Hard-Wall Confinement of a Fractional Quantum Hall Liquid
HHard-Wall Confinement of a Fractional Quantum Hall Liquid
E. Macaluso and I. Carusotto INO-CNR BEC Center and Dipartimento di Fisica, Universit ` a di Trento, 38123 Povo, Italy (Dated: September 25, 2017)We make use of numerical exact diagonalization calculations to explore the physics of ν = 1 / 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 /r FQH 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 / a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p 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 THEORETICALMODELA. The model Hamiltonian
We consider a 2D system described by the Hamiltonian H = H + H int + H conf , where H = N (cid:88) i =1 ( p i + A ) M (1)is the kinetic energy of particles experiencing an effectiveorthogonal magnetic field B = ∇ × A = B ˆ e z , H int = (cid:88) i 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 a m , which respectively create andannihilate particles in the LLL state of angular momen-tum m and real-space wave function ϕ m ( r, φ ) = 1 l B √ πm ! e imφ (cid:18) r √ l B (cid:19) m e − r l B (4)with magnetic length l B = (cid:112) (cid:126) c/B .This approximation is valid if we restrict to the low-energy physics of the system and we assume that thecyclotron energy (cid:126) ω C = (cid:126) B/m is far larger than all otherenergy scales set by the potential energy V ext and thecharacteristic interaction energy V ≡ g int / l B . Withinthis approximation the Hamiltonian terms can be writtenin second-quantization terms as: H = ε (cid:88) m a † m a m (5) H int = g int πl B (cid:88) αβγρ Γ( α + β + 1) √ α ! β ! γ ! ρ ! δ ( α + β,γ + ρ ) ( α + β +2) a † α a † β a γ a ρ (6) H conf = (cid:88) m U m a † m a m = (cid:88) m V ext m ! γ ↑ (cid:18) m + 1 , R ext l B (cid:19) a † m a m (7)where ε = (cid:126) ω C / t ) denotes the Euler gamma functionΓ( t ) ≡ (cid:90) ∞ x t − e − x d x, (8)whereas γ ↑ ( t, R ) is the so-called upper incomplete gammafunction γ ↑ ( t, R ) ≡ (cid:90) ∞ R x t − e − x d x. (9)Note that since we are neglecting excitation to higherLandau levels, all N -particle states have the same kineticenergy ε N , which effectively drops out of the problem.As a consequence we neglect the kinetic term and wefocus on the Hamiltonian ˜ H ≡ H − H = H int + H conf . B. The confinement potential Within the LLL approximation, the confinement po-tential is summarized by its value U m on each single-particle state of angular momentum m . This quantity isplotted in Fig. 1 for a few choices of V ext and R ext . In therecent work [31], this dependence was assumed to fulfillthe condition U m − (cid:28) U m (cid:28) U m +1 , but in practice this prova prova dss R ext = 5 V ext = 10R ext = 4.8 V ext = 10R ext = 4.8 V ext = 2 -3 -2 -1 FIG. 1. Angular momentum-dependence of the confinementpotentials U m . 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 U m ’s does not fulfill the extremelysteep condition. condition does not appear to be so simple to satisfy witha realistic potential.For sufficiently large R ext , we can in fact approximate U m (cid:39) V ext m ! (cid:18) R ext l B (cid:19) m exp (cid:18) − R ext l B (cid:19) , (10)so that having a large ratio U m U m − (cid:39) R ext ml B (11)requires a wide disk radius R ext (cid:29) √ ml B , much largerthan the FQH droplet we intend to prepare. On the otherhand, for √ ml B (cid:38) R ext , the potential U m smoothlyapproaches its limiting value V ext .Given the exponential factor in (10), simultaneouslyhaving an overall appreciable potential U m and a verysteep m -dependence (11) requires a very large potentialstrength V ext : 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 ∆ U m / U m for a smallvariation ∆ R ext can be estimated to be a large number (cid:12)(cid:12)(cid:12)(cid:12) ∆ R ext R ext (cid:20) m − R ext l B (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) (cid:39) R ext l B ∆ R ext l B . (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 | n , n , n , . . . (cid:105) number states with n m =0 , , . . . , 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 m max 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. V ext = 0 and/or R ext → ∞ -, 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 the ν = 1 / V – the so-called Laughlin gap.In more detail, the ν = 1 / L L = N ( N − 1) amongall the states forming such a degenerate ground state andis described by the celebrated wave function: ψ L ( { z i } ) ∝ (cid:89) i 24 26 28 30 32 34 366.0066.0086.016.0126.0146.0166.018 Laughlin stateQuasi-particle state Edge excitations Singlequasi-holestate0.180.160.060.080.10.120.14 FIG. 2. Spectrum of many-body states of a N = 6 parti-cle system experiencing a harmonic confinement of frequency (cid:126) υ = 2 . × − V . 