Entanglement entropy and Schmidt number as measures of delocalization of $\alpha$ clusters in one-dimensional nuclear systems

0.3 MB·English

by  Yoshiko Kanada-En'yo

KUNS-2541
Entanglement entropy and Schmidt number as measures of delocalization of α clusters
in one-dimensional nuclear systems
Yoshiko Kanada-En’yo
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
We calculated the von Neumann entanglement entropy and the Schmidt number of one dimen-
tional(1D)clusterstatesandshowedthattheseareusefulmeasurestoestimateentanglementcaused
by delocalization of clusters. We analyze system size dependence of these entanglement measures
inthelinear-chainnαstatesgivenbyTohsaki-Horiuchi-Schuck-R¨opkewavefunctionsfor1Dcluster
gasstates. WeshowthattheSchmidtnumberisanalmostequivalentmeasurestothevonNeumann
entanglement entropy when the delocalization of clusters occurs in the entire system but it shows
differentbehaviorsinapartiallydelocalizedstatecontaininglocalized clustersanddelocalizedones.
5 It means that the R´enyi-2 entanglement entropy, which relates to the Schmidt number, is found
1 to be almost equivalent to the von Neumann entanglement entropy for the full delocalized cluster
0 systembutit is lesssensitive tothepartially delocalized clustersystem than thevonNeumannen-
2 tanglement entropy. Wealso propose anew entanglement measurewhich has ageneralized form of
n theSchmidtnumber. Sensitivityofthesemeasuresofentanglementtothedelocalization ofclusters
a in low-density regions was discussed.
J

7
2 I. INTRODUCTION

]
h Nuclear many-body systems are self-bound systems of four species of Fermions, spin up and down protons and
t neutrons. There, a variety of phenomena arise originating in many-body correlations. One of the remarkable spatial
-
l correlations is ”cluster” which is a subunit composed of strongly correlating nucleons. A typical cluster in nuclear
c
systems is an α cluster, which is a composite particle consisting of four nucleons. If there is no correlation between
u
nucleons in a nucleus, all nucleons behave as independent particles and the nucleus is an uncorrelated state with a
n
[ clear Fermi surface written by a single Slater determinant wave function. However, in reality, correlations between
nucleons areratherstrongand αclusters areoften formedatthe surface inparticularin lightnuclei. Once α clusters
1
are formed, delocalization of clusters occurs to gain the kinetic energy of center of mass motion (c.m.m.) of clusters
v
in some situations. The delocalization of clusters involves many-body correlations beyond a Slater determinant.
9
3 Inrealisticnuclearsystems,a degreeofthe (de)localizationofclustersdepends onthe competition(balance)ofthe
6 kinetic energy and potential energy of clusters and is strongly affected also by the Pauli blocking between nucleons
6 in clusters and a core nucleus. The delocalization limit of α clusters is an α cluster gas state, where all α clusters
0 movealmostfreely likea gas. Suchacluster gashas beenpredictedto appearinthe second0+ state of12C [1,2]. To
.
1 describe cluster states of delocalizedα clusters, a new type of cluster wave function, the so-called“Tohsaki-Horiuchi-
0 Schuck-R¨opke” (THSR) wave function, has been introduced by Tohsaki et al. [2]. The THSR wave function is
5 essentiallybasedonαclustersinacommonGaussianorbithavingarangeofthesystemsize,andsuitabletodescribe
1 generalcluster gas states of n α clusters. Indeed, it has been shown that 8Be(0+) and 12C(0+) can be described well
: 1 2
v by the THSR wave functions of 2α and 3α, respectively. Since the optimized THSR wave functions for these states
i have much larger ranges of the Gaussian orbit of clusters than the cluster size, 8Be(0+) and 12C(0+) are interpreted
X 1 2
as gas-like cluster states of 2α and 3α [2–5].
r The THSR wave function has been extended to apply to 20Ne, and found to be able to describe also 16O+α states
a
in 20Ne [6, 7]. Recently, Suhara et al. have proposed that this concept of the α-cluster gas is applicable also to one
dimension (1D) cluster motion in linear-chain nα structures [8]. They have proposed the 1D-THSR wave functions
andshownthatthe1D-THSRwavefunctionswiththeoptimizedGaussianrangescandescribewelltheexactsolutions
of the linear-chain3α and 4α states. Their result somewhat contradicts to the the conventionalpicture of the linear-
chainnαstructuresthatspatiallylocalizedαclustersarearrangedin1Dwithcertainintervals[9]. Eventhoughtheir
model restricted in 1D is not enough to settle the problem of stability of the linear-chain states in realistic nuclear
systems in 3D, their work provides a new picture of 1D cluster states and may lead to a understanding of cluster
phenomena in nuclear many-body systems. Moreover, the 1D cluster state is an academically interesting problem of
quantum many-Fermion systems.
Todistinguishbetweenlocalizationanddelocalizationofcompositeparticles(clusters)inmicroscopicwavefunctions
ofFermion(nucleon)systems,oneshouldcarefullyconsidertheantisymmetrizationeffectofnucleonsbetweenclusters.
