Many-anyon trial states

The problem of bounding the (abelian) many-anyon ground-state energy from above, with a dependence on the statistics parameter which matches that of currently available lower bounds, is reduced to studying the correlation functions of Moore-Read (Pfaffian) and Read-Rezayi type clustering states.


Introduction
The importance of the concept of quantum statistics for our understanding of observed collective phenomena in nature cannot be overstated. While fermions stack according to the Pauli principle to form a Fermi sea, with its implications for atomic structure, conduction bands in solids, etc., bosons can sit together to display amplified single-particle behavior, as manifested by Bose-Einstein condensation and the coherent propagation of light. Upon restricting to two spatial dimensions on the other hand, it turns out to be logically conceivable to have other types of identical particles than bosons and fermions, satisfying braid statistics instead of permutation statistics, and which have been given the name 'anyons' [1][2][3][4][5] (see also [6, p. 386]). These have the property that under continuous exchange of two particles their wave function changes not merely by a sign ±1, but any complex phase factor e iαπ is allowed, where the real number α is known as their 'statistics parameter'. Moreover, for logical consistency one has to keep track of any topological winding of the particles during their exchange. For instance, if one particle moves around in configuration space in such a way as to enclose p other particles in a complete (counterclockwise, say) loop, a phase 2pαπ must arise, while if two particles are exchanged once and in the process p other particles are enclosed, the phase must be (1 + 2p)απ. All such topological complications vanish in the case of α = 0 (bosons) and α = 1 (fermions). The concept has also been extended from phases (abelian) to unitary matrices (non-abelian), but we shall here stick to the simpler (though not at all simple) abelian case. Furthermore, instead of demanding that the wave function changes its phase according to the above form of topological boundary conditions, also known as the 'anyon gauge picture', one may equivalently model such phases by means of attaching magnetic flux to ordinary identical particles, i.e. bosons or fermions, resulting in a magnetic many-body interaction. This is then called the 'magnetic gauge picture' for anyons. We refer to the extensive reviews [7][8][9][10][11][12][13][14][15][16] for a more complete background on the topic.
The idea of particles with attached magnetic flux is fundamental to the fractional quantum Hall effect (FQHE) [17][18][19][20][21], which is a manifestation of a strongly correlated many-body state of electrons subject to planar confinement and a strong transverse magnetic field. More recently it has been proposed that similar effects apply to trapped bosonic atoms in artificial magnetic fields [22][23][24][25][26], and to graphene [27,28]. The quasiparticles arising in the FQHE of electrons are predicted to be anyons with a corresponding fractional statistics parameter. However, as discussed in [19,Section 9.8.2], there has been some confusion in the literature concerning the exact values of α involved. This can be traced to different conventions but also to the fact that the statistics parameter has hitherto only been defined indirectly via the operation of adiabatic braiding and the computation of a corresponding Berry phase, as first outlined in [29]. Only very recently has an effective Hamiltonian for anyons been derived [30], which shows unambiguously how they may arise in a FQHE context and what the statistics parameter then should be, confirming that fermions against the background of a fermionic Laughlin state with odd exponent 2p+1 effectively couple to Laughlin quasiholes to behave as emergent anyons with α = 2p/(2p + 1) (see also [31]).
Despite the concept of anyons having been around now for almost four decades, a satisfactory understanding of their physics is still lacking. Due to their complicated many-body interaction (or geometry) it has not been possible to solve the anyon Hamiltonian for its complete spectrum or even its ground state, except in the two-particle case where it can be reduced to a one-particle problem and thus be solved analytically [1,4,32], while in the three-and four-particle cases it has been studied numerically [33][34][35][36]. Nevertheless, as is evident from the vast body of literature (the author can count more than 300 papers on the topic), there has been a fair amount of progress on the many-anyon problem, most of which is based on various approximations. One of the most discussed is average-field theory (see e.g. [37,38,10,39,16] for review), where the individual anyons are replaced by their average magnetic field, something which is arguably reasonable in a sufficiently dense regime. Other approximations assume either a very strong external magnetic field, thereby reducing to lowest-Landau-level anyons which turn out to be solvable with a connection to Calogero-Sutherland models [40,14], or in the case of the free dilute gas, that it is sufficient to only take two-particle interactions into account [32,41]. It has however been stressed that real progress in understanding the anyon gas cannot be made without knowledge of the true many-body spectrum.