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 V . additional angular momentum ∆ L . The degeneracy ofsuch states is given by the number of partitions of theinteger ∆ L .For low additional angular momentum ∆ L (cid:28) 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 / n QH excitations at positions ξ , . . . , ξ n is de-scribed by the wave function: ψ n − qh ( { z i } , { ξ i } ) ∝ (cid:18) N (cid:89) i =1 n (cid:89) j =1 ( z i − ξ j ) (cid:19) ψ L ( { z i } ) . (15)In the following, in order to preserve radial symmetry,we will restrict ourselves to states presenting QHs in theorigin ξ i = 0 for all i = 1 , . . . , n , so that ψ n − qh, ( { z i } , { ξ i = 0 } ) ∝ (cid:18) (cid:89) i z ni (cid:19) ψ L ( { z i } ) . (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 = υL z 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 / 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 / J αλ 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 = [ λ , λ , . . . ], 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-called Laplace-Beltrami operator [39]: H αLB = (cid:88) i ( z i ∂ i ) + 1 α (cid:88) i 1] corresponds to the sym-metrized monomial M [4 , , , ≡ Sym( z z z z ). In thisperspective, Jacks have a peculiar expansion in terms ofsymmetrized monomials M µ ’s: J αλ = M λ + (cid:88) µ (cid:31) λ c λµ ( α ) M µ , (18)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 [ . . . , λ i , . . . , λ j , . . . ] togenerate another one - called descendant - of the form[ . . . , λ i − δm, . . . , λ j + δm, . . . ] (with λ i − δm ≥ λ j + δm ).The corresponding coefficients c λµ ( 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 k particles 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 − andsuitable root partition [34]. Among these bosonic FQHstates, we focus here on the ν = 1 / k = 1 , r = 2) and the wave function (13) turnsout to be given by the Jack polynomial with α = − λ = [2 N − , N − , . . . , m corresponds to a monomial z m .So, any bosonic many-particle state | n , n , n , . . . (cid:105) ob-tained by occupying each LLL wave function of angularmomentum m with a well-defined number n m particlescan be described as a symmetrized monomial M λ , where λ ≡ [ λ , λ , . . . ] 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 ≥ m = 4 orbital, two in the m = 2 orbital, one in the m = 1 orbital and the remaining in the m = 0 orbitalcan be written as |N − , , , , (cid:105) = Sym( z z z z ) ≡M [4 , , , . A. Laughlin states and their excitations in terms ofJacks Based on this connection, all rotationally symmetriceigenstates of the Hamiltonian | ψ (cid:105) = (cid:88) i c i | n , n , n , . . . (cid:105) i , (19)correspond to homogeneous symmetric polynomials inthe particle coordinates of degree equal to the well-defined total angular momentum L = (cid:88) m m ¯ n m , (20)where ¯ n m ≡ (cid:104) ψ | a † m a m | ψ (cid:105) . 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 ( z i − z j ) when the i -th and the j -th particles approach each other. In particular, onecan expand (cid:81) i 2] + [2] = [12 , , , , 2] and λ b = Ω + η b = [10 , , , , 2] + [1 , 1] = [11 , , , , m occupied orbitals into orbitalswith even higher single-particle angular momenta. Inthe concrete example above, λ a is obtained by movingone particle from the m = 10 orbital to the m = 12orbital, while λ b 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 = − n -QH wavefunction (16) is given by | n . . . (cid:105) , which can beobtained starting from the Laughlin one by moving eachparticle from the m orbital to the m + n orbital. The rootpartition corresponding to such a configuration is there-fore λ n ≡ Ω + κ n , where Ω denotes again the root parti-tion associated with the ν = 1 / n-QH partition κ n is the sequence obtained by repeating N times the same integer n , i.e. κ n ≡ [ 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 J α Ω+ η = J βη J α Ω , (21)in which α = − r − , β = r +1 and J α Ω denotes the Jackrepresenting the ν = 1 /r Laughlin wave function. In ourspecific case of ν = 1 / β = 2 / S ( { z i } ) in (14) in terms of Jacks. Thepolynomial S ( { z i } ) can in fact be expanded in the ba-sis of Jacks with Jack parameter β = 2 / η ’s. Each of these edge state wave functions J βη ψ L { z i } can then be written as a single Jack using(21) and therefore they can be easily constructed usingthe expansion (18), J − η = J / η J − = J / η ψ L ( { z i } ) . (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 / S ( { z i } ) pre-factors on thedifferent basis of EJs of the form J β = νη , where ν = 1 /r is 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, (cid:10) J νη (cid:12)(cid:12) J νµ (cid:11) = j µ ( ν ) δ η,µ for N → ∞ , (23)under the Laughlin scalar product (cid:104) φ | χ (cid:105) ≡ (cid:90) C N d { z i } [ φ ( { z i } )] ∗ χ ( { z i } ) | ψ L ( { z i } ) | , (24)the associated edge state wave functions turn out to bealso orthogonal in the limit of N → ∞ . While use ofthese β = ν EJs J νη 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 = 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 V ext and R ext . 