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Local exclusion for intermediate and fractional statistics Douglas Lundholm1,∗ and Jan Philip Solovej2,∗ 1Institut Mittag-Leffler, Aurav¨agen 17, SE-182 60 Djursholm, Sweden 2Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark A local exclusion principle is observed for identical particles obeying intermediate/fractional ex- change statistics in one and two dimensions, leading to bounds for the kinetic energy in terms of the density. This has implications for models of Lieb-Liniger and Calogero-Sutherland type, and impliesanon-triviallower boundfortheenergyoftheanyongas wheneverthestatistics parameter is an odd numerator fraction. We discuss whether thisis actually a necessary requirement. 3 PACSnumbers: 05.30.Pr,03.65.Db,03.75.Hh 1 0 2 I. INTRODUCTION formulation of these models). n As a different approach, we would like to stress in the a The majority of interesting phenomena in many-body followingthatthe effects ofexclusionarealsoencodedin J quantum mechanics are in some way associated to the theLieb-Thirringinequality[11]formany-particleenergy 6 fundamental concept of identical particles and statistics. forms. For the case of identical spinless fermions in an 1 Elementary identical particles in three spatial dimen- externalpotentialV ind-dimensionalspace,itstatesthat sions are either bosons, obeying Bose-Einstein statistics, there is a uniform bound for the energy of a normalized ] h or fermions, obeying Fermi-Dirac statistics. The former N-particle state ψ: p are usually represented using wave functions which are - N−1 t symmetric under particle permutations, while the latter ψ,Hˆψ λ C V (x)1+d/2ddx, (1) n implement Pauli’sexclusionprinciple by exhibiting total h i≥ − | k| ≥ − dZ | − | a Xk=0 anti-symmetry under particle interchange. On the other u q hand, for point particles living in one and two dimen- with the N-particle Hamiltonian operator [ sions there are logical possibilities different from bosons 2 atincds [f1e–r3m].ioAnslt,hsoou-cgahllfiedrstinrteegrmareddeidataesoorffpruarcetiloynaaclasdteamtisic- Hˆ =Tˆ0+Vˆ = N (cid:18)−21∇2j +V(xj)(cid:19), v Xj=1 interest — filling the loopholes in the arguments leading 0 2 to the two standard permutation symmetries — these the conventions ~ = m = 1, V± := (V V )/2, and ± | | 5 have recently become a reality in the laboratory, with a positive constant C . The inequality (1) incorporates d 2 the adventof trapped bosonic condensates [4] and quan- Pauli’sexclusionprinciple via the intermediate sumover 5. tum Hall physics [5], and thereby the discoveries of ef- the negative energy levels λk of the one-particle Hamil- 0 fective models of particles (or quasi-particles) that seem tonian ˆh = 1 2 +V(x). It furthermore incorporates 2 toobeythese generalizedrulesforidenticalparticlesand the uncertain−ty2∇principle, andis infactequivalentto the 1 statistics. Wereferto[6–8]forextensivereviewsonthese kinetic energy inequality : v topics. Xi unAdeltrhstoouogdhinnotne-rimntseroafcstiinnggleb-opsaorntsicalendHiflebremrtiosnpsaacreeswanedll hψ,Tˆ0ψi ≥ (dd(+2/2C)d1+)22//dd Z ρ(x)1+2/dddx, (2) r operators,the samecannotbe saidaboutparticlesobey- a involving the one-particle density ρ of ψ; normalized ing these generalized interchange statistics. Namely, de- ρ(x)ddx = N. In dimension d = 3, the expression spite some