Emergence of Topological and Strongly Correlated Ground States in trapped Rashba Spin-Orbit Coupled Bose Gases

6.4 MB·English

by  B. Ramachandhran

Emergence of Topological and Strongly Correlated Ground States in trapped
Rashba Spin-Orbit Coupled Bose Gases
B. Ramachandhran1, Hui Hu2, and Han Pu1
1Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA
2ARC Centres of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy,
Swinburne University of Technology, Melbourne 3122, Australia
(Dated: December 11, 2013)
We theoretically study an interacting few-body system of Rashba spin-orbit coupled two-
componentBosegasesconfinedinaharmonictrappingpotential. WesolvetheinteractingHamilto-
nianatlargeRashbacouplingstrengthsusingExactDiagonalizationscheme,andobtaintheground
3 statephasediagramforarangeofinteratomicinteractionsandparticlenumbers. Atsmallparticle
1 numbers, weobservethatthebosonscondensetoanarrayoftopologicalstateswithn+1/2quan-
0
tum angular momentum vortex configurations, where n=0,1,2,3... At large particle numbers, we
2
observetwodistinctregimes: atweakerinteractionstrengths,weobtaingroundstateswithtopolog-

n ical and symmetry properties that are consistent with mean-field theory computations; at stronger
a interaction strengths, we report the emergence of strongly correlated ground states.
J

3 PACSnumbers: 05.30.Jp,03.75.Mn,71.70.Ej,71.45.Gm,03.75.Lm

]
s I. INTRODUCTION and non-abelian vector potentials. For example, authors
a
in Ref. [12] consider an SO-coupling Hamiltonian in the
g
- Ultracold atomic gases offer an exceptional platform presenceofareal(abelian)magneticfieldandattemptto
t simulatethephysicsoftraditionalquantumHallsystems;
n to explore many-body quantum phenomena due to out-
a standing experimental control over interatomic interac- (b) one that preserves T symmetry, and which can be
u showntobegauge-equivalenttoaHamiltonianinapure
tions,systemgeometry,densityandpurity[1]. Numerous
q non-abelian vector potential. In this work, we study an
research groups have, for example, successfully demon-
.
t SO-coupling Hamiltonian of the latter class, and discuss
a strated the manifestation of few-body bound states and
the emergence of ground states with unique topological
m superfluidstatesinBoseandFermigasesintrappedatom
and correlation properties.
experiments[2,3]. Furthermore,phenomenalexperimen-
-
d tal progress has been achieved with atomic gases loaded In this manuscript, we study an interacting few-body
n in optical lattices to emulate traditionally condensed- system of two-component Bose gases confined in a two-
o matter phenomena like superfluid-insulator transition, dimensional (2D) isotropic harmonic trapping poten-
c
anti-ferromagnetism, and frustrated many-body systems tial with Rashba SO-coupling. The manuscript is or-
[
[4–6]. However, due to the neutral nature of atomic ganized as follows: In Sec. II, we outline the model

1 gases,mostexperimentalsystemswerelimitedtoexplor- Rashba SO-coupling Hamiltonian and discuss various
v ingquantumphenomenathatwouldoccurintheabsence symmetries. We show that the Hamiltonian is gauge-
0
of electromagnetic fields. Recently, even this limitation equivalenttoparticlessubjecttoapurenon-abelianvec-
0
wasovercome, whenlaserfieldswereusedtosuccessfully tor potential that preserves T symmetry. Then, we con-
8
0 generate effective magnetic and electric fields in neutral sider the non-interacting limit of this Hamiltonian, and
. atoms [7]. The introduction of (synthetic) gauge fields discuss single-particle solutions at small and large SO-
1
in ultracold neutral atomic systems has thus opened the coupling strengths. We proceed to discuss the imple-
0
3 possibility of exploring a whole new set of phenomena mentation of Exact Diagonalization (ED) scheme to ob-
1 that would manifest in the presence of abelian and non- tain the low-energy eigenstates of the interacting Hamil-
: abelian vector potentials [8]. tonian in the regime of interest to us - at large SO-
v
i In the presence of synthetic gauge fields in trapped ul- coupling strengths. Then, we introduce various analysis
X tracold bosonic systems, experimental evidence for spin- techniques, namely:- energy spectrum, density distribu-
r orbit (SO) coupling with equal Rashba and Dressel- tion,single-particledensitymatrix,pair-correlationfunc-
a haus type strengths was reported in a seminal paper tion,reducedwavefunction,entanglementspectrum,and
[9]. Recently, commendable experimental progress has entanglement entropy. Each technique would offer its
alsobeenachievedtowardssimulatingSO-couplinginul- unique perspective to the overall understanding of the
tracold fermionic systems [10], a phenomenon critical to ground state properties.
the simulation of certain topologically insulating states In Sec. III, we discuss the phase diagram and analyze
incondensed-mattersystems[11]. InthepresenceofSO- the ground state properties of the interacting Hamilto-
coupling,agenericHamiltonianmaybebroadlyclassified nianatdifferentparticlenumbersN,andatvariedinter-
intwoclasses: (a)onethatbreaksT (time-reversal)sym- atomic interaction strengths. At small particle numbers
metry, and which can be shown to be gauge-equivalent withN =2,weillustratetheuniquetopologicalandsym-
to a Hamiltonian in the combined presence of abelian metryproperties ofground states. Inthe relativelylarge
2
particlenumberscenariowithN =8,weobservethatthe possible regime of strong interactions where a (cid:39)a,a ,
z ↑↓
ground states fall into two distinct regimes: (a) at weak one needs to include confinement-induced resonance in
interactionstrengths(mean-field-likeregime),weobserve thecalculationof2Dinteractionstrengthsg andg [13].
↑↓
ground states with topological and symmetry properties In harmonic traps, it is natural to use the trap units;
that are also obtained via mean-field theory computa- that is, to take (cid:126)ω as the unit for energy, and the
⊥
tions; (b) at intermediate to strong interaction strengths harmonic oscillator length a = (cid:112)(cid:126)/(Mω ) as the
⊥ ⊥
(strongly correlated regime), we report the emergence of unit for length. This is equivalent to setting (cid:126) =
strongly correlated ground states. We proceed to illus- k = M = ω = 1. For the SO-coupling, we intro-
B ⊥
trate the topological, symmetry and strong correlation duce an SO-coupling length a = (cid:126)2/(Mλ ) and con-
λ R
properties of these ground states. Finally in Sec. IV, we sequently define a dimensionless SO-coupling strength
summarize and present concluding remarks. λ = a /a = (cid:112)(M/(cid:126)3)λ /√ω . In a recent ex-
SO ⊥ λ R ⊥
periment [9], a spinor (spin-1) Bose gas of 87Rb atoms
with F = 1 ground state electronic manifold is used to
II. THEORETICAL FRAMEWORK create SO-coupling, where two internal "spin" states are
selected from this manifold and labelled as pseudo-spin-
up and pseudo-spin-down. This gives an effective spin-
A. System under study
1/2 Bose gas. In this SO-coupled spin-1/2 BEC, λ
SO
is about 10. In a typical experiment for 2D spin-1/2
We study a two-component Bose gas confined in
87RbBECs[14],theinteratomicinteractionstrengthsare
a 2D isotropic harmonic trapping potential: V(ρ) =
aboutg(N−1)≈g (N−1)=102 ∼103((cid:126)ω a2). These
Mω2(x2+y2)/2 = Mω2ρ2/2. We consider the Rashba ↑↓ ⊥ ⊥
⊥ ⊥ coupling strengths, however, can be precisely tuned by
SO-coupling term, that couples pseudo-spin-1/2 degree
properly choosing the parameters of the laser fields that
of freedom and linear momentum, of the form: V =
SO lead to the harmonic confinement and the SO-coupling.
