Pinning effect and QPT-like behavior for two particles confined by a core-shell potential

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by  Pier Paolo Marchisio

Pinning effect andQPT-likebehaviorfortwo particles confined by a core-shell potential
P.P. Marchisio,1,∗ J.P. Coe,1,2,† and I. D’Amico1,‡
1 Department of Physics, University of York, York YO10 5DD, United Kingdom.
2DepartmentofChemistry,SchoolofEngineeringandPhysicalSciences,Heriot-WattUniversity,Edinburgh,EH144AS,UK
(Dated:today)
Westudythegroundstateentanglement, energyandfidelitiesofatwo-electronsystemboundedbyacore-
shell potential, where the core widthis varied continuously until it eventually vanishes. This simple system
displaysarichandcomplexbehavior: asthecorewidthisvaried,thissystemischaracterizedbytwopeculiar
transitionswhere,fordifferentreasons,itdisplayscharacteristicssimilartoafew-particlequantumphasetran-
2
sition.Thefirstoccurrencecorrespondstosomethingakintoasecondorderquantumphasetransition,whilethe
1
secondtransitionismarkedbyadiscontinuity,withrespecttothedrivingparameter,inthefirstderivativesof
0
quantitieslikeenergyandentanglement. Thestudyofthissystemallowstoshedlightonthesuddenvariation
2
ofentanglementandenergyobservedinRef.1. Wealsocomparethecore-shellsystemwithasystemwherea

n corewellisabsent: thisshowsthat,evenwhenextremelynarrow,thecorewellhasarelevant‘pinning’effect.
a Interestingly,dependingonthepotentialsymmetry,thepinningofthewavefunctionmayeitherhalveordouble
J
thesystementanglement (withrespect totheno-core-wellsystem) whentheground stateisalreadybounded

7 totheouter(shell)well. Intheprocesswediscussthesystemfidelityandshowtheusefulnessofconsidering
1 theparticledensityfidelityasopposedtothemorecommonlyused–butmuchmoredifficulttoaccess–wave-
functionfidelity.Inparticularwedemonstratethat–forground-stateswithnodelessspatialwavefunctions–the

] particledensityfidelityiszeroifandonlyifthewavefunctionfidelityiszero.
r
e
h
t I. INTRODUCTION was regarded as something potentially akin to a QPT but in
o
thefew-particlecase.
.
t
a In order to understand this phenomenon, in this paper we
The realization of the importance of entanglement trig-
m
will study systems related to Ref. 1 and characterized by
gered a rethink in the way one can understand and quantify
- some quantumprocesses. Indeed, quantuminformationthe- rectangular-like confining potential. We will focus on how
d
ground-stateentanglement,energy,andfidelitiesareaffected
n ory(QIT)hasstemmedfromtheapplicationofentanglement
byvaryingthepotentialcorewidthandshowthatthesesimple
o andthesuperpositionprincipletotheprocessingandtransmis-
systemsencompassindeedarichandcomplexbehavior. The
c sionofdata,[2]anditisnowacknowledgedthatentanglement
[ system we will mainly concentrate on is given by two elec-
canplayacentralroleinthedescriptionandunderstandingof

1 quantumphase transitions(QPTs).[3–5] In QIT and QPTs it tronstrappedwithinacore-shellpotential,whosecorereduces
inwidthuntiliteventuallydisappears(seeFig.1). Thismay
v isimportanttodeterminehowaquantumstatechangesunder
1 quantumoperationsorbyvaryingexternalparameters.Thefi- representa(core-shell)quantumdotwithanexternally-driven
4 delity[2,6]–extensivelyusedinQITtoassessthe‘closeness’ confiningpotential:quantumdotsareoneofthemostpromis-
5 inghardwareforthephysicalrealizationofQITdevices,[15–
ofdifferentquantumstates–maynaturallyencompasstheef-
3 23] hence, our findings may be of interest for QIT applica-
fect of a driving parameter on a system, and, as such, it has
1. beenproposedasakeytoolinunderstandingQPTs.[7–9]En- tions. The system groundstate is initially boundto the core
0 well, but will become bound to the outer well (or shell) as
tanglementandfidelitycanthenprovideacommonlanguage
2 the core width is reduced to zero and the outer well width
forQITandQPTs.[10–12]ThedefinitionofaQPThasbeen
1 increases. We willshow thatthe correspondingsharp entan-
widenedbysome authorsto includechangesinthe quantum
:
v stateoffew-particlesystemssuchassinglet-triplettransitions glementvariationischaracterizedbytwoverydifferenttransi-
i tions. Thefirstpresentselementsakintoasecond-orderQPT
X in a single quantum dot.[13] Few-particle systems have also
andisassociatedwiththetransitionofthegroundstate from
beenusedtocharacterizethepredictivepowerofQPTindica-
r thecoretotheouterwell;thesecondismarkedbyadisconti-
a tors for a system undergoinga QPT in the thermodynamical
nuityintheenergyandentanglementderivativeswithrespect
limit.[14]
tothedrivingparameter,andwedemonstratethatitisdueto
Inpreviouswork,[1]itwasshownthatthetransitionfrom
the peculiarities of the confining potential. We will also ex-
acore-shelltoadoublewellpotentialinducesasuddenvaria-
plicitlydiscusstheimplicationsofthesefindingsforthesys-
tionofboththeentanglementandtheenergyoftwoelectrons
temdescribedinRef.1.
initiallyconfinedwithinthecorewell.Thisvariationbecomes
Ouranalysisisimportantinthecontextoflocalsensitivity
sharper as the confiningpotentialbecomesharder, i.e. more
analysis.Inparticular,duetothepivotalrolethattheentangle-
similar to a rectangular-like potentials. This steep variation
mentplaysinseveral[24]quantumprotocols(suchasquantum
algorithms[25],quantumteleportation[26]andsomequantum
cryptography protocols[27]) here we report on the sensitiv-
ityoftheentanglementwithrespecttosmallvariationsofthe
∗Electronicaddress:[email protected]
†Electronicaddress:[email protected] externalparameter,[28,29]characterizetheregionofthepa-
‡Electronicaddress:[email protected] rameterspaceoverwhichtheentanglementshowsthesteepest
2
variation and, consequently, ascertain the possibility of em-
ployingthe potentialvariationsasentanglement‘switch’. In
fact ourcalculationsshow that the presenceof an innercore 2V for Wiw >|x|
0 2
hasastrongpinningeffectontheentanglementevenwhenthe
VDIW(x;R<w)=V for (cid:12)Wow(cid:12) >|x|≥ Wiw
gogsyrnleosmttuhenmeednssty)s(tswastytehemmeinmsgceeaotolmrrmiecpaeadstyrryeysd,tbeiottmoum)nthadoeyrtcoidonortfurhaebecslpteooeuiintttsedhrievnrwaglheusaleyll.vs(teaeDsmtyhempewemeintnhdettoairnnuigc-t and 00 othe(cid:12)(cid:12)(cid:12)(cid:12)rw2ise(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) 2 (cid:12)(cid:12)(cid:12)(4)
corewell. Thismightpotentiallybeexploitedtoinducesharp
V for Wow >|x|
variations (switch) of the entanglement by modifying small V (x;R≥w)= 0 2 (5)
DIW 0 otherwise.