When clusters largely overlap with each other, motion of clusters is strongly affected by the Pauli blocking between
nucleons inother clusters. As a resultofPauliblockingeffect, when the systemsize is as smallasor smallerthan the
cluster size, clusters can not move freely and the system becomes equivalent to a localized cluster state. In the case
of a large system size, where the overlapbetween clusters is small, clusters can move rather freely. It means that the
2
delocalization occurs not in high-density regions but in low-density regions. Indeed, we have investigated the cluster
motion of an α cluster around the 16O and shown that the delocalization of the cluster occurs in a long tail part far
from the core [10]. Also in the 1D nα states, Suhara et al. have shownpeak structures in the density distributions of
the linear-chan 3α and 4α states suggesting partial localization at high density inner regions even in the 1D gases of
3α and 4α.
Toestimatemany-bodycorrelationscausedbythedelocalizationofclustersweneedameasureofcorrelationsthatis
freefromtheantisymmetrizationeffect. Thereareseveralentanglementmeasuresthatareessentiallybasedondensity
matrix and can be used to estimate many-body correlations in quantum many-body systems. The Schmidt number
[11]isoneofthemeasures,andthevonNeumannentanglemententropy[12]isanothermeasure. Theformerhasbeen
proposed by Grobe et al. in 1994 [11], and applied to measure entanglement, i.e., many-body correlations in such
systems as atomic and nuclear systems (see, for example, Refs. [13–15] and references therein). The latter has been
introduced by Bennett et al. in 1996 [12], and widely used in various fields such as condensed matter and quantum
field theory (see Refs. [16–20] and references therein). In the previous paper, we have calculated the entanglement
entropyfortheone-bodydensitymatrixin1Dclusterstatestomeasuretheentanglement(correlations)causedbythe
delocalization of clusters, and found that the delocalization occurs in low-density regions but is relatively suppressed
in high density regions [21].
Let us consider uncorrelated and correlated states of Fermion systems. An uncorrelated state is given by a Slater
determinant wave function, for which the one-body density matrix ρˆ is a projector ρˆ2 = ρˆ in the single-particle
Hilbertspace. Itmeansthattheeigenvalues ρ ofone-bodydensitymatrixsatisfiesρ2 =ρ ,i.e., ρ =1or0. ρ is the
l l l l l
occupationprobabilityofthe single-particlebasisthatdiagonalizesthe one-body densitymatrixanditis nothing but
the Schmidt coefficients in the Schmidt decompositionfor the one-body density matrix. Both ofthe Schmidt number
and entanglemententropy are entanglement measureswhich can estimate how the Schmidt coefficients deviates from
the condition ρ2 = ρ for uncorrelated states. However, these measures have different dependences on the one-body
l l
density matrix, i.e., the eigen values ρ .
l
In this paper, we calculate the entanglement entropy and the Schmidt number of 1D cluster states and make it
clear whether these entanglement measures are useful to investigate the delocalization of clusters. We analyze the
system size dependence of these measures in 1D α-cluster states given by the 1D-THSR wave functions. We show
that the entanglement entropy and the Schmidt number are almost equivalent measures when the delocalization of
clusters occurs in the entire system but they show different behaviors in partially delocalizedcluster states composed
oflocalizedclustersanddelocalizedones. We alsoproposeanew entanglementmeasurewhichhasageneralizedform
of the Schmidt number.
This paperisorganizedasfollows. We describethe entanglementmeasuresdefinedbythe one-bodydensity matrix
inSectionII.SectionIIIdiscussespropertiesofthesemeasuresofidealstatesinatoymodel. InsectionIV,wecalculate
these measures in 1D nuclear systems of α clusters and discuss sensitivity of these measures to the delocalization of
clusters. The paper concludes with a summary in section V.
II. MEASURES OF ENTANGLEMENT
TheentanglemententropyandSchmidtnumberarethemeasuresofentanglementinquantummany-bodysystems,
which are defined by the density matrix. In this section, we describe these measures and also propose a new measure
by extending the Schmidt number. We also define spatial distributions of these entanglement measures.
A. One-body density matrix
In the present paper, we use only the one-body density matrix, which has been often discussed in nuclear systems
[22], though more general density matrices are used to define measures of entanglement. For a wave function Ψ(A)
| i
of an A-particle state, the matrix element of the one-body density in an arbitrary basis is given as
ρ = Ψ(A) c†c Ψ(A) , (1)
pq h | q p| i
where c† and c are the creation and annihilation operators, respectively. The one-body density matrix is regarded
p p
as the matrix element of the one-body density operator ρˆ for the wave function Ψ(A),
Ψ
ρˆ = p ρ q . (2)
Ψ pq
| | i h |
pq
X
3
The one-body density matrix can be diagonalized by a unitary transformation of single-particle basis
(D†ρD)ll′ = ρlδll′, (3)
a†l = Dl′lc†l′, (4)
l′
X
where
ρ = Ψ(A) a†a Ψ(A) , (5)
l h | l l| i
0 ρ 1 (6)
l
≤ ≤
is the eigen value of the one-body density matrix and means the occupation probability of the single-particle state
l in the wave function Ψ(A). ρ corresponds to so-called Schmidt coefficients in the Schmidt decomposition for the
l
one-body density matrix. The trace of the one-body density matrix ρ equals to the particle number A as
A = Trρ= ρ . (7)
l
l
X
Note that the normalization of the one-body density matrix is not a unit but is the particle number A. In the
present paper, the entanglement entropy and the Schmidt number are defined by thus defined one-body density
matrix normalized as Trρ=A.