−iλ (σˆ ∂ −σˆ ∂ ),whereλ istheRashbaSO-coupling
R x y y x R
strength and σˆ are 2×2 Pauli matrices. The model
x,y,z
Ham´iltonian for the interacting system is then given by: B. Gauge-equivalent form of H
0
H= dr[H +H ],
0 int
(cid:20) (cid:126)2∇2 (cid:21) A generic single-particle Hamiltonian may be written
H = Ψ† − +V (ρ)+V −µ Ψ, (1) intheformH =(p−A)2/2M,wherep=(cid:126)kisthepar-
0 2M SO g
ticlemomentumandkisthewave-vector. Thevectorpo-
(cid:88)
H = (g/2) Ψ†Ψ†Ψ Ψ +g Ψ†Ψ Ψ†Ψ , (2) tentialAmaypossiblyhavecomponentsinbothphysical
int σ σ σ σ ↑↓ ↑ ↑ ↓ ↓
spaceand spinspace. Dependinguponthe commutation
σ=↑,↓
properties of the components of A, we may hence have
where r = (x,y) and Ψ = [Ψ (r),Ψ (r)]T denotes the anabelianornon-abeliantypevectorpotential. Thepri-
↑ ↓
spinor Bose field operators. The chemical potential µ is marymotivationbehindderivingagauge-equivalentform
´to be determined by the total number of bosons N (i.e., istomapourmodelHamiltonianH0 ontoHg,andhence
drΨ†Ψ=N). Forsimplicity,wehaveassumedthatthe derive the nature of A. It is conceivable that depend-
intra-component interaction strengths are equal, so that ing upon the nature of H0, A could comprise of purely
g =g =g. The Hamiltonian is invariant under sym- abelian components, or purely non-abelian components,
↑↑ ↓↓
metry operations associated with the anti-unitary time- or a combination of both.
reversal operator T = iσˆyC, and the unitary parity op- In order to map H0 onto Hg, it suffices to compare
erator P = σˆzI, where C and I perform complex conju- Hg with the terms −(cid:126)2∇2/2M −iλR(σˆx∂y −σˆy∂x) in
gationandspatialinversionoperationsrespectively. The H0. The latter terms may actually be rewritten as
Hamiltonian is also invariant under the combined PT |p|2/2M+λ (kˆ σˆ −kˆ σˆ ). For a two-component Bose
R y x x y
operator, which is unitary since operators P and T anti- gas confined in a 2D isotropic harmonic trap, we have
commute, i.e., since [P,T] = 0. We further note that a two-component vector potential A, with A ,A be-
+ x y
Rashba SO-coupling term breaks inversion symmetry. ing 2 × 2 matrices. Comparing H with H , we ex-
0 g
In experiments, the two-dimensionality can be real- pect Ax ∝ σˆy and Ay ∝ −σˆx. Specifically, it can be
ized by imposing a strong harmonic potential V(z) = shown that the vector potential is A = (Ax,Ay,0) =
Mω2z2/2 along axial direction in such a way so that ((cid:126)Mω )1/2λ (σˆ ,−σˆ ,0). In trap units, we then sim-
z ⊥ SO y x
µ,kBT (cid:28) (cid:126)ωz. For the realistic case of 87Rb atoms, ply have A = λSO(σˆy,−σˆx,0). The term involving |A|2
the interaction strengths can be calculated from the two isaconstant,andcanbegaugedoutwithoutlossofgen-
s-wav√e scattering lengths a (cid:39) 10√0aB and a↑↓, using erality. Therefore,thestrengthofthenon-abelianvector
g = 8π((cid:126)2/M)(a/a ) and g = 8π((cid:126)2/M)(a /a ), potential proportionally determines the strength of SO-
z ↑↓ ↑↓ z
respectively. Here, az =(cid:112)(cid:126)/(Mωz) is the characteristic coupling. It is further evident that [Ax,Ay] (cid:54)= 0, and
oscillatorlengthinz-direction,anda istheatomicBohr that A is a pure non-abelian vector potential. Further-
B
radius. Note that throughout this work, we consider in- more, the T operator commutes with the SO-coupling
teraction strengths such that a (cid:29) a,a . In another term λ (kˆ σˆ − kˆ σˆ ). In essence, the model Rashba
z ↑↓ R y x x y
3
SO-couplingHamiltonianinEqn.(1)isgauge-equivalent (a)λSO= 0.1
toparticlessubjecttoapurenon-abelianvectorpotential 0.5
that preserves T symmetry. Proposals to realize vector )
ρ
potentials of similar forms have been addressed by mul- (
↑
tiple groups [8, 15–17]. φ
−0.5
C. Single-particle solutions 1 (b)λSO2= 20
0.5
We solve the model Hamiltonian H in the absence of )
ρ
interatomicinteractionsandobtainthesingle-particleso- (
↑
lutions. Rewriting the H component in Eqn. (1), the φ
0 −0.5
single-particle wavefunction φ(r)=[φ (r),φ (r)]T with
↑ ↓
energy (cid:15) is given by
1 2
(cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) ρ
H −iλ (∂ +i∂ ) φ φ
osc R y x ↑ =(cid:15) ↑ ,
−iλ (∂ −i∂ ) H φ φ
R y x osc ↓ ↓
(3) Figure1: (coloronline). Plots(a)and(b)showwavefunctions
whereH ≡−(cid:126)2∇2/(2M)+V (ρ). Inpolarcoordinates φ↑(ρ) of single-particle states in the n=0 manifold at small
osc
(ρ,ϕ),wehave−i(∂ ±i∂ )=e∓iϕ[±∂/∂ρ−(i/ρ)∂/∂ϕ]. and large SO-coupling strengths respectively. m = 0 (solid
y x black), m = 1 (dotted red), m = 2 (dash-dotted black) and
The single-particle wavefunction takes the form
m=3 (dashed red).