effort in this direction [9, 10], many-particle quantumstates for intermediate andfractionalexchange Ron the r.h.s. of (2) may be recognized as the kinetic en- ergy approximation from Thomas-Fermi theory. It is in statistics have in general not admitted a simple descrip- this caseconjectured[11]that (2) holds with exactly the tion in terms of single-particle states restricted by some Thomas-Fermi expression on the right. The best known exclusion principle. The reason for this difficulty is that result is, however, a factor (3/π2)1/3 smaller [12]. The the general symmetry of the wave function under par- bounds (1) and (2) need to be weakened in the case of ticle interchange is naturally modeled using pairwise or weaker exclusion. In the case that each single-particle many-body interactions,hence leaving the much simpler state can be filled q times (e.g. in models with q spin realmofsingle-particleHamiltonians(andalsointroduc- states, or cp. Gentile intermediate statistics [13]) the ingothermathematicaldifficultiesaswell,alreadyatthe r.h.s. oftheinequalitites(1)resp. (2)aretobemultiplied by q resp. q−2/d. Bosons can then be accommodated by q =N, yielding trivial bounds as N . ∗ Work partly done while visiting Forschungsinstitut fu¨r Mathe- In this Letter we wish to report on→a∞new set of Lieb- matik,ETHZu¨rich(D.L.),resp. InstitutMittag-Leffler(J.P.S.). Thirring-typeinequalitiesforintermediateandfractional 2 statistics,whichfollowfromacorrespondinglocal version whichapproachonetakestoquantization[2,6,18],iden- of the exclusion principle, applicable to such interacting ticalparticlesin1Dcanagainbemodelledasbosons,i.e. systems. Ourapproachisverymuchinspiredbythework wave functions symmetric under the flip r r of any 7→ − [14] of Dyson and Lenard (see also [15]), who used only two relative particle coordinates r := x x , together j k − suchalocalformofthePauliprincipletorigorouslyprove with a local interaction potential, singular at r = 0 and thestabilityofordinaryfermionicmatterinthebulk(the either of the form δ(r) or 1/r2. We write inequalities (1) and (2) were subsequently invented by α(α 1) Lieb and Thirring to simplify their proof). Although V (r):=2ηδ(r), V (r):= − , (4) the numerical constants resulting from our method are S H r2 comparatively weak, we believe the forms of our bounds withstatisticsparametersη,α R,forthecorresponding to be conceptually very useful, and as a result we also ∈ casesofSchr¨odinger-resp. Heisenberg-typequantization. learn something non-trivial about the elusive anyon gas. These correspond to the choices of boundary conditions The majority of the results presented here were derived for the wave function ψ at the boundary r = 0 of the in full mathematical detail in [16, 17], to which we refer configuration space the interested reader for detailed reference. ∂ψ =ηψ, at r =0+, resp. ψ(r) rα, as r 0+. ∂r ∼ → II. IDENTICAL PARTICLES IN ONE AND TWO Hereη =0resp. α=0representbosons(Neumann b.c.) DIMENSIONS while η = + resp. α = 1 represent fermions (Dirich- ∞ let or analytically vanishing b.c.; see [17]). Suggested by Werecalltheconventionalmodelsforintermediateand such pairwise boundary conditions, one may define [19] fractional exchange statistics for scalar non-relativistic thetotalkineticenergyforanormalizedcompletelysym- quantum mechanical particles in one and two spatial di- metric