regionsoftheconfiningpotential. (cid:26) (cid:12) (cid:12)
(cid:12) (cid:12)
Finally, in the spirit of density-functional theory,[30] we
V hasacompactrepresentationthroughtheHeaviside
DIW
will study whether the particle density can be used to track
stepfunction,
the system’s ground state behavior via a particle-density fi-
delity. We note that, from an experimental point of view,
thedensityisamoreaccessiblequantitythanthefullsystem V (x;R)=Viw(x;R)+Vow(x;R), (6)
DIW
wavefunction;ourresultsshowthat,atleastforthesystemat
hand,theparticle-densityfidelitydeliversinformationsimilar where
tothewavefunctionfidelity. Importantlywewilldemonstrate
that, for ground-states with nodeless spatial wavefunctions, Viw(x;R)≡V [θ(x+(w−R)/2)θ(−x+(w−R)/2)]
0
theparticle-densityfidelityiszeroifandonlyifthewavefunc- (7)
tionfidelityiszero. and Vow(x;R) = Viw(x;−R) describe the inner and the
outerwell, respectively. Eq.(6) isequivalentto Eqs.(4)and
(5)ifweassignθ(0)=0. Thisisconsistentwithconsidering
theHeavisidestepfunctionθ(x)as,forexample,thelimit(in
adistributionsense[31])forp→∞of
II. SYMMETRICPOTENTIAL,MODELSYSTEMS
1−e−(px)2
Wewillfirstconcentrateonsystemswithasymmetriccon- θp(x)= 1+e−mpx , (8)
finingpotential(seeFig.1).
We considerthree one-dimensionalsystems, each consist- where p and m are positive integers. With p ∼ 10 and
ingoftwointeractingelectronsboundbyaconfiningpotential m ∼ 20, we get a smooth, ‘softer’ version of VDIW. As
andwhoseHamiltonianineffectiveatomicunitsis p → ∞ arguments similar to the ones developed in Ref. 1
seem to suggest a discontinuity in the entanglemententropy
2 1 ∂2 andenergyderivatives(andhencesomethingreminiscentofa
H = −2∂x2 +Vi(xj,R) +U(x1,x2). (1) QPTinthefew-particleregime). Thechosenparametrization
j=1" j # forthepotentialwillhelpustobetterunderstandthislimit.
X
Here we set U(x ,x ) = δ(x − x ) to represent a con-
1 2 1 2
tact Coulomb repulsion between the electrons. V (x ,R)
i j
are the confiningpotentialscharacterizingthe three systems, B. Benchmarksystem
i=DIW,OWOandDW,seebelow.
Theconfiningpotentialofthe‘outerwellonly’(OWO)sys-
temisgivenbyV ≡Vow,seeinsetofFig.1. Weusethis
OWO
systemasabenchmark.
A. Systemwitha‘disappearing’innerwell
The potential of the ‘disappearing’inner well (DIW) sys-
C. Core-shelltodoublewellsystem
tem, V (x;R), ischaracterizedbyaninner(core)andan
DIW
outershellwell,seeFig.1. AstheparameterRincreases,the
innerwellwidth,Wiw,becomesnarrowerandtheouterwell This is the rectangular-likelimit of the system considered
width,Wow,increasesas inRef.1. Asthedrivingparameterchanges,thispotentialis
modifiedfromacore-shelltoadouble-wellpotential.Theex-
w−R forR<w plicitexpressionofthispotentialintherectangular-likelimit
Wiw(R)= (2)
0, forR≥w canbewrittenas
(cid:26)
Wow(R)=w+R. (3)
V (x;R)=V θ(−x+(w−R)/2)θ(x+(3w−R)/2)
DW 0
TakingV0 asthedepthoftheouterwell,wecanwrite +θ(cid:2)(−x+(3w−R)/2)θ(x+(w−R)/2) .(9)
(cid:3)
3
HerethetransformationsR =2w−Randd=w/2give we will refer to the parameter region around R as the ‘mi-
DW c
the control parameter and the inter-well distance as used in grationregion’: forthesevaluesofthedrivingparameterthe
Ref.1,respectively. system wavefunction is the most sensitive to driving param-
Forthesubsequentcalculations,unlessotherwisestated,we eter changes. Here the electronwavefunction‘expands’into
use w = 5 a , where a is the Bohr radius, and V = −10 theouterwelland,asaconsequenceofthis,thesystemshows
0 0 0
Hartree. themostinterestingbehavior. Thisregionofhighsensitivity
isrelativelynarrowandinfactforR > 5a thegroundstate
0
0 energybecomesaveryslowly-varying,decreasingfunctionof
Wow R.
e) −5 |V0 | Thefirstderivativeofthegroundstateenergywithrespect
Hartre−10 Wiw taot Rthe=dr5ivain0,gbpuatraitmisetsemr,odoEth0/edlsRew, hdeisrpela(syeseaFidgi.sc2oBn)t.inTuhitiys
V (DIW 0 discontinuity is found in the first derivatives with respect to
−15 −10 |V0 | R of all the quantities we consider. dE0/dR has a maxi-
mum at R = 4.7a . From Fig. 2B (inset and main panel)
−20 0
−20 −8 −4 0 4 8 we see that at first the shrinking of the inner well increases
−8 −6 −4 −2 0 2 4 6 8 thegroundstateenergywithanincreasing“speed”.However,
x(a) inthemigrationregionthechangeintheground-stateenergy
0
rapidlyslowsdown:inthisregionthewavefunctionisstarting
FIG.1:PotentialVDIW versusxforR=4a0.Inset:sameasmain to spread into the largerouterwell, hence movingtowardsa
panelbutforR=5a0,forwhichDIWandOWOsystemscoincide. regimewhereE isalmostconstantwithR.
0
The second derivative of E with respect to R displays a
0
markedminimumatR=4.90a andaninfinitediscontinuity
0
atR=5a ,seeFig.2C.
