We investigate the entanglement between a spin and its environment in impurity systems which exhibit a second-order quantum phase transition separating a delocalized and a localized phase for the spin. As an application, we employ the spin-boson model, describing a two-level system (spin) coupled to a sub-Ohmic bosonic bath with power-law spectral density, J(omega) proportional to omega(s) and 0 < s < 1. Combining Wilson's numerical renormalization group method and hyperscaling relations, we demonstrate that the entanglement between the spin and its environment is always enhanced at the quantum phase transition resulting in a visible cusp (maximum) in the entropy of entanglement. We formulate a correspondence between criticality and impurity entanglement entropy, and the relevance of these ideas to nanosystems is outlined.
Background: One of the fundamental problems of quantum information science is quantum entanglement. Recently, quantum phase transition in nuclear systems is studied in connection with quantum entanglement.Purpose: In this paper, the use of entanglement entropy as a suitable signal for the study of quantum phase transition in the even-even and odd-
A nuclei is investigated. The effect of the coupling of a single fermion to a boson core on entanglement entropy is studied.Method: By use of the affine
SU(1,1) Lie algebra and through the Schmidt decomposition in the framework of the interacting boson model (IBM) and interacting boson-fermion model (IBFM), entanglement entropy in the even-even and odd-
A are obtained. The entanglement entropy is used for tracking and studying the shape phase transition in these nuclei.Results: The entanglement entropy in the IBM and the IBFM is calculated. The entanglement entropy values of the low-lying states of
122−−134Xe,102−−110Pd, and
123−−133Xe were calculated and analyzed. It is found that entanglement entropy is a suitable order parameter to detect shape phase transition in nuclear systems.Conclusions: The obtained results indicate that the entropy of entanglement is sensitive to the shape-phase transition between spherical and $\ensuremath{\gamma}$-unstable regions in nuclei. The results show that entanglement entropy is a powerful tool for identifying shape phase transitions in nuclei. It is found that the coupling of the single fermion with angular momentum
j to the even-even system does not change the geometry imposed by the boson core performing the transition and only the entanglement entropy values have been shifted by the addition of the odd particle with respect to the even case.
We investigate the entanglement properties of an ensemble of atoms interacting with a single bosonic field mode via the Dicke (superradiance) Hamiltonian. The model exhibits a quantum phase transition and a well-understood thermodynamic limit, allowing the identification of both quantum and semiclassical many-body features in the behavior of the entanglement. We consider the entanglement between the atoms and the field, an investigation initiated in Lambert et al. [Phys. Rev. Lett. 92, 073602 (2004)]. In the thermodynamic limit, we give exact results for all entanglement partitions and observe a logarithmic divergence of the atom-field entanglement, and discontinuities in the average linear entropy.