(cid:20) φ (ρ) (cid:21)eimϕ
φ (r)= ↑ √ , (4)
m φ↓(ρ)eiϕ 2π 2 20 (a)λSO= 0.1
/
O
with well-defined total angular momentum j , that is
z 2S
a sum of orbital and spin angular momenta. In gen- λ
+ 10
eral, we may denote the energy spectrum as (cid:15) , where
nm
m
n=(0,1,2...) is the quantum number for the transverse
n
(radial) direction. ǫ
Thesingle-particlewavefunctionφm(r)isaneigenstate 2 7 (b)λSO= 20
of the unitary P operator: /
O
2S 5
(cid:20) φ (ρ) (cid:21)eimϕ λ
Pφm(r)=σz(−1)m −φ↓↑(ρ)eiϕ √2π =(−1)mφm(r). +m 3
n 1
The T symmetry preserved by the Hamiltonian results ǫ
in a two-fold degeneracy (Kramer doublet) of the en- −15 −10 −5 0 5 10 15
ergy spectrum: any eigenstate φ(r) = [φ (r),φ (r)]T m
↑ ↓
is degenerate with its time-reversal partner Tφ(r) =
[φ∗(r),−φ∗(r)]T. This symmetry is preserved even in Figure 2: (color online). Plots (a) and (b) show energy
↓ ↑
the presence of interatomic interactions, as the terms in spectrum of single-particle states at small and large SO-
interacting Hamiltonian H are T-invariant. The su- coupling strengths respectively: n = 0 → 7 (bottom→top)
int
perposition state, of φ (r) and its time-reversal partner and m=−16→+15. While energies of states within each n
m
state, is an eigenstate of the unitary PT operator: are represented by a specific symbol, it is evident that states
with higher n have progressively higher energies.
PT[φ (r)+Tφ (r)]=(−1)m+1[φ (r)+Tφ (r)].
m m m m
smallandlargeSO-couplingstrengths. FromFig.2(a),it
We solve the single-particle spectrum by adopting a isevidentthattheenergyspectrumisstronglydispersive
numerical basis-expansion method, details of which are inmatsmallSO-couplingstrengths,withalargeoverlap
outlinedinourearlierwork[18]. InFig.1,weshowwave- between the energies of single-particle states with differ-
functions of single-particle eigenstates at representative ent radial quantum number n. Qualitatively, the energy
valuesofsmallandlargeSO-couplingstrengths. Itisevi- spectrum at small SO-coupling strengths may be under-
dentthatalargerSO-couplingstrengthleadstoincreased stood as a weak perturbation of the harmonic oscillator
oscillationsandincreasedlocalizationatradiidetermined energylevelsofthetwopseudo-spincomponents. Onthe
by |m| in the radial direction. Corresponding wavefunc- other hand, we observe from Fig. 2(b) that the energy
tions φ (ρ) also have similar characteristics. In Fig. 2, spectrumisweaklydispersiveornearly flatinmatlarge
↓
weshowtheenergyspectrumforsingle-particlestatesat SO-coupling strengths. For the range of m shown here,
4
thereisnooverlapbetweentheenergiesofsingle-particle states in the basis, the dimension of Hilbert space is
states belonging to different radial quantum numbers n, D = (N +M −1)!/N!(M −1)!. With M = 24, for ex-
i.e, each n manifold represents single-particle states la- ample, D =300 for N =2, and D =7888725 for N =8.
belled by their azimuthal angular momenta m with no The dimension of Hilbert space grows dramatically with
overlap with adjacent n manifolds. Furthermore, the system size and hence, for practical purposes, we limit
harmonictrappingpotentialmaybequalitativelyunder- our configuration to a finite size. We observe that the
stood as a weak perturbation to the energy spectrum at solution becomes essentially exact when we consider a
large SO-coupling strengths of the corresponding trans- sufficient number of single-particle states. To solve the
lationally invariant system. problem at hand, it is convenient to work with the SO
The localized nature of the wavefunctions in Fig. 1(b) single-particle basis:
and the weakly dispersive nature of the single-particle
(cid:20) (cid:21) (cid:20) (cid:21)
energy spectrum in Fig. 2(b) are characteristics that jus- Φ(r)=(cid:88) φ↑nm(r) a ≡ (cid:88) φ↑i(r) a , (5)
φ (r) nm φ (r) i
tify a comparison of the single-particle basis states at ↓nm ↓i
nm i≡nm
large SO-coupling strengths with 2D Landau Level (LL)
structures in magnetic fields. In Ref. [16], the authors wherethefieldoperatoraiisrelatedtothesingle-particle
discuss the mapping between H0 and 2D LL Hamilto- state [φ↑nm(r),φ↓nm(r)]T. Then, Eqns. (1) and (2) sim-
nianinarigorousfashionandgeneralizetheterminology ply become
of LLs as ‘topological single-particle level structures la- (cid:88) (cid:88)
H= (cid:15) a†a + V a†a†a a , (6)
beledbyangularmomentumquantumnumberswithflat i i i ijkl i j k l
or nearly flat spectra’ [16]. Making use of this general- i ijkl
ization, we term the n = 0 manifold as the lowest LL
where (i,j,k,l) collectively denotes (n,m), and V =
structure (LLL), n=1 manifold as the next highest LL, ijkl
(g/2)[V↑↑ +V↓↓ ]+g V↑↓ with
and so on. As seen in Fig. 2(b), the radial quantization ijkl ijkl ↑↓ ijkl
ˆ
generatesenergygapsbetweenadjacent LLsof theorder
of trap energy (cid:126)ω , i.e., of order unity in trap units. V↑↑ = drφ∗ (r)φ∗ (r)φ (r)φ (r)
⊥ ijkl ↑i ↑j ↑k ↑l
To summarize, we emphasize that the generalized LLs ˆ
discussed here are created by a truly non-abelian vec- V↓↓ = drφ∗ (r)φ∗ (r)φ (r)φ (r) (7)
tor potential, i.e., in the absence of any real (abelian) ijkl ˆ ↓i ↓j ↓k ↓l
magnetic fields. The strength of Rashba SO-coupling
V↑↓ = drφ∗ (r)φ∗ (r)φ (r)φ (r).