0
III. RESULTSFORENTANGLEMENTANDENERGY The behaviorsof the Coulomb energyhUi, and of the ki-
(DIWANDOWOSYSTEMS) netic energy hTi are plotted in the upper panel of Fig. 3,
whereh...iindicatestheground-stateexpectationvalue. For
the DIW potential, both display a maximumlocated at R =
Tocalculatetheground-stateproperties,wedirectlydiago-
4.47a (corresponding to an inner to outer well ratio of
nalize the HamiltonianEq. (1), by writingits eigenfunctions 0
0.058). The ratio between the Coulomb and the kinetic in-
Ψ as a linear combinationof single-particlebasis functions
k
teractions,Fig.3B,providesanunambiguoussignatureofthe
andtruncatingthecorrespondingexpansionas
migration point R , whereas no particular structure emerges
c
from the visual inspection of both Coulomb and kinetic en-
M M ergyseparately,Fig.3A.
Ψ (x ,x )= a η (x ;ω)η (x ;ω), (10)
k 1 2 j1,j2;k j1 1 j2 2
jX1=1jX2=1
B. Entanglement
whereη (x;ω)aretheeigenfunctionsoftheone-dimensional
j
harmonic oscillator with angular frequency ω. A single-
We calculate the spatial entanglement[32] using the von
particlebasissizeofM = 50withω = 2ensuresgoodcon-
NeumannentropyS andthelinearentropyL,
vergenceoftheresultsatanyR.
We calculatethe particle-particlespatialentanglement[32]
S =−Trρ log ρ , (11)
and the ground-state energy of the system for 4a ≤ R ≤ red 2 red
0
8a0. For the DIW system R = 4a0 correspondsto a core- L=Tr(ρred−ρ2red)=1−Trρ2red, (12)
shell structure with the two electrons confined in the inner
well,whileforR≥5a0wehaveVDIW =VOWO. whereρred =TrA|ΨihΨ|isthereduceddensitymatrixfound
by tracing out the spatial degrees of freedom of one of the
twoparticles(subsystem‘A’)andΨistheground-state. We
consideralsothepositionspace-informationentropyS ,
A. Energy n
S =− n(x)lnn(x)dx, (13)
First we consider the ground state energy E of the DIW n
0
system (solidlinein Fig.2A)againstthe benchmark(OWO, Z
dashedline). wheren(x)isthesystemparticledensity.
As R becomes larger, the inner well narrows and the en- ForapurebipartitestatethevonNeumannentropySisthe
ergyofthetwo-electronstateincreases,untiltheelectronsare unique function that satisfies all the entanglement measure-
eventually‘forced’into the outerwell. The groundstate en- ment conditions,[33, 34] while the linear entropy L is com-
ergy leaves the inner well at R ≡ R = 4.96a . This cor- putationallyconvenientandquantifiestheentanglementinthe
c 0
respondstoaninnertoouterwellratioof0.0039. Hereafter, sense thatit givesan indicationof the numberandspreadof
4
4
A) A) DIW <T> OWO <T>
−20 DIW <U> OWO <U>
E (Hartree)0−−2284 E (Hartree)0−−−222210 ODWIWO >; <T> (Hartree) 23 U>; <T> (Hartree) 00000.....12345
−32 <U 1 < 0
4.9 4.95 5 5.05
4.8 4.9 5 5.1
R(a0) R (a0)
−36 0
4 4.5 5 5.5 6 6.5 7 7.5 8 4 4.5 5 5.5 6 6.5 7 7.5 8
R(a0) R (a)
25 0
B) DIW 0.5
Hartree/ ) a0 112050 (Hartree/ ) a0 12120000 OWO T> 0 0.4.45 B) 000 0...344.5826
(R dR >/< 0.34
dE /d0 5 dE /0 00 44..88 44..99R(a0) 55 55..11 <U 0.35 0.3 4.8 4.9 5 5.1
0 0.3 R (a0) DIW
OWO
−5
4 4.5 5 5.5 6 6.5 7 7.5 8 0.25
R (a) 4 4.5 5 5.5 6 6.5 7 7.5 8
0
40 R (a)
C) DIW 0
OWO 0.6
C) DIW
2 2 2ad E /dR (Hartree/ )00−−84 000 2 2 2ad E /dR (Hartree/ )00−−−−−−11848422 00000000 44..88 44..99R (a) 55 5.1 L , S, Sn,rr 00 ..024 VonS NpeaucLemi−naIennanfro EE Ennnttrrtoroopppyyy
0
−120 −0.2
4 4.5 5 5.5 6 6.5 7 7.5 8
R (a0) 4 4.5 5 5.5 6 6.5 7 7.5 8
R (a)
0
FIG.2: GroundstateenergyE0(panelA),firstandsecondderiva-
tive of E0 with respect to R (panel B and C, respectively) for the FIG.3: PanelA:Coulombenergy,hUi,andkineticenergy,hTi,for
theDIWandOWOpotentialsasafunctionofthedrivingparameter
DIW(solidline)andOWO(dashedline)potentialsasfunctionsof
thedrivingparameterR.Inallthethreepanelstheinsetzoomsonthe R. Inset: asmain panel, but intheneighborhood of the migration
‘migrationregion’withRc = 4.96a0indicatedbyaverticaldotted pointRc,markedbyaverticaldottedline. PanelB:Ratiobetween
line. the Coulomb interaction energy and the kinetic energy, hUi/hTi,
versusRforboththeDIWandOWOpotential. Inset: detailsofthe
‘migrationregion’ withthevertical dottedlineindicatingthepoint
termsintheSchmidtdecompositionofthestate.Theposition- R = Rc. PanelC:ThevonNeumann(S,dottedline),andrescaled
linear(Lr,solidline)andspace-information(Sn,r,dashedline)en-
spaceinformationentropyS canbeconsideredasanapprox-
n tropies as functions of R for the DIW system. The rescaling was
imationto S whenoffdiagonaltermsareneglected[32]and choseninsuchawaythatLr andSn,r areequaltoS atR = 8a0.
iswrittenintermsoftheparticledensity,soitcouldbemore ThisresultsinLr =3.04LandSn,r =0.15Sn.
easilyanddirectlyaccessedbyexperiments.