Ifawavefunction Ψ(A) isaSlaterdeterminant,ρ =1foroccupiedsingle-particlestatesandρ =0forunoccupied
l l
| i
single-particle states. It means that the one-body density operator ρˆ satisfies ρˆ2 = ρˆ and is a projector in the
Ψ Ψ Ψ
single-particle Hilbert space if Ψ(A) is a non-entangled state given by a Slater determinant wave function.
B. Measures of entanglement
1. Entanglement entropy
The vonNeumann entanglemententropy has been introduced by Bennett et al. and provedto be an entanglement
measure in quantum many-body systems (see, for instance, Refs. [15, 18] and references therein). The entanglement
entropy is defined by the von Neumann entropy of one of the reduced density matrices and called “von Neumann
entanglement entropy”, which we call the “entanglement entropy” unless otherwise noted. In the present paper,
we consider the entanglement entropy only for the one-body density matrix of Fermion systems. The entanglement
entropy that is defined by the one-body density matrix is given as,
S = Trρlogρ= ρ logρ . (8)
l l
− −
l
X
The entanglement entropy is zero if a wave function Ψ(A) is a Slater determinant, because ρ = 1 for occupied
l
| i
single-particle states and ρ = 0 for unoccupied single-particle states. It means that a system has non-zero positive
l
value of the entanglement entropy only if the system contains many-body correlations beyond a Slater determinant,
i.e., if the system is entangled.
2. Schmidt number
Another entanglement measure is the so-called Schmidt number, which has been introduced by Grobe et al. and
usedto measuremany-bodycorrelationsinatomicandnuclearphysics[13–15]. TheSchmidtnumberK isdefinedas,
A A
K = = , (9)
ρ2 Trρ2
l l
which estimates the number of states involvedin the SPchmidt decomposition. K equals one if a wavefunction Ψ(A)
| i
is a Slater determinant, and K is greater than one for entangled states. In analogy to the entanglement entropy,
which is generated by the entanglement, it is useful to consider the quantity K 1,
−
A (ρ ρ2)
K 1= 1= l l− l . (10)
− ρ2 − ρ2
l 1 P l l
P P
4
It is clear that K 1 = 0 only for non-entangled states because ρ2 = ρ , i.e., ρˆ2 = ρˆ is satisfied only if a state is
− l l Ψ Ψ
givenbyaSlaterdeterminant. Itmeansthatanon-zeropositivevalueofK 1isgeneratedbyentanglement,thatis,
−
many-body correlations beyond a Slater determinant. One of the merits of the K number is that it is given by Trρ2
which can be calculated by the matrix element ρ in an arbitrary basis without the diagonalizationof the one-body
pq
density matrix.
We should comment that the logarithm of the Schmidt number is nothing but the R´enyi entanglement entropy of
order 2 (R´enyi-2 entanglement entropy) for the one-body density matrix,
logK = SRenyi, (11)
2
1 Trρξ
SRenyi = log . (12)
ξ 1 ξ { A }
−
Itisknownthat,intheξ 1limit,theR´enyientanglemententropybecomesequaltothevonNeumannentanglement
→
entropy. In this paper, we discuss the Schmidt number instead of R´enyi-2 entropy though they are equivalent
entanglement measures.
3. Extension of Schmidt number
Inthepresentpaper,weproposeanentanglementmeasurewhichisregardedasageneralizedversionoftheSchmidt
number. As shownin Eq.(10), the originofnon-zerocontributionsin K 1is partially occupiedsingle-particlestate
with 0 < ρ <1. Ignoring the total scaling factor 1/ ρ2, the contribut−ion W(ρ ) (the weight function) of a single-
l l l l
particle state with the occupation probability 0 < ρ < 1 in K 1 has the ρ dependence, W(ρ ) = ρ ρ2, which is
P − l l l− l
different from the weightfunction W(ρ )= ρ logρ in the entanglemententropy S. The former has relatively small
l l l
−
weights for single-particle states with small occupation probability ρ < 1/2 compared with the latter although the
l
occupied ρ = 1 and unoccupied ρ = 0 single-particle states have no weight in both cases (see the ρ dependence of
l l
the weight functions in Fig. 1). We here introduce an alternative weight function, W(ρ )=ργ(1 ρ ), and define an
l l − l
entanglement measure K by extending the entanglement measure K as follows,
γ
ργ Trργ
K = l l = , (13)
γ ρ(1+γ) Trρ(1+γ)
Pl l
ργ(1 ρ )
K 1 = Pl l − l , (14)
γ − ρ(1+γ)
P l l
where the parameter γ is a positive constant. In thePcase of γ = 1, the K number becomes consistent with the
γ
Schmidt number K. Similarly to the known entanglement measures (S and K 1), K 1 equals to zero only for
γ
− −
a Slater determinant, whereas a non-zero positive value of K 1 is generated by entanglement, that is, many-body
γ
−
correlations beyond a Slater determinant. Note that the diagonalization of the density matrix is required to obtain
the K number for a non-integer γ differently from the K number. In this paper, we choose γ = log2 which gives
γ
the weight function W(ρ ) = ρlog2(1 ρ ) having ρ dependence similar to W(ρ ) = ρ logρ for the entanglement
l l − l l l − l l
entropy in 0.001.ρ 1 (see Fig. 1).