wave function ψ describing N identical particles mensions. As mentioned in the introduction, there are on the full real line R to be T := ψ,Tˆ ψ where by now a number of standard references for their back- S/H h S/H i ground and derivations, which we will accordingly skip 1 N ∂2 here. We will mainly follow the notation in [6], with Tˆ := + V (x x ). (5) technical details addressed in [17]. S/H −2Xj=1 ∂x2j 1≤jX<k≤N S/H j − k Identicalparticlesin2D,anyons,havethepossibilityto pick up an arbitrary but fixed phase eiαπ upon continu- In other words, the Schr¨odinger case is nothing but oussimpleinterchangeoftwoparticles[2,3]. Astandard the Lieb-Liniger model for one-dimensional bosons with waytomodelsuch(abelian)anyons,intheso-calledmag- pairwise Dirac delta interactions [20], while the Heisen- netic gauge, is by means of bosons in R2 together with berg case corresponds to the homogeneous part of the a statisticalmagnetic interactiongivenby the vectorpo- Calogero-Sutherland model with inverse-square interac- tential tions [21]. For the following results we will restrict to η 0 (Schr¨odinger type intermediate statistics) and (x x )I ≥ A =α j − k , α R (mod 2), α 1 (Heisenberg type ‘superfermions’) for which the j Xk6=j |xj −xk|2 ∈ sta≥tistics potentials (4) are nonnegative, i.e. repulsive. It is useful to be able to speak about the expected wherexI denotesa90◦ counter-clockwiserotationofthe kinetic energyof a wavefunction on some localregionQ vector x. This attaches to every particle an Aharonov- ofspace(typically acube inthe following). For fermions Bohm point flux of strength 2πα, felt by all the other or bosons we naturally define this quantity to be particles. The kinetic energyfor N suchparticles is thus given by T := ψ,Tˆ ψ , 1 N A h A i N T0Q := 2Xj=1ZRdN |∇jψ|2χQ(xj)dx, 1 Tˆ := D2, (3) A 2 j where χQ 1 on Q and χQ 0 on the complement Qc. Xj=1 Analogous≡ly for anyons, ≡ where D = i +A , and ψ is represented as a com- j j j N pletely symm−et∇ric square-integrable function on (R2)N. TQ := 1 D ψ 2χ (x )dx, The case α = 0 then corresponds to bosons, and α = 1 A 2Xj=1ZR2N | j | Q j to fermions. The case of identical particles confined to move in and for Schr¨odinger and Heisenberg type intermediate only one spatial dimension is special and in some sense statistics, TQ := S/H degenerate, since particles cannot be interchanged con- tinuously without colliding. In quantum mechanics this N 1 nweacveessfituantcetsiosnomaet cthhoeicceoollfisbioonunpdoairnytsc.ondDietpioennsdfinorgtohne 2Xj=1ZRN |∂jψ|2+Xk6=jVS/H(xj −xk)|ψ|2χQ(xj)dx.   3 Note that if the full space Rd has been partitioned into Wecallthefollowingobservationalocalexclusionprin- a family of non-overlapping regions Q then the total ciple for generalized exchange statistics since it implies k { } kinetic energy decomposes as T = TQk . thatthelocalkineticenergyisnonzerowheneverwehave 0/S/H/A k 0/S/H/A morethanoneparticle,andhencethattheparticlescan- We will furthermore write T = T to dePnote the free F 0 kinetic energy for the particular case of fermions in R3. not occupy the same single-particle state (which on a local region would be the zero-energy ground state). Lemma 1 (Local exclusion principle) Given any fi- III. LOCAL EXCLUSION nite interval Q R of length Q, we have for η 0 ⊂ | | ≥ The starting point for our energy bounds will be the ψ¯Tˆ ψdx (n 1)ξS(η|Q|)2 