In Fig. 3B, the linear, von Neumann, and position-space
information entropy are plotted as a function of R for the ticledensityuniquelydeterminesalltheground-stateproper-
DIWsystem. LandSn havebeenrescaledsothattheyhave ties of the system, so in principle the ground-stateentangle-
the same value of S at R = 8a . All quantities show the ment for this system could be written as a functional of the
0
samequalitativebehavior,LandS rescalingalmostperfectly density;theoverallsimilaritybetweenS –explicitlywritten
n
onto each other. In particular, all quantities show a non- asafunctionalofthedensity–andthetwoentanglementmea-
differentiable point at R = 5a0 and present a minimum lo- sures S and L reinforces the idea that pertinent information
cated in the same R region. However, the minimum of Sn can be extracted from the electron density. As for the DIW
(R = 4.45a0) is nearer to the maximum of hUi than the systemLcanberescaledverywellontoS,wewillcontinue
minima of the other two entropies (R = 4.51a0 for L and usingthecomputationallyconvenientlinearentropyL.
R=4.52a0forS),andismorepronounced. ThelinearentropyoftheDIW,LDIW,andoftheOWOsys-
BytheHohenberg-Kohntheorem,[30]thegroundstatepar- tem are comparedin Fig. 4A. L displays three regions.
DIW
5
A) ofhUicorrespondsheretoaminimumoftheentanglementas
0.16
DIW theseextremaoccurwhentheelectronsaremostconfinedand
OWO hence in an almost factorized state.[1] The decrease of hUi
0.12
0.12 isasignatureofthewavefunctionspillingintotheouterwell
and,consequently,ofanincreasinginfluenceofCoulombcor-
L 0.08 0.08
L relations in shaping the wavefunction with a corresponding
0.04 increase of the entanglement. In the migration region (with
0.04
0 hUi(R )beingapproximately7%ofitsmaximumvalue),the
c
4.8 4.9R (a) 5 5.1 wavefunctiondensity is substantiallyspreadwithin the outer
0 0
well, andsmall variationsof R producelargechangesin the
4 4.5 5 5.5 6 6.5 7 7.5 8
entanglement.
R (a)
0
WenotethatinRef.35,asystemsimilartoDIW,withanin-
2.5 B) DIW
OWO nerwellshrinkinginwidthbutneverdisappearing,wasstud-
2 ied. In this a case no discontinuity in any derivative of the
22
relevantquantitieswerefound.
dR 1.5 L/dR 11
dL/ 1 d
00 C. ComparisonbetweentheDIWandtheOWOpotentials
0.5 44..88 44..99 55 5.1
R (a)
0
0 InFigs.2Aand4AE0andLareplottedforboththeOWO
andDIWpotential. AtR = R themany-bodyground-state
4 4.5 5 5.5 6 6.5 7 7.5 8 c
R (a0) isboundedtotheouterwell,andinparticularE0DIW(Rc) ≈
50 0.99EOWO(R ). In contrast, L (R ), which is marked
C) DIW 0 c DIW c
OWO byadotinFig.4,isapproximatelyhalfofthecorresponding
40 5500
entanglementvalueintheOWOcase. Weunderlinethathere
30 2R 3300 theinnerwellhasafinitedepthbuttheinnertoouterwellratio
2dR L/d isonly0.0039,so, froma geometricpointofview,the inner
2d L/ 20 2d 1100 wellshouldbenegligible.HoweverintheregionRc ≤R<5
10 −10 thebehaviorofhUi,withhUi >hUi ,suggeststhat
44..88 44..99R (a) 55 55..11 theelectronsremainstronglyDpiInWnedtothOeWinOnerwellregion
0 0
eventhoughthewidthofthelatterisbasicallynegligible. As
−10 aconsequenceaverynarrowinnerwellisabletomodifythe
4 4.5 5 5.5 6 6.5 7 7.5 8
R (a) distributionoftheelectronsinsuchawaythattheirentangle-
0
mentishighlyandnon-linearlyreduced. This‘pinningprop-
FIG.4: PanelA:Linearentropy(L)asafunctionofRfortheDIW erty’ of the entanglementmightopen possibilities of rapidly
andOWOpotentials. ThedotsindicatesthevalueofLDIW atR= andefficientlymodifyingtheentanglementinananostructure
Rc.PanelsBandC:firstandsecondderivativeswithrespecttoRof system.
thethelinearentropyasafunctionofR.Inallthreepanels,theinset
representstherespectivefunctionaroundthemigrationpointRc,the
latterbeinghighlightedbytheverticaldottedline.
IV. GROUND-STATEWAVEFUNCTIONAND
PARTICLE-DENSITYBEHAVIOR
Thefirst is characterizedbya slow variationin entropywith InFig.5thehighsensitivityofthesystemwavefunctionto
a shallow minimum at R = 4.51a0. In the second region small changes of the driving parameter in the migration re-
(4.9a0 .R<5a)theentropyincreasesveryrapidly,andfi- gionisexplicitlydemonstrated. Thefigurein factshowsthe
nallyforR >5a0 theentropyincreaseslinearlywithR. The wavefunctioncontourplots for R = 4.5a0 (∼ maximumof
first derivative of LDIW (Fig. 4B) presents a shoulder-like hUiDIW,panelA),R = Rc (‘migration’point,panelB)and
structure connecting the first and second regions; then, after R = 5a (value at which the inner well disappears, panel
0
theboostintherateofchangeoftheentropy,thederivativehas C).Thewavefunctionbecomesmoreandmoreconfineduntil
a finite discontinuityat R = 5a0. The second derivativeof R ≈ 4.5a0, for which it displays a single maximum (panel
LDIW presentstwomaximaatR=4.89a0andR=4.99a0, A). A further reduction of the inner well width induces the
andaminimumatR=4.92a0.Ithasaninfinitediscontinuity wavefunction to leak into the outer well (compare scales on
atR=5a0 axis of panels A and B). Around R ≈ Rc the wavefunction
The rate of change of the entropy shown in Fig. 4 is the starts to separate into two lobes, butremainslargestclose to
result of the competing effects of the confinement strength theinnerwell(pinningeffect).AsRincreasesbeyondR ,the
c
and of the Coulomb repulsion. However, since it is the ra- shape of the wavefunction displays two well-defined lobes,
tio betweenthese two factorswhich governsthe responseof reflecting the effect of the electron-electron repulsion com-
thesystemtoavariationofthedrivingparameter,amaximum bined with the diminished confinement strength (panel C).
6
The wavefunction width and height though remain roughly differencewiththewavefunction,thechangeinthenumberof
constant, compare panels B and C. We note that the wave- peaksoftheparticledensityassociatedtothedisappearanceof
functionshape appearsto change“smoothly”as R increases thecentralmaximumcanthenbeassociatedwiththediscon-
and, in particular, no detectable change in the geometry of tinuity in the derivativesof energyand entanglementcaused
thewavefunctionseemstotakeplaceatR = 5a ,wherethe bythedisappearanceoftheinnerwell.
