The generalized density matrix in the Lipkin model

M. Tohyama, Peter Schuck, Shun-Jin Wang

Denis Lacroix A

A density matrix functional theory is constructed semi-empirically for the two-level Lipkin model. This theory, based on natural orbitals and occupation numbers, is shown to provide a good description for the ground state energy of the system as the two-body interaction and particle number vary. The application of density matrix functional theory to the Lipkin model illustrates that it could be a valuable tool for systems presenting a shape phase transition such as nuclei. The improvement of one-body observables description as well as the interest for energy density functional theory are discussed.

Karl Blum

Lu Jia, V. G. Zelevinsky

The collective bosonic Hamiltonian is derived from the microscopic nucleonic Hamiltonian by the generalized density matrix method. Independent parameters in the collective Hamiltonian are fixed completely, solutions are given in detail. The random phase approximation corresponds to the harmonic potential of the current approach. The full solution (very close to the exact diagonalization) is obtained over the whole region of parameters including and beyond the instability point of the random phase approximation. The method is tested in the simple Lipkin model.

Karl Blum

José A Carrasco, Federico Finkel, Artemio González-López +2 more

Abstract We introduce a new class of generalized isotropic Lipkin–Meshkov–Glick models with spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of type. We evaluate in closed form the reduced density matrix of a block of L spins when the whole system is in its ground state, and study the corresponding von Neumann and Rényi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as when L tends to infinity, where the coefficient a is equal to ( m − k )/2 in the ground state phase with k vanishing magnon densities. In particular, our results show that none of these generalized Lipkin–Meshkov–Glick models are critical, since when their Rényi entropy R q becomes independent of the parameter q . We have also computed the Tsallis entanglement entropy of the ground state of these generalized Lipkin–Meshkov–Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when . Finally, in the case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of . This is also true in the case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an ( m + 1)-simplex in whose vertices are the weights of the fundamental representation of .