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 restrictto the R ext (cid:38) R cl case, where R cl (cid:39) (cid:112) N /ν √ l B denotesthe semiclassical radius of a N -particle FQH droplet withfilling factor ν . While this assumption guarantees thatthe U m 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 / 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 25 30 35 40 4566.0026.0046.0066.0086.016.012 Quasi-particlestate Twoquasi-holestateSinglequasi-holestateLaughlin stateEdgeexcitations 25 30 35 40 4566.0026.0046.0066.0086.016.012 Laughlin stateEdgeexcitationsQuasi-particlestate Singlequasi-holestate Twoquasi-holestate25 30 35 40 455.99866.0026.0046.0066.0086.016.012 Laughlin state Single quasi-hole state Twoquasi-holestateQuasi-particlestate Edgeexcitations 24 26 28 30 32 34 36 38 4066.0026.0046.0066.0086.016.0126.0146.0166.018 Laughlin stateQuasi-particle state Singlequasi-holestateEdge excitations00.020.040.060.080.10.12 00.020.040.060.080.10.12-0.0200.020.040.060.080.10.12 00.020.040.060.080.10.120.140.160.18 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 V ext = 20 V and R ext = 4 . √ l B and with V ext = 100 V and R ext = 5 . √ l B , respectively. In both cases the ground state is given by the Laughlin one and for a fixed value of ∆ L different 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 with V ext = 30 V and R ext = 5 . √ l B : in this case all low-lying excited states up to L = 34 have energies below the bulk excitationgap so that the first four energy branches are visible. Panel d) corresponds to a stronger HW confinement with V ext = 100 V and R ext = 4 . √ l B 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- 30 35 40 4566.0016.0026.0036.0046.0056.0066.0076.0086.0096.01 0 1 2 3 4 5 6 700.020.040.060.080.10.120.140.160.180.2 L=34 2nd excitedL=34 3rd excitedJack EP=[2,1,1]Jack EP=[2,2] L=35 2nd excitedL=35 3rd excitedJack EP=[2,1,1,1]Jack EP=[2,2,1] L=35 4th excitedL=35 5th excitedJack EP=[3,1,1]Jack EP=[3,2] L=36 5th excitedL=36 6th excitedL=36 7th excitedJack EP=[3,1,1,1]Jack EP=[3,2,1]Jack EP=[3,3] L=38 13th excitedJack EP=[4,4] L=38 13th excitedJack EP=[4,4]L=38 13th excited L=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] L=38 13th excitedL=38 13th excited Vext=15V0Jack 2/3 EP=[4,4]Jack 1/2 EP=[4,4] L=35 2nd excitedL=35 3rd excitedJack 2/3 EP=[2,1,1,1]Jack 2/3 EP=[2,2,1] L=35 4th excitedL=35 5th excitedJack 2/3 EP=[3,1,1]Jack 2/3 EP=[3,2] L=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 V ext = 30 V and R ext =5 . √ l B 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 / β = 2 / V ext = 15 V .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 = L L + 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 φ η ( { z i } ) = J / η ψ L ( { z i } ) , (25)and the k -th energy branch contains all states whose wavefunctions can be obtained by multiplying ψ L ( { z i } ) byJacks with partitions of the form η = [ η , η , η , . . . ] = [ k, η , η , . . . ] . (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 = L L + | 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 η ≥ η ≥ · · · ≥ η N . (27)This implies that the maximum additional angular mo-mentum that can be obtained by considering EPs with η = k is ∆ L = N k 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 Ω + 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 inenergy can be related to the value of the second element η 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 = [ η , η , η , . . . ] = [ k, j, η , . . . ] , (28)with the exception of the 1-st sub-branch in which thereis also the Jack with partition η = [ k ] having η = k andno η . 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 η i -, 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 R ext → ∞ or V ext → 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 η have increasing energies. For instance among the two L = 34 eigenstates in the second energy branch [see Fig. EP (cid:10) J βη ψ L (cid:12)(cid:12) Ψ (cid:11) (cid:10) J νη ψ L (cid:12)(cid:12) Ψ (cid:11) (cid:10) J βη ψ L (cid:12)(cid:12) J νη ψ L (cid:11) η = [1 , , , 1] 0.9991 0.9991 1.00000.9994 0.9994 η = [2 , , 1] 0.9783 0.9936 0.98780.9798 0.9953 η = [2 , 2] 0.9896 0.9902 0.99080.9914 0.9922 η = [4 , 4] 0.9473 0.9501 0.94520.9669 0.9733TABLE I. Overlaps of the numerical ED eigenstates | Ψ (cid:105) withthe corresponding trial wave functions (22) and (25) for asystem of N = 6 particles. Values in black refer to a HWconfinement of parameters R ext = 5 . √ l B and V ext = 30 V while those in red to the R ext = 5 . √ l B and V ext = 15 V case. η = 1 sub-branch has a lower energythan the one in the η = 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 η 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 η 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 L z 