0
non-differentiablepointsoftheentropyandenergiesareboth
located. 1.4 R=4.9
1.2 0.3 R=4.95
A) DIW: R = 4.5 (a) 0.26 R=4.998
1−.15 0 11..48 −1ay ( ) 0 0 .18 00..2128 −2 0 2 R=5
a()0−0 .05 001..260 Densit 00..64
x2 0.2
0.5 0.2
1 0
−6 −4 −2 0 2 4 6
1.5
−1.5 −1 −0.5 0 0.5 1 1.5 x ( a )
x (a) 0
1 0
B) DIW: R=4.96 (a)
−8 0
FIG. 6: Density n(x;R) for the DIW potential plotted against the
−6 0.25 positionxforfourdifferentvaluesofR(aslabeled). Inset:zoomof
−4 00..125 mainpanelforR≈5.
−2
) 0.1
a(0 0 0.05
x2 2 0
4
V. FIDELITYOFTHEGROUND-STATEWAVEFUNCTION
6
8
−8 −6 −4 −2 0 2 4 6 8 The fidelity between two states quantifies their similarity
x (a)
C) DIW: R = 5 (a) 1 0 andassuchhasbeenextensivelyusedinquantuminformation
−8 0
theory.[2]Morerecentlythefidelityhasbeenintroducedasa
−6 0.25
methodforthecharacterizationofQPTs:[7,8,36]asignature
−4 0.2
0.15 ofQPTisanabruptchangeinthewavefunction,[37]andthis
−2
a()0 0 00..015 suggests the evaluation of the fidelity between states across
x2 2 0 the critical point as a good choice for the identification of a
4 QPT. Here we will use this method to better understand the
6 systembehaviorinthemigrationregion.
8 Foroursystemtheground-statefidelityisgivenby
−8 −6 −4 −2 0 2 4 6 8
x (a)
1 0
F(R ,R )=|hψ(x ,x ;R )|ψ(x ,x ;R )i|; (14)
1 2 1 2 1 1 2 2
FIG.5: (coloronline)Contourplotofthewavefunctionagainstthe
particles’positionsx1 andx2 fortheDIWpotentialatR = 4.5a0 following Eq. (10) we then calculate it as F(R1,R2) =
(4∼.96maa0xi(mpaunmeloBf)h,UaniDdIRW=an5dam0i(npiominutmatowfLhi,cphaWneDliwAIW),R==0,Rpacn=el (cid:12)terpMjr1e,tje2d=1inajt1w,jo2,nco(Rm1p)laemj1e,jn2t,nar(yRw2)a(cid:12)y.s,F[3(8R]1a,nRd2a)cccaonrdbineginto-
C). (cid:12)P (cid:12)
(cid:12)this we will considertwo different(cid:12)sets of valuesfor R1 and
R .
2
Asalreadymentioned,theparticledensityn(x;R) should Inquantuminformationtheory,thefidelitycanbeseenasa
uniquelycapturethesystemground-statebehavior,sowewill generalizationofameasureofsimilaritybetweentwoclassi-
now check if the density shape is more susceptible than the calprobabilitydistributions.[2]Letustake R = R , where
1 0
wavefunctiontotheshrinkinganddisappearanceoftheinner ψ(R ) is the reference state, then the fidelity is the overlap
0
well. In Fig. 6 the density is plotted for various values of between this initial state and the wavefunction ψ(R) calcu-
R. Weseethat,asRincreases,theheightofthecentral(and lated as R = R varies in the parameterspace. The fidelity
2
only)peakdiminishes. ForR ≈ R thedensitydevelopstwo F(R ,R)clearlydependsonthechoiceofthereferencestate.
c 0
shouldersandatR = 5a thecentralpeakdisappearsandis Thefactthattheminimumoftheentanglementcorrespondsto
0
replaced by two peaks which are symmetric around the ori- aquasi-productstate(seeFig.4, panelA),whichevolvesto-
gin. This can more clearly be seen in the inset. At least for wards a highly entangled state as R increases, suggests as a
thesystemathand,thepinningfromtheinnerwellhasamore naturalchoiceR = 4.52a ,correspondingtotheminimum
0 0
clear-cuteffectontheshapeofthedensitythanontheshape ofthelinearentropy.
of the wavefunction, as in particular it determines the pres- Alternatively,thefidelitycanbeseen asa geometricalob-
enceorabsenceofacentralpeakfortheparticledensity. At ject connected to the Fubini-Study distance between quan-
7
tum states,[8] where the square distance between infinitesi- functionsandN particlescanbeapproximatedas
mallyclosestatescanbeapproximatedasds2 ≈2(1−F).
FS
In this case the fidelity is calculatedbetween two wavefunc- δR2 ∂ψ(x ...x ;R) 2
1 N
F(R,R+δR)≈1− dx ...dx ,
tionsdependingoninfinitesimallydifferentparameters,ψ(R) 2 ∂R 1 N
and ψ(R + δR). At the critical point, where there is an Z(cid:18) (cid:19) (15)
abrupt change in ψ, this function has a minimum and pos- shows a discontinuity at R = 5a , in accordance with the
0
sibly a discontinuity. In Fig. 7, the fidelities F(R ,R) and discontinuityfoundinthederivativesofallthequantitiesdis-
0
cussedsofar.
1 VI. FIDELITYOFTHEPARTICLEDENSITY
A)
0.8 R 0
F(R ,R)0 00..46 dF(R ,R)/d0−−42 cthaasFteo,trhtehaep1a-fiprdtaiercltliietcyldemesnyasysitteybmeiswwnir(tihtxte;cnRon)intr=otlerp|mψar(saxmo;fRett)eh|re2Rdsoew,nseinithytahvaiess
0.2 ODWIWO 4.8 4.9R (a0 )5 5.1 F(WRe1,mRa2y)g=eneralinz(ext;hRis1t)on(ax‘;dRen2)sditxy.fidelity’byusingthe
R p
0 densityarisingfromN-particlesystems,
4.5 4.7 4.9 5.1 5.3 5.5 5.7 5.9 6
R (a)
0
n(x,R)=N |ψ(x,x ...,x ;R)|2dx ...dx . (16)
2 n 2 n
Z
1
B) 1 anddefiningthe‘densityfidelity’as
0.99996
R) 0.99996 0.99992 1
R+ δ 0.99988 Fn(R1,R2)= N n(x;R1)n(x;R2)dx. (17)
R, 0.99992 4.8 4.9 5 5.1 Z p
F( DIW R (a0) Fn(R1,R2) has the properties expected from a fidelity, that
0.99988 OWO is0 ≤ Fn(R1,R2) ≤ 1anditmeasurestheoverlapbetween
particledensitiesasthedrivingparameterRisvaried.Wewill
4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 alsodemonstratethatF (R ,R )vanishesif andonlyif the
R (a) n 1 2
0 correspondingwavefunctionfidelityF(R ,R )vanishes.