ψ 2dx, (9) following local consequence of the Pauli exclusion prin- Z S ≥ − Q2 Z | | Qn | | Qn ciple for fermions, which was used by Dyson and Lenard intheirproofofstabilityofmatter[14]: Letψ be awave and for α 1 ≥ functionofnspinlessfermionsinR3, i.e. anti-symmetric w.r.t. every pair of particle coordinates, and let Q be a ψ¯Tˆ ψdx (n 1)ξH(α)2 ψ 2dx, (10) cube of side length l. Then, for the contribution to the Z H ≥ − Q2 Z | | Qn | | Qn free kinetic energy with all particles in Q, while for a square Q R2 with area Q and any α R 1 n ξ2 ⊂ | | ∈ 2ZQnXj=1|∇jψ|2dx ≥ (n−1)lF2 ZQn|ψ|2dx, (6) 21Z n |Djψ|2dx ≥ (n−1)cCQα2,n Z |ψ|2dx, (11) QnXj=1 | | Qn where ξ =π/√2. In other words,the energy is nonzero F with c = 0.179. It then follows for the expected kinetic for n 2 and grows at least linearly with n. In [14], Q ≥ energy on a d-dimensional cube Q with volume Q that wasreplacedbyaballofradiusl and√2ξ bythesmall- | | F est positive root of the equation (d2/dξ2)(sinξ/ξ) = 0. ξ2 The inequality (6) follows by expanding ψ in the eigen- TQ S/H/A/F ρ(x)dx 1 , (12) functions of the Neumann Laplacian on Q, or by the S/H/A/F ≥ |Q|2/d (cid:18)ZQ − (cid:19)+ method below at the cost of a slightly weaker constant ξF. where ξS2/H/A/F stands for ξS(η|Q|)2, ξH(α)2, cCα2,N, Now, for the 1D case we introduce ξ (ηl) resp. ξ (α) resp. ξ2, with corresponding dimension d=1,1,2,3. S H F to be the smallestpositive solutions of ξtanξ =ηl, resp. (d/dξ)(ξ1/2J(ξ)) = 0, where J is the Bessel function of Let us consider the proof for the Heisenberg case. order α 1/2. These ξS/H arise as quantization condi- Using the separation of the center-of-mass n j∂j2 = tions for−the wave function upon considering the Neu- ( ∂ )2+ (∂ ∂ )2,the(Neumann)kinetPicenergy j j j<k j− k mann problems foPr n 2 pParticles on an interval Q=[a,b] is ≥ ( ∂2+V (r))ψ =λψ, ∂ ψ =0 (7) − r S/H r |r=±l ψ¯Tˆ ψdx H Z Qn inthepairwiserelativecoordinater onaninterval[ l,l], − 1 yielding a lowest bound for the energy λ=ξ2 /l2. ψ¯ (∂ ∂ )2+V (x x ) ψdx S/H j k H j k ≥Z (cid:18)−2n − − (cid:19) In the case of anyons we define QnXj<k 2 Cα,n :=p∈{0,m1,.i.n.,n−2}mq∈iZn|(2p+1)α−2q|, (8) ≥ nXj<kQnZ−2 QZ [−δ(RZ),δ(R)ψ¯] (cid:0)−∂r2+VH(r)(cid:1)ψdrdRdx′ which measures the fractionality of the parameter α and 2 ξH(α)2 ψ 2drdRdx′, (13) arises from a local pairwise relative magnetic Hardy in- ≥ n Z Z δ(R)2 Z | | equality in 2D [16]. In (8), the minimization over even Xj<kQn−2 Q [−δ(R),δ(R)] integers 2q has to do with the underlying bosonic sym- where R := (x + x )/2, r := x x , x′ = metry of the wave function, while the odd integer 2p+1 j k j k − (x ,...,(cid:30)x ,...,(cid:30)x ,...,x ), and δ(R) := 2min R dependsonthenumberpofotheranyonsthatcanappear 1 j k N {| − a, R b . We then use (13) and δ(R)−2 Q−2 inside a two-anyoninterchangeloop(with the additional | | − |} ≥ | | to obtain (10). Inserting the partition of unity 1 = +1stemmingfromthestatisticsfluxoftheinterchanging pair itself). Note that for bosons Cα=0,n 0 while for A⊆{1,...,N} l∈AχQ(xl) l∈/AχQc(xl) into the expres- fermions C 1. ≡ sPion for TQ wQe then obtaiQn (cp. [14–17]) α=1,n ≡ H 4 N 1 THQ =XA ZRN jX∈A 2|∂jψ|2+(j6=X)k=1VH(xj −xk)|ψ|2lY∈AχQ(xl)lY∈/AχQc(xl)dx   1  ∂jψ 2+ VH(xj xk)ψ 2 dxl dxl ≥ Z Z 2 | | − | | XA (Qc)N−|A| Q|A| jX∈A j6=Xk∈A lY∈A lY∈/A   ξ (α)2 ξ (α)2 N ≥XA (|A|−1) H|Q|2 Z(Qc)N−|A|ZQ|A||ψ|2lY∈AdxllY∈/Adxl = H|Q|2 