Download - Hard-Wall Con nement of a Fractional Quantum Hall LiquidHard-Wall Con nement of a Fractional Quantum Hall Liquid E. Macaluso 1and I. Carusotto 1INO-CNR BEC Center and Dipartimento

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Hard-Wall Confinement of a Fractional Quantum Hall Liquid

E. Macaluso1 and I. Carusotto1

1INO-CNR BEC Center and Dipartimento di Fisica, Universita di Trento, 38123 Povo, Italy(Dated: September 25, 2017)

We make use of numerical exact diagonalization calculations to explore the physics of ν = 1/2bosonic fractional quantum Hall (FQH) droplets in the presence of experimentally realistic cylindri-cally symmetric hard-wall potentials. This kind of confinement is found to produce very differentmany-body spectra compared to a harmonic trap or the so-called extremely steep limit. For arelatively weak confinement, the degeneracies are lifted and the low-lying excited states organizethemselves in energy branches that can be explained in terms of their Jack polynomial represen-tation. For a strong confinement, a strong spatial deformation of the droplet is found, with anunexpected depletion of its central density.

I. INTRODUCTION

Fractional quantum Hall (FQH) states have been firstobserved in two-dimensional electron gases in the pres-ence of strong magnetic fields [1]. Since then, theyhave been one of the most active branches of quantumcondensed-matter physics, with a rich variety of intrigu-ing phenomena and a close connection to the topologi-cal properties of the underlying many-body states [2, 3].Very recently, the interest for these systems has beenrenewed by long-term perspectives in view of quantuminformation processing applications [4].

In parallel to these advances in electronic systems, frac-tional quantum Hall physics is receiving a strong atten-tion also from the communities of researchers workingon quantum gases of ultra-cold atoms [5] and in quan-tum fluids of light [6]. Even though they are electricallyneutral particles, both atoms and photons can displaymagnetic effects when subject to the so-called syntheticor artificial magnetic fields.

The first proposal in this direction has been to makea trapped atomic cloud to rotate at a fast angular speedand exploit the mathematical analogy between the Cori-olis force and the Lorentz force on charged particles [7].Later on, researchers have focussed on dressing the atomswith suitably designed optical and magnetic fields so toassociate a Berry phase to their motion [8, 9]. In thelast years, this has led to the observation of some amongthe most popular models of topological condensed-matterphysics, such as the Hofstadter-Harper model [10, 11],the Haldane model [12], as well as the so-called spin Halleffect [13] and the nucleation of quantized vortices by asynthetic magnetic field [14].

Also in the optical context, topologically protectededge states related to the integer quantum Hall ones havebeen observed in suitably designed magneto-optical pho-tonic crystals [15], in optical resonator lattices [16] as wellas in arrays of waveguides [17], while non-planar macro-scopic ring cavities have been demonstrated to supportLandau levels for photons [18].

In both atomic and optical systems, the present ex-perimental challenge is to push the study of systemsexperiencing artificial magnetic fields into a regime of

strongly interacting particles where strongly correlatedstates are expected to appear, in primis fractional quan-tum Hall states [2, 3]. While too high values of the systemtemperature are one of the main difficulties encounteredby atomic realizations, photonic systems are facing thechallenges [6, 19] of generating sufficiently large photon-photon interactions mediated by the optical nonlinearityof the underlying medium and dealing with the intrin-sically driven-dissipative nature of the photon gas. Apromising solution to the former issue based on coher-ently dressed atoms in a Rydberg-EIT configuration hasbeen investigated in [20]. Different pumping schemes togenerate quantum Hall states of light have been investi-gated in [21–23].

With respect to electronic systems, atomic and pho-tonic systems are expected to offer a much wider flexi-bility and a more precise control on the external poten-tial confining the FQH droplet and, consequently, on theproperties of its edge. Early theoretical works on thesesystems have focussed on harmonic confinements [24–28]for which, however, the smoothness of the confining po-tential hinders a clear distinction between bulk and edge.Only recently researchers, motivated by the realization ofa flat-bottomed traps for ultra-cold atoms [29] and by theflexibility in designing optical cavities [18] and arrays ofthem [30], have started investigating hard-wall (HW) po-tentials and the peculiar many-body spectral propertiesthey induce in the excitation modes of the FQH droplet.Along these lines, a so-called extremely steep limit hasbeen considered in [31], characterized by a marked hier-arcy of the confinement potential experienced by the se-quence of lowest-Landau-level single-particle orbitals. Inparticular, it was shown that the eigenstates of a ν = 1/rFQH droplet correspond under such a idealized confine-ment to certain Jack polynomials, from which one canextract an analytic expression for their energies.

In the present work, we provide a general study of theeffect of a general and experimentally realistic HW poten-tial on ν = 1/2 bosonic FQH droplets. Using a numericalapproach based on exact diagonalization, we character-ize the ground state of the confined system as well as itslow-lying excited states depending on the confinementpotential. In a weak confinement regime, a classification

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of excited states based on their representation in terms ofJack polynomials is proposed. The energy ordering of thebranches and sub-branches of this spectrum and the rela-tive energies of states within the same sub-branch are dis-cussed and physically explained. This analysis confirmsthe presence of signicant deviations from the standardchiral Luttinger liquid theory of edge excitations [32] asfirst predicted in [31] for an idealized extremely steeplimit, but also highlights crucial qualitative differencesfrom this latter work. In the strong confinement regime,a peculiar deformation of the cloud with a marked den-sity depletions at its center is pointed out and physicallymotivated.

The structure of the article is the following. In Sec.IIwe introduce the system Hamiltonian and we review thebasics of Laughlin states and of their low-lying excita-tions in the unconfined and harmonic confinement cases.In Sec.III we briefly review the main features of Jackpolynomials and of their application to fractional quan-tum Hall physics. The main new results of this work arereported in Secs.IV and V for the weak and the strongconfinement cases, respectively. Conclusions are finallydrawn in Sec.VI.

II. PHYSICAL SYSTEM AND THEORETICALMODEL

A. The model Hamiltonian

We consider a 2D system described by the HamiltonianH = H0 +Hint +Hconf , where

H0 =

N∑i=1

(pi + A)2

2M(1)

is the kinetic energy of particles experiencing an effectiveorthogonal magnetic field B = ∇×A = B ez,

Hint =∑i<j

gint δ(2)(ri − rj), (2)

describes contact interactions between the particles ofstrength gint and finally

Hconf =

N∑i=1

Vext θ(|ri| −Rext) (3)

represents a radially symmetric hard-wall confining po-tential, which confines the particles in a disk-shaped re-gion. Here, the potential height Vext and the disk radiusRext can set at will, while θ(x) denotes the Heaviside stepfunction.

For the sake of convenience, we choose the so-calledsymmetric gauge for the vector potential - i.e. A =B (−y/2, x/2, 0) - which makes cylindrical rotationalsymmetry manifest and guarantees that all many-bodyeigenstates have a well-defined angular momentum. For

sufficiently large magnetic fields B, we can restrict our-selves to the lowest Landau level (LLL) and use the lad-der operators a†m and am, which respectively create andannihilate particles in the LLL state of angular momen-tum m and real-space wave function

ϕm(r, φ) =1

lB√

2πm!eimφ

(r√2lB

)me− r2

4l2B (4)

with magnetic length lB =√

~c/B.This approximation is valid if we restrict to the low-

energy physics of the system and we assume that thecyclotron energy ~ωC = ~B/m is far larger than all otherenergy scales set by the potential energy Vext and thecharacteristic interaction energy V0 ≡ gint/2l

2B . Within

this approximation the Hamiltonian terms can be writtenin second-quantization terms as:

H0 = ε0

∑m

a†mam (5)

Hint =gint2πl2B

∑αβγρ

Γ(α+ β + 1)√α!β!γ!ρ!

