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arXiv:2206.10702 · Physical Review C

Entanglement entropy between short range correlations and the Fermi sea in nuclear structure

E. Pazy

Published May 19, 2023• 30 citations
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Abstract

The nuclear structure orbital entanglement entropy of short range correlation (SRC) is calculated based on the nuclear scale separation, specifically, the entanglement between the SRC orbitals and the rest of the system. It should be stressed that this is a single nucleon and not a pair entanglement entropy between the proton and neutron. The entanglement arises from the probability for a nucleon to occupy a momentum state above the Fermi momentum. The momentum space of the nucleus separates into two parts such that nucleons can occupy the mean-field part of the wave function, i.e., the Fermi sea (FS) and separately the high-momentum SRC part. The orbital entropy obtained is between these two parts where essentially two momentum subspaces are defined, one containing all the low momentum FS states and the other containing the high-momentum part as a SRC ``orbital'' state. For the calculation of the decoupling of low and high momenta which was established by the similarity normalization group the SRC is employed and viewed as a further orbital which can be multiply occupied. Since the probability of the occupation of a single SRC is given by the nuclear contact a simple general expression of the orbital entanglement entropy for SRC by employing the generalized contact formalism is obtained. This general formula for the SRC orbital entanglement entropy of a nuclear structure in terms of the nuclear contact allows one to obtain the scaling of the entropy in terms of the mass number, $A$. From this formula it is evident that, unlike the entanglement entropy of many quantum systems which scales with the surface area, the orbital entanglement entropy associated with the SRC in large nuclei is linearly dependent on $A$, i.e., it is shown to be extensive.