ZRN Xj=1χQ(xj)−1|ψ|2dx,   whereinthelaststepweagainusedthepartitionofunity. the r.h.s. is bigger for less constant density, but scales This proves (12) in the α 1 Heisenberg case. The with the number of particles only as N (in contrast to ≥ Schr¨odinger case follows similarly, while the anyon case the Lieb-Thirring inequality). requires the application of a magnetic Hardy inequal- Combining local uncertainty with local exclusion, and ity (also giving rise to a local repulsive potential whose cleverly splitting the space into smaller cubes depending strength is measuredby (8)), and we refer to [16, 17] for on the density, one can then prove the following energy the proofs. bounds: The constants ξ2 of proportionality in (12) ap- S/H/A/F Theorem 2 (L-T inequalities for 1D Schro¨dinger) pear as lower bounds on the strength of local exclusion, For η 0 and could e.g. be compared with the global constant of ≥ proportionalityinHaldane’sgeneralizedexclusionstatis- tics [9]. For the case of anyons, the constant ξ2 C2 TS CS ξS(2η/ρ∗(x))2ρ(x)3dx, (15) A ∝ α,N ≥ ZR is actually N-dependent, and it is clear from the defini- tion (8) that this constant can become identically zero for some constant 3 10−5 C 2/3, where ρ∗(x) is S · ≤ ≤ for sufficiently large N if α is an even numerator (re- definedas thesupremumoftheaveraged density ρ/Q Q | | duced) fraction. However, we have shown in [16] that over all intervals Q containing x. In particularR, if ρ is forα=µ/ν anodd numerator fraction, the limiting con- homogeneous, e.g. ρ∗ γρ¯for some γ >0, then stant is non-zero and equal to lim C = 1/ν. It ≤ N→∞ α,N hence also becomes weaker with a bigger denominator ν T C ξ (2η/(γρ¯))2 ρ(x)3dx, (16) S S S inthe statisticsparameter. Forirrationalα the constant ≥ ZR is non-zero for all finite N, but the limit is again zero. and if ρ is supported on an interval of length L T /L C ξ (2η/(γρ¯))2ρ¯3, ρ¯:=N/L. (17) S S S IV. LIEB-THIRRING-TYPE INEQUALITIES ≥ The inequalities (1) and (2) for fermions combine the Pauliexclusionprinciplewiththeuncertaintyprincipleto produce non-trivial and useful bounds for the energy as the number of particles N becomes large. We shall com- plement the local form of the exclusion principle above with the followinglocal form of the uncertainty principle on a d-dimensional cube Q, valid for the free kinetic en- ergy of any bosonic wave function ψ, and hence applica- bleinourcasesofintermediatestatisticsafterdiscarding thepositivestatisticspotentialsor,inthecaseofanyons, using the diamagnetic inequality D ψ ψ : j j | |≥|∇ | || ρ1+2/ddx ρdx TQ c Q c Q . (14) 0/S/H/A ≥ 1R( ρdx)2/d − 2RQ2/d Q | | R The constants c > c > 0 only depend on d. Mathe- 2 1 matically, (14) is a form of Poincar´e-Sobolev inequality, FIG.1. PlotofξS(t)(solid)andarctanpt+4t2/π2 (dashed) and we refer to [16, 17] for details and proofs. Note that as a function of t≥0. 