δ(α+β,γ+ρ)

2(α+β+2)a†αa

†βaγaρ

(6)

Hconf =∑m

Um a†mam

=∑m

Vextm!

γ↑

(m+ 1,

R2ext

2l2B

)a†mam (7)

where ε0 = ~ωC/2 is the kinetic energy of the (massivelydegenerate) LLL, Γ(t) denotes the Euler gamma function

Γ(t) ≡∫ ∞

0

xt−1e−xdx, (8)

whereas γ↑(t, R) is the so-called upper incomplete gammafunction

γ↑(t, R) ≡∫ ∞R

xt−1e−xdx. (9)

Note that since we are neglecting excitation to higherLandau levels, all N -particle states have the same kineticenergy ε0N , which effectively drops out of the problem.As a consequence we neglect the kinetic term and wefocus on the Hamiltonian H ≡ H −H0 = Hint +Hconf .

B. The confinement potential

Within the LLL approximation, the confinement po-tential is summarized by its value Um on each single-particle state of angular momentum m. This quantity isplotted in Fig. 1 for a few choices of Vext and Rext. In therecent work [31], this dependence was assumed to fulfillthe condition Um−1 Um Um+1, but in practice this

3

0 5 10 15 20 25 30 35 40 45

0

1

2

3

4

5

6

7

8

9

10 prova prova dss Rext

= 5 Vext

= 10

Rext

= 4.8 Vext

= 10

Rext

= 4.8 Vext

= 2

0

10

20

30

40

50

60

70

80

90

100

8 10 1210-3

10-2

10-1

100

FIG. 1. Angular momentum-dependence of the confinementpotentials Um. The inset shows a magnified log-scale viewon the region of m values corresponding to the highest occu-pied single-particle orbitals in a N = 6 Laughlin state. Thepotential parameters considered here are those typically usedfor the confinement of N = 6 particle systems, for which thequite slow increase of the Um’s does not fulfill the extremelysteep condition.

condition does not appear to be so simple to satisfy witha realistic potential.

For sufficiently large Rext, we can in fact approximate

Um 'Vextm!

(R2ext

2l2B

)mexp

(−R

2ext

2l2B

), (10)

so that having a large ratio

UmUm−1

' R2ext

2ml2B(11)

requires a wide disk radius Rext √

2mlB , much largerthan the FQH droplet we intend to prepare. On the otherhand, for

√2mlB & Rext, the potential Um smoothly

approaches its limiting value Vext.Given the exponential factor in (10), simultaneously

having an overall appreciable potential Um and a verysteep m-dependence (11) requires a very large potentialstrength Vext: physically, this is due to the fact that theHW potential only acts on the far tail of the LLL wavefunction. However, having a remote, but strong HW po-tential makes the system very sensitive to weak variationsof the HW parameters. For instance, the relative vari-ation of the confinement potential ∆Um/Um for a smallvariation ∆Rext can be estimated to be a large number∣∣∣∣∆RextRext

[2m− R2

ext

2l2B

]∣∣∣∣ ' RextlB

∆RextlB

. (12)

These arguments on the difficulty of fulfilling the con-dition assumed in [31] are a further motivation for our

numerical calculations including a realistic form of theconfinement potential.

The numerical calculations reported in this work arebased on a direct exact diagonalization (ED) of thesecond-quantized Hamiltonian on a truncated Fock spacespanned by |n0, n1, n2, . . .〉 number states with nm =0, 1, . . . ,N particles in the m-th LLL orbital. To be pre-cise, before diagonalizing the Hamiltonian matrix we setto zero all entries associated with basis elements havingtotal number of particles lower than N and/or angularmomentum quantum number different from those of in-terest.

Furthermore, we can take advantage of the descriptionin terms of Jack polynomials [33–36] as an heuristic toolto conveniently choose the cutoff mmax on the single par-ticle LLL orbitals to be included in the calculation. Jackpolynomials indeed allow to know the number of single-particle orbitals needed for the description of the Laugh-lin state and its edge and quasi-hole excitations in theabsence of confinement. Since the external confinementhas the effect of shrinking the cloud, we do not expectthat the description of the eigenstates of the total Hamil-tonian including confinement will require any additionalorbital. As a further check, the accuracy of the numericalresults has been ensured by verifying their independenceon the cut-off.

C. Laughlin states in the absence of confinement

In absence of any confinement - i.e. Vext = 0 and/orRext → ∞ -, the eigenstates of a system of contact-interacting bosonic 2D particles experiencing an effectiveorthogonal magnetic field are well-known from the the-ory. In particular, such unconfined system is character-ized by a widely degenerate ground state, containing theν = 1/2 Laughlin state, as well as its quasi-hole (QH) andedge excitations (EEs). All such states are characterizedby a vanishing interaction energy and are separated fromexcited states (including quasi-particle (QP) excitations)by a bulk excitation gap proportional to the interactionenergy V0 – the so-called Laughlin gap.

In more detail, the ν = 1/2 Laughlin state has thelowest angular momentum value LL = N (N − 1) amongall the states forming such a degenerate ground state andis described by the celebrated wave function:

ψL(zi) ∝∏i<j

(zi − zj)2 e−∑

i |zi|2/4l2B , (13)

in which zk = xk − i yk denotes the position of the k-thparticle in the complex plane. In their general form, thewave functions of states in the ground state manifold canbe written as

ψedge(zi) ∝ S(zi)ψL(zi), (14)

where ψL(zi) denotes the Laughlin wave function (13)and S(zi) is a generic homogeneous symmetric poly-nomial in the particle coordinates whose degree gives the

4

24 26 28 30 32 34 366.006

6.008

6.01

6.012

6.014

6.016

6.018

Laughlin state

Quasi-particle state

Edge excitations

Single

quasi-hole

state

0.18

0.16

0.06

0.08

0.1

0.12

0.14

FIG. 2. Spectrum of many-body states of a N = 6 parti-cle system experiencing a harmonic confinement of frequency~υ = 2.5 × 10−3 V0. While the ground state is the Laughlinstate, the energies of the low-lying excited states is propor-tional to the total angular momentum: all EEs with the sameangular momentum are thus degenerate and the dynamicsshows a single characteristic frequency v. The width of thebulk Laughlin gap to the lowest quasi-particle states is con-trolled by the interaction energy V0.

additional angular momentum ∆L. The degeneracy ofsuch states is given by the number of partitions of theinteger ∆L.