≤
It should be commented that logK is given by the R´enyi entanglement entropy of order γ and 1+γ as,
γ
logK = (1 γ)SRenyi+γSRenyi. (15)
γ − γ 1+γ
C. Spatial distributions of entanglement measures
To investigate the spatial distribution of the ”important single-particle states” that contribute to the non-zero
entanglement entropy, we have defined the spatial distribution s(r) of the entanglement entropy, which have been
introduced in the previous paper [21] as,
S = ( ρ logρ )= s(r)dr, (16)
l l
−
l Z
X
s(r) = ( ρ logρ )φ∗(r)φ (r). (17)
− l l l l
l
X
5
0.4
ρW() 0.2
-ρlogρ
ρ(1-ρ)
ρlog2(1-ρ)
0
0 0.5 1
ρ
1
0.1
0.01
ρW() 0.001
0.0001 -ρlogρ
1e-05 ρ(1-ρ)
ρlog2(1-ρ)
1e-06
1e-06 0.001 1
ρ
FIG. 1: Weight functions W(ρ)=−ρlogρ,ρ(1−ρ),and ρlog2(1−ρ) for S, K, and K , respectively.
log2
Here the factor ρ logρ is contribution of the single-particle state l in S, and φ∗(r)φ (r) means the density
distribution in l−alnd itlis normalized as φ∗(r)φ (r)dr = 1. The|reifore, the distrlibutilon s(r) reflects spatial
| i l l
distributions of the important single-particle states l that contribute to the total entanglement entropy, whereas it
is hardly affected by almost occupied single-pRarticle|stiates having ρ 1. The expression of S with s(r) is analogous
l
≈
to that of the particle number with the density distribution,
A = ρ = ρ(r)dr, (18)
l
l Z
X
ρ(r) = ρ φ∗(r)φ (r). (19)
l l l
l
X
Note that the distribution s(r) is not quantity determined only by local information at the position r. It is different
from the density distribution ρ(r) which is determined only by the local information.
We also define the distributions κ(r) and κ (r) for K 1 and K 1, respectively, to see spatial distributions of
γ γ
− −
the ”important single-particle states” in total amount of the measures,
(ρ ρ2)
K 1 = l l− l
− ρ2
P l l
= κP(r)dr, (20)
Z
1
κ(r) = (ρ ρ2)φ∗(r)φ (r)
′ρ′2 l− l l l
l l l
X
K
= P(ρ ρ2)φ∗(r)φ (r), (21)
A l− l l l
l
X
and
ργ(1 ρ )
K 1 = l l − l
γ − ρ1+γ
P l l
= κP(r)dr, (22)
γ
Z
1
κ (r) = ργ(1 ρ )φ∗(r)φ (r). (23)
γ ρ1+γ l − l l l
l l′ l′
X
P
6
Similarlytos(r),thesedistributions,κ(r)andκ (r),showspatialdistributionsoftheimportantsingle-particlestates
γ
l that contribute to the total amount of K 1 and K 1, respectively. Note that these distributions are not local
γ
| i − −
quantities.
III. S, K, AND K OF IDEAL STATES IN A TOY MODEL
γ
In this section, we discuss behaviors of S, K, and K for correlated (entangled) states composed of delocalized
γ
clusters in a toy model. Here, γ in the K number is assumed to be 0<γ <1.
γ
Letus considern speciesofparticles. Forinstance,n =2forspinupanddownFermions,andn =4forspinup
f f f
and down protons and neutrons. We use the label σ for species of Fermions such as σ =p , p , n , n for nuclear
↑ ↓ ↑ ↓
systems. We consider an A-body system containing the same number n = A/n of σ particles. We assume that the
f
systemis symmetricforthe exchangeofspecies,thatis,single-particleorbitalsareoccupiedbyallspeciesofparticles
withanequalweight. Thenthedensitymatrixisdiagonalwithrespecttoσ,andallspeciesofparticleshavethesame
occupation probability ρ = ρ independent from σ. We consider the one-body density matrix and operator in the
σl l
reduced space with the dimension n and define the entanglement measures, S, K, and K , for a species of particles
γ
by using the reduced matrix of the one-body density. The total entanglement entropy S and the total numbers
total
K and K aregivenby the measuresfor eachspecies as S =n S, K =K, andK =K . In this
total γ,total total f total γ,total γ
paper, we discuss S, K, and K for a species of particles.