5 FIG. 2. Plot of ξH(α) as a function of α≥0. FIG. 3. A sketch of Cα=limN→∞Cα,N as a function of α. Compare with Lieb-Liniger [20], where for a free sys- The reasonfor the more technical forms (15) and (18) tem in the thermodynamic limit N,L with fixed ascomparedto(2)and(20)isthelocaldependenceofthe density ρ¯, T /L 1e(2η/ρ¯)ρ¯3 with e→(t)∞ t,t 1, strengthofexclusionintheSchr¨odingercase,respectively e(t) π2,t S →(se2e also [22]). A good nu∼merica≪l ap- the possibility for arbitrarily strong exclusion (α → ∞) → 3 → ∞ in the Heisenberg case. We sketch a proof only for the proximationtoξ isgivenbyξ (t) arctan t+4t2/π2 S S simpleranyoniccase,andreferto[16,17]fordetails. For ≈ for all t 0 (see Fig. 1). p anapplicationofthesamemethodtofermionsin3Dand ≥ the generalization to q spin states we refer to [24] where Theorem 3 (L-T inequalities for 1D Heisenberg) a model allowing for point interactions was considered. For α 1 and arbitrary intervals Q such that the ex- ≥ Letusforsimplicityassumeρtobesupportedonsome pected number of particles ρ 2 Q ≥ square Q in the plane which we proceed to split into R 0 3 four smaller squares iteratively, organizing the resulting ρdx THQ ≥ CHξH(α)2(cid:16)RQQ2 (cid:17) , (18) sbuebasrqruaanrgeesdQsointahattreQe0Ti(ssfieenaFlilgy.c4o)v.eTrehdebpyroacecdolulerectcioann | | Q T of non-overlapping squares marked B s.t. 2 B withaconstant1/32 C 2/3. Itfollows inparticular ∈ ≤ H ρ < 8, and Q T marked A s.t. 0 ρ < 2, that if ρ is confined t≤o a len≤gth L and N 2 then Q ∈ A ≤ Q ≥ Rand s.t. at least one B-square is at the topmoRst level of T /L C ξ (α)2ρ¯3, ρ¯:=N/L. (19) everybranchofthetreeT. OntheB-squaresweuse(12) H H H ≥ togetherwith(14)toobtain(withc′ >0somenumerical k We have ξH(1) = π/2 and, asymptotically, ξH(α) α constants) ∼ as α (see Fig. 2). Compare with Calogero and tShuetrhme→roldany∞nda[m21ic],lwimheitreNo,nLe finds,TaHn/dLw→ithπ62aαp2pρ¯l3icaintiothnes TAQ ≥Cα2,N(cid:18)c′1Z ρ2+ cQ′2 (cid:19), Q∈TB. (22) → ∞ Q | | of Thomas-Fermi theory [23]. The local bound (18) can The A-squaresarefurthergroupedinto asubclassA on e.g. beappliedwiththeadditionofanexternalpotential 2 which the density is sufficiently non-constant, ρ2 > forobtaininglowerestimatesforthe groundstateenergy Q (alongwithestimatesforthecorrespondinggroundstate 2c2( ρ)2/Q for Q T T , so that by (14R) c1 Q | | ∈ A2 ⊆ A density; see [17] for examples). R c Theorem 4 (L-T inequalities for anyons) For any TAQ > 41 Z ρ2, Q∈TA2, (23) α R Q ∈ andasubclassA onwhich ρ2 2c2( ρ)2/Q. One T C C2 ρ(x)2dx, (20) 1 Q ≤ c1 Q | | A ≥ A α,NZR2 can then use the structure oRf the splittinRg of squares to prove that, for the set (Q ) of all such A -squares for some constant 10−4 C π. It follows that if A1 B 1 A which can be found by stepping backwards in the tree T ≤ ≤ α=µ/ν is a reduced fraction with odd numerator µ and from a fixed B-square Q and then one step forward, ρ is supported on an area L2 then B TA/L2 ≥ CAνρ¯22, ρ¯:=N/L2. (21) Q∈AX1(QB)ZQρ2 ≤kX∞=032cc124k|2Q2B| = 3c21c2|Q1B|. (24) 6 whereLdenotesthetotalangularmomentumofthestate Q0 ψ. NotethatifL= α N (whichcouldoccurforcertain A A N andfractionalαa−slo(cid:0)n2g(cid:1)asther.h.s. isaneveninteger) A A then this bound reduces to the bosonic bound for the A A energy. A B A A B A A A B A B A Itis well-knownthat energylevelsand degeneraciesin this model depend very non-trivially on both N and α, A A B A AAAB and we can point out certain similarities in the limiting graph of C (see Fig. 3) with known features in spec- α,N FIG.4. ExampleofasplittingofQ0andacorrespondingtree tra for N = 2,3,4 and corresponding extrapolations to T of subsquares. For the B-square at level 3 in the tree, the large N [2, 7, 25, 26]. It is e.g. intriguing to compare set A(Q) of all associated A-squares (cp. A1(Q) in the text) this graph — which can be obtained by cutting out a consists of 