strength, and in-turn the flatness of the single-particle ijkl ↑i ↓j ↑k ↓l
energyspectracanbeexperimentallycontrolledbyusing
laser fields. At large SO-coupling strengths, as shown We perform the ED calculation in Fock space and
for λ = 20, we obtain a nearly flat single-particle en- the Hamiltonian H can be written as a matrix of di-
SO
ergy spectra. In a non-interacting two-component Bose mension D2, naively accounting for the possibility of
gas, quantum statistics obviates the occurrence of corre- inter-coupling every Fock state [19]. It is clear from
lated states in a spectra that is not perfectly flat, due to the single particle solutions discussed in Eqn. (3), that
potential condensation of all the particles in the lowest the single-particle term (cid:15)ia†iai contributes only to diag-
energy single-particle states, identified by j = ±0.5, of onal entries of the Hamiltonian matrix, while the inter-
z
the LLL (n = 0 manifold). However, in the presence action term V a†a†a a contributes to off-diagonal en-
ijkl i j k l
of inter-particle interactions, nearly flat energy spectra triesaswell. Theenumerationofoff-diagonalentriescan
is sufficiently abled to act as an interesting playground be enormously simplified by accounting for a symmetry
to allow for the emergence of strongly correlated ground preservedbyH: conservationoftotalangularmomentum
(cid:80)
states. We now proceed to introduce the ED scheme to J = j , as readily seen from Eqn. (6). If an entry
z N z
solve the interacting Rashba SO-coupled Hamiltonian at V is to be nonzero, we must have m +m =m +m
ijkl i j k l
large SO-coupling strengths. in Eqn. (7). Using only the radial wavefunction, we have
(provided m +m =m +m ),
i j k l
ˆ
1 ∞
D. Interacting few-body problem - Exact V↑↑ = ρdρφ (ρ)φ (ρ)φ (ρ)φ (ρ)
ijkl 2π ↑i ↑j ↑k ↑l
Diagonalization scheme ˆ0
1 ∞
V↓↓ = ρdρφ (ρ)φ (ρ)φ (ρ)φ (ρ) (8)
We solve the interacting Rashba SO-coupled Hamil- ijkl 2π ↓i ↓j ↓k ↓l
ˆ0
tonian H in Eqns. (1) and (2) within the Configuration 1 ∞
V↑↓ = ρdρφ (ρ)φ (ρ)φ (ρ)φ (ρ).
Interaction alias Exact Diagonalization scheme. In this ijkl 2π ↑i ↓j ↑k ↓l
0
scheme, we expand the interacting many-body Hamil-
tonian in an appropriate single-particle basis (configu- This enables one to visualize the Hamiltonian in block-
ration) to obtain the solution. The solution becomes diagonal form, i.e., each block is a manifold compris-
exact when we consider an infinite number of single- ing of Fock states with a fixed Jz. Hence, the term
particle states. With N bosons and M single-particle V a†a†a a can only couple states within the same
ijkl i j k l
5
manifold,thereforeresultinginasparseHamiltonianma- Fock states: Ψ = (cid:80)nd α Φ , where n is the dimen-
G p=1 p p d
trix. We solve this sparse matrix to identify the low en- sionofgroundstateJ manifoldandα isthecoefficient
z p
ergy states of the system. of the Fock state Φ . As discussed in Sec. IIA, the in-
p
AsdiscussedinSec.IIA,theHamiltonianHpreserves teracting Hamiltonian H is invariant under two unitary
T symmetry. In a certain LL, the energies of states symmetry operations, P and PT. With the knowledge
labelled j and −j are equal and hence, we need to of ground state wavefunction Ψ , we are now equipped
z z G
consider both positive and negative angular momentum to determine if the ground state is an eigenstate of P or
states in the single-particle configuration. This has two PT operator.
majorimplications: (a)computationalintensityincreases
tremendously, and (b) a given configuration would never
be sufficient to obtain a complete J manifold, where 2. Density distribution and single-particle density matrix
z
all contributing single-particle states are included. We
note here that the latter issue does not arise when the WiththeknowledgeofΨ ,weareequippedtoextract
G
Hamiltonian breaks T symmetry, as in studies of rotat- variouspropertiesofthegroundstate. Wederivedensity
ingtrappedgasesorgasessubjecttorealmagneticfields distribution from the expectation value of single-particle
[20,21]. Inthesestudies,itwassufficienttoconsideronly density operator, written in second-quantized form as
positive j states and hence obtain complete J mani-
z z
folds. In the limit of large SO-coupling strengths, if the ρˆ(r)=(cid:88)(cid:104)φ (r’)|δ(r−r’)|φ (r)(cid:105)a†a , (9)
i j i j
interaction strengths are such that the energy contribu-
ij
tion from H is less than unity (in trap units), we may
int
restrict ourselves to the lowest n = 0 manifold. Within where |φ (r)(cid:105) is the single-particle state identified by in-
i
this LLL approximation, we may consider a sufficient dex j in the LLL [21]. In our case, we also have an
z
numberofsingle-particleeigenstatestoobtainessentially additional index to denote up- and down- spin compo-
exact low energy eigenstates. nents. Since J is a good quantum number, the opera-
z
tora†a selectsonlyonesingle-particlestatewithinLLL
i j
approximation. As a consequence, it does not contain
E. Analysis techniques information about products of different amplitudes and
loses information about interference pattern [21]. Hence,
EDschemeenablesustosolvetheRashbaSO-coupled the density distribution solely preserves the information
Hamiltonian H and obtain the ground state phase dia- on individual densities:
gram at various interaction strengths and particle num-
M
bers. The ground states have interesting topological, (cid:88)
n(r)=(cid:104)Ψ |ρˆ(r)|Ψ (cid:105)= |φ (r)|2 O , (10)
symmetry and strong correlation properties. Here, we G G i i
i=1
outline the details of various techniques that we use to
analyze these properties. where O is the total ground state occupation of the
i
single-particle state |φ (r)(cid:105) [21]. Within the LLL ap-
i
proximation, O are essentially eigenvalues of the diag-
i
1. Energy spectrum onal single-particle density matrix. Since single-particle
states in Eqn. (4) are eigenstates of P operator, it is ev-
First step in our analysis is to identify the total angu- ident that the density distributions n(r) would be cylin-
larmomentummanifoldJ towhichthegroundstatebe- drically symmetric. For example, representative plots of
z
longs. As discussed earlier, the Hamiltonian matrix has Oi as a function of jz, and plots of density distributions
a block-diagonal form, with each block identified by its are shown in Figs. 6 and 9.