γ
For simplicity, A particles are assumed to stay on sites in a space. The number of available sites (single-particle
states) is m and we use the label k for the jth site. Let us first consider a system of A=n particles. If an A-body
j f
stateisanidealstateofindependentparticles,thewavefunctioncanbewrittenbyasimpleproductofsingle-particle
wave functions
Ψ(1,2,...,n )=ψ(1)ψ(2) ψ(n ), (24)
f f
···
ψ(i)= c(k)φ (i). (25)
k
k=k1X,k2,...,km
Forinstance,c(k)isconstantasc(k)=1/√mforafreegasstate. Forsuchannon-entangledstate, S =0, K 1=0,
and K 1 = 0 because the one-body density operator is given as ρˆ = ψ ψ and satisfies ρˆ2 = ρˆ . A−nother
γ − Ψ | ih | Ψ Ψ
exampleisa“localizedcluster”systemofacluster,whereallparticlesarelocalizedatonesite k toformacomposite
j
particle (a cluster) at k . The wave function is given as
j
nf
Ψ(1,2,...,n )= φ (i). (26)
f kj
i=1
Y
This wave function for a localized clusters also has zero measures, S = 0, K 1 = 0, and K 1 = 0, because the
γ
− −
one-bodydensity operatoris givenasρˆ = φ φ andsatisfiesagainρˆ2 =ρˆ . Namely,the localizedcluster wave
Ψ | kjih kj| Ψ Ψ
function is a non-entangled state.
We consider a state of a delocalized cluster in a strong correlationlimit,
1 nf nf nf
Ψ(1,2,...,n )= φ (i)+ φ (i)+ φ (i) , (27)
f √m( k1 k2 ··· km )
i=1 i=1 i=1
Y Y Y
where the composite particle composed ofA=n particles moves freely in the entire space with an equalprobability
f
1. This is a highly entangled (strongly correlated) state, where, if a particle is observed at a certain site, all other
m
particlesarealwaysobservedatthe samesite. This isastrongcouplinglimit andanexampleofadelocalizedcluster.
The one-body density operator,
m
1
ρˆ = k k , (28)
Ψ j j
m| ih |
j=1
X
corresponds to the Schmidt decomposition with the common Schmidt coefficients, 1/m. We get S = logm, K = m,
and K = m. In general, if eigen values of ρ are constant ρ = 1/m (l = 1,...,m ) for a given number (m ) of
γ l l V V V
states and ρ =0 for l>m , we get
l V
S = logm , eS =m , (29)
V V
K = m , (30)
V
K = m . (31)
γ V
7
It indicates that eS, K, and K equal to the number m of states involved in the Schmidt decomposition. m is
γ V V
regarded as the effective volume size. Note that, for a Slater determinant, eS, K, and K equal to 1 indicating that
γ
the effective volume size is one which can not be decomposed.
NextweconsideranA-particlesystemofn=A/n clusters. Heren isthe number ofclusters(composite particles)
f
formed by n constituent Fermions. For a state of n localized clusters at n sites k = k ,...,k , the wave function
f j1 jn
is given as,
n−1
Ψ(1,2,...,A) = (n!)−nf/2 φ (hn +1) φ (hn +n )
A( kj1 f ··· kj1 f f )
h=0
Y
= (n!)−nf/2 φ (1) φ (n ) φ (A n +1) φ (A) . (32)
A kjh ··· kj1 f ······ kjn − f ··· kjn
n o
Since the occupation probability is ρ = 1 for occupied single-particle states k ,...,k and ρ = 0 for unoccupied
l j1 jn l
single-particle states, the system has the zero value of the measures, S =0, K 1=0, and K 1=0.
γ
− −
Let us consider a state of n clusters in the delocalized limit where all clusters are delocalized and move freely in
a volume size m like a gas. For this state of delocalized clusters, the occupation probability is ρ = n/m for
V l V
l = 1,...,m . m should not be less than n because of the Pauli principle of n Fermions. The measures of this
V V
delocalized cluster system are,
m m
S = nlog V , eS/n = V , (33)
n n
m
V
K = , (34)
n
m
V
K = . (35)
γ
n
It indicates that K and K are consistent with eS/n. neS/n, nK, and nK equal to the number m of the states
γ γ V
involved in the Schmidt decomposition and they estimate the effective volume size of the delocalization of clusters.
For the case of m = n = A/n , all m single-particle states are completely occupied by A particles and clusters
V f V
can not move at all. The state is equivalent to the localized cluster, and it has zero value, S = 0, K 1 = 0, and
−
K 1 = 0, of entanglement measures. In case m is larger than n, the measures, S, K 1, and K 1, become
γ V γ
− − −
positive indicating that the delocalization of clusters occurs and the system becomes an entangled state. In the case
thatm is muchlargerthann,the systemcorrespondsto a low-densitycluster gasanditis ahighly entangledstate.
V
Finally, we consider the case of a partial delocalization that n 1 clusters are localized to form a core and only
−
the last cluster is delocalized. This situationcorrespondsto an α cluster movingalmost freely arounda corenucleus.