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 † m a m operators 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 U m are considered. For in-stance, for the realistic confinement potential considered0 EP=[2,1,1] EP=[2,1,1]EP=[2,2] EP=[2,2]EP=[2,1,1] EP=[2,2] 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] 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 † m a m 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] and [2 , , , , 1] and [2 , , 1] and finally panel c)reports a † m a m matrix elements evaluated by considering EJs with EPs [3 , , 1] and [3 , 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, (cid:104) φ η | H conf | φ µ (cid:105) (cid:112) [ (cid:104) φ η | H conf | φ η (cid:105)(cid:104) φ µ | H conf | φ µ (cid:105) ] (29)takes respectively the (very small) values 0 . . . φ η and φ µ 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 ψ − qh ( { z i } , ξ = 0) ∝ (cid:18) (cid:89) i z i (cid:19) ψ L ( { z i } ) . (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 ¯ n m ≡ (cid:104) ψ | a † m a m | ψ (cid:105) 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 ¯ n m 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 (cid:39) N = 6 forthe ∆ L = 0 Laughlin state slowly moves towards lower m ’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 the1 Quasi-hole state 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 ¯ n m ≡ (cid:104) ψ | a † m a m | ψ (cid:105) of single-particle LLL orbitalscalculated on Jack trial wave functions for states in the k = 1energy branch [38]. The markedly different shapes of ¯ n m shown in panels a)-g) for the different states within the lowest k = 1 energy branch can be used to explain their ordering inenergy. In panel h) we compare the ¯ n m of states within the k = 2 energy branch as predicted by Jack trial wave functionswith different β = 2 / ν = 1 / n m 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] arecharacterized by very different values of ¯ n m for m = 11 , m U m , 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. ¯ n , ¯ n and ¯ n -are very similar for the single-QH state and for the neigh-boring Jack with EP η = [1 , , , , 1] and L = 35. Actu-ally those occupation numbers are slightly lower for thislatter state. This explains why for large enough R ext 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 R ext 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 V ext - weslightly reduce the HW radius R ext and can be attributedto the fact that the η = [1 , , , , 1] Jack polynomialhas higher ¯ n m =8 , values than the single-QH state. Asa result, for lower values of R ext 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 < η = [1 , , , , 1] Jack polynomial. Therefore, in light ofwhat we have just said, we expect that for even smallervalues of R ext 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 R ext -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 V ext there exists infact a finite interval of R ext 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 . V ), 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 R ext , the dominantcontribution to H conf comes from large m terms corre-sponding to larger U m 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.2 FIG. 7. Upper panel: energetic behavior of the Laughlinstate and its single-QH and edge excitations as function of theHW radius R ext for fixed values of the HW strength V ext =10 V 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 of R ext 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 R ext 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 EP η = [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 the η = [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 V ext 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 V ext , 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 Laughlin stateLaughlin state fitQH stateQH state fit2-QH state2-QH state fitQP stateQP state fit FIG. 8. Solid lines: energies of the Laughlin state and its QHand QP excitations as function of the HW potential strength V ext for fixed values of the potential radius R ext = 4 . √ l B 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. 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 R ext for fixed potential strength V ext = 100 V and particle number N = 6. As we cansee, more we reduce the potential radius R ext - 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 V ext of the same order as before, but muchsmaller disk radii R ext . For such high values of V ext , thestrong confinement condition can be reached as soon as R ext (cid:38) R cl . 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 / L L = N ( N − L < L L 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 by4the 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 R ext , 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 ≈ VI. 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