unique J value. It is evident that each of these blocks
z
can essentially be diagonalized independently. The en-
ergyspectrumcomprisesofenergyeigenvaluesfromeach 3. Pair-correlation function
block, and the lowest eigenvalue and its corresponding
Jz may be readily associated with the ground state. De- Pair-correlation functions help us analyze the inter-
generacies in the energy spectrum naturally reflect the nal structure of the ground states. We write the
degeneracies in the ground state. For example, a typical pair-correlation operator (not normalized) in second-
energy spectrum plot is shown in Fig. 3. quantized form [21],
Dimension of Fock space in the ground state J mani-
z
(cid:88)
fold will be much smaller when compared to the Hilbert ρˆ(r,r )= φ∗(r)φ∗(r )φ (r)φ (r )a†a†a a . (11)
0 i j 0 k l 0 i j l k
space dimension D. For a given parameter set, once we
ijkl
identify the ground state J manifold, we can extract
z
the coefficients of all Fock states from the correspond- In our case, we also have an additional index to denote
ing eigenvector. In essence, we may then represent the up- and down-spin components. For instance, we may
ground state wavefunction as a sum of all contributing compute pair-correlation functions that determine the
6
conditionalprobabilitytofindanup-spinoradown-spin, we intend to probe the ground state correlation prop-
when an up-spin component is assumed to be present at erties that specifically stem from the presence of inter-
a fixed point r , i.e., (cid:104)n (r )n (r)(cid:105) or (cid:104)n (r )n (r)(cid:105) re- particle interactions. To achieve this goal, we take cues
0 ↑ 0 ↑ ↑ 0 ↓
spectively. Wemaychooser tobeawayfromtheorigin, from seminal papers in Ref. [23]. We choose a proper
0
but with a substantial amplitude of n(r). Due to angu- single-particle basis comprising of the set of eigenstates
lar momentum conservation, the condition i+j = k+l inEqn.(4)ofthesingle-particleHamiltonianH . Insuch
0
must further be fulfilled. Computing the expectation a single-particle basis, entanglement in the ground state,
value of ρˆ(r,r ) with respect to Ψ , we obtain the pair- or any non-degenerate energy eigenstate, occurs specifi-
0 G
correlation function as cally due to the presence of interactions [23].
The first step in discussing any entanglement measure
(cid:88)(cid:88)
ρ(r,r )= α∗α φ∗(r)φ∗(r )φ (r)φ (r ) is to partition the system and compute entanglement
0 p p(cid:48) i j 0 k l 0
ijkl pp(cid:48) properties between different subsystems. As discussed
(cid:104)Φ |a†a†a a |Φ (cid:105). (12) in Sec. IIC, similar to 2D LL orbitals, the single-particle
p i j l k p(cid:48) eigenstates at large SO-coupling strengths are fairly lo-
calizedinnature. Thiswarrantsustoconsiderpartition-
When the wavefunction Ψ is an eigenstate of PT oper-
G ing the system in orbital space [24]. The T symmetry
ator, pair-correlationfunctionillustratethegroundstate
preserved by the Hamiltonian H naturally prompts us
symmetry properties. Furthermore, they reveal the cor-
to partition the orbitals into two subsystems: positive
relationsbetweenup-anddown-spincomponentsinreal-
j states (subsystem A) and negative j states (subsys-
space. Pair-correlation functions at representative inter- z z
temB). WewritethegroundstatewavefunctioninFock
action strengths are shown in Figs. 6 and 9.
space as Ψ = (cid:80)nd α Φ , where Φ is represented as
G p=1 p p p
| n n ....n n (cid:105). Here, n represents the oc-
−jc −(jc−1) jc−1 jc jz
cupation number of the single-particle eigenstate j , and
4. Reduced wavefunction z
as discussed in Sec. IID, a finite size cut-off is made at
a certain value j ≡ j for computational feasibility.
c z,c
We shall now discuss techniques to analyze if the Now, we proceed to compute the bipartite entanglement
ground states possess vortex structures with distinct properties between subsystems A and B, i.e., between
topological properties. One identifying property is the the positive and negative j states respectively.
z
presenceofquantizedvaluesofskyrmion number, asdis- Orbital entanglement spectrum:- With the knowledge
cussedinourearlierwork[18]. However,thisrequiresthe of Ψ , we compute the entries of the density matrix ρˆ
G
computation of ground state wavefunction in real-space, for the ground state as
a computationally prohibitive task for the bosonic few-
particle system under study. Here, we discuss a viable (cid:104)n(cid:48) ....n(cid:48) |ρˆ|n ....n (cid:105)=α α∗, (14)
approachtoidentifythetopologicalnatureoftheground −jc jc −jc jc p p
state by computing the reduced wavefunction [22]: where the generic density operator is ρˆ=|Ψ (cid:105)(cid:104)Ψ |.
G G
Now, we compute the reduced density matrix (RDM)
Ψ(r,r∗,...,r∗ ) ρˆ by tracing out the degrees of freedom of subsystem
ψ (r)= 2 N . (13) A
rwf Ψ(r∗,r∗,...,r∗ ) B, meaning ρˆ =Tr ρˆ. As shown in Ref. [23], occupa-
1 2 N A B
tionnumbersactasdistinguishabledegreesoffreedomin
Reduced wavefunction ψ (r) is computed with respect characterizing entanglement in a finite system of identi-
rwf
to one particle, here particle with index 1, while the re- calquantumparticles. Henceinourstudy, RDMiscom-
mainingN−1particlesareplacedattheirmostprobable puted by tracing out the occupation of all the negative
locations r∗ [22]. In our case, we also have an additional j states from the density matrix:
i z
index to denote up- and down-spin components. With
(cid:48) (cid:48)
ψ (r) and ψ (r) known, we can now extract phase (cid:104)n ....n |ρˆ (1/2,..,j )|n ....n (cid:105)= (15)
rwf,↑ rwf,↓ 1/2 jc jc c 1/2 jc
informationandcomputeadistincttopologicalquantity, (cid:88) (cid:48) (cid:48)
(cid:104)n ..n n ..n |ρˆ|n ..n n ..n (cid:105)
vorticity, i.e., the number of phase slips from +π to −π −jc −1/2 1/2 jc −jc −1/2 1/2 jc
along a closed contour. An integer-valued vorticity is an n−jc..n−1/2
unambiguous way of establishing that the ground state
The RDM ρˆ has a block-diagonal structure, with each
is topological in nature with a distinct vortex structure. A
block characterized by the total angular momentum JA
For example, typical phase plots revealing different vor- z
that corresponds only to particles in subsystem A. The
ticities are shown in Figs. 6 and 9.
block-diagonal structure allows us to compute all the
eigenvalues of the RDM using full-diagonalization tech-
niques. Orbital entanglement spectrum (OES), termed
5. Entanglement measures sobecausethepartitionisdefinedinorbitalspace,isthe
plot of entanglement pseudo-energies ξ as a function of
i
We compute entanglement measures to analyze corre- JA. Here, ξ =−lnρA, with ρA being the eigenvalues of
z i i i
lation properties of various ground states. In particular, RDM ρˆ [25]. It is evident that ξ with smaller magni-
A i
7
tudes maximally contribute to the ground state proper- has a half-quantum vortex configuration. We may as
ties. well construct a zero angular momentum PT-eigenstate,
PlotsofOES revealinformationabouttheoccupation from an equal superposition of opposite angular m√omen-
of various Fock states in a given ground state manifold, tum P-eigenstates: Φ = (Φ ±T Φ )/ 2. In
PT,jz=0 P P
andin-turnthecorrelationpropertiesofthegroundstate. the absence of interactions, either of the P-eigenstates
IfvariousFockstatesΦ inthegroundstateJ manifold or the superposition PT-eigenstate are degenerate. In
p z
have similar magnitudes of α , it results in similar RDM addition, any arbitrary superposition of the degenerate
p
eigenvalues of ρA, and in-turn, similar magnitudes of ξ . P-eigenstates, which in principle need not be a PT-
i i
Thus, if an OES plot reveals that ξ values are degener- eigenstate, will also be a degenerate ground state.