In this partially localized case of a delocalized cluster around a core, the occupation probability is ρ = n/m for
l V
l =1,...,m and ρ =1 for the n 1 single-particle states occupied by constituent particles of the core. Then, the
V l
−
measures of this partially delocalized system are
S = logm , eS/n =m1/n, (36)
V V
nm
V
K = , (37)
(n 1)m +1
V
−
(n 1)mγ +m
K = − V V . (38)
γ (n 1)mγ +1
− V
In the low-density limit of the large m , we get
V
n
K , (39)
→ n 1
−
K m1−γ. (40)
γ → V
Let us compare the fully delocalized cluster state and the partially delocalized cluster state in the large m limit
V
(the large volume size limit). The former state corresponds to a dilute cluster gas, where all clusters are delocalized
moving freely like a gas, and is a highly entangled state. For the fully delocalized cluster state, all the entanglement
measures are sensitive to m as given in Eqs. (33), (34), and (35). That is, eS/n, K, and K equal to m /n and
V γ V
equivalently good indicators to measure the delocalization of clusters. However, for the latter case of the partial
delocalization, which corresponds to a delocalized cluster around a core, eS/n, K, and K are not equivalent but
γ
show different dependences on m . As clearly shown in Eqs. (36), (39), and (40), eS/n and K increases as the
V γ
m increases, but K becomes constant and does not depend on m in the large m limit. It indicates that S and
V V V
8
K can be useful measures sensitive to the partial delocalization in subsystem, but K is insensitive to the partial
γ
delocalization. The reason for the insensitivity of K is the significant contribution from fully occupied single-particle
states with ρ = 1 in the denominator of the definition of the K number, which makes the contribution from the
l
delocalized part minor. From another point of view, it is found that K can be a good probe to clarify whether the
delocalization of clusters occurs in the entire system or not. m dependences of eS/n and K are the powers of 1/n
V γ
and1 γ,respectively. Moregenerally,forthepartiallydelocalizedsystemthatiscomposedofn delocalizedclusters
g
−
andn localizedclusters,them dependencesofeS/n andK inthelargem limitarethepowersofn /nand1 γ,
0 V γ V g
−
respectively. Here, n and n are the numbers of delocalized and localized clusters, respectively, and n +n = n.
g 0 0 g
This means that, in the case n /n > 1 γ of a small fraction of delocalized clusters, K is more sensitive to the
g γ
−
delocalization of clusters than eS/n, whereas, in the case of n /n 1 γ, the K number has the m dependence
g γ V
≈ −
similar to eS/n.
IV. APPLICATION TO 1D NUCLEAR SYSTEMS OF α CLUSTERS
In the present paper, we use the delocalized cluster wave functions in 1D for the linear-chain nα states which are
investigatedin the previous paper [21]. We also adopt the α+(2α) wave functions for a state of an α cluster around
a 2α core. We analyze the entanglementmeasures, S, K, andK , and also their spatialdistributions, and discuss
log2
the system size dependence of these measures.
A. Localized and delocalized α cluster wave functions in 1D
We herebriefly explainthe adoptedmodel wavefunctions for(de)localizedcluster states in1D.More details ofthe
modelwavefunctionsaredescribedinthepreviouspaper[21]. Weconsiderintrinsicwavefunctionsofthelinear-chain
nα-clusterstatesalignedtothexaxis. Itmeansthatthe(de)localizationofαclustersaredefinedforα-clustermotion
along the x axis.
For a localized nα-cluster wave function, we use the BB wave function [23] as,
1
Φnα(R ,...,R ) = ψα ψα , (41)
BB 1 n √A!A R1··· Rn
ψα = φ0sχ (cid:2)φ0sχ φ0sχ(cid:3) φ0sχ , (42)
R R p↑ R p↓ R n↑ R n↓
i i i i i
1
φ0s = (πb2)−3/4exp (r R )2 . (43)
−2b2 − i
(cid:20) (cid:21)
ψα is the four-nucleonwavefunction ofthe ith α cluster expressedby the (0s)4 harmonic oscillator(ho)shell-model
R
i
configuration localized around the spatial position R . χ is the spin-isospin part and φ0s is the spatial part of the
i
single-particlewavefunction. For1Dclusterstates,the positionparameterR issettobeR =(R ,0,0),andthe1D
i i i
BB wave function is expressed as Φnα(R ,...,R ). The parameter b for the α-cluster size is chosen to be b=1.376
BB 1 n
fm same as in Ref. [8]. Note that a single BB wave function is a localized cluster wave function written by a Slater
determinant of single-particle wave functions and it has exactly zero values of the measures, S = 0, K 1 = 0, and
−
K 1 = 0. General wave functions for 1D nα systems can be written by linear combination of BB wave functions
γ
Φnα−(R ,...,R ).