In a recent series of works [42][43][44][45][46][47], the question concerning the manyanyon ground state has been investigated in the light of modern mathematical methods. Interestingly, it was found that the ground-state energy for the free ideal anyon gas can be non-trivially bounded from below, but only under the assumption that α is an odd-numerator rational number (in contrast to electron FQHE which typically involves odd-denominator filling factors). To settle the issue whether this is the true behavior or rather just an artifact of the method used to obtain the bounds, one also needs to bound the energy from above using suitable trial states. This however turns out to be a very difficult problem for anyons, contrary to the more common situation where finding the upper bound is the easier part.
We shall here proceed in the setting of abelian anyons with no external magnetic field (which is indeed relevant in the FQHE context; cf. e.g. [15]), and propose, building on [43], that good variational ground states for the many-anyon problem are given by clustering states of the Moore-Read and Read-Rezayi type that have already been studied for some time in the context of special (proposedly non-abelian) regimes of the FQHE. In particular, these types of states seem to give a much lower energy for even-numerator α than for odd numerators, and offer a corresponding picture of condensation, respectively, a reduced Fermi sea of anyons. In view of the above considerations, such a picture could in the context of the FQHE potentially have far-reaching consequences.
2 The many-anyon ground-state energy For concreteness and for easier comparison with the results which are available in the literature, we consider as our starting point anyons confined in a harmonic oscillator potential. In the magnetic gauge, the Hamiltonian operator for N non-relativistic ideal 1 anyons with mass m in a harmonic trap with frequency ω ≥ 0, and in units such that = 1, iŝ where denotes the magnetically coupled momentum operator of particle j at position x j ∈ R 2 . Each particle sees an Aharonov-Bohm magnetic flux 2πα attached to every other particle, as is given explicitly by the magnetic po- For reference we take the HamiltonianĤ N to act on bosonic states Ψ ∈ L 2 sym ((R 2 ) N ), and there is associated with the free kinetic energy operator T α a natural subspace (form domain) D N α consisting of the states Ψ which have a finite expectation value for their kinetic energy (see [44,Sec. 2.2] and [46, Sec. 1.1] for details). The case α = 0 then corresponds to bosons and α = 1 to fermions, with D N α=0 = H 1 sym (R 2N ) and D N α=1 = U −1 H 1 asym (R 2N ) the Sobolev spaces of symmetric/antisymmetric square-integrable functions having square-integrable first derivatives. In the latter case we have used the singular gauge transformation U −1 ∇ x j U = iA j , with and with the coordinates here represented by z j = x j + iy j ∈ C, to explicitly switch from fermions to their bosonic representation via flux attachment. The same transformation can be used to show that the full spectrum must be periodic in α up to any even integer 2q, by composing Ψ with U −2q which preserves symmetry. Also, one could equivalently have chosen to model everything in terms of fermions with statistics parameter β := α − 1.
The order a = α N involved is given by the 'fractionality' of α as measured by This expression has the peculiar many-body limit [42,Prop. 5] (see Figure 1) In particular, the lower bound in (3) depends on α as √ α * = ν −1/2 for small odd-numerator fractions and tends to zero with N for even-numerator and irrational numbers. In addition to the above, for an arbitrary state Ψ with fixed total angular momentum L ∈ Z one also has the bound [50,51,13] where, if L = −α N 2 (i.e. an average angular momentum of −α per particle pair) would be achieved exactly on some state, then the remaining bound is just the ground-state energy for bosons. As discussed in [50], since E 0 (N ) = O(N 3/2 ) the bound (5) implies that as the number of particles increases there will be more and more level crossings in the ground state to different angular momenta, resulting in a qualitative picture (see [50, Fig. 1]) with some interesting features in common with Figure 1. Note that for every finite N , the ground-state curve is conjectured to be continuous in α and piecewise smooth, however the number of such pieces grows like √ N .