For low additional angular momentum ∆L N ,the edge excitations (EE) can be understood as area-preserving shape deformations of the FQH droplet [37].Another remarkable class of states among those in (14)are quasi-hole (QH) excitations, characterized by frac-tional charge and anyonic statistics [2] and correspond-ing to density depletions of a half of a particle withina Laughlin state. In general a ν = 1/2 Laughlin statepresenting n QH excitations at positions ξ1, . . . , ξn is de-scribed by the wave function:

ψn−qh(zi, ξi) ∝( N∏i=1

n∏j=1

(zi − ξj))ψL(zi). (15)

In the following, in order to preserve radial symmetry,we will restrict ourselves to states presenting QHs in theorigin ξi = 0 for all i = 1, . . . , n, so that

ψn−qh,0(zi, ξi = 0) ∝(∏

i

zni

)ψL(zi). (16)

In the absence of confinement, all these QHs and EEsare degenerate with the Laughlin ground state. On theother hand, it is known that the introduction of an exter-nal confining potential allows to at least partially removethis degeneracy, in a way which depends on both the po-tential shape and on the confinement parameters. For

instance, a harmonic potential [see Fig. 2] introducesan energy shift ∆E = υLz proportional to the angularmomentum and to the harmonic confinement frequency.As a consequence, this confinement is able to remove thedegeneracy between states with different angular momen-tum and to provide a non degenerate ground state in theLaughlin form (13). However, excited states with thesame angular momentum remain degenerate.

As we shall see in the following of this work, the hard-wall confinement (3) is instead able to completely removethe degeneracy of the lowest energy states of a ν = 1/2FQH liquid and to induce a rich structure in the energyvs. angular-momentum diagram of many-body states.To capture the physics underlying this organization, Jackpolynomials will be an essential tool.

III. BASICS OF JACK POLYNOMIALS

In the previous section, we have briefly mentioned Jackpolynomials - or simply Jacks - as a useful tool to prop-erly choose the cut-off on the single-particle orbitals tobe included in the numerical calculation wave functions.As we shall see in the following, their power goes far be-yond this as they allow to easily build useful trial wavefunctions to describe FQH states. In this section we shallreview those main features that will be used in Sec.IV forthe interpretation of the energy spectra of ν = 1/2 FQHdroplets.

In general, Jacks Jαλ are homogeneous symmetric poly-nomials identified by a rational parameter α - called Jackparameter - and by a root partition λ. Here, by parti-tion we mean a non-growing sequence of positive inte-gers, λ = [λ1, λ2, . . .], so that the sum of the integers inthe sequence corresponds to the number that gets parti-tioned. The degree of the Jack is given by this number,indicated by |λ|. Furthermore, Jacks have been foundto exhibit clustering properties [38] and also to corre-spond to some of the polynomial solution of the so-calledLaplace-Beltrami operator [39]:

HαLB =

∑i

(zi∂i)2 +

1

α

∑i<j

zi + zjzi − zj

(zi∂i − zj∂j), (17)

where ∂i ≡ ∂/∂zi.Partitions of length N can be biunivocally associated

to symmetrized monomials in N ≥ N variables in thefollowing way: the partition [λ1, λ2, λ3, . . . , λN ] corre-

sponds to the monomial Sym(zλ11 zλ2

2 zλ33 . . . zλN

N ) whereall other variables zN+1 . . . zN appear with degree 0. Forexample, the partition [4, 2, 2, 1] corresponds to the sym-metrized monomial M[4,2,2,1] ≡ Sym(z4

1z22z

23z

14). In this

perspective, Jacks have a peculiar expansion in terms ofsymmetrized monomials Mµ’s:

Jαλ =Mλ +∑µλ

cλµ(α)Mµ, (18)

5

where µ runs over all partitions that can be obtainedfrom the root partition λ through all possible sequencesof squeezing operations [38]. Under such an operation,one starts from a parent partition [. . . , λi, . . . , λj , . . . ] togenerate another one - called descendant - of the form[. . . , λi−δm, . . . , λj +δm, . . . ] (with λi−δm ≥ λj +δm).The corresponding coefficients cλµ(α) can be computedby means of a recursive construction algorithm [33, 40].

Jack polynomials with negative α appear in the the-ory of the quantum Hall effect. In this context the ad-missible root configurations are given by some of the so-called (k, r) admissible root configurations. The (k, r)admissibility means that there can not be more than kparticles into r consecutive orbitals. In particular thebosonic Read-Rezayi k series of states has been provento be given by single Jacks of parameter α = −k+1

r−1 and

suitable root partition [34]. Among these bosonic FQHstates, we focus here on the ν = 1/2 Laughlin state forwhich (k = 1, r = 2) and the wave function (13) turnsout to be given by the Jack polynomial with α = −2 androot partition λ = [2N − 2, 2N − 4, . . . , 2].

To better understand this connection, it is useful toestablish a link between a partition and a distributionof particles among the different LLL orbitals. Neglect-ing the ubiquitous Gaussian factor and the normaliza-tion constant, each LLL single-particle wave function (4)of angular momentum m corresponds to a monomial zm.So, any bosonic many-particle state |n0, n1, n2, . . .〉 ob-tained by occupying each LLL wave function of angularmomentum m with a well-defined number nm particlescan be described as a symmetrized monomialMλ, whereλ ≡ [λ1, λ2, . . . ] is a sequence of positive integers – i.e. apartition – which indicates the LLL wave functions oc-cupied by the different particles in descending order. Forexample, states of N ≥ 4 particles having one particle inthe m = 4 orbital, two in the m = 2 orbital, one in them = 1 orbital and the remaining in the m = 0 orbitalcan be written as |N − 4, 1, 2, 0, 1〉 = Sym(z4

1z22z

23z

14) ≡

M[4,2,2,1].

A. Laughlin states and their excitations in terms ofJacks

Based on this connection, all rotationally symmetriceigenstates of the Hamiltonian

|ψ〉 =∑i

ci |n0, n1, n2, . . .〉i , (19)

correspond to homogeneous symmetric polynomials inthe particle coordinates of degree equal to the well-defined total angular momentum

L =∑m

mnm, (20)

where nm ≡ 〈ψ|a†mam|ψ〉. In addition, one can checkthat the Laughlin state (13) - neglecting the Gaussian

term - satisfies (17) for α = −2, which reflects the factthat (13) vanishes as (zi − zj)

2 when the i-th and thej-th particles approach each other. In particular, onecan expand

∏i<j(zi − zj)2 and check that the obtained

expansion is exactly the one in (18) with root partitionΩ = [2N −2, 2N −4, . . . , 2]. This means that by knowingthe Jack parameter α = −2 and the root configurationMΩ ≡ |1 0 1 0 . . . 1 0 1〉 one can construct the full Laugh-lin state simply applying the squeezing operation and therecursive formula for the coefficients.

This formulation in terms of Jacks can be extendedto both QH and EE excited states above the Laughlinstate. Concerning EEs, studies have demonstrated thatin general their wave functions (14) can not be expressedas single Jacks, but are linear combinations of a finiteset of them [36]. In particular, edge states of angularmomentum L = LL + ∆L, can be expressed as linearcombination of those α = −2 Jacks whose partitions areobtained by adding to the root partition Ω of the Laugh-lin state any partition η of the additional angular momen-tum |η| = ∆L of maximum length equal to the numberN of particles, namely λ = Ω + η. Here, the sum of twopartitions λ = [λ1 . . . λl] and [µ1 . . . µm] of lengths l ≥ mis defined as a partition of length l whose elements areλ + µ = [λ1 + µ1 . . . λm + µm, λm+1 . . . λl] (extension tothe l < m case is straightforward by requiring the sumto be a commutative operation).