i
ate or nearly degenerate, this is a clear manifestation of
the correlated nature of the ground state. On the other
hand, if the OES plot reveals that the values of ξi are In the presence of inter-particle interactions, the
distinctly non-degenerate, the ground state is clearly not ground state is not anymore determined solely by the
correlated. For example, representative OES plots are energy contribution of the non-interacting part of the
shown in Figs. 6, and 9. HamiltonianH . Dependinguponthestrengthsofg and
0
Entanglement entropy:-PlotsofOES revealthewhole g , the energy contribution from the interacting part
↑↓
spectrum of eigenvalues of the RDM and help us un- of the Hamiltonian H also plays a crucial role. This
int
derstand the correlation properties of the ground state. competitioncanbebetterunderstood,especiallyatlarge
However, it is sometimes useful to extract just a single SO-coupling strengths, by analyzing the single-particle
representative quantity from the RDM [26]. Entangle- wavefunctions and energy. As shown in Fig. 2(b), energy
mententropy(EE)issuchameasurethatcanbereadily contributions due to H tries to keep the particles in
0
oabntdaiinseddeffirnomedtahse SseAt o=f e−igtern[ρvˆaAlulneρsˆAρ]Ai =of−th(cid:80)e RiρDAiMlnρρˆAAi ,. sfotartreespwulistihveloiwnteerrvaacltuioenofstarnegnugltahrs,meonmeregnytacojnzs.idHeorwateivoenrs,
A higher entropy value means that the ground state is due to H tries to keep the particles as far away from
int
more homogeneously spread in Fock space, i.e., a larger each other as possible. This in-turn means that the par-
numberofFockstatesΦp makesubstantialcontributions ticles tend to occupy states with larger value of angular
towards the ground state. A distinct advantage of an momenta, since they have a larger localization radii as
EE plot is that we are able to look at entropy values for shown in Fig. 1(b). In essence, the ground state of the
a whole range of interaction strengths in a single plot, interactingmany-bodyHamiltonianisdeterminedbythe
andthereby,understandcorrelationpropertiesofvarious competition between the H and H terms.
0 int
phases. For example, representative EE plots are shown
in Figs. 4, 5, 7, and 8.
Insummary,densitydistribution,eigenvaluesofsingle-
The simplest scenario where the competition between
particle density matrix, pair-correlation function and re-
theH andH terms,in-turntheeffectofinter-particle
duced wavefunction would help us identify various sym- 0 int
interactions, clearly manifests is in an interacting prob-
metry and topological properties of the ground states.
lemwithN =2particles. Forthisreason,wediscussthe
Computation of RDM from proper single-particle basis
results for N =2 particles and analyze the ground state
enablesustoextractvariousentanglementmeasuresand
properties in greater detail, before proceeding to larger
allowustoanalyzecorrelationpropertiesthatspecifically
particle numbers. We solve the interacting few-body
stem from inter-particle interactions.
Hamiltonian H at large SO-coupling strengths using ED
scheme within LLL approximation. The computational
intensity, especially at large interaction strengths, limits
III. RESULTS AND DISCUSSION thefeasibilityofthisschemetotheorderofN =8parti-
cles [29]. In an earlier mean-field study on homogeneous
AsdiscussedinSec.IIC,intheabsenceofinteractions, two-componentBosegas[30], itwasshownthatthepar-
all particles would simply condense into the two lowest ticles condense into either a single plane-wave state (for
energysingle-particleeigenstatesintheLLLidentifiedby g >g )oradensity-stripestate(forg <g ). Similarly,
↑↓ ↑↓
quantum numbers j = ±0.5. This is due to the weak, in our earlier related work on trapped two-component
z
but finite, dispersion in j present in the single-particle Bose gas [31], depending on the relative magnitudes of
z
energy spectrum shown in Fig. 2(b). The P-eigenstate, g and g , we show that states with distinct topological
↑↓
identified by j = +0.5, is represented by wavefunction andsymmetrypropertiesemergeinthemean-fieldphase
z √
Φ = [φ (ρ),φ (ρ)eiϕ]T/ 2π. It has a half-quantum diagram. Takingcuesfromtheseresults,inthisstudy,we
P ↑ ↓
vortex configuration, as the spin-up component stays in solve for the ground state wavefunction at various inter-
the s-state and the spin-down component is in the p- action strengths, however fixing the relative magnitude
state[18,27,28]. Theresultingspintextureofthistopo- g /g at0.5or1.5. Inthissection,wepresenttheresults
↑↓
logical state is of skyrmion type [18]. The degenerate atdifferentparticlenumbersN,andanalyzethetopolog-
time-reversed P-eigenstate, identified by j = −0.5 and ical, symmetry and correlation properties of the ground
z √
represented by TΦ = [φ (ρ)e−iϕ,−φ (ρ)]T/ 2π, also states using various techniques discussed in Sec. IIE.
P ↓ ↑
8
3.5
2 0.50065
/ 0.51 P 1 P 2 P 3 P 4
O 2.5
λ2S 0.5006 N PT
−0.5 0 0.5 /
+ z1.5
J
N
/ 0.5 0.5
E 0 (a)
(a)g =0.001; g↑↓/g =0.5
0.5008 P 1 P 2 P 3 P 4
2 1
/O 0.51 E PT
λ2S 0.5007−0.5 0 0.5 E0.5
+
N 0.5 (b)
/ 0
E (b)g =0.001; g↑↓/g =1.5 0.05 0.1 g 0.2 0.3
−2 −1 −0.5 0 0.5 1 2
J /N
z Figure5: (coloronline). Plotsof(a)groundstateJz/N man-
ifolds and (b) entanglement entropy, as a function of interac-
Figure3: (coloronline). Energyspectrumforextremelyweak tion strength g with λSO =20,N =2,g↑↓/g=1.5.
interaction strengths with λ = 20 and N = 2. Here, each
SO
marker(red)representsthelowestenergyeigenvalueofaspe-
cific block diagonal with a fixed value of J . Since energy
z other hand, we observe from Fig. 3(b) that the ground
eigenvalues are very close, we identify the ground state ener-
state is degenerate in J /N = ±0.5 manifolds. In ei-
giesbycircled(black)markersandfurther,showthezoomed- z
ther scenario, in Fig. 3(b), we determine that Ψ is an
in plots in the inset. G
eigenstate of P operator. It is evident that, even in the
presence of extremely weak interaction strengths, the in-
3.5 teracting Hamiltonian picks either a P-eigenstate or a
PT-eigenstate to be the ground state. Furthermore, it
2.5
N is clear that the ground state is sensitive to the relative
/z1.5 PT P 3 P 4 magnitudes of g↑↓ and g.