On the other hand, an approach that has been used extensively in the literature to understand the anyon gas (see the reviews and references therein) is to employ the 'average-field approximation' 2 Here it has been assumed that the anyons see each other via an approximately constant magnetic field B(x) ∼ 2πα (x), with (x) the local density of particles, and hence they each have a lowest-Landau-level energy |B|/(2m) ∼ π|α| /m. In [45] it has been shown rigorously that such an approximation is indeed correct in the limit of almost-bosonic anyons (i.e. α → 0; see also [47]) in a confining trap, however one needs to be careful with what is meant exactly with the approximation and how such a limit is performed since strictly speaking the anyons cannot be ideal but extended (see [52] and Section 6 below). Also note that the periodicity for ideal anyons mentioned above is not naturally implemented in (6), so we must at least expect to replace α with its periodization α 2 from (4). The main question raised from the bounds (3) and (5) is whether for certain α such that α * α 2 -most notably for even-numerator rational α such as α = 2/3, for which α * = 0 -and for particular states such that L ∼ −α N 2 , the true ground-state energy E 0 (N ) could be considerably lower than the one (6) expected from average-field theory. Building on [43], we shall here explore the possibility that this is actually the case.

A local exclusion principle for anyons
In order to understand the origin of the peculiar dependence of the above energy bounds on α * , and of the form of the corresponding proposed trial states, we first need to briefly discuss the findings in [42,43,46] of a local exclusion principle for anyons.
Normally when one talks about an exclusion principle for identical particles one has in mind an occupation picture, where only a limited number of particles can sit in each distinguishable one-body state. The prime example is of course the usual Pauli exclusion principle for fermions, although various extensions have also been discussed in the literature [48,49,53,40,[54][55][56] (the role of such generalized exclusion in the context of anyons has also been reviewed in [57]). Sometimes the notion is generously extended to concern exclusion of coincident points in the configuration space, and anyons have often been pointed out to obey such exclusion, either because it is required for their topological definition or because their singular (for ideal anyons) magnetic interaction forces the wave function to vanish on the diagonals, just like fermions do by means of antisymmetry (see also [58]). However, note that such a notion of exclusion also applies to the hard-core Bose gas whose thermodynamic ground-state energy vanishes in the dilute limit (in analogy with the non-interacting gas) in two and higher dimensions [59][60][61][62][63][64].
For anyons whose statistics is generated by a true many-body interaction (or a very complicated geometry), a stronger notion of exclusion is required, and such a notion has been developed in [42,43,46] in the form of an effective repulsive long-range pair interaction. Namely, recall that the effect of the statistics is a change of phase of the anyonic wave function by e iαπ whenever two particles are interchanged via a simple continuous loop in configuration space, or a total phase (1 + 2p)απ whenever such an exchange loop at the same time encloses p other particles. On the other hand, particles are also allowed to have pairwise relative angular momenta, and such momenta are by continuity restricted to only change the phase by an even multiple of π. One way of viewing this condition (in the magnetic gauge picture) is that the many-body wave function is modeled as a bosonic (or fermionic) one and thus it needs to be antipodal-symmetric (resp. antisymmetric) with respect to the relative coordinate, hence π-periodic (resp. anti-periodic) in the relative angle. Another way to see it (in the anyon gauge or geometric picture) is that the wave function is a section of a locally flat complex line bundle with the topological continuity condition that its phase around such a loop should jump by (1 + 2p)απ plus 2π times an arbitrary winding number. Assuming then that the relative momentum or winding is the even integer −2q if the particles are orbiting in a reversed (for α > 0) direction in order to cancel as much of the magnetic or topological phase as possible, we arrive at a total phase (2p + 1)α − 2q times π per exchange for the particle pair. In the kinetic energy, any nonzero remainder phase of this sort gives rise to a centrifugal-barrier repulsion where r denotes the relative distance of the particle pair, and we have taken the infimum over all possibilities p ∈ {0, 1, . . . , N −2} and q ∈ Z to obtain the lower bounds in terms of α N ≥ α * . In the case of α = µ/ν being an arbitrary reduced fraction with an odd numerator µ and a positive denominator ν, it turns out using simple number theory [42,Prop. 5] that this phase mismatch can never be completely cancelled, and in fact α * = 1/ν > 0. On the other hand, if µ is an even number it is evident that cancellation is possible for particular values of p and q, and therefore α * = 0. For irrational values of α, one can use that any such number can be approximated arbitrarily well by both even-and odd-numerator rational numbers and hence α * = 0 (although note that a very large N may be required in such a process).
A geometric interpretation of the potential (7) is that there is non-trivial curvature (magnetic flux) sitting at each of the enclosed particles but effectively seen at the center r = 0 of the particle pair, and its presence is felt by the kinetic energy in the form of an effective potential. The situation is from this perspective analogous to that of a free quantum particle moving on a cone [65][66][67], here with its apex angle depending on α and the number of enclosed particles.