Is then immediate to check that the Jack polynomialsobtained in this way correspond to partitions satisfying|λ| = |Ω| + |η| = LL + ∆L = L, so the correspond-ing states indeed have the required angular momentum.Furthermore, this choice for the possible edge partitions(EPs) η’s is in agreement with the degeneracy predictedby the single branch chiral Luttinger theory of edge states[32].

As an example, consider the ∆L = 2 edge excitationsof the N = 6 particle ν = 1/2 Laughlin state. In thiscase there are two possible EPs given by ηa = [2] andηb = [1, 1]. Therefore the ∆L = 2 edge states trial wavefunctions can be obtained as linear combinations of JacksJ

(−2)λa and J

(−2)

λb , where the admissible root partitions areλa = Ω + ηa = [10, 8, 6, 4, 2] + [2] = [12, 8, 6, 4, 2] andλb = Ω + ηb = [10, 8, 6, 4, 2] + [1, 1] = [11, 9, 6, 4, 2]. Interms of root configurations this means moving particlesoccupying the highest-m occupied orbitals into orbitalswith even higher single-particle angular momenta. Inthe concrete example above, λa is obtained by movingone particle from the m = 10 orbital to the m = 12orbital, while λb is obtained by moving one particle fromthe m = 10 orbital to the m = 11 one and another fromthe m = 8 orbital to the one with m = 9.

The situation is simpler for the state displaying n QHexcitations located at z = 0, whose wave function is givenin (16). In contrast to the generic EEs considered above,this state can be expressed as a single α = −2 Jackpolynomial: the root configuration for the n-QH wavefunction (16) is given by |0n1 0 1 . . . 1 0 1〉, which can beobtained starting from the Laughlin one by moving each

6

particle from the m orbital to the m+n orbital. The rootpartition corresponding to such a configuration is there-fore λn ≡ Ω + κn, where Ω denotes again the root parti-tion associated with the ν = 1/2 Laughlin state and then-QH partition κn is the sequence obtained by repeatingN times the same integer n, i.e. κn ≡ [n, n, . . . , n].

B. Edge Jacks

The recent work [36] has proven an interesting rela-tion between Jacks with EPs η’s as their roots - whichwe will call edge Jacks (EJs) - and the Jacks with rootpartitions of the form λ = Ω+η discussed in the previoussubsection. In particular, they are related by

JαΩ+η = Jβη JαΩ , (21)

in which α = − 2r−1 , β = 2

r+1 and JαΩ denotes the Jack

representing the ν = 1/r Laughlin wave function. In ourspecific case of ν = 1/2 Laughlin states, we have β = 2/3.

Relation (21) suggests a description of generic edgestates described by a homogeneous symmetric polyno-mial pre-factors S(zi) in (14) in terms of Jacks. Thepolynomial S(zi) can in fact be expanded in the ba-sis of Jacks with Jack parameter β = 2/3 and differentpartitions η’s. Each of these edge state wave functionsJβη ψLzi can then be written as a single Jack using(21) and therefore they can be easily constructed usingthe expansion (18),

J−2Ω+η = J2/3

η J−2Ω = J2/3

η ψL(zi). (22)

A drawback of this procedure is that wave functions ofthis form for different edge partitions η’s are not orthog-onal for β = 2/3.

This potential difficulty can be overcome by using analternative expansion of the S(zi) pre-factors on thedifferent basis of EJs of the form Jβ=ν

η , where ν = 1/ris the usual FQH filling factor and η is one of the EPsdiscussed above [31]. Since such Jacks with β = ν areorthogonal in the thermodynamic limit,⟨

Jνη∣∣Jνµ⟩ = jµ(ν) δη,µ for N →∞, (23)

under the Laughlin scalar product

〈φ|χ〉 ≡∫CN

dzi[φ(zi)]∗χ(zi) |ψL(zi)|2, (24)

the associated edge state wave functions turn out to bealso orthogonal in the limit of N → ∞. While use ofthese β = ν EJs Jνη has the great advantage of leadingto edge state wave functions that are orthogonal in thethermodynamic limit, its drawback is that these wavefunctions in general can not be written as single Jacks -as it instead happens for β = 2

r+1 EJ’s via eq.(21). Inthe next section we will make use of these wave functionsto interpret the result of our numerical ED calculations.

IV. WEAK CONFINEMENT REGIME

After reviewing the basic concepts of fractional quan-tum Hall physics and of Jack polynomials, in the next twosections we are going to present and discuss our ED nu-merical results for the low-lying part of the many-bodyspectrum for different values of the HW potential pa-rameters Vext and Rext. Different regimes can be dis-tinguished depending on the value of these parameters,in particular we identify a weak confinement regime (dis-cussed in this section) and a strong confinement regime(considered in the next Sec.V). In both cases we restrict

to the Rext & Rcl case, where Rcl '√N/ν

√2lB denotes

the semiclassical radius of a N -particle FQH droplet withfilling factor ν. While this assumption guarantees thatthe Um components of the confinement potential are agrowing function of m, it includes a wider set of poten-tials than the extremely steep HW limit of [31].

The weak confinement regime is characterized by aweak mixing of the Laughlin state and its EE and QHexcited states with states above the Laughlin gap, e.g.quasi-particle excitations. In this regime, as one can seein Fig. 3 the ground state of the system is non-degenerateand is very close to the ν = 1/2 Laughlin state (13). Onthe other hand, the HW potential pushes the low-lyingexcited states to higher energies and removes the degen-eracy between EEs with the same additional angular mo-mentum ∆L. In particular, for a given ∆L, the numberof EEs with energies lying below the bulk excitation gapdepends on the potential parameters. Indeed more we in-crease the potential strength –or we reduce the potentialradius– more EEs end up having energies lying above thebulk Laughlin excitation gap. As a consequence, only forthe lowest values of ∆L it is possible to resolve all EEs[see Fig. 3 a)-c)].

As a first step of our study of the FQH liquid undera weak HW confinement, in Sec.IV A, we will presenta classification of the many-body states into branchesand sub-branches and we will highlight its relation tothe EP of the corresponding Jack polynomials along thelines of [31]: the excellent overlap with the analytic trialwavefunction is remarkably associated to significant de-viations from the chiral Luttinger liquid theory of theedge, as visible in both the degeneracy of states and intheir ordering within a given sub-branch. More detailson the ordering of the different sub-branches are givenin Sec.IV B: discrepancies from the extremely steep limitof [31] are pointed out and a unexpected energy-crossingbetween states sharing the same total angular momentumhighlighted for varying trap parameters. In the followingSec.IV C, we shall discuss how the dispersion of stateswithin a given subbranch can be widely controlled viathe HW potential parameters. As a striking example, weshall present a regime in which the single-QH state is thefirst excited state. Finally, in IV D, the compressibilityof the FQH liquid and of its quasi-hole and quasi-particleexcited states is characterized in terms of the dependenceof the eigenstate energies on the confinement potential