J Figs. 4(a), 5(a): We solve the interacting Hamil-
0.5 tonian H at various interaction strengths and identify
(a)
0 correspondinggroundstatemanifoldsJ /N inFigs.4(a)
z
and 5(a). It is evident from the phase diagram that de-
pending on g and g , the ground states belong to differ-
1 PT P 3 P 4 ↑↓
ent J /N manifolds. Furthermore, we determine if the
E z
E groundstatewavefunctionΨG isaneigenstateofPT op-
0.5 erator, andtherebyidentifywhetherthestatebelongsto
P or PT symmetry phase. In a broader sense, it is evi-
(b)
0 dent that a ground state in PT symmetry phase belongs
0.05 0.1 0.2 0.3
g to J /N = 0 manifold, while ground states in various
z
J /N (cid:54)= 0 manifolds belong to P symmetry phase. EE
z
plots in Figs. 4(b) and 5(b) reveal correlation properties
Figure4: (coloronline). Plotsof(a)groundstateJ /N man-
z
in various phases. For pedagogical purposes, before we
ifolds and (b) entanglement entropy, as a function of interac-
tion strength g with λ =20,N =2,g /g =0.5. For rep- explainthefeaturesinEE plots,wefirstdiscussthesym-
SO ↑↓
resentative interaction strengths denoted by circled (black) metry, topological and correlation properties of ground
markers, we illustrate the ground state properties in Fig. 6. states.
In Fig. 6, we illustrate density distributions, eigenval-
ues of single-particle density matrix, orbital entangle-
A. N =2 ment spectrum, pair-correlation functions and reduced
wavefunctions at representative interaction strengths
AsdiscussedinSec.IIE1,weanalyzetheenergyspec- withinvariousJz/N manifoldsofFig.4(a). Usingasimi-
trum to identify the ground state angular momentum larlineofreasoning,wemayunderstandthepropertiesof
manifold J , or equivalently, J /N. In Fig. 3(a), we no- groundstatesinFig.4(b). Letusnowproceedtodiscuss
z z
tice that the ground state belongs to J /N = 0 mani- various plots shown in Fig. 6.
z
fold. We further determine that the ground state wave- Figs. 6(a1) → 6(a4): In this top row, we dis-
function Ψ is an eigenstate of PT operator. On the cuss the ground state properties of the PT eigenstate
G
9
(a1)g=0.001 1 (a2) (a3) (a4)
g↑↓/g=0.5 8 0.2 0.8
ρ() 6 Jz/N =0 Oi0.5 ξi 4 y 0.4
n 3
−0.2
0 0 0
0.25 0.5 −5.5 −1.5 1.5 5.5 00.5 1.5 2.5 3.5 −0.2 x 0.2
(b1)g=0.065 1 (b2) (b3) (b4) 1.2
8
g↑↓/g=0.5 0.2
ρn() 23 Jz/N =0 Oi0.5 ξi 4 y 00..48
−0.2
1
0 0 0
0.25 0.5 −5.5 −1.5 1.5 5.5 00.5 1.5 2.5 3.5 −0.2 x 0.2
(c1)g=0.07 1 (c2) (c3)
8
g↑↓/g=0.5
) 3 Jz/N =2.5
ρn( 2 Oi0.5 ξi 4
1
0 0 0
0.25 0.5 −5.5 2.5 5.5 00.5 1.5 2.5 3.5
(d3)
(d1)g=0.26 1 (d2) 8
g↑↓/g=0.5
ρn() 23 Jz/N =3.5 Oi0.5 ξi 4
1
0
0 0 00.5 1.5 2.5 3.5
0.25 0.5
ρ −5.5 j 3.55.5 JA/N
z z
Figure 6: (color online). Plots in each row illustrate the ground state properties at a representative interaction strength of
Fig. 4(a). In the first column (from left), we show density distributions of spin-up component n (ρ) (solid green) and of
↑
spin-downcomponentn (ρ)(dashedred). Inthesecondcolumn, weshoweigenvaluesO ofsingle-particledensitymatrixasa
↓ i
function of angular momentum j of the single-particle states |φ (r)(cid:105). In the third column, we show corresponding OES plots
z i
ofentanglementpseudo-energiesξ asafunctionofJA/N,theaverageangularmomentumofsubsystemA. Inthelastcolumn,
i z
weshowcontourplots(a4)and(b4)thatarenormalizedpair-correlationfunctions(cid:104)n (r )n (r)(cid:105),withr denotedbya(yellow)
↑ 0 ↓ 0
marker. Phase plots (c4) and (d4) are derived from reduced wavefunction ψ (r), which is computed by fixing one of the two
c,↓
particles at their most probable locations and their corresponding radii are indicated by (yellow) markers. The closed dashed
(blue) contour is a guide to the eye, that allows us to count the number of phase slips.
in J /N = 0 manifold at g = 0.001 of Fig. 4(a). As pair-correlation function (cid:104)n (r )n (r)(cid:105) of this PT eigen-
z ↑ 0 ↓
shown in Fig. 6(a1), the cylindrically symmetric density state. Thisplotillustratedtheconditionalprobabilityto
distributionsn (ρ)andn (ρ)overlap. BeingaPT eigen- findadown-spin,whenanup-spincomponentisassumed
↑ ↓
state, it is evident from Fig. 6(a2) that the positive and tobeatafixedpointr ,andrevealsthepresenceofcorre-
0
negativeangularmomentumstatesareequallyoccupied. lated regions (magnitude closer to 1) and anti-correlated
Furthermore, the time-reversal partner states identified regions (magnitude closer to 0). This plot illustrates the
by quantum numbers j = ±0.5 are predominantly oc- correlationspresentbetweenup-spinanddown-spincom-
z
cupied. As expected, from the corresponding OES plot ponents that are not revealed by the cylindrically sym-
inFig.6(a3), weobservethatthepredominantcontribu- metric density distributions.
