We introduce a new class of generalized isotropic Lipkin–Meshkov–Glick models with su(m+1) spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of su(m+1) type. We evaluate in closed form the reduced density matrix of a block of Lspins 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 a log L 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 L-->∞ 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 su(m+1) Lipkin–Meshkov–Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when m-k≥3. Finally, in the su(3) 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 su(3). This is also true in the su(m+1) case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an (m + 1)-simplex in R^m whose vertices are the weights of the fundamental representation of su(m+1).
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 .
Abstract For general symmetric multi-qubit systems, the behavior of one- and two-qubit entanglement for Dicke, spin coherent and parity-adapted (even and odd) spin coherent states is determined. These quantum correlations are quantified by linear and von Neumann entropies of the corresponding one- and two-qubit reduced density matrices of the multi-qubit system. These states play a fundamental role in the study of Hamiltonian systems written in terms of collective generators of the angular momentum algebra like, for example, the Lipkin–Meshkov–Glick (LMG) model. Here we shall use these entanglement measures as a signature to characterize the different quantum phases that appear in these models.
Abstract We introduce a class of generalized Lipkin–Meshkov–Glick (gLMG) models with s u ( m ) interactions of Haldane–Shastry type. We compute the partition function of these models in closed form by exactly evaluating the partition function of the restriction of a spin chain Hamiltonian of Haldane–Shastry type to subspaces with well-defined magnon numbers. As a byproduct of our analysis, we obtain strong numerical evidence of the Gaussian character of the level density of the latter restricted Hamiltonians, and study the distribution of the spacings of consecutive unfolded levels. We also discuss the thermodynamic behavior of a large family of s u ( 2 ) and s u ( 3 ) gLMG models, showing that it is qualitatively similar to that of a two-level system.
Lipkin-Meshkov-Glick (LMG) models describe a set of
N qubits which are located at the corners of an
N-dimensional simplex and embedded in an external magnetic field. Because of its high coordinate number, quantum correlations in the models are quite nontrivial. For instance, in the large-
N limit, two-site entanglement concurrence vanishes for any magnetic field. Multisite global entanglement and generalized global entanglement also tend to vanish in some quantum regions. We characterize the quantum correlations in the LMG model with multipartite nonlocality. In LMG models with anisotropy in the $x\ensuremath{-}y$ plane, for any fixed magnetic field, the ground-state nonlocality (denoted by
S) is found to scale as ${log}_{10}\mathcal{S}\ensuremath{\sim}aN+b$, with
a and
b the fitting parameters. For LMG models which are isotropic in the $x\ensuremath{-}y$ plane, nevertheless, the nonlocality for each Dicke state scales as ${log}_{10}\mathcal{S}\ensuremath{\sim}a\frac{1}{N}+b$, with
b>0. Signals for the quantum phase transitions of the models will also be discussed. These results indicate that multipartite nonlocality captures some key ingredient of the ground-state quantum correlations in the models. In addition to the ground states, we also study multipartite nonlocality in the LMG models at finite temperatures. In the anisotropic models, the thermal stability of the nonlocality at low temperatures is determined by the energy gap. For the isotropic models, nevertheless, as the temperature increases, the nonlocality presents a sharp valley which is followed by a round peak. The behavior is far beyond the above-mentioned ``thermal stability vs energy gap'' picture. The underlying mechanics is the ``contribution inversion'' of low-lying energy states in the thermal-state nonlocality $\mathcal{S}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{T})$; that is, in a low-temperature region, the first-excited state ``defeats'' the ground state and plays a dominant role in thermal-state nonlocality.
We establish a relation between several entanglement properties in the Lipkin-Meshkov-Glick model, which is a system of mutually interacting spins embedded in a magnetic field. We provide analytical proofs that the single-copy entanglement and the global geometric entanglement of the ground state close to and at criticality behave as the entanglement entropy. These results are in deep contrast to what is found in one- dimensional spin systems where these three entanglement measures behave differently.