7

25 30 35 40 456

6.002

6.004

6.006

6.008

6.01

6.012

Quasi-particle

stateTwo

quasi-hole

state

Single

quasi-hole

stateLaughlin state

Edge

excitations

25 30 35 40 456

6.002

6.004

6.006

6.008

6.01

6.012

Laughlin state

Edge

excitations

Quasi-particle

state

Single

quasi-hole

state

Two

quasi-hole

state

25 30 35 40 455.998

6

6.002

6.004

6.006

6.008

6.01

6.012

Laughlin state

Single quasi-hole state

Two

quasi-hole

state

Quasi-particle

state

Edge

excitations

24 26 28 30 32 34 36 38 406

6.002

6.004

6.006

6.008

6.01

6.012

6.014

6.016

6.018

Laughlin state

Quasi-particle state

Single

quasi-hole

state

Edge excitations

0

0.02

0.04

0.06

0.08

0.1

0.12

0

0.02

0.04

0.06

0.08

0.1

0.12

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

FIG. 3. Many-body spectra for N = 6 particle system in the weak confinement regime experiencing different HW confiningpotentials. While the chiral Luttinger liquid theory well captures the number of energy branches and sub-branches for eachvalue of ∆L, the characteristic linear relation between energy and angular momentum breaks down for all choices of HWparameters.Panels a) and b) correspond to a HW confinement with Vext = 20V0 and Rext = 4.95

√2 lB and with Vext = 100V0

and Rext = 5.25√

2 lB , respectively. In both cases the ground state is given by the Laughlin one and for a fixed value of ∆Ldifferent non-degenerate EEs can be resolved below the bulk excitation gap. In these spectra also the two different sub-branchesof the second EE energy brach can be easily distinguished. Panel c) corresponds to an even weaker HW confinement withVext = 30V0 and Rext = 5.25

√2 lB : in this case all low-lying excited states up to L = 34 have energies below the bulk excitation

gap so that the first four energy branches are visible. Panel d) corresponds to a stronger HW confinement with Vext = 100V0

and Rext = 4.95√

2 lB for which the ground state and first excited one are given by the Laughlin state and the single-QH state,respectively. In this case, for a given value of ∆L < N there is just one non-degenerate branch of excited states below the bulkexcitation gap.

strength. A. Classification of states in terms of Jacks

The global organization of the EEs is therefore easiestunderstood in Fig. 3 c) where the confinement is weak-est: looking at the full spectrum, instead of focusing on aspecific value of ∆L, we can see that EEs organize them-

8

30 35 40 456

6.001

6.002

6.003

6.004

6.005

6.006

6.007

6.008

6.009

6.01

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=34 2nd excitedL=34 3rd excitedJack EP=[2,1,1]Jack EP=[2,2]

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=35 2nd excitedL=35 3rd excitedJack EP=[2,1,1,1]Jack EP=[2,2,1]

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=35 4th excitedL=35 5th excitedJack EP=[3,1,1]Jack EP=[3,2]

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=36 5th excitedL=36 6th excitedL=36 7th excitedJack EP=[3,1,1,1]Jack EP=[3,2,1]Jack EP=[3,3]

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=38 13th excitedJack EP=[4,4]

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=38 13th excitedJack EP=[4,4]

L=38 13th excited

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=34 2nd excitedL=34 3rd excitedJack 2/3 EP=[2,1,1]Jack 2/3 EP=[2,2]Jack 1/2 EP=[2,2]Jack 1/2 EP=[2,1,1]

0.17

0.16

0.15

0.14

1.5 2 2.5

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=38 13th excitedL=38 13th excited Vext=15V0Jack 2/3 EP=[4,4]Jack 1/2 EP=[4,4]

1 2

0.15

0.16

0.17

0.18

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=35 2nd excitedL=35 3rd excitedJack 2/3 EP=[2,1,1,1]Jack 2/3 EP=[2,2,1]

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=35 4th excitedL=35 5th excitedJack 2/3 EP=[3,1,1]Jack 2/3 EP=[3,2]

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2L=36 5th excitedL=36 6th excitedL=36 7th excitedJack 2/3 EP=[3,1,1,1]Jack 2/3 EP=[3,2,1]Jack 2/3 EP=[3,3]

FIG. 4. a) Eigenstates of a N = 6 particle system experiencing an HW potential of parameters Vext = 30V0 and Rext =5.25√

2 lB with energies lying below the bulk excitation gap and angular momenta L ≤ 45. b)-f) Comparison between thedensity profiles of some of the eigenstates depicted in a) and the associated trial wave functions constructed starting from EJsof parameter ν = 1/2 - dashed lines - and/or β = 2/3 - dotted lines with markers -. As we can see, the eigenstate descriptionin terms of Jacks is so good that discrepancies in the density profiles are hard to see in most cases. In order to displayhow deviations of the numerical eigenstates from the Jacks trial wave functions can be further reduced by considering weakerconfinements, panel f) shows also the density profile - red solid line - for a lower value of the HW strength, i.e. Vext = 15V0.Insets in b) and f) show that on these scales differences between trial wave functions associated to the same EP but a differentJack parameter are not visible.

selves in a sort of energy branches: the k-th branch startsfrom an angular momentum L = LL + k and ends witha state presenting k QHs at the origin. In between, itsplits into k sub-branches.

Although such a structure of the spectrum could seemquite complicated, it can be completely explained interms of Jack polynomials. In this and the next sub-sections, we are in fact going to see that in the weak con-finement regime the EEs turn out to be well describedby the wave functions

φη(zi) = J1/2η ψL(zi), (25)

and the k-th energy branch contains all states whose wavefunctions can be obtained by multiplying ψL(zi) byJacks with partitions of the form

η = [η1, η2, η3, . . . ] = [k, η2, η3, . . . ]. (26)

As a first step, simple angular momentum argumentsbased on the observed extension of the branch supportthis statement. As a Jack of partition λ = Ω + η has an-gular momentum L = LL+ |η|, the lowest ∆L allowed by

the form (26) is the one associated with η = [k], namely∆L = k. At the same time partitions are ordered se-quences of positive integers and therefore they must sat-isfy

η1 ≥ η2 ≥ · · · ≥ ηN . (27)

This implies that the maximum additional angular mo-mentum that can be obtained by considering EPs withη1 = k is ∆L = Nk and that it comes from the partition[k, . . . , k] corresponding to the k-QH state.

Within this picture, the monotonically growing energyof the bands as a function of k is easily understood asthe main contribution to the confinement energy is givenby the outermost particle, whose LLL wave functions ispeaked on a larger ring. On the k-th branch, the outer-most occupied orbital of the configuration is indeed theone of angular momentum Ω1 + k, namely the one asso-ciated to the particle that was moved by k orbitals in theoutward direction from the outermost occupied orbital ofthe Laughlin root configuration.

Along the same lines, the order of the sub-branches in

9

energy can be related to the value of the second elementη2 of the partition. In particular the j-th sub-branch ofthe k-th energy branch is composed of the states whosetrial wave functions can be constructed from Jacks havingpartitions

η = [η1, η2, η3, . . . ] = [k, j, η3, . . . ], (28)

with the exception of the 1-st sub-branch in which thereis also the Jack with partition η = [k] having η1 = k andno η2. This interpretation of the different sub-branches,together with the constraint (27), is further confirmed bythe fact that the number of states in a given sub-branchdepends only on the sub-branch index j and on the num-ber of particles N - which fixes the maximum number ofpartition elements ηi -, but not on the branch index k.Subtle features related to this ordering of levels will behighlighted in more detail in Sec.IV B and compared tothe conclusions of [31] for the extremely steep HW limit.

Our interpretation of the branches is numerically val-idated in the plots of Fig.4, which successfully comparethe density profiles predicted by the trial wave functionsin (22) to the one numerically predicted by ED. The devi-ations are a consequence of the mixing with quasi-particleexcitations induced by the confinement as well as of thenon-orthogonality of wave functions associated with dif-ferent EPs of the same additional angular momentum∆L. Note that the calculation of the density profilesshown in this picture is made significantly simpler by theexpression of the edge state wavefunction in terms of sin-gle Jacks via (22).