DOI
10.1103/physrevc.107.054308

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The nuclear structure orbital entanglement entropy of short range correlation (SRC) is calculated based on the nuclear scale separation, specifically, the entanglement between the SRC orbitals and the rest of the system. It should be stressed that this is a single nucleon and not a pair entanglement entropy between the proton and neutron. The entanglement arises from the probability for a nucleon to occupy a momentum state above the Fermi momentum. The momentum space of the nucleus separates into two parts such that nucleons can occupy the mean-field part of the wave function, i.e., the Fermi sea (FS) and separately the high-momentum SRC part. The orbital entropy obtained is between these two parts where essentially two momentum subspaces are defined, one containing all the low momentum FS states and the other containing the high-momentum part as a SRC ``orbital'' state. For the calculation of the decoupling of low and high momenta which was established by the similarity normalization group the SRC is employed and viewed as a further orbital which can be multiply occupied. Since the probability of the occupation of a single SRC is given by the nuclear contact a simple general expression of the orbital entanglement entropy for SRC by employing the generalized contact formalism is obtained. This general formula for the SRC orbital entanglement entropy of a nuclear structure in terms of the nuclear contact allows one to obtain the scaling of the entropy in terms of the mass number, AAA. From this formula it is evident that, unlike the entanglement entropy of many quantum systems which scales with the surface area, the orbital entanglement entropy associated with the SRC in large nuclei is linearly dependent on AAA, i.e., it is shown to be extensive.
The Short-Range Correlations between nucleons in nuclei is regarded as a complex system. We investigate the relationship between the orbital entanglement entropy of SRCs Sij in nuclear structures and Tan contact cij, and find that the orbital entanglement entropies and Tan contacts corresponding to proton-proton SRC pairs and neutron-proton SRC pairs in nuclei demonstrate a scaling relation. More specifically, the proportionality of entanglement entropy between proton-proton pairs and neutron-proton pairs is directly related to the ratio of nuclear contacts within the atomic nucleus, demonstrating an approximate ratio of 2.0. Our research suggests that this scaling relationship should hold true for all symmetric nuclei, furthermore, we offer a possible explanation for this phenomenon.
For quantum many-body systems with short-range correlations (SRCs), the intimate relationship between their magnitude, the behavior of the single-particle occupation probabilities at momenta larger than the Fermi momentum, and the entanglement entropy is a new qualitative aspect not studied and exploited yet. A large body of recent condensed matter studies indicates that the time evolution of the entanglement entropy describes the nonequilibrium dynamics of isolated and strongly interacting many-body systems, in a manner similar to the Boltzmann entropy, which is strictly defined for dilute and weakly interacting many-body systems. Both theoretical and experimental studies in nuclei and cold atomic gases have shown that the fermion momentum distribution has a generic behavior n(k)=C/k4n(k)=C/{k}^{4}n(k)=C/k4 at momenta larger than the Fermi momentum, due to the presence of SRCs, with approximately 20% of the particles having momenta larger than the Fermi momentum. The presence of the long momentum tails in the presence of SRCs changes the textbook relation between the single-particle kinetic energy and occupation probabilities, ${n}_{\text{mf}}(k)=1/{1+exp\ensuremath{\beta}[\ensuremath{\epsilon}(k)\ensuremath{-}\ensuremath{\mu}]}$ for momenta very different form the Fermi momentum, particularly for dynamics processes. SRCs induced high-momentum tails of the single-particle occupation probabilities increase the entanglement entropy of fermionic systems, which in its turn affects the dynamics of many nuclear reactions, such as heavy-ion collisions and nuclear fission.
Abstract Quantum entanglement offers a unique perspective into the underlying structure of strongly-correlated systems such as atomic nuclei. In this paper, we use quantum information tools to analyze the structure of light and medium-mass berillyum, oxygen, neon and calcium isotopes within the nuclear shell model. We use different entanglement metrics, including single-orbital entanglement, mutual information, and von Neumann entropies for different equipartitions of the shell-model valence space and identify mode-entanglement patterns related to the energy, angular momentum and isospin of the nuclear single-particle orbitals. We observe that the single-orbital entanglement is directly related to the number of valence nucleons and the energy structure of the shell, while the mutual information highlights signatures of proton–proton and neutron–neutron pairing, as well as nuclear deformation. Proton and neutron orbitals are weakly entangled by all measures, and in fact have the lowest von Neumann entropies among all possible equipartitions of the valence space. In contrast, orbitals with opposite angular momentum projection have relatively large entropies, especially in spherical nuclei. This analysis provides a guide for designing more efficient quantum algorithms for the noisy intermediate-scale quantum era.
Background: Entanglement plays a central role in a diverse array of increasingly important research areas, including quantum computation, simulation, measurement, sensing, and communication. Extensive suites of investigations have been performed to better understand entanglement in atomic and molecular quantum many-body systems, while the exploration of entanglement in the structure of nuclei and their reactions is presently in its infancy.Purpose: The goal of this work is to begin investigating the entanglement properties of nuclei from first-principles nuclear many-body calculations. We attempt to identify common features and emergent structures of entanglement that could ultimately lead to new and natural many-body schemes. With an eye toward quantum accelerators in future hybrid-supercomputers, criteria for partitioning nuclear many-body calculations into quantum and classical components may provide advantages in future large-scale computations. Along the way we look for explanations of the relative success of phenomenological models such as the nuclear shell model, and for better ways to match to low-energy nuclear effective field theories and lattice QCD calculations to nuclear many-body techniques that are based upon entanglement.Method: We explore the entanglement between single-particle states in 4He^{4}\mathrm{He}4He and 6He^{6}\mathrm{He}6He. The patterns of entanglement emerging from different single-particle bases are compared, and possible links with the convergence of observables are explored, in particular, ground-state energies. The nuclear wave functions are obtained by performing active-space no-core configuration-interaction calculations using a two-body nucleon-nucleon interaction derived from chiral effective field theory. Entanglement measures within single-particle bases exhibiting different degrees of complexity are determined, in particular, harmonic oscillator (HO), Hartree-Fock (HF), natural (NAT) and variational natural (VNAT) bases. Specifically, single-orbital entanglement entropy, two-orbital mutual information, and negativity are studied.Results: The entanglement structures in 4He^{4}\mathrm{He}4He and 6He^{6}\mathrm{He}6He are found to be more localized within NAT and VNAT bases than within a HO basis for the optimal HO parameters we have worked with. In particular a core-valence structure clearly emerges from the full no-core calculation of 6He^{6}\mathrm{He}6He. The two-nucleon mutual information shows that the VNAT basis, which typically exhibits good convergence properties, effectively decouples the active and inactive spaces.Conclusions: Measures of one- and two-nucleon entanglement are found to be useful in analyzing the structure of nuclear wave functions, in particular the efficacy of basis states, and may provide useful metrics toward developing more efficient schemes for ab initio computations of the structure and reactions of nuclei, and quantum many-body systems more generally.
Abstract We examine mode entanglement and correlation of two fermionic particles analytically and numerically. We study the one- and two-mode entropies and a global characteristic, the one-body entanglement entropy considering angular momentum coupled state with a single configuration. We show that with rearrangement of the single-particle orbitals the Slater decomposition can be obtained which can also be applied for configuration mixing subject to special restriction. With the help of the Slater decomposition, we derive analytical expressions for the entanglement measures, which becomes very simple for zero total angular momentum. The Slater decomposition allows us to define associated modes, and it turns out that they have identical one- and two-mode entropies. Furthermore, we show that specific single configurations describe maximally entangled states when the total angular momentum is zero. The numerical shell model study of two valence neutrons in the sd shell is revealed that the one-body entanglement entropy of the ground state is close to the maximal value, and the associated modes have the largest mutual information.
The self-consistent Green’s function method within the ladder approximation provides a microscopic description of correlated nuclear systems which properly treats the nucleon-nucleon correlations induced by the short-range and tensor components present in any realistic interaction. These correlations produce a sizable depletion of low momenta below the Fermi surface as well as the occupation of high momenta in the nuclear ground state. A few representative results for nuclear matter are presented to illustrate the present progress in the application of this method to nuclear systems.