1 2
We note that a density fidelity has been proposed for lat-
FIG.7: PanelA:fidelityF(R0,R)vsRforR0 = 4.52a0. Inset:
firstderivativeofF(R0,R)vsR. PanelB:fidelityF(R,R+δR) tice systems and linked with QPTs in Ref. 39. We initially
vsRwithδR=0.002.Inset:asforthemainpanel,butzoomingon calculate the density fidelity with respect to R0. Fn(R0,R)
theregionofthefidelityminimum. TheplotsrefertotheDIWand shows a non-differentiablepoint at R = 5a corresponding
0
OWO potential (aslabeled); theverticaldotted linecorresponds to to the disappearance of the inner well (Fig. 8A); its deriva-
Rc. tivein respectto Ris plottedinthe inset. We notethesimi-
laritybetweenthebehaviorofF (R ,R)andF(R ,R)and
n 0 0
between their derivatives, the main difference being that the
F(R,R+ δR) are plotted as a function of R (panel A and residualfidelityforR> 5a islargerforthedensitythanfor
0
B,respectively). F(R ,R)displaysthreedistinctregimes,in thewavefunction.
0
accordancewiththebehaviorofalltheotherquantitiesstud- We show F (R,R+ δR), where δR = 0.002, in Fig. 8
n
ied so far. In particular, we see that for 4.8a . R < 5a (panel B). As for the corresponding wavefunction fidelity,
0 0
we have a dramatic decrease of the fidelity: the wavefunc- here the discontinuity at R = 5a appears directly in the
0
tion is rapidly changing from a quasi-product state towards ‘densityfidelity’. AgainthebehaviorofF (R,R+δR)and
n
a triplet-like entangled state[1] (compare Fig. 5, panels A F(R,R+δR)areverysimilar(compareFig.8Bwiththein-
and C). The derivative dF(R ,R)/dR presents a minimum set of Fig. 7B), with the density preservinga slightly higher
0
at R ≈ 4.92a0, near the migrationpointRc. For R > 5a0 fidelityatitsminimum,whichoccursatR=4.92a0
the fidelity is almost constant and drastically reduced, with In the DIW system, viewing the density seems to more
F(R0,R = 5) ≈ 0.19: in this region the wavefunction clearlyandreadilydisplaythefastchangesinthegroundstate
is nearly orthogonal to the reference state. We note that propertiescorrespondingtothediscontinuityinthederivatives
F(R0,R)isnotdifferentiableatR=5a0. ofE0andLthanviewingthewavefunction(seecommentsto
Figs. 5 and 6). This may be due to the lesser formal com-
The behavior of F(R,R + δR) (Fig. 7B) shows that the plexity of the density, which is always a function of a sin-
mostsignificantchangesin the wavefunctionare confinedto gle position vector – x in the present case – as opposed to
themigrationregion,aroundamarkedminimumatR=4.93, thecomplexmany-bodywavefunction,afunctionofN posi-
againveryclosetoR . F(R,R+δR),whichforrealwave- tion vectors whose parameter space is clearly more difficult
c
8
1 positive, so for any two such N-particle ground-state wave-
A) DIW
functionsψ andψ wemaydefinetherealpositivefunction
1 2
x;R)) 0.8 0
x;R ),n(0 −−12 d F n ( n (x;R 0 ),n(x;R))/dR f1,2(x)≡Z ψ1ψ2dx2...dxN. (18)
F (n(n 0.6 −3 Foranyfixedxthisdefinesaninnerproduct,aspositivedef-
initenessissatisfiedbyf (x) = n(x)/N > 0forfiniteex-
1,1
4.8 4.9 5 5.1
ternalpotentials.TheCauchy-Schwarzinequalitycanthenbe
0.4
4.5 4.6 4.7 4.8 4.9 5 writtenas
B) DIW
1
ψ ψ dx ...dx
1 2 2 N
δ+ R)) 0.99998 ≤ |ψ1|2dxR2...dxN |ψ2|2dx2...dxN 12 . (19)
R
R),n(x; integratin(cid:16)gRbothsideswithrespRecttoxleadsto (cid:17)
x; 0.99996
n(
F (n F ≤Fn (20)
0.99994 4.8 4.85 4.9 4 .95 5 5.05 5.1 andsoifFn tendstozero,somustF.
R(a0) For a general wavefunction, a fidelity of zero does not
imply a density fidelity of zero as for example two excited
FIG.8: PanelA: DensityfidelityFn(R0,R), R0 = 4.52a0, for state wavefunctionsmay both be non-zeroin some finite re-
theDIWsystem,plottedagainsttheparameterR. Inset: Derivative
gion of space but still be orthogonal. In addition, if we
ofFn(R0,R)withrespecttoRplottedagainstR(R0 = 4.52a0).
comparewavefunctionsarisingfromdifferentformsofinter-
PanelB:Fn(R,R+δR)withδR =0.002,plottedagainstthepa-
rameterRfortheDIWsystem. particleinteractions,sayattractiveandrepulsive,then,again,
the density cannot always discriminate between orthogonal
wavefunctions. This can be explicitly seen by considering
to analyse and visualize. In addition the particle density fi- the limiting case of infinite inter-particle attraction or repul-
delity is able to predict all the other notable features of the sion. Let us consider two particles in one dimension: in
wavefunctionfidelity,suchastheminimumoccurringaround the case of infinite attraction their wavefunction will sat-
R ≈ Rc. Asnoted,minimainthefidelityF(R,R+δR)are isfy ψA(x1,x2) = 0 if x1 6= x2, while for infinite repul-
associatedtoabruptchangesinthewavefunctionandmaysig- sion we have ψR(x1,x2) = 0 if x1 = x2, otherwise both
naltheoccurrenceofa QPT,so, in accordancewithRef. 39, ψ > 0. Clearly we obtain ψAψRdx1dx2 = 0. Let us
our results suggests that the density fidelity may be used as now consider the single particle densities. In general it will
analternativetothewavefunctionfidelitytounderstandbrisk be nR(x) = |ψR(x,x1)|2dRx1 > 0. To ensure normal-
changes in the ground state and hence to study QPTs. This ization, ψA(x1,x2)2 = φ1(x1)2δ(x1 −x2), with φ1(x1) it-
R
isinlinewiththeHohenberg-Kohntheoremwhichinitssim- selfnormalized,givingacorrespondingsingleparticledensity
plest form shows that for non-degenerate ground-states, the nA(x) = |φ1(x)|2 > 0. Itfollowsthattherelateddensityfi-
densityuniquelydeterminesthemany-bodywavefunctionand delity n (x)n (x)dxisdifferentfromzero.