In some panels, we have added the corresponding den-sity profiles calculated using the trial wave function (25)(in others, the curves for the two kinds of Jacks are in-distinguishable on the scale of the figure). The agree-ment with the numerics is once again excellent. As thesetrial wave functions are orthogonal in the thermodynamiclimit, we expect that the remaining deviations will disap-pear if one considers N →∞ and vanishingly weak HWpotential Rext →∞ or Vext → 0.

A further confirmation of our interpretation comesfrom the overlap between the numerical eigenstates andthe Jacks trial wave functions: as one can see in TableI, the overlap is very good with the edge Jacks of (22).This overlap gets even closer to 1 when the Jacks of (25)are considered [41] and/or when a weaker confinement isconsidered.

B. Order of sub-branches

In the previous sub-section we have presented a gen-eral criterion for the ordering in energy of the differentbranches and of the different sub-branches within a givenbranch. As clearly shown in Fig. 4 a) sub-branches as-sociated with higher values of the second EP element η2

have increasing energies. For instance among the twoL = 34 eigenstates in the second energy branch [see Fig.

EP⟨Jβη ψL

∣∣Ψ⟩ ⟨JνηψL

∣∣Ψ⟩ ⟨Jβη ψL

∣∣JνηψL⟩η = [1, 1, 1, 1] 0.9991 0.9991 1.0000

0.9994 0.9994η = [2, 1, 1] 0.9783 0.9936 0.9878

0.9798 0.9953η = [2, 2] 0.9896 0.9902 0.9908

0.9914 0.9922η = [4, 4] 0.9473 0.9501 0.9452

0.9669 0.9733

TABLE I. Overlaps of the numerical ED eigenstates |Ψ〉 withthe corresponding trial wave functions (22) and (25) for asystem of N = 6 particles. Values in black refer to a HWconfinement of parameters Rext = 5.25

√2lB and Vext = 30V0

while those in red to the Rext = 5.25√

2lB and Vext = 15V0

case.

4 b)], the one in the η2 = 1 sub-branch has a lower energythan the one in the η2 = 2 sub-branch.

Despite this behavior persists for all HW parametersvalues we have considered, it is crucial to note that thisresult is in contrast to what was found in [31] for theextremely steep HW limit where sub-branches associatedwith increasing η2 values are characterized by decreasingenergies.

This disagreement is easily understood by noting howthe energy shifts in the extremely steep HW limit onlydepend on the highest m occupation number, while, aswe have seen, this is no longer the case for more realisticconfining potentials: in particular, the observed orderingof the sub-branches in the two cases is explained by thefact that increasing values of η2 correspond to slightlylower occupations of the highest m orbital but also muchhigher occupations of the second highest one [see Fig. 6h)].

This energetic behavior characterizing sub-branchesbelonging to the same energy branch, together with theobservation that the eigenstates are the same in differ-ent confining regimes, suggests that in the transition be-tween the two regimes the eigenstates in the different sub-branches cross in energy without mixing. While such acrossing would be obviously protected by rotational sym-metry for eigenstates with different angular momenta, itis quite unexpected for same Lz eigenstates which wouldtypically mix in a non-trivial and potential-dependentway.

From our calculations, it however appears that this isnot the case and the Jacks (25) remain precise eigenstatesof the Hamiltonian for all considered confinement poten-tials. To further corroborate this statement, we havestudied the value of the matrix elements of the a†mamoperators contributing to the confinement energy (7): asone can see in Fig. 5, their off-diagonal matrix elementsin the Jack basis turn out to be much smaller in magni-tude than the diagonal ones and, even more remarkably,have a markedly oscillating dependence on m.

As a result, they easily average to very small valueswhen realistic potentials Um are considered. For in-stance, for the realistic confinement potential considered

10

0 2 4 6 8 10 12 14 16-0.2

0

0.2

0.4

0.6

0.8

1EP=[2,1,1] EP=[2,1,1]EP=[2,2] EP=[2,2]EP=[2,1,1] EP=[2,2]

0 2 4 6 8 10 12 14 16

EP=[2,1,1,1] EP=[2,1,1,1]EP=[2,2,1] EP=[2,2,1]EP=[2,1,1,1] EP=[2,2,1]

0 2 4 6 8 10 12 14 16

EP=[3,1,1] EP=[3,1,1]EP=[3,2] EP=[3,2]EP=[3,1,1] EP=[3,2]

FIG. 5. Matrix elements of the operators a†mam evaluated on trial wave functions (25) describing states having the sameangular momentum eigenvalue and belonging to the same energy branch but different sub-branches. In particular: panel a)shows results for EJs with EPs [2, 1, 1] and [2, 2], panel b) concerns EJs with EPs [2, 1, 1, 1] and [2, 2, 1] and finally panel c)reports a†mam matrix elements evaluated by considering EJs with EPs [3, 1, 1] and [3, 2]. In all cases the off-diagonal matrixelements in the Jack basis are much smaller than the diagonal ones and also characterized by an oscillating dependence on m.Both these features lead to very small values once averaged on the different m’s and provide an explanation for the unexpectedenergy crossing with no mixing that take place when passing from our realistic potential regime to the extremely steep limitof [31].

in Fig.4(a), the rescaled expectation value,

〈φη|Hconf |φµ〉√[〈φη|Hconf |φη〉〈φµ|Hconf |φµ〉]

(29)

takes respectively the (very small) values 0.0033, 0.0202and 0.0119 for the φη and φµ wave functions correspond-ing to the Jacks considered in the three panels of Fig.5.

C. Fine structure of the spectrum

In the previous subsections we have proposed a hier-archical classification of the states in terms of the firstentries of the EJ root partition. While this criterionallows to classify the order in energy of the differentbranches and sub-branches, it does not offer much in-sight on the physical mechanisms underlying the relativeenergy of states within a given branch or sub-branch. Asthe derivative of the energy dispersion with respect to an-gular momentum determines the angular speed of a per-turbation propagating along the edge of the disk-shapedcloud, an experimental measurement of the surface dy-namics and its collective modes may provide a valuableinsight on the nature of the edge excitations and on its de-viation from standard chiral Luttinger liquid theory [32].

A complete physical interpretation of the numericalresults is a very complicate task, in this section we mainlyfocus our attention on the possibility of having the single-QH excitation state

ψ1−qh(zi, ξ = 0) ∝(∏

i

zi

)ψL(zi). (30)

as the first excited state above the Laughlin groundstate. The interest of this specific feature stems from itsmarked deviation from the chiral Luttinger liquid theoryof edge states (which predicts a monotonical increase ofthe eigenenergies as a function of ∆L) as well as from thepotential utility of a massive population of QH states inview of studies of anyon physics. In Fig.3 d), we illustratea set of confinement parameters for which this is indeedthe case.

This peculiar energetic behavior of the lowest energyexcited states in the weak confinement regime can beexplained by looking at the expectation value of occu-pation numbers nm ≡ 〈ψ|a†mam|ψ〉 taken on trial wavefunctions corresponding to states in the k = 1 energybranch. While for stronger confinement potentials oneshould take into account the reorganization of the stateswithin the manifold of non-interacting states as well asthe possible mixing with quasi-particle states above theLaughlin gap, such a simple analysis based on the oc-cupation numbers is expected to be accurate at a linearresponse level in the weak confinement limit.