8elements, while for thetwo B-squaresat level 2, wedgeofslopeν fromthe upper half-planeateveryeven A(Q) coincide and has 4 elements. numerator rational point µ/ν on the horizontal axis — with the general structure indicated in Fig. 1 in [25]. It was in [25] argued by perturbation theory that the Inotherwordstheenergyonallsubsquareswithconstant behavior for the exact ground state energy as N low density is dominated by that from exclusion on the is approximately E √αωN3/2 for α 0 and →E ∞ B-squares. We therefore find from (22) that 0 ∼ ∼ 0 ∼ √8ωN3/2/3 for α 1, requiring L= α N +O(N3/2) ∼ − 2 by (26). In any case, as long as E0 grow(cid:0)s(cid:1)slower than TQB C2 c′ ρ2+c′ ρ2, quadraticallywith N, we areledto look for sequencesof A ≥ α,N 1Z 3 Z ground states ψ with L= α N to leading order in N.  QB Q∈AX1(QB) Q  Note that there are also c−ons(cid:0)tr2a(cid:1)ints on ψ coming from and hence, since all A -squares are covered in this way, the bosonic symmetry. 1 T = TQ TQ C C2 ρ2 for Motivated by the Laughlin states in the fractional A Q∈T A ≥ Q∈TB∪TA2 A ≥ A α,N Q0 quantumHalleffect[27],wecouldforparticularN =νK some nPumerical conPstant C >0. R A consider trial wave functions of the form ψ =Φψ , with For (17) and (21) we use ρpdx Np Q 1−p, and α Q0 ≥ | 0| for (21) we used the fact thRat lim C = 1/ν for ν N→∞ α,N such odd numerator fractions and zero otherwise. ψα := zjk −α (z¯jk)µ ϕ0(xj) | | jY<k σX∈SNqY=1(j,kY)∈σ.Eq j∈Yσ.Vq (27) V. APPLICATION AND DISCUSSION for evennumeratorfractions α=µ/ν [0,1], andψ := α ∈ The bound (21) provides a non-trivial lower bound ν K−1 z −α (z¯ )µ ϕ (x ) for the energy per unit area for an ideal gas of anyons | jk| jk k j∈σ.Vq with odd-fractional statistics parameter α. The numeri- jY<k σX∈SNqY=1(j,kY)∈σ.Eq k^=0 calconstantC 10−4inthisboundhasin[17]beenim- (28) A provedto 0.05≥9,which is still quite far from the exact for odd numerators µ, where Φ should regularize the semiclassic≥alconstantπ for the two-dimensionalspinless short-scalebehavior(duetothesingularJastrowfactor). fermion gas. In any case, this non-trivial bound raises We write zjk :=zj zk for the complex relative particle − the veryinterestingquestionwhether suchLieb-Thirring coordinates,ϕk denotetheeigenstatesoftheone-particle inequalities are in fact not valid for even numerator and Hamiltonian hˆ = 1∆+V and of which we may form −2 irrationalα. Wegivesomemotivationforwhythiscould a Slater determinant kϕk, while Eq and Vq are sets of be the case by considering the following observations. edgesandverticesofνVdisjointcompletegraphsinvolving By minimizing the lower bound for the energy follow- K particleseach,andwhichweactonbypermutationsσ ing from (20) in terms of ρ it is straightforward [17] to tosymmetrizethestates. Twopossiblechoicesofregular- obtain a bound izing symmetric functions Φ, giving rise to the expected pairwise short-scale behavior z α in ψ, could be jk ∼| | 1 8C T + Vˆ E AC ωN3/2 (25) A h iψ ≥ 0 ≥ 3r π α,N Φ = z 2α(r2+ z 2)−α, (29) r0 | jk| 0 | jk| jY<k for the ground state energy E of N anyons placed in 0 a harmonic oscillator potential V(x) = ω2 x2. For odd with a parameter r >0, or the parameter-free (but less 2 | | 0 numeratorαthisimprovestheboundgivenin[25](which smooth) is also valid for arbitrary α): N ν−1 N(N 1) Φ= z α, (30) E0 ≥ ω(cid:18)N +(cid:12)(cid:12)L+α 