In [42] and [46] it has been shown rigorously by means of a family of magnetic Hardy inequalities that such an effective pairwise inverse-square 'statistical repulsion' (7) indeed arises in the many-anyon system. Although the effect is in some sense local and weighted only linearly in the number of particles (in contrast to a usual pair-interaction term in the Hamiltonian), it is still of long-range type and sufficiently strong to produce a 'degeneracy pressure' (represented concretely in the form of Lieb-Thirring inequalities; cf. [68,43]) and consequently non-trivial energy bounds in terms of α * for the ideal or dilute anyon gas, such as (3). We also stress that the method used to obtain the effective pair potential (7), which was introduced in [42] and developed to encompass more general situations in [46], is well suited for numerical investigations of lower bounds to the ground-state energy.

Constructing anyonic trial states
In order to match the available lower bounds for E 0 (N ) from above, it was in [43] suggested to study variational trial states of the form with Φ ∈ L 1 loc,sym a locally integrable regularizing factor, N = νK a suitable sequence of particle numbers and, in the case of α being an even-numerator reduced fraction α = µ/ν ∈ [0, 1], while for odd numerators µ, Here z jk := z j − z k are the pairwise relative complex coordinates with the usual identification C z j ↔ x j ∈ R 2 , and we have grouped, or 'colored', the particles into ν different colors where G q = (V q , E q ) denotes the complete graph over each such group of |V q | = K vertices=particles (cf. Figure 2). The symmetrization S over all the particles then amounts to symmetrization over all such colorings, and can be viewed as passing from a set of Figure 2: An illustration of a coloring of N = 12 particles with ν = 3 colors into K = 4 clusters. Each colored edge corresponds to one unit −µ of pairwise angular momentum. Also shown is the contribution to the magnetic potential αA j and the current J j of particle j due solely to the cluster V * 1 .
distinguishable particles (by color) to indistinguishable (cf. [69]). The ϕ k , k = 0, 1, 2, . . ., are the eigenstates (ordered by increasing energy) of a corresponding one-body Hamiltonian and in (9) we have formed the Slater determinant of the K first such states in the variables of each color group V q to obtain matching symmetry.
It is known that these states possess clustering properties, so that for example the symmetric polynomial 3 (together with a confining factor such as the Gaussian) exhibits clusters of ν particles. This, as well as other interesting and useful properties of such symmetric polynomials, have also been observed by means of an identification with correlators of certain conformal field theories (see e.g. [74,75]) and with Jack polynomials [76]. For instance, one has that if the positions of ν particles (i.e. one cluster) are identified, then (also compare to Figure 2) In particular, f N then vanishes whenever ν + 1 or more particles are brought together. Furthermore, if proceeding in this way to group all particles into disjoint clusters V * q , q = 1, . . . , K, with |V * q | = ν (think of complete graphs G * q = (V * q , E * q ) dual to G q in a sense), and then identifying their positions, say z j = ζ q for j ∈ V * q , one obtains a Laughlin state with exponent ν 2 α, Note that this clustering behavior matches very well with both the form of the magnetic potential A j and the Jastrow prefactor in ψ α . Namely, the attractive Jastrow factor contracts the clusters and balances with the intercluster repulsion coming from the Jack polynomial f N (z) in such a way that each particle x j sees from any other cluster V * q / ∈ j, say located at y ↔ ζ at a large distance r = |r| = |z j − ζ|, the attraction k∈V * q |z jk | −α ∼ r −να = r −µ and at the same time the effective repulsion ∼ |z j −ζ| µ = r µ from (12). Also, the total contribution to the magnetic potential seen by particle x j from this cluster is αA j (x j ; x k∈V * q ) ∼ ναr −⊥ , while at the same time the particle has an orbital angular momentum around the cluster with an opposite current contribution J j = −iψ −1 α ∇ j ψ α (x j ; x k∈V * q ) ∼ −µr −⊥ , again thanks to (12) (and complex conjugation). One should also observe that (see Figure 2), due to the balance between Jastrow attraction and Jack repulsion, clusters are formed out of particles with different colors, i.e. in different groups V q . Furthermore, every particle has exactly one edge in E q going to exactly one particle in every other cluster, namely the particle of the same color, and this is what gives the orbital angular momentum contribution (z jk ) µ . In this way Figure 3: A sequence of three particle configurations chosen to illustrate the properties of the trial states Ψ, here with α = 2/3, N = 12, and Φ = Φ r 0 =1.3 as defined in (17). We fix the positions of 11 particles in two 3-clusters (dashed circles), two 2-clusters (dash-dotted circles), and a single particle at three intermediate positions between the 2-clusters (dotted circles). Then arg Ψ (top row) and |Ψ| 2 (bottom row) are plotted as functions of the remaining particle position on the square [−10, 10] 2 . One may note that each 3-cluster binds two vortices, which produce the desired orbital angular momentum and also cancel the attraction from the cluster very effectively. The attraction is much stronger from a 2-cluster than from a single particle, but entirely absent from 3-clusters.