The distribution of the occupation numbers nm amongdifferent m single particle orbitals for growing total an-gular momentum in the first k = 1 branch of excitedstates is shown in Fig. 6. As we increase ∆L, the broadcentral peak of occupation visible around m ' N = 6 forthe ∆L = 0 Laughlin state slowly moves towards lowerm’s while becoming less pronounced. At the same time,another peak appears at the high-m edge of the distri-bution and moves towards lower m while becoming morepronounced until it transforms into a central peak in thesingle-QH state. Remarkably, the m-distribution of thislatter state is quite similar to the one of the Laughlinstate, just shifted by one unit of m. Even though the

11

0 5 10

0

0.2

0.4

0.6

0.8

1

1.2Laughlin state

0 5 10

0

0.2

0.4

0.6

0.8

1

1.2L=31 EP=[1]

0 5 10

0

0.2

0.4

0.6

0.8

1

1.2L=32 EP=[1,1]

0 5 10

0

0.2

0.4

0.6

0.8

1

1.2L=33 EP=[1,1,1]

0 5 10

0

0.2

0.4

0.6

0.8

1

1.2L=34 EP=[1,1,1,1]

0 5 10

0

0.2

0.4

0.6

0.8

1

1.2L=35 EP=[1,1,1,1,1]

0 5 10

0

0.2

0.4

0.6

0.8

1

1.2Quasi-hole state

0 5 10

0

0.2

0.4

0.6

0.8

1

1.2L=34 EP=[2,2]L=34 EP=[2,1,1]

L=34 2/3 EP=[2,1,1]L=34 2/3 EP=[2,2]L=34 1/2 EP=[2,1,1]L=34 1/2 EP=[2,2]

FIG. 6. Panels from a) to g) show the average occupationnumbers nm ≡ 〈ψ|a†mam|ψ〉 of single-particle LLL orbitalscalculated on Jack trial wave functions for states in the k = 1energy branch [38]. The markedly different shapes of nmshown in panels a)-g) for the different states within the lowestk = 1 energy branch can be used to explain their ordering inenergy. In panel h) we compare the nm of states within thek = 2 energy branch as predicted by Jack trial wave functionswith different β = 2/3 and ν = 1/2 parameters: the differ-ences are minor and the global behavior of nm as functionof m turns out to be mostly determined by the EP. In par-ticular, trial wave functions with EP’s [2, 2] and [2, 1, 1] arecharacterized by very different values of nm for m = 11, 12.Together with the fact that the potential energy is dominatedby the highest-m Um, this explains the different ordering ofsub-branches found in our calculations compared to the ex-tremely steep limit of [31].

features of the m-distribution shift towards small m’s forgrowing ∆L, the overall center of mass of the distributiongrows as expected as ∆L.

From these curves, we can notice that values of the

highest m’s occupation numbers - i.e. n9, n10 and n11 -are very similar for the single-QH state and for the neigh-boring Jack with EP η = [1, 1, 1, 1, 1] and L = 35. Actu-ally those occupation numbers are slightly lower for thislatter state. This explains why for large enough Rext theenergy shift due to the HW potential is almost the samefor these two states and in particular why the L = 35edge state is -by a short difference- the first excited state.Indeed for very large values of Rext all contributions to(7) from lower m states can be neglected and the occupa-tion numbers of the highest m state fully determine theconfining energy, as discussed in detail in [31].

On the other hand, it is also possible to obtain spectrain which the single-QH state is the first excited one, asshown in Fig. 3 d). This happens when -at fixed Vext- weslightly reduce the HW radius Rext and can be attributedto the fact that the η = [1, 1, 1, 1, 1] Jack polynomialhas higher nm=8,9 values than the single-QH state. Asa result, for lower values of Rext the energy contributiongiven by these m’s may overcome the one relative to thehigher m’s, so that the energy shift caused by the HWconfinement on the L = 35 edge state may be biggerthan that on the single-QH state. Fig. 6 shows alsothat for m < 7 the single-QH state has values of theoccupation numbers which are higher than those of theη = [1, 1, 1, 1, 1] Jack polynomial. Therefore, in light ofwhat we have just said, we expect that for even smallervalues of Rext the single-QH state will cease to be thefirst excited one in favor of lower angular momentumedge states.

This physics is summarized in the plots of the Rext-dependence of the energies of the Laughlin state and ofits QH and edge excitations that are shown in the upperpanel of Fig.7 : for a fixed value of Vext there exists infact a finite interval of Rext values in which the single-QH state is indeed the first excited one. At the sametime, it is important to stress the fact that despite forthese trap parameters the EE energies are not negligiblerespect to the Laughlin gap (0.1V0), the overlap betweennumerical eigenstates and η = [1, . . . ] Jack polynomialsremain extremely high [see Fig.7, lower panel] meaningthat the first excited state is really the single-QH one.

Also the energetic behavior characterizing the otheredge states - i.e. those with 32 < L < 35 - can be ex-plained in terms of the occupation numbers plotted inFig. 6. Indeed we can note that Jacks describing statesin the k = 1 energy branch associated with increasing val-ues of the additional angular momentum ∆L, are char-acterized by occupation numbers which show peaks atlower m’s. As for sufficiently large Rext, the dominantcontribution to Hconf comes from large m terms corre-sponding to larger Um values, it is completely reasonablethat higher ∆L states have lower energies, in agreementwith [31]. Quite strikingly, note that a decreasing en-ergy with ∆L means that wavepackets of edge excita-tions propagate backwards with respect to the cyclotronorbits, in disagreement with the usual chiral Luttingerliquid theory.

12

4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 5.60.975

0.98

0.985

0.99

0.995

1

1.005

Laughlin state

L = 31 edge state

L = 32 edge state

L = 33 edge state

L = 34 edge state

L = 35 edge state

Quasi-hole state

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

4.76 4.765

0.03

0.0305

0.031

0.0315

FIG. 7. Upper panel: energetic behavior of the Laughlinstate and its single-QH and edge excitations as function of theHW radius Rext for fixed values of the HW strength Vext =10V0 and of the particle number N = 6. The inset showsthe existence of an interval in which the single-QH state isthe first excited state. Lower panel: overlap between theexact numerical eigenstates and the Jack trial wave functionsdescribing the 1st energy branch. Despite for large values ofRext the overlap between the lowest energy L=36 eigenstateand the single-QH wave function is of the order of 1, it remainsvery high also for Rext values for which the L=36 state is thefirst excited one.

Despite this explanation is in complete agreement withwhat we observe for L ≥ 32, it seems to be odd withthe energetic behavior of the L = 31 Hamiltonian eigen-state corresponding to the global dipole-like motion ofthe cloud. The Jack associated with such a states has EPη = [1] and it has the highest occupation of the m = 11LLL wave function. However, this Jack has the peculiar-ity of having all the other high m occupation numberslower than those of both the single-QH state and theη = [1, 1, 1, 1, 1] Jack, which explains the observation ofenergies for the L = 31 state which are similar to theones of the L = 35 and of the single-QH states.

