Scaling of the von Neumann entropy in a two-electron system near the ionization threshold

Abstract

The critical behavior of entanglement near a quantum phase transition has been studied intensely over the past few years. Few-body quantum systems show critical behavior near the ionization point. We investigate the scaling properties of the von Neumann entropy for an atomiclike system near the ionization threshold. Using finite size scaling methods we calculate the critical charge and the critical exponent associated to the von Neumann entropy. The parallelism between the behavior of entanglement near a quantum phase transition and the behavior of the von Neumann entropy in a critical few-body quantum system is analyzed.

Dynamics of Entanglement for Two-Electron Atoms

“…(4). Since the state |Ψ has spatial and spin degrees of freedom, both remain factorized, the von Neumann entropy of the reduced density operator becomes 19,23 …”

“…17 The von Neumann entropy has been used to study the dynamics of entanglement between two trapped atoms, 18 the entanglement between two (spin-1/2) fermions in a cylindrical harmonic trap, 19 the entanglement for one-dimensional spin systems in external time-dependent magnetic fields [20][21][22] and the scaling properties of entanglement shared between the two electrons of an atomic like system near the ionization threshold. 23 Gittings and Fisher 24 showed that the von Neumann entropy for the reduced density matrix of half the system can be used as an entanglement measure for the case of indistinguishable particles. Huang and Kais showed that entanglement, as measured by the von Neumann entropy, can be used as a measure of electron-electron correlation in quantum chemistry calculations.…”

Quantum entanglement for two electrons in the excited states of helium-like systems

“…Hofer used Gaussian wave functions to examine basis set convergence for electron-electron entanglement in helium-like systems [26]. Osenda and Serra have investigated the effect of critical charge on the ground and excited states for the S-wave model of helium atom [27,28]. Tichy et al [29] has reviewed the recent developments on entanglement for atomic and molecular systems.…”

Entanglement Computation in Atoms and Molecules

“…In two papers, [24], [25], the authors computed the von Neumann entropy of a two electron system, a heliumlike atom, near the ionization threshold. Their model considers again a simplified Hamiltonian, namely one of the form:…”

“…This means that the models uses a spherical average of the repulsion between electrons. In [24] the von Neumann entropy of the ground state only of Helium is computed, focusing on its dependence on the "coupling" parameter and on the search of its critical values. In [25] a similar model is used for the lowest energy triplet state.…”