BB 1 n
For a delocalized cluster wave function, we use the 1D-THSR wave functions of nα, which have been proposed by
Suhara et al. to describe the linear-chain 3α- and 4α-cluster states in 12C and 16O systems [8]. The 1D-THSR wave
functions are given by linear combination of BB wave functions with a Gaussian weight as,
n R2
Φnα (β)= dR dR exp i Φnα(R ,...,R ). (44)
1D-THSR 1··· n (− β2) BB 1 n
Z i=1
X
Iftheantisymmetrizationisignored,theΦnα (β)expressesthenαstatewhereallαclustersareconfinedinthey
1D-THSR
and z directions while they move in the x direction in the Gaussian orbitwith the range parameter β. β corresponds
to the system size of the 1D nα state. In case of the system size β is as small as or smaller than the α-cluster size
b, the 1D-THSR wave function is approximately equivalent to a localized cluster wave function given by a Slater
determinant because of the antisymmetrization effect. As β increases, the delocalization of α clusters occurs. When
β is large enough comparedwith the α-cluster size b, the system goes to a dilute 1D α-cluster gas where n α clusters
move almost freely like a gas in the x direction.
9
Inthepracticalcalculation,theR integrationisapproximatedbysummationonmeshpointsinafiniteboxasdone
i
in the previous paper. For 2α, 3α, and 4α systems, we make a correction of the total c.m.m. to eliminate a possible
artifactfromβ dependence in the totalc.m.m. as describedin the previous paper. It means that the 1D-THSR wave
function of 1α without the c.m.m. correction expresses a system of an α cluster bound in an external field, which is
not a realistic state of an isolate nucleus, whereas those of 2α, 3α, and 4α with the c.m.m. correction correspond to
self bound nα systems with linear-chain structures predicted in nuclear states such as excited states of 12C and 16O.
In this paper, we discuss the entanglement measuresfor 1α with no c.m.m. correction(nc) and those for 2α, 3α, and
4α with the c.m.m. correlation. We also show the enganlement entropy for 2α with no c.m.m. correction, just for
comparison.
We also use the 1D-THSR wave function of α+(2α) for the case of an α cluster around the 2α core,
R2
Φα-(2α) (β) = dR exp 1 Φ3α(R ,R =+ε,R = ε), (45)
1D-THSR 1 −β2 BB 1 2 3 −
Z (cid:26) (cid:27)
where the 2α core is located at the origin and an α cluster is distributed around the core with a Gaussian weight.
Whenβ islargeenoughcomparedwiththeα-clustersizeb,thewavefunctiondescribesadelocalizedαclusteraround
the 2α core at the origin. This wave function is associatedwith the partially delocalized cluster wave function in the
toy model discussed previously. We use a small value of ε=0.02 fm to describe the 2α core almost equivalent to the
h.o. (0s)4(0p )4 configuration. This wave function has been originally introduced in previous paper to describe the
x
α+16O cluster states in 20Ne.
We calculate numerically the one-body density matrices for these 1D cluster wave functions, and obtain the mea-
sures,S,K,andK fromtheeigenvaluesρ oftheone-bodydensitymatrices. Thedetailedmethodofthepractical
log2 l
calculationisdescribedinthepreviouspaper. Spatialdistributions,s(r),κ(r),andκ (r)andthedensitydistribution
γ
ρ(r) are integrated out along y and z axes, and we get distributions, s(x), κ(x), κ (x), and ρ(x), projected onto the
γ
x axis.
B. nα cluster states
We analyze S, K, and K of the 1D nα states written by the 1D-THSR wave functions, Φnα (β). Figure
log2 1D-THSR
2 shows the system size dependence of the entanglement entropy of the 1α state with no c.m.m. correction (1α ),
nc
and 2α, 3α, and 4α states with the c.m.m. correction,as well as that of a 2α state with no c.m.m. correction(2α ),
nc
S/n, eS/n, and n(eS/n 1) are plotted as functions of the dimensionless system size β/b. In the β/b = 0 limit, the
−
entanglement entropy equals zero, because the 1D-THSR wave functions are equivalent to localized nα cluster wave
functions in this limit. As the β/b increases, the delocalization of clusters occurs and the entanglement entropy is
generated. eS/n for the 1α state increases almost linearly to β/b as expected from a naive picture that a cluster
nc
moves freely in a finite volume, which is decomposed into m β/b states by the quantum decoherence in the one-
V
∼
bodydensitymatrix. Similarlytothe1α state,S for2α,3α,and4αstatesincreaseswiththeincreaseofthesystem
nc
size, indicating that the entanglement entropy is generated as the delocalization of clusters is enhanced. eS/n for 2α,
3α, and 4α states also shows almost linear dependence to β/b. The n(eS/n 1) plots in Figs. 2(c) and 2(d) show a
−
goodcorrespondenceof the β dependence ofthe entanglemententropybetween 1α and 2α , and thatbetween 2α,
nc nc
3α, and 4α states. The values of n(eS/n 1) for 2α, 3α, and 4α are relatively smaller compared with those for 1α
nc
−
and 2α because of the c.m.m correction performed for the 2α, 3α, and 4α states.
nc
Let us compare other measures, K and K , with the entanglement entropy. Figure 3 shows the system size
log2
dependence of eS/n, K, and K of 1α , 2α, 3α, and 4α states. K and K shows the β/b dependence quite
log2 nc log2
similar to that of eS/n except for global normalization factors. This result indicates that the entanglement entropy,
K, and K , can be approximately equivalent measures for the cluster delocalization of nα states. It is consistent
log2
with the naive expectation from the analysis of the delocalized cluster states in the toy model discussed in the
previous section. It means that neS/n, nK, and nK estimate the number of the states involved in the Schmidt
log2
decomposition as shown in Eqs. (33), (34), and (35).