there is a natural cancellation between magnetic flux and angular momentum on the level of each individual particle. The same holds in the case of the odd-numerator states, which however have an additional repulsion and possibly angular momentum coming from the Slater determinant in (9).
In the fully clustered picture (13), becomes the necessary gauge transformation U −νµ to remove the overall statistical effects of the clusters (here we have ν copies of the even integer µ because there are ν particles moving in each cluster, each seeing a flux να = µ from every other cluster). Note that the role of Φ is to regularize the singular short-scale dependence of ψ α arising upon bringing particles very close together (the Jastrow factor in ψ α diverges with each pair like |z jk | −α ), and in (14) we have assumed that this has effectively removed the singular factor (j,k)∈E * q |z jk | −α from within each cluster. We also note that the states (8) and (9) naturally generalize for α ∈ Z to the correct gauge copies Ψ = U −α Ψ 0 of the bosonic resp. fermionic ground states, Ψ 0 = ⊗ N ϕ 0 resp. Ψ 0 = ∧ N −1 k=0 ϕ k , in this case leaving out the need for the regulator Φ. With the assumption that the total energy increases with the total degree of one-particle states ϕ k , the odd-numerator states clearly have a higher energy than the even-numerator ones, simply enforced by the symmetry constraint. One needs to explain, however, why one cannot just take the same states ψ α but shifted to a reducible fraction α = µ/ν = µ /ν , with µ = kµ, ν = kν, k ≥ 2. Note first that the necessary properties of the states may not be valid for such fractions and indeed there are certain assumptions on irreducibility in the context of Jack polynomials [76], but let us proceed anyway with the discussion, aiming for a better understanding of the argument.
Within the class of even or odd states we expect that the size of clusters, i.e. the denominator ν resp. ν , dictates the energy of the regulator Φ which therefore favors the irreducible case k = 1. Also note that we cannot shift from an even state to an odd one by extending the fraction by k in this way, but one could certainly take k to be even and thereby shift an odd state into an even one, and thus argue that the energy should then become lower (consider for example α = 2/6 instead of 1/3 or, for α = 1, Cooper pairs instead of a single Slater determinant of fermions). To argue against this possibility we need to study the pairwise structure of the states closer.
First note that the pairwise relative angular momentum for any (shifted or not) even state ψ α=µ /ν always comes in multiples of µ . The statistical repulsion (7) then always gives a positive energy unless for some p, q ∈ {0, 1, 2, . . .} (1 + 2p)α = qµ ⇒ qν = 1 + 2p, which requires ν to be odd, so that k = 1 (and ν is already odd if µ was even). We also note that if any ν-particle clusters happen to be enclosed in the 2-anyon exchange loops, they each contribute ν to p, that is 2να to the magnetic phase, and at the same time a matching −2µ to the relative momentum (one −µ for each of the anyons orbiting around the cluster). Furthermore, on a length scale such that a typical pair of particles has exactly one multiple of µ as orbital angular momentum, a full cancellation with the magnetic phase demands p = (ν − 1)/2 (this is indeed an integer in the even-numerator case), i.e. about half of the particles of a cluster are enclosed. On the very smallest scales, i.e. much less than the average interparticle distance, we can accept a phase mismatch and strong repulsion, to be controlled by Φ, in analogy with e.g. the hard-core Bose gas whose energy vanishes logarithmically with low density in two dimensions [61,63]. Figure 4: A similar plot as in Figure 3 with the same state Ψ. Now three 3-clusters are fixed, one particle takes three different positions along the horizontal axis, while the relative coordinate of the remaining pair of particles is plotted on the square [−20, 20] 2 with the center of mass of the pair fixed at the origin. In relative coordinates the picture is antipodal-symmetrized, and one may also note that on a circle with radius equal to the nearest 3-cluster distance (dotted) the total phase circulation is 4π. In this way the probability of finding a third particle within an interparticle distance may possibly be increased without a significant cost in exclusion energy.