R A
so all the propertiesof the system.[30] We pointoutthatthe However, if we assume the requirements needed for stan-
R p
particledensityisamucheasierquantitytocalculate(andto dardDFT,i.e. groundstateandsameinterparticleinteraction,
experimentallyaccess)thanthefullmany-bodywavefunction. thenwecanarguethatthedensityfidelitycandetectorthogo-
Assuchtheuseofthefidelitydensitymightbecomeofgreat nal, nodelessground-statewavefunctions. Thelackofnodes
helpinunderstandingphenomenasuchasQPTs. Similarlyits inthegroundstatesmeansthatwecanchooseaphasesothat
characteristicsashighlightedabovesuggestthatitcouldbea bothourwavefunctionsarenevernegative. Hereafidelityof
usefultoolforlocalsensitivityanalysis. zerocorrespondstothehypotheticalsituationwhenthewave-
functionsdonotoverlapatall. Whentheinterparticleinter-
action is fixed this lack of overlap arises because the wave-
A. One-to-onecorrespondencebetweenvanishingofground functions are spatially distinct, and so the densities will not
stateparticledensityfidelityandwavefunctionfidelity overlap. Henceforground-stateswithnodelessspatialwave-
functions,thedensityfidelityiszeroifandonlyifthespatial
We will now demonstrate the important property that, for wavefunctionfidelityiszero.
systemswithfiniteexternalpotentialsandground-stateswith
nodelessspatial wavefunctions, the density fidelity is zero if
and only if the ground-state spatial wavefunction fidelity is VII. QPT-LIKETRANSITION(SYMMETRICSYSTEMS)
zero.
The nodelessspatial ground-statewavefunctionof a time- As pointed out previously, a minimum in F(R,R+ δR)
independentHamiltonianmayalwaysbetakentoberealand mayhighlightaQPTandcertainlywitnessesarapidchangein
9
thewavefunction.Inourcasethe,minimuminF(R,R+δR) VIII. ORIGINOFTHEDISCONTINUITIESOBSERVED
observedinFig. 7correspondsto the transitionbetweentwo ATR=w
separatesetsofgroundstates;thefirstsetboundedbythein-
ner and the second set bounded by the outer well. Fig. 7A
It was demonstrated in Ref. 40, 41 that a discontinuity in
showsthatthistransitionisbetweenstatesthatarealmostor-
the first (second) derivative of the ground-state energy with
thogonal. As the width of the inner well is reduced, the en-
respect to the driving parameter – a signal of a QPT – may
ergy gap between these set of states reduces: this transition
correspondtoadiscontinuityinthe(derivativeofthe)ground-
hassomeofthecharacteristicsofasecond-orderQPT.
stateentanglement.
This is apparent when looking at the ground state energy
derivatives: d2E0/dR2 presentsinthisregionamarkedmin- WhatweobserveinthepresentcaseatR = wisinsteada
imum,whichinturncorrespondstoaninflectionpointinthe discontinuityinthesameorderderivativesoftheground-state
energy first derivative. If this were a full-fledged QPT, this energyand entanglement. Moreover, all the other quantities
inflectionpointwouldhave a verticaltangent, and hencethe understudy, such as hUi or S , are non-differentiableat the
n
minimumind2E0/dR2wouldbecomeadivergency. same point. A similar situation was speculated for the limit
AsdiscussedinRefs40and41,asecond-orderQPTshould p → ∞inRef.1. Herewewouldliketoclarifytheoriginof
besignaledbyacorrespondingstructureinthefirstderivative thediscontinuitiesweobserve.
oftheentanglement. Thefirstderivativeoftheentanglement
Firstofallwecanextractfromthefidelitiesimportantinfor-
entropypresentsindeedastructure(ashoulder)whosewidth
mationonthegroundstatewavefunctionbehavioratR = w:
can be defined by the first maximum-minimum structure in
thecontinuityofF(R ,R)showsthattheground-statewave-
d2L/dR2,i.e4.89a ≤R≤4.92a (seeFig.4,panelsBand 0
0 0 functionis continuous,on average, at R = w (see Fig. 7A).
C): this shoulder indeed frames the region of the minimum
Howeverthe discontinuity of ∂F(R ,R)/∂R| indicates
of d2E /dR2. The bulk of the wavefunctionchange should 0 R=w
0 a discontinuousderivativefor the wave function at the same
occur in the region of the minimum of F(R,R + δR): in
point(insetofFig.7A).Thediscontinuityof∂ψ(x;R)/∂Rat
Fig.9wethenpresentthewavefunctionatR=4.91a (panel
0 R=wisconfirmedbythediscontinuityofF(R,R+δR)at
A) and R = R = 4.96a (panel B). The plots confirm a
c 0 thesamepoint,seeEq.(15).
quite substantial change in the wavefunction, which smears
over the upper well as R increases, changing from a single, To understandthe abovepicturewe considertheHamilto-
pointedpeaktowardsatwo-lobegeometry. nianforageneralpotential
Asforthecaseofafinite-sizesystemwhichwouldundergo
a QPTin thethermodynamiclimit,[42]the transitionwe ob-
serveinthewavefunctionoccursovera(small)parameterre-
H(R)=H + V(x ,R), (21)
0 j
gion and slightly away from the expected ‘critical’ value of
j
thedrivingparameter,i.e.,forR.R . X
c
whereH = T +U isindependentfromthedrivingparam-
0
0.5 A) DIW: R=4.91 a eterR,T isthekineticenergy,andU istheelectron-electron
0
0.4
interaction.FromtheHellmann-Feynmantheoremwehave
0.3
0.2
0.1 dE(R) ∂V(x ;R)
0 = ψ(x;R) j , ψ(x;R) , (22)
dR ∂R
j (cid:28) (cid:12) (cid:12) (cid:29)
−8 X (cid:12) (cid:12)
−6 (cid:12) (cid:12)
−x42 − (2 a 0 )0 2 4 6 8 −8 −6 −4 −2x1 0( a 0 ) 2 4 6 8 wpwahirtehtircerleexssp.e=cCt(otxon1st,eh.qe.up.exanrNtalym),erietfepr∂(cid:12)reRVse,(nxthtjse;ntRhte)h/ics∂o(cid:12)RdoirsdicsionndatitisencsuoointfytitnchuoeouNulds
propagatetodE/dR.