We also compare the spatial distributions of these measures, s(x), κ(x), and κ (x) of 1α in Fig. 4. The
log2 nc
densitydistributionisalsoshown. Asdescribedpreviously,s(x), κ(x), andκ (x)reflectspatialdistributionsofthe
log2
important single-particle states that give non-zero contributions to total measures S, K 1, K 1, respectively.
log2
− −
Note that the shape, in particular, the spatial broadness of distributions is of importance. but their global scaling
(normalization)isnotsomeaningful. Itisfoundthats(x)andκ (x)showquitesimilardistributionstoeachother.
log2
TheyaremorebroadlydistributedthanthedensitydistributionindicatingthatS andK 1aregeneratedinlow-
log2
−
densityregionsbutrelativelysuppressedinhigh-densityregions. Differentlyfroms(x)andκ (x), theenhancement
log2
in low-densityregionsand the suppressionin high-density regionsarerelatively weakin κ(x). As a result, the spatial
10
extentofκ(x)isnotasremarkableasthatofs(x)andκ (x). ThisdifferenceinthespatialdistributionsofS,K 1,
log2
−
K 1 comes from the different weight functions W(ρ), namely, S and K 1 are relatively sensitive to single-
log2 log2
− −
particle states with low occupation probability compared with K 1 as already shown in Fig. 1. The distributions
−
s(x), κ(x), and κ (x) for the 2α, 3α, and 4αstates areshownin Fig. 5 as wellas density distribution. Similarly to
log2
the 1α state, s(x) andκ (x) aremore broadlydistributed than the density distributionand alsoslightly broader
nc log2
than κ(x), indicating againthat S and K 1 are generatedin low-density regions but suppressed in high-density
log2
−
regions.
Inthepresentresult,wefindthattheS,K andK canbeusefulmeasurestoestimatetheentanglementcausedby
log2
thedelocalizationofclustersinthe1Dnαstatesgivenbythe1D-THSRwavefunctions. Asthesystemsizeincreases,
thedelocalizationofclustersdevelopsandnon-zeroS,K 1,andK 1aregenerated. Inthespatialdistributions
log2
− −
of these entanglement measures, significant contributions come from low-density regions than high-density regions.
Quantitatively, the Schmidt number (K) is less sensitive to low-occupation probability single-particle states than
the entanglement entropy (S) and the K number resulting in the less broad distribution of κ(x) than s(x) and
log2
κ (x), which are remarkably broader than the density distribution.
log2
1 .25 12ααnncc 00..68 234ααα
S/n 1 (a) S/n 0.4 (d)
0.5 0.2
0 0
0 1 2 3 4 5 0 1 2 3 4 5
β/b β/b
4 .455 12ααnncc 1 .28 23αα
S/ne 23 ..355 (b) S/ne 11..46 4α(e)
2
1.5 1.2
1 1
0 1 2 3 4 5 0 1 2 3 4 5
β/b β/b
S/nn(e-1) 123 ...1234555 12αα(nnccc) S/nn(e-1) 012 ...12555 234ααα(f)
0.5
0 0
0 1 2 3 4 5 0 1 2 3 4 5
β/b β/b
FIG.2: Systemsizeβ/bdependenceoftheentanglemententropyS andeS/ninthe1D-THSRwavefunctionsofnα. n(eS/n−1)
is also shown. The 1D-THSR wave functions with the total c.m.m. correction are used for 2α, 3α, and 4α systems, and that
with no c.m.m. correction is used for the 1α system (1αnc). The results of the 1D-THSR wave function with no c.m.m.
correction for 2α system (2αnc) are also shown for comparison.
C. an α cluster around a 2α core
WeinvestigateS,K,andK ofα+(2α)clusterwavefunctions. Weusethe1D-THSRwavefunctionΦα-(2α) (β)
log2 1D-THSR
in Eq. (45) for the α+(2α) states associated with the partially delocalized cluster wave function. We also adopt the
parity projected BB wave function with the fixed α-(2α) distance d as used in the previous paper as,
Φα-(2α),+(d) = (1+Pˆ )Φ3α(R =d,R =+ε,R = ε)
BB r BB 1 2 3 −
= Φ3α(R =d,R =+ε,R = ε)
BB 1 2 3 −
+Φ3α(R = d,R =+ε,R = ε), (46)
BB 1 − 2 3 −
where Pˆ is the parity transformation operator. Note that Φα-(2α),+(d) is given by the linear combination of two
r BB
Slater determinants. This corresponds to the symmetry restored state where the parity symmetry is broken in the
intrinsic state because of the cluster development.
Figure 8 shows S, K, and K as well as eS/n for the 1D THSR wave function and the parity-projectedBB wave
log2
function of the α+(2α) system. Looking at the result of the parity-projected BB wave function, we find that, as d