In the case α = 2/3 it is natural that as the scale then increases a bit we first see exactly one enclosed anyon, p = 1, of the ν = 3 cluster of which the particle pair is taken, and then additional full clusters on the scale of the average interparticle distance (cf. [46, Fig. 3] and Figure 4). Hence this allows for a full cancellation of the exchange phase over large scales in this particular state, thus reducing its overall statistical repulsion on large regions of the configuration space. However, in general it seems that there needs to be a delicate balance between Φ and ψ α in order to obtain such special probability distributions, and this remains the least understood aspect of these trial states at the moment (see also [77]).

Ideal anyons in a harmonic trap
At this point one might worry about actually computing (or at least bounding) the energy of the proposed trial states. Fortunately, however, it turns out that ψ α given in (8) for even-numerator α is an exact (but singular) eigenfunction of the harmonic oscillator HamiltonianĤ N with (see [78][79][80]13] where the degree of the state is, by (11), Being a singular eigenfunction means that the identity (15) holds wherever ψ α is smooth, namely outside of the fat diagonal of the configuration space Since ψ α is not a true eigenstate (normalizable and in the domain) of the operatorĤ N , its formal energy being is not a contradiction to (5). Also, by contrast, note that for ψ α with oddnumerator α deg ψ α ∼ ν although in this case ψ α does not have the structure of an exact eigenfunction since it involves a polynomial in both z j andz j . Two possible choices of regularizing symmetric functions Φ mentioned in [43], giving rise to the expected pairwise short-scale behavior ∼ |z jk | α in Ψ, could be with a parameter r 0 > 0 to be optimized over, or the parameter-free (but less smooth) where nn k (j) denotes the kth nearest neighbor of particle j (among particles of the set A ⊆ {1, 2, . . . , N } if instead writing nn k (j; A)). However, as seen below, an ansatz closer to the Bijl-Jastrow form or the Dyson form [60] Φ D = f (|z 2 1 |)f (|z 3 nn 1 (3;1,2) |) . . . f (|z N nn 1 (N ;1,2 as used for 2D Bose gases with suitable two-particle correlations f (see [64]), could be better. We note that taking Φ 0 as in (18) as a regulator raises the degree of the state Ψ = Φ 0 ψ α for even numerators to formally (if (15) were still to hold) produce the energy which by (11) exactly matches the lower bound (5). We also note that for odd-numerator α and magic (i.e. shell-filling) numbers K, which matches both the average-field approximation (6) for α = α * and, for small α * and up to the value of the numerical constant, the improved rigorous lower bound (3) that was recently obtained by means of a Lieb-Thirring inequality [46] as a consequence of the statistical repulsion (7). Using the above observations, we then shift the problem of estimating the ground-state energy E 0 (N ) from above for even-numerator fractions to concern only the energy of the regulator Φ: where α is an even-numerator fraction. Then Proof. By taking an appropriate approximating sequence in [44,Lemma 3]) we may assume without loss of generality that Φ ∈ C ∞ c (R 2N \ ) (smooth with compact support outside ). Then where, using that αA j is real and a partial integration, Hence, and by (15) it then remains to prove that the last line is zero. Expanding the derivative and collecting the terms, and making another partial integration (now for Φ 2 = |Φ| 2 ∈ C ∞ c (R 2N \ )), one finds that it equals where J[u] := i 2 (u∇ū −ū∇u). Finally we may use that ∇ · A = 0 and the eigenfunction equation (15) and its complex conjugate to show that j ∇ j · (J j [ψ α ] + αA j |ψ α | 2 ) = 0 on R 2N \ , which proves the proposition.
The energy in the state Ψ thus depends solely on the correlations of the weight |ψ α | 2 and its balance with the regulator Φ, which is required to vanish sufficiently fast as particles come together. However, if these correlations effectively turn out to decay faster than the average interparticle spacing then, in analogy with dilute hard-core bosons [61,63], there may be room for a smaller energy (note that Φ should not have Dirichlet but rather Neumanntype boundary conditions at the interparticle scale).

The R-extended anyon gas
That exact eigenstates for the many-anyon problem can be found at all is far from trivial, and the reason for it to hold for the above states is that they satisfy a remarkable simplifying identity. Here we shall consider this identity in detail and greater generality, in the context of the extended anyon gas.