tiontothegroundstateisfromtheentanglementpseudo-
energyξ atJA/N =+0.25. FromFigs.6(a2)and6(a3), Figs. 6(b1) → 6(b4): In this second row, we dis-
i z
itisclearthatthethemaximallycontributingFockstate cuss the ground state properties of the PT eigenstate in
is Φ =| n = 1,n = 1(cid:105), which explains J /N = 0 manifold at g = 0.065 of Fig. 4(a). As dis-
PT jz=−0.5 jz=+0.5 z
the overlapping density distributions of n (ρ) and n (ρ) cussed with reference to Fig. 6(a1), the density distribu-
↑ ↓
in Fig. 6(a1). In Fig. 6(a4), we plot the (normalized) tions n (ρ) and n (ρ) overlap in Fig. 6(b1). It is evident
↑ ↓
fromFig.6(b2)thatthetime-reversalpartnerstatesiden-
10
tified by j =±0.5 and j =±1.5 are almost equally oc- Fig. 6(d3), we observe that the ground state is predomi-
z z
cupied. From the corresponding OES plot in Fig. 6(b3), nantlyoccupiedbyξ atJA/N =3.5. However,itmaybe
i z
we observe that the ground state is equally occupied by conceived from Fig. 6(d2) that the ground state has con-
ξ at JA/N = +0.25 and +0.75. From Figs. 6(b2) and tributions from various Fock states, for example: Φ =|
i z P
6(b3), it is clear that the the maximally contributing n = 2(cid:105) or Φ =| n = 1,n = 1(cid:105) or
jz=+3.5 P jz=+1.5 jz=+5.5
Fock states are Φ =| n = 1,n = 1(cid:105) and Φ =| n = 1,n = 1(cid:105). It is clear that with
PT jz=−0.5 jz=+0.5 P jz=+2.5 jz=+4.5
Φ =| n = 1,n = 1(cid:105). To illustrate the increasing inter-particle interaction strengths, the parti-
PT jz=−1.5 jz=+1.5
internal structure of this PT eigenstate and the corre- cles distribute themselves in higher angular momentum
lations between up-spin and down-spin components, we manifolds. Furthermore,itisevidentthatthenetorbital
show the pair-correlation function in Fig. 6(b4). angular momentum of spin-up component in the ground
stateis+3andthatofspin-downcomponentis+4. Cor-
Figs. 6(c1) → 6(c4): In this third row, we discuss
respondingly, the phase plot of down-spin component in
thegroundstatepropertiesoftheP eigenstateinJ /N =
z
Fig. 6(d4) reveals a vorticity of 4. For convenience, we
+2.5 manifold at g = 0.07 of Fig. 4(a). While the cor-
identify this P eigenstate as P4. We further note that
responding ground state is degenerate in J /N = ±2.5
z
the phase plots of down-spin components derived for P1
manifolds, we restrict our discussion to J /N = +2.5
z
and P2 eigenstates in Fig. 5(a) exhibit a vorticity of 1
manifold without loss of generality. The cylindrically
and 2 respectively.
symmetric density distributions n (ρ) and n (ρ) are dis-
↑ ↓
tinct, as shown in Fig. 6(c1). In this P eigenstate, there Figs. 4(b), 5(b): Asnotedearlier,OES preservesthe
is an inherent asymmetry in the occupation of positive wholespectrumofeigenvaluesoftheRDM,andhenceal-
and negative angular momentum states. This is evident lows us to extract information about the occupation of
fromtheplotofsingle-particledensitymatrixeigenvalues Fock states with different subsystem angular momenta
O in Fig. 6(c2). This explains the presence of distinct JA. WithourunderstandingofOES plotsinFigs.6,we
i z
density distributions in Fig. 6(c1). Furthermore, we ob- now proceed to explain various features observed in EE
serveapeakintheoccupationofeigenstateidentifiedby plotsofFigs.4(b)and5(b). (i)Thepresenceofdistinctly
j = +2.5 in Fig. 6(c2). From the corresponding OES different slopes suggests the presence of distinct correla-
z
plotinFig.6(c3),weobservethatthegroundstateispre- tion properties in ground states within various phases.
dominantly occupied by ξ at JA/N =2.5. To illustrate (ii)Withineachphase,EE increasesmonotonouslywith
i z
the internal structure of this P eigenstate, we show the increasingg. AsdiscussedinSec.IIE5,thisresultsfrom
phaseplotderivedfromthereducedwavefunctionψ (r) an increasingly homogeneous distribution of Fock states
c,↓
in Fig. 6(c4). To better understand this phase plot, we in the ground state J /N manifold, and in-turn an in-
z
take cues from plots in Figs. 6(c2) and 6(c3). Though creased correlation. For example, to illustrate this fea-
we observe from Fig. 6(c3) that the ground state is pre- turewithinthePT symmetricphaseinFig.4(b),wemay
dominantly occupied by ξ at JA/N = 2.5, it may be compare OES plots in Fig. 6(a3) and 6(b3) and observe
i z
conceived from Fig. 6(c2) that the ground state has con- an increased homogeneity in distribution of Fock states.
tributions from various Fock states, for example: Φ =| (iii) The presence of nearly degenerate ξ values results
P i
n = 2(cid:105) or Φ =| n = 1,n = 1(cid:105) or in a reduction in the slope of EE. While this feature is
jz=+2.5 P jz=+1.5 jz=+3.5
Φ =|n =1,n =1(cid:105). Fromtherepresenta- observed at larger interaction strengths within the PT
P jz=+0.5 jz=+4.5
tionofsingle-particleeigenstatesinEqn.(4),itisevident symmetricphaseofFig.4(b), theOES plotinFig.6(b3)
that the net orbital angular momentum of spin-up com- helpsusunderstandthis. (iv)TransitiontoaP symmet-
ponent in the ground state is +2 and that of spin-down ric phase is marked by a sharp reduction in the value of
componentis+3. Correspondingly,thephaseplotofthe EE [32]. To better understand this feature, we compare
down-spin component in Fig. 6(c4) reveals a vorticity of OES plots in Fig. 6(b3) and 6(c3) and observe a sharp
3. Wenoteherethatthevorticityisthenumberofphase reduction in homogeneity of ξ values, accompanied by
i
slips from +π to −π, i.e., when the shadowing changes a substantial drop in the minimum value of ξ . In sum-
i
from white to black. For convenience, we identify this P mary,weemphasizethattheknowledgeofOES helpsus
eigenstateasP3,where3isthevorticityofthedown-spin understand various features exhibited by EE plots.
component.
In summary, it is evident that the interacting Hamil-
Figs. 6(d1) → 6(d4): Inthislastrow,wediscussthe tonian picks either a P-eigenstate or a PT-eigenstate to
ground state properties of the P eigenstate in J /N = be the ground state. The ground state is sensitive to the
z
+3.5 manifold at g =0.26 of Fig. 4(a). The correspond- relative magnitudes of g and g. J /N plots allow us
↑↓ z
ing ground state is degenerate in J /N = ±3.5 mani- to identify various P and PT symmetry phases in the
z
folds, while we restrict our discussion to J /N = +3.5 interacting system. With the analysis of density distri-
z
manifold. As expected for a P eigenstate, the density butions,single-particledensitymatrixandreducedwave-
distributionsn (ρ)andn (ρ)showninFig.6(d1)aredis- functions,weillustrategroundstatesymmetryandtopo-
↑ ↓
tinct. In addition to the asymmetric occupation of posi- logical properties. We assert that the bosons condense
tiveandnegativeangularmomentumstatesinFig.6(d2), into an array of P-symmetric topological ground states
we observe a peak occupation of eigenstate identified that have n+1/2 -quantum angular momentum vortex
by j = +3.5. From the corresponding OES plot in configuration, with n = 0,1,2,3. With the analysis of
z