Starting from these intriguing numerical results, on-

going work is trying to understand the physical originof the deviations from the chiral Luttinger liquid theory,so to disentangle finite-size effects and highlight the in-teresting edge physics, including nonlinear effects in theedge dynamics that were anticipated in [42] and may beresponsible for the mixing of different quantum states ofthe Luttinger liquid.

D. (In)compressibility of the states

As a further interesting feature of the Laughlin stateand of its low-lying excited states, it is interesting to com-plete our study with a short discussion of their responseto an increase of the HW potential strength Vext as away to measure their compressibility. In view of futureexperimental studies particular, this strategy may pro-vide access to one of the most celebrated properties ofFQH liquids.

As one can observe in Fig. 8, the energies of the Laugh-lin state and of the quasi-particle (QP) state grows al-most linearly in the external potential strength, confirm-ing the expected incompressible behavior.

On the other hand, the energies of QH excited statesgrows less than linearly as a function of Vext, which wit-ness the ability of these states to rearrange themselvesin response to the confinement. This is manifestation oftheir finite compressibility of the state. As expected, the

0 1 2 3 4 5 65

5.002

5.004

5.006

5.008

5.01

5.012

5.014

5.016

5.018Laughlin stateLaughlin state fitQH stateQH state fit2-QH state2-QH state fitQP stateQP state fit

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

FIG. 8. Solid lines: energies of the Laughlin state and its QHand QP excitations as function of the HW potential strengthVext for fixed values of the potential radius Rext = 4.65

√2 lB

and of the number of particles N = 5. Dashed lines are linearfits: the more accurate this fit, the weaker the compressibilityof the state. As expected, the compressibility dramaticallyincreases when QH’s are inserted in the fluid, while it is notaffected by the initial presence of QP’s.

13

FIG. 9. Behavior of the Laughlin state (first row), the single-QH state (second row) and the single-QP state (third row) densityprofiles as function of the HW radius Rext for fixed potential strength Vext = 100V0 and particle number N = 6. As we cansee, more we reduce the potential radius Rext - from left to right -, more the densities of the Laughlin state and its excitationsdecrease at the center. Densities are normalized such that their spatial integrals over the whole 2D plane recovers the totalnumber of particles N .

larger the number of QHs, the stronger this compressibil-ity. A similar compressible behavior is also found for EEstates (not shown in the figure).

While these results are restricted to relatively weakconfinement potentials that are not able to generate amassive number of extra quasi-particles in the fluid, inthe next Section we will see how a strong compressionof the cloud is able to distort the density profile of thecloud in quite unexpected ways.

V. STRONG CONFINEMENT REGIME

After having discussed the weak confinement regimewhere the physics takes place within the non-interactingstate manifold, we now turn our attention to the strongconfinement regime where significant mixing with quasi-particle states above the Laughlin gap can occur and thedensity distribution of the FQH droplet is sizably com-

pressed in space.

With no loss of generality, we focus on HW potentialstrengths Vext of the same order as before, but muchsmaller disk radii Rext. For such high values of Vext, thestrong confinement condition can be reached as soon asRext & Rcl. The study of this case exhausts the range ofconfinement regimes and completes the physics of a FQHliquid confined in a HW potential.

For sufficiently strong confinements, the lowest en-ergy many-body states have angular momentum eigen-values lower than the ν = 1/2 Laughlin state one LL =N (N − 1): Even though for no (or weak) confinementthe energies of all eigenstates of low angular momentumL < LL lie above the bulk Laughlin gap, the confine-ment potential has in fact a much weaker effect on thesestates than on the spatially more extended states of theLaughlin family, so their relative order in energy can beswapped.

This physically expectable result is accompanied by

14

the surprising behavior of the density profiles of theeigenstates that is illustrated in Fig. 9 for the Laugh-lin state and its QH and QP excited states: the moreone squeezes the system by reducing Rext, the more thevalue of the density at the center decreases. This appar-ently counterintuitive behavior can be explained if onetakes into account the invariance under rotation of theHamiltonian and the associated conservation of the totalangular momentum.

Each eigenstate of the Hamiltonian can in fact be writ-ten as a linear combination of elements of the occupationnumber basis (19) sharing the same angular momentum.As a result, a large population in high m orbitals mustbe compensated by a high population in the low m or-bitals as well. Since the effect of the confinement poten-tial is strongest on the high-m orbitals, minimization ofthe confinement energy leads to a reorganization of theeigenstates in favor of those configurations that show areduced occupation of high-m orbitals and, consequently,of low-m orbitals as well. As the density at the center ofthe cloud is mostly determined by low-m states, the me-chanical compression of the droplet from outside leads toa marked depletion of the central region as well, as visiblein the r ≈ 0 region of the right-most panels of Fig. 9.

VI. CONCLUSIONS

In this work we have made use of numerical exact diag-onalization calculations to study the effect of a realistichard-wall confinement on a ν = 1/2 fractional quantumHall droplet of bosons with contact interactions.

We have found that the physics of such systems is farricher than that of harmonically confined systems andof HW models in the extremely steep limit consideredin [31]. The more general and realistic potential we haveconsidered removes the degeneracy between edge excita-tions with the same angular momentum, so that many-body states organize themselves in energy branches thatcan be interpreted in terms of Jack polynomials. Theenergy ordering of states can be controlled via the po-tential parameters in a much wider way than originallyexpected on the basis of a chiral Luttinger liquid theory.In particular, parameters for which the single-quasi-holestate is the first excited state above the Laughlin groundstate have been identified and level crossings unexpect-edly with no mixing have been found. Under a strong

confinement potential, the density profile of the dropletresults dramatically modified with the surprising appear-ance of a density depression at the center. Interestingexperimental consequences of our theoretical results havealso been discussed.

Together with the extremely precise one-to-one corre-spondence between the energy eigenstates and the Jackstrial wave functions, these features make a weak HWconfinement one of the most appealing regimes whereto perform experimental investigation of fractional quan-tum Hall physics in atomic of photonic systems. Ultra-cold atoms in different traps and photons in suitably de-signed cavities or cavity arrays are in fact the subjectof intense studies as quantum simulators of many-bodyphysics as well as prospective candidates where to ob-serve and study advanced topological many-body states,potentially with quantum information applications.

From a theoretical point of view, the description ofeigenstates in terms of Jack polynomials appears to bea powerful tool in view of larger scale numerical calcu-lations, but may also contribute to the physical under-standing of unexpected features such as the the energyanticrossing discussed in IV B. Furthermore, it could helpoptimization of protocols to generate fractional quantumHall states in a driven-dissipative context as recently pro-posed in [23], to understand the physical origin of themarked deviation of the numerically observed spectrafrom the chiral Luttinger liuqid theory, to unravel theresponse of FQH droplets to time-dependent potentials,as well as to investigate more complex ring geometriesin which excitations of the inner and outer edges canbe strongly mixed and quasi-hole tunneling phenomenacan occur. All these issues are the subject of on-goingresearch.

VII. ACKNOWLEDGMENTS

Continuous discussions and early stage technical sup-port from R. O. Umucalılar are warmly acknowledged.This work was supported by the EU-FET Proactive grantAQuS, Project No. 640800, and by the AutonomousProvince of Trento, partially through the project “Onsilicon chip quantum optics for quantum computing andsecure communications” (“SiQuro”). IC acknowledgesthe Kavli Institute for Theoretical Physics, University ofCalifornia, Santa Barbara (USA) for the hospitality andsupport during the early stage of this work.

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