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arXiv:0704.3130

Hartree-Fock Approximation and Entanglement

L. Martina, Giulio Soliani

Published Apr 24, 2007• 2 citations
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Abstract

The relation between the correlation energy and the entanglement is analytically constructed for the Moshinsky's model of two coupled harmonic oscillators. It turns out that the two quantities are far to be proportional, even at very small couplings. A comparison is made also with the 2-point Ising model.
DOI
10.48550/arxiv.0704.3130

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Our goal is to clarify the relation between entanglement and correlation energy in a bipartite system with infinite dimensional Hilbert space. To this aim, we consider the completely solvable Moshinsky's model of two linearly coupled harmonic oscillators. Also, for small values of the couplings, the entanglement of the ground state is nonlinearly related to the correlation energy, involving logarithmic or algebraic corrections. Then, looking for witness observables of the entanglement, we show how to give a physical interpretation of the correlation energy. In particular, we have proven that there exists a set of separable states, continuously connected with the Hartree–Fock state, which may have a larger overlap with the exact ground state, but also a larger energy expectation value. In this sense, the correlation energy provides an entanglement gap, i.e. an energy scale, under which measurements performed on the 1-particle harmonic sub-system can discriminate the ground state from any other separated state of the system. However, in order to verify the generality of the procedure, we have compared the energy distribution cumulants for the 1-particle harmonic sub-system of the Moshinsky's model with the case of a coupling with a damping Ohmic bath at 0 temperature.
In both molecular physics and condensed matter theory, deeper understanding of the correlation energy density ϵc(r){ϵ}_{c}(\mathbf{r})ϵc​(r) remains a high priority. By adopting L\"owdin's definition of correlation energy as the difference between the exact and the Hartree-Fock values, here we propose two alternative routes to define this. One of these involves both exact and Hartree-Fock (HF) wave functions, while the second requires a coupling constant integration. As an exact analytical example of the first route, we treat the two-electron model atom of Moshinsky, for which both confinement potential and interactions are harmonic. Though the correlation energy density ϵc(r){ϵ}_{c}(\mathbf{r})ϵc​(r) is known analytically, we also investigate numerically its relation to the exact ground-state density in this example.
It has been known for decades that black holes behave as thermodynamic objects and, as such, have an entropy, which was acquired in exact form as the Bekenstein-Hawking entropy. However, the origin of this entropy, and in particular its proportionality to the area of the horizon, remained unexplained. Motivated by this mystery, several authors examined the possibility of entanglement entropy as a source of black hole entropy, by considering a scalar ?eld in the ground state and\ntracing out the degrees of freedom inside an imaginary sphere, analogous to the interior of the black hole. The resulting entanglement entropy between the two regions turned out to scale with the area of their mutual boundary. In the simplest model the scalar ?eld is represented by a system of coupled harmonic oscillators. After an introduction to the phenomenon of quantum entanglement and the associated entanglement entropy, we will review the problem of two coupled oscillators in the ground state and derive the entanglement entropy using an approach in Fock space, as opposed to the standard method involving integrals in positon space. This will also\nallow us to look at some excited states of the system. Subsequently we will try to extend this approach to a system of N coupled harmonic oscillators and derive the entanglement entropy between an inner and outer region of oscillators.