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arXiv:1803.08883 · Physical Review A

Fermionic entanglement in superconducting systems

Martina Di Tullio, N. Gigena, R. Rossignoli

Published Jun 11, 2018• 22 citations
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Abstract

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 $\overline{k}$. In contrast, the entanglement associated with the reduced density matrix of four single-particle modes $k,\overline{k}, {k}^{\ensuremath{'}},{\overline{k}}^{\ensuremath{'}}$, 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 $k,{k}^{\ensuremath{'}}$ 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. 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.

DOI
10.1103/physreva.97.062109

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We study the ground state properties of the one-dimensional extended Hubbard model at half filling from the perspective of its particle reduced density matrix. We focus on the reduced density matrix of two fermions and perform an analysis of its quantum correlations and coherence along the different phases of the model. Specifically, we study its (i) entanglement entropy, (ii) ${\ensuremath{\ell}}_{1}$ norm of coherence, (iii) irreducible two-body cumulant matrix, and (iv) entanglement spectrum. Our results show that these different properties are complementary to each other depending on the phase of the system, exhibiting peculiar behaviors such as discontinuities and maximum or minimum values at the quantum phase transitions, thus providing a qualitative view of the phase diagram of the model. In particular, in the superconducting region, we obtain that the entanglement spectrum signals a transition from a dominant singlet (SS) to triplet (TS) pairing ordering in the system. Moreover, from the analysis of the dominant eigenvector in the reduced state, we can relate the SS-TS transition to the spatial separation between the fermion pairs in the two different pairing orderings. The entanglement gap is also able to highlight a transition---at a few-body level---in the ground state wave function, not discussed previously in the literature. While other quantifiers are less sensitive to few-body defects in the wave function, the entanglement gap can work as a magnifying glass for these, capturing such small fluctuations.
The ground-state entanglement of a single particle of the N-harmonium system (i.e., a completely-integrable model of NNN particles where both the confinement and the two-particle interaction are harmonic) is shown to be analytically determined in terms of NNN and the relative interaction strength. For bosons, we compute the von Neumann entropy of the one-body reduced density matrix by using the corresponding natural occupation numbers. There exists a critical number NcN_cNc​ of particles so that below it, for positive values of the coupling constant, the entanglement grows when the number of particles is increasing; the opposite occurs for N>NcN > N_cN>Nc​. For fermions, we compute the one-body reduced density matrix for the closed-shell spinned case. In the strong coupling regime, the linear entropy of the system decreases when NNN is growing. For fixed NNN, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.
We examine the mode entanglement and correlation of two fermionic particles. We study the one- and two-mode entropy and a global characteristic, the one-body entanglement entropy. We consider not only angular momentum coupled states with single configuration but use the configuration interaction method. With the help of the Slater decomposition, we derive analytical expressions for the entanglement measures. We show that when the total angular momentum is zero specific single configurations describe maximally entangled states. It turns out that for a finite number of associated modes the one- and two-mode entropies have identical values. In the shell model framework, we numerically study two valence neutrons in the sdsdsd shell. The one-body entanglement entropy of the ground state is close to the maximal value and the associated modes have the largest mutual information.