2− (cid:12)(cid:12)(cid:19), (26) jY=1kY=1| jk(j)| (cid:12) (cid:12) (cid:12) (cid:12) 7 withk(j)denotingthekthnearestneighborofparticlej. all known exact eigenstates there is this simple corre- These states ψ have L= α N +αν−1N (for (27) and spondence between the degree and the energy. It is − 2 2 for certain magic numbers K(cid:0) in(cid:1) (28)) and the property an interesting fact that adding the degree of Φ in the that only up to ν particles can be selected in each term nearest-neighbor form (30) produces ω(1+αν−1)N, i.e. 2 without involving a repulsive factor (z¯ )µ from an edge exactly the r.h.s. of (26) for the above value of L, jk in for some q, allowing for the formation of groups of speaking for a low energy for even numerator fractions. q E ν anyonswithintegerstatisticsfluxµ. Namely,whilethe On the other hand, the degree of the odd numerator Jastrowfactoractstoattractallparticles,thisattraction states (28) necessarily grows with K as K3/2 due ∼ is on large scales exactly balanced whenever a group of to the Slater determinant. While the resulting energy ν non-repelling anyons has formed, since an anyon xj E = ω(N + degψα) ∼ ων(N/ν)3/2 satisfies but does far outside the group, seeing the total attractive factor not match the bound (25) exactly w.r.t. α, a corre- (r−α)ν =r−µ where r is the distance from the group, sponding picture of ideal anyons forming essentially free ∼is also repelled by at least one anyon x in the group, ν-anyon groups with fermionic type statistics would ac- k with a factor z¯ µ rµ from that corresponding edge tually match the form of the bound (21), involving the jk in . Thisbal|anc|ec∼ouldacttodistributetheanyons,on reduced density ρ¯/ν =K/L2. q E the average,in such groups of ν. Furthermore, the total We finallyremarkthatthere arealsomanyinteresting contributionfromsuchagrouptothestatisticspotential connections between the forms of the fractions appear- A seenbythedistantparticlex wouldbe ναrI/r2 = j j ing here and those of fractionally charged quantum Hall µrI/r2, while the particle also has an or∼bital angular quasiparticles [30]. Another question concerns possible momentum µaroundthegroup(duetothatsameedge relations with q-commutation relations, with q = eiαπ to x and p−hase of (z¯ )µ) with velocity µrI/r2, k jk [31]. ∼ − again leading to an exact cancellation. ACKNOWLEDGMENTS The forms (27) and (28) bring out a structural dif- ference between even and odd numerators µ. The limit α=1of (28)isthefermionicgroundstateinthebosonic Supportfromthe DanishCouncil forIndependent Re- representation, while the states in (27) are (modulo the searchaswellasfromInstitutMittag-Leffler(Djursholm, Jastrow factor) actually found to be exactly the Read- Sweden)isgratefullyacknowledged. D.L.wouldalsolike Rezayi states [28] for fractional Quantum Hall liquids to thank IHE´S, FIM ETH Zurich, and the Isaac New- in their bosonic form. The state (27) is an exact but tonInstitute(EPSRCGrantEP/F005431/1)forsupport singular (requiring the regularization by Φ) eigenstate andhospitalityviaanEPDIfellowship. WethankE.Ar- of the Hamiltonian with energy E = ω(N + degψ ), donne, G. Felder, J. Fr¨ohlich, H. Hansson, J. Hoppe, E. α where degψ = αν−1N is the total degree of the Langmann and R. Seiringer for comments and discus- α − 2 non-Gaussian part of the wave function (cp. [29]). In sions. 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