B) DIW: R = 4.96 a
0.2 0
We notethatin thecase ofa firstorderQPT,the disconti-
0.1 nuityindE/dRshouldarisefromthewavefunction,andnot
0 fromthepotential. Inthepresentcase,whilethefidelityindi-
catesa continuouswavefunctions,notonlydE/dR, butalso

alltheotherquantitiesusedasindicatorsforaQPT,presenta
−8
−6−4 pointofnon-analyticityatR = w = 5a0. Itishenceneces-
x2−(2 a 0 )0 2 4 6 8 −8 −6 −4 −2 x1 0( a 0 ) 2 4 6 8 asaffreycttothuendoetrhsetranqduahnotwitiaesdoisfcoinntteirneusitt,yainndthineppoatretinctuialalrwiof,ulidn
contrast with the situation in Refs. 41, 40, it would produce
discontinuitiesinthefirstderivativeofbothenergyandentan-
FIG. 9: Ground-state wavefunction plotted against the particles’
positionsx1andx2fortheDIWpotentialatR=4.91a0,panelA, glemententropies.
andR=Rc =4.96a0,panelB. Byconsideringthetime-independentSchrödingerequation
associatedtoEq.(21)wecanwrite
10
The behavior of the entanglement and its derivatives in this
limitisshowninFig.10.
Tψ(x;R) Uψ(x;R) 2
− ψ(x;R) − ψ(x;R) +E(R)= V(xj;R). (23) 0.6 A) DW
j=1 0.6
X 0.5 L(R )
DW
0.4
Eq. (23) is well-definedfora nodelessgroundstate wave- 0.4
0.2
functionandcanbeseen asafamilyofequationslabeledby
thecontinuousparameterR. Eq.(23)showsthatifthepoten- L 0.3 02 2.5 3
tial is discontinuous only at a set of points of measure zero 0.2 RD W(a0)
then, at most, it may only directly cause ψ and/or Tψ to be
0.1
discontinuous on that same set of points. Hence, for finite
discontinuities,thesediscontinuitieswillnotpropagatetoany 0
4 4.5 5 5.5 6
integratedquantitiessuchasexpectationvalues.Wethencon-
R (a)
tinuebyassumingthatψandTψare,atworst,discontinuous 0
overasetofpointsofmeasurezero.WedifferentiateEq.(23) 6 B) DW
66
5
withrespecttoR,anduseEq.(22)toobtain 44 dL/dR
4 22
R
1 ∂ψ(x;R) [Tψ(x;R)]∂ψ(x;R) L/d 3 00 44..99 55 5.1
− ψ(x;R)T ∂R + ψ2(x;R) ∂R d 2 R (a0)
∂V(x ;R) ∂V(x ;R) 1
= j − ψ(x;R) j ψ(x;R) .
∂R ∂R 0
Xj (cid:20) (cid:28) (cid:12)(cid:12) (cid:12)(cid:12) (cid:29)(cid:21) 4 4.5 5 5.5 6
(cid:12) (cid:12) (24) R (a)
(cid:12) (cid:12) 0
200
Finitediscontinuitiesinthepotentialmaymeanthatitsderiva- 200 DW
tive with respect to R will comprise delta functions, and 150 100 d2 L/dR2
hence that, unless accidental cancellations occur, these dis- 100 0
scououmnsteainttuhRiatite=tshewR˜yi.lalTrpehresounpc,ahgoantthetahtteoh∂rhiVg∂hV(tx-(hjxa;jnR;dR)/s)∂i/dR∂eRioifis.EdqLis.ec(to2un4st)inwause-- 22d L/dR 5 00 −10 04.9 R ( a 05) 5.1
havetwodiscontinuousfunctionswithrespecttoR,butasthe −50
secondtermdoesnotdependonx,theright-handsideisac-
tually discontinuousat (x;R˜) for all or almost all values of −100 C)
x since no accidental cancellations can hold for all x. This 4 4.5 5 5.5 6
R (a)
0
meansthatthelefthandsideofEq.(24)willpresentthesame
discontinuities. As ψ(x;R) and Tψ are at least continuous FIG.10:Upperpanel:EntanglemententropyLplottedagainstRfor
almost everywhere with respect to x at (x;R˜), this implies the potential of Eq. (9) and w = 5a0. Inset: same as main panel
that ∂ψ(x;R)/∂R has indeed to be discontinuousat (x;R˜) butplottedinrespecttoRDW foraneasiercomparisonwithRef.1.
foralloralmostallxandhencethefirstderivativeinrespect Middlepanelandlowerpanel: dL/dRandd2L/dR2,respectively,
toRofanyfunctionalofψwillbeingeneraldiscontinuousat vsRforthepotentialofEq.(9)andforw=5a0.Theinsetspresent
thetransitionregionandtheverticaldottedlinethemigrationpoint
R=R˜. Thisisexactlywhatweobserve.
Rc =4.98a0forthepotentialEq.(9).
In Appendix A1 we illustrate these points by explicitly
analyzing the effect of the finite discontinuity at the point
For the DW potential Eq. (9), the plot of F(R ,R)
0
(x=0;R=w)inV .
DIW (Fig. 11B, with R = 4.76a the minimum of L for this
0 0
In the next section we will instead consider a counter-
system) confirms that the reference state, almost factorized
example for which, due to an accidental cancellation,
and bounded to the inner well, is practically orthogonal to
h∂V(x ;R)/∂Ri – and hence all first derivatives in respect
j thetriplet-likegroundstatereachedafterthetransitiontothe
toR–remainscontinuouseveninthepresenceofdiscontinu-
double well potential.[1] However a QPT-like transition oc-
itiesinV(x ;R)similartotheonesoftheDIWpotential.
j curs only when the ground state bounded to the inner well
migratestotheouterwell,seetheshoulderindL/dRandthe
minimum of d2E /dR2 in Figs. 10B and 11A, respectively.
0
IX. CORE-SHELLTODOUBLEWELLPOTENTIAL The subsequent transition to a double-well potential merely
furtherisolates the two lobes of the wavefunctionfromeach
InRef.1itwasspeculatedthat,intherectangular-likepo- other. Thesharpincreaseinentanglementwhichcorresponds
tential limit, the observed sharp transitions in energies and to thisfurtherchange, doesnotthensignalanyfurtherQPT-
entanglementwoulddisplaynon-analyticitiesasthepotential likepoint,asconfirmedbytheabsenceofadditionalstructures
changesfromacore-shellstructuretoadoublewellpotential. ind2E /dR2andF(R,R+δR)(seeFig.11AandFig.11C).
0