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We examine the fermionic entanglement in the ground state of the fermionic Lipkin model and its relation with bipartite entanglement. It is first shown that the one-body entanglement entropy, which quantifies the minimum distance to a fermionic Gaussian state, behaves similarly to the mean-field order parameter and is essentially proportional to the total bipartite entanglement between the upper and lower modes, a quantity meaningful only in the fermionic realization of the model. We also analyze the entanglement of the reduced state of four single-particle modes (two up-down pairs), showing that its fermionic concurrence is strongly peaked at the phase transition and behaves differently from the corresponding up-down entanglement. We finally show that the first measures and the up-down reduced entanglement can be correctly described through a basic mean-field approach supplemented with symmetry restoration, whereas the concurrence requires at least the inclusion of RPA-type correlations for a proper prediction. Fermionic separability is also discussed.
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The dynamics of the one-tangle and the concurrence is analyzed in the Lipkin-Meshkov-Glick model which describes many physical systems such as the two-mode Bose-Einstein condensates. We consider two different initial states which are physically relevant and show that their entanglement dynamics are very different. A semiclassical analysis is used to compute the one-tangle which measures the entanglement of one spin with all the others, whereas the frozen-spin approximation allows us to compute the concurrence using its mapping onto the spin squeezing parameter.
We examine distinct measures of fermionic entanglement in the exact ground state of a finite superconducting system. It is first shown that global measures such as the one-body entanglement entropy, which represents the minimum relative entropy between the exact ground state and the set of fermionic gaussian states, exhibit a close correlation with the BCS gap, saturating in the strong superconducting regime. The same behavior is displayed by the bipartite entanglement between the set of all single particle states $k$ of positive quasimomenta and their time reversed partners $bar{k}$. In contrast, the entanglement associated with the reduced density matrix of four single particle modes $k,bar{k}$, $k,bar{k}$, which can be measured through a properly defined fermionic concurrence, exhibits a different behavior, showing a peak in the vicinity of the superconducting transition for states $k,k$ close to the fermi level and becoming small in the strong coupling regime. In the latter such reduced state exhibits, instead, a finite mutual information and quantum discord. And while the first measures can be correctly estimated with the BCS approximation, the previous four-level concurrence lies strictly beyond the latter, requiring at least a particle number projected BCS treatment for its description. Formal properties of all previous entanglement measures are as well discussed.
The statistical mechanics characterization of a finite subsystem embedded in an infinite system is a fundamental question of quantum physics. Nevertheless, a full closed form { for all required entropic measures} does not exist in the general case even for free systems when the finite system in question is composed of several disjoint intervals. Here we develop a mathematical framework based on the Riemann-Hilbert approach to treat this problem in the one-dimensional case where the finite system is composed of two disjoint intervals and in the thermodynamic limit (both intervals and the space between them contains an infinite number of lattice sites and the result is given as a thermodynamic expansion). To demonstrate the usefulness of our method, we compute the change in the entanglement and negativity namely the spectrum of eigenvalues of the reduced density matrix with our without time reversal of one of the intervals. We do this in the case that the distance between the intervals is much larger than their size. The method we use can be easily applied to compute any power in an expansion in the ratio of the distance between the intervals to their size. {We expect these results to provide the necessary mathematical apparatus to address relevant questions in concrete physical scenarios, namely the structure and extent of quantum correlations in fermionic systems subject to local environment.
We establish a set of nonequilibrium quantum phase transitions in the Lipkin-Meshkov-Glick model under monochromatic modulation of the inter-particle interaction. We show that the external driving induces a rich phase diagram that characterizes the multistability in the system. Interestingly, the number of stable configurations can be tuned by increasing the amplitude of the driving field. Furthermore, by studying the quantum evolution, we demonstrate that the system exhibits a set of quantum phases that correspond to dynamically stabilized states.
We investigate the entanglement measures of tripartite W-State and GHZ-state in noninertial frame through the coordinate transformation between Minkowski and Rindler. First it is shown that all three qubits undergo in a uniform acceleration $a$ of W-State, we find that the one-tangle, two-tangle, and $pi$-tangle decrease when the acceleration parameter $r$ increases, and the two-tangle cannot arrive to infinity of the acceleration. Next we show that the one qubit goes in a uniform acceleration $a_{1}$ and the other two undergo in a uniform acceleration $a$ of GHZ-state, we find that the two-tangle is equal to zero and $N_{B_I (A_I C_I)} = N_{C_I (A_I B_I)} eq N_{A_I (B_I C_I)}$, but one-tangle and $pi$-tangle never reduce to zero for any acceleration.
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