Proton-neutron entanglement in the nuclear shell model Calvin W. Johnson

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Proton-neutron entanglement in the nuclear shell

model

Calvin W. Johnson

E-mail: cjohnson@sdsu.edu

Oliver C. Gorton

San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-1233

Abstract. We compute the proton-neutron entanglement entropy in the

interacting nuclear shell model for a variety of nuclides and interactions. Some

results make intuitive sense, for example that the shell structure, as governed by

single-particle and monopole energies, strongly aﬀects the energetically available

space and thus the entanglement entropy. We also ﬁnd a surprising result: that

the entanglement entropy at low excitation energy tends to decrease for nuclides

when N6=Z. While we provide evidence this arises from the physical nuclear force

by contrasting with random two-body interactions which shows no such decrease,

the exact mechanism is unclear. Nonetheless, the low entanglement suggests that

in models of neutron-rich nuclides, the coupling between protons and neutrons

may be less computationally demanding than one might otherwise expect.

Submitted to: J. Phys. G: Nucl. Part. Phys.

arXiv:2210.14338v2 [nucl-th] 17 Mar 2023

Proton-neutron entanglement in the nuclear shell model 2

1. Introduction

The structure of atomic nuclei exhibits a mixture of simple and complex behaviors.

What is meant by ‘simple’ can be subtle, but typically it means the behavior can be

described by far fewer degrees of freedom than that required by modeling the nucleus

as a collection of Ainteracting nucleons; examples of simplicity include algebraic

models [1] and mean-ﬁeld pictures [2]. Of course, one must acknowledge that models

themselves are not physical observables. Furthermore, complex models can mimic

simpler ones, for example quasidynamic symmetries [3, 4, 5], where a Hamiltonian

mixes symmetries yet observables such as spectra and ratios of transition strengths

are consistent with ‘simpler’ symmetry-respecting models.

Entanglement is a concept describing whether the observable coordinates of a

quantum system are independent; whether measurement of one generalized coordinate

q1inﬂuences future measurements of another coordinate q2of a system ψ(q1, q2, ...)[6,

7]. Such correlations can be described by the entanglement entropy, a concept which

has become popular in recent years due to increasing interest in quantum information

and the potential of quantum computing [8, 9]. It is trivial to write down states

which are either separable (not entangled) or in a superposition of separable states

(entangled), but the creation of entangled states in nature relies on the existence of

an interaction that mixes the relevant degrees of freedom.

Here we consider the entanglement between the proton components and

neutron components of conﬁguration-interaction models of nuclei. Other recent

work in entanglement entropy in nuclei addressed single-particle and seniority-mode

entanglement [10, 11] as well as orbital entanglement revealing shell closures [12]; we

note the ﬁrst two papers reference unpublished versions of the research reported here.

Although it does not directly correspond to the work here, we point out previous

analyses of nuclear conﬁguration-interaction wave functions using ‘entropy,’ such as the

conﬁguration information entropy [13, 14], which is simple but basis dependent, and

the invariant correlation entropy [15], which is much more complicated to compute.

By contrast, because of the way our conﬁguration interaction code constructs the

wave functions, extraction of the wave function amplitudes in terms of proton-neutron

coeﬃcients is straightforward, a signiﬁcant motivation for our approach.

In section 2 we lay out the basic framework of shell-model conﬁguration-

interaction calculations. In section 3, we deﬁne entanglement entropy as well as related

concepts. We then provide examples of entanglement entropies for a variety of cases

and show how much of the behavior for ground state entropies can be understood

through standard concepts in nuclear structure physics. A persistent phenomenon,

however, is not so easily explained: realistic ground states of nuclides with N6=Z

tend to have signiﬁcantly smaller entanglement entropies than those with N=Z.

We also show trends for entropies for all states. We can show this is related to some

components of realistic nuclear forces by contrasting them with results using random

interactions. While the mechanism for suppressing the entanglement eludes us, it

is nonetheless worth reporting, not only as an apparently robust yet unexplained

phenomenon, but also because it has a practical consequence: the low-lying states

of neutron-rich nuclides have fewer nontrivial correlations between the proton and

neutron components. This, in turn, suggests a practical approach for such nuclides,

one which we are currently developing.

Proton-neutron entanglement in the nuclear shell model 3

2. The nuclear conﬁguration-interaction shell model

We ﬁnd low-lying states of a nuclear Hamiltonian by the conﬁguration-interaction

method in a shell-model basis [16, 17, 18]. Any many-body Hamiltonian can be

written in second quantization formalism as a polynomial in creation and annihilation

operators [2]:

H=X

iˆa†

iˆai+1

ijkl

Vijklˆa†

iˆa†

jˆalˆak,(1)

where iare single particle energies and Vijkl are the two-body interaction matrix

elements. The single-particle operators ˆa†

icreate spin-1/2 nucleons in simple harmonic

oscillator states with quantum numbers: ni(radial quantum number), li(orbital

angular momentum), and ji(total angular momentum). Many-body states are

constructed as antisymmetrized products of these single particle states.

To make calculations tractable, we limit the number of single particle valence

states. For example, several of our calculation assume a ﬁxed 16O core and allows

valence nucleons in the 1s1/2-0d3/2-0d5/2orbits, colloquially known as the sd shell;

we also work in the pf-shell (40Ca core with valence orbits 1p1/2,3/2-0f5/2,7/2) and

the combined sd-pf shells. Starting from a ﬁnite single-particle valence space yields

a ﬁnite many-body basis [18]:

|Ψi=X

cα|αi,(2)

where we use the occupation representation of Slater determinants, that is, of the form

|αi=Qiˆa†

i|0i.In particular we work in the M-scheme, which means the total Jzor

Mof all basis states is ﬁxed to the same value. For our calculations here we construct

all possible valence conﬁgurations with ﬁxed M. Furthermore, we factorize the basis

into proton (π) and neutron (ν) components, so that we can write:

|αi=|µπi⊗|σνi.(3)

This enables calculation of the proton-neutron entanglement entropy.

The parameters iand Vijkl in Eq. (1) are input parameters of the Hamiltonian.

For details see the reviews in [16, 17, 18]. For our calculations we used both high-

quality empirical interactions ﬁtted separately in each model space to experimental

spectra, as well as schematic interactions known to capture many features of nuclear

structure, and randomly generated parameters. Using Eq. (1) and the factorized basis

(2) one can compute [18, 19] the matrix elements of the Hamiltonian in the many-body

basis, Hα,β =hα|ˆ

H|βi.Then the time-independent Schr¨odinger equation becomes a

simple matrix eigenvalue problem: H~c =E~c.

3. Entanglement entropy

The entanglement entropy is a fundamental tool in quantum information science [6,

8, 9]. Here we brieﬂy review the development found in those sources.

For a pure quantum state |Ψi, the density operator is ˆρ=|ΨihΨ|; in a basis

{|αi}, i.e., Eq. (2), the density matrix elements are ρα0α=c0

αc∗

α. Because this is

idempotent, ρ2=ρ, and thus has either 0 or 1 as eigenvalues, the von Neumann

entropy, S=−tr(ρlog ρ) vanishes.

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Proton-neutronentanglementinthenuclearshellmodelCalvinW.JohnsonE-mail:cjohnson@sdsu.eduOliverC.GortonSanDiegoStateUniversity,5500CampanileDrive,SanDiego,CA92182-1233Abstract.Wecomputetheproton-neutronentanglemententropyintheinteractingnuclearshellmodelforavarietyofnuclidesandinteractions.Someresults...

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Proton-neutron entanglement in the nuclear shell model Calvin W. Johnson

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