture the universal effects of long-range interactions and yet be exactly solvable in the canonical ensemble. In this work we consider the quantum version of the HMF model in one dimension and provide a classification of all possible stationary solutions of its generalized Gross-Pitaevskii equation (GGPE), which is both nonlinear and nonlocal. The exact solutions are Mathieu functions that obey a nonlinear relation between the wave function and the depth of the mean-field potential, and we identify them as bright solitons. Using a Galilean transformation these solutions can be boosted to finite velocity and are increasingly localized as the mean-field potential becomes deeper. In contrast to the usual local GPE, the HMF case features a tower of solitons, each with a different number of nodes. Our results suggest that long-range interactions support solitary waves in a novel manner relative to the short-range case.