By an 'R-extended anyon' we mean that we have replaced the singular Aharonov-Bohm flux on each anyon by a uniform field on a disk of finite radius R > 0. In other words, we replace (2) by (cf. [81,52,82,45,30,46]) so that, where 1 D(y,R) denotes the indicator function on a disk of radius R centered at y. Note that this form for the magnetic interaction is actually the natural one from the perspective of emergent anyons [30], for which the size R is implied by the experimental conditions. There is also a natural dimensionless parameter in the problem given by the ratio of the size of the magnetic flux to the average interparticle distance,γ := R¯ 1/2 . This has been called the 'magnetic filling ratio' in [52,83,46]. Based on the long-range local exclusion principle (7) and further shortrange magnetic bounds that arise only in this extended context, it was shown rigorously in [46] that a homogeneous gas of such R-extended anyons satisfies in the thermodynamic limit (on a box of side length L → ∞ and with fixed average density¯ = N/L 2 ) a universal bound for the energy per particle of where C is a positive universal constant and (see Figure 5) This bound interpolates between a dilute regime where the effect of the statistical repulsion dominates (note that there is also, even for α ∈ 2Z \ {0}, a strictly positive interaction energy which vanishes with the density similarly to that of a hard-core 2D Bose gas), and a dense regime where the dependence on α matches that which is expected from average-field theory (6).
In [45] it was shown that for R-extended anyons in an external trap V and in a limit such that the fillingγ is high but the statistics parameter α small ('almost-bosonic' anyons; see also [47]), the average-field approximation is a correct description in the sense that the particles become identically distributed in a self-generated magnetic field. However, given the linear dependence on the strength of the magnetic field |α| (our lower bound (22) is valid for any α ∈ R) for high densities, and the periodicity in α for ideal anyons, there must be some non-trivial interpolation between these two regimes.
Let us now introduce a convenient notation for an associated scalar (super)potential (cf. [45]) We then have the following property, which is essentially a result concerning supersymmetry of the corresponding Pauli operator. It has been discussed in that context in [73,81], however we will here supply a different proof.
In particular, Ψ ± are for R = 0 generalized zero-energy eigenfunctions of the N -anyon kinetic energy operatorT α considered on R 2N \ .
Proof. We find it convenient to work with G(C 2 ), the complex Clifford algebra over R 2 , and write for the π/2-rotation x ⊥ = (xe 1 + ye 2 ) ⊥ = xI, i.e. multiplication from the right with the pseudoscalar I = e 1 e 2 . Note that and thus Furthermore, where the fundamental simplifying identity used is that for any x, y ∈ C 2 x(1 ± iI) · y(1 ± iI) = 0, since I 2 = −1 and xI · yI = x · y. We have also used ∇ · ∇I = ∇ · ∇ ⊥ = 0, and that for z ± = x ± iy and f : C → C analytic and ∆f = 0.
In the R-extended case we therefore take as our trial states Ψ = Φψ α with the Jastrow factor in (8) and (9) replaced by e −α j<k w R (x j −x k ) , and in the case of the homogeneous gas the ϕ k are taken to be the eigenstates of the Neumann Laplacian on the square Q L of side length L (thus ϕ 0 ≡ L −1 ). Note that these states are regular even without the factor Φ, since e −αw R (x) = |x| −α , |x| > R, R −α e α 2 (1−|x| 2 /R 2 ) , |x| ≤ R.
However, in order to obtain the correct balance for a low total energy, and to take the appropriate limits, we expect that an additional regulator is still necessary. In particular, in the dilute limitγ = R¯ 1/2 → 0 the Jastrow factor describes an attraction which needs to be turned into a short-range repulsion, as illustrated by the below reformulation of the energy in terms of Φ. The proof is almost identical to that of Proposition 1, where the use of the identity (15) is replaced by Proposition 2.
Proposition 3. Assume Φ ∈ H 1 0 (Q N L ; R) is such that Ψ = Φψ α ∈ D N α and (∇Φ)ψ α ∈ L 2 (Q N L ), where α ∈ [0, 1] is an even-numerator fraction. Then In the dilute limit, in which the scattering length of the soft-disk potential αW R becomes relatively small, this again seems to be able to produce a low energy for even-numerator states. Also, for odd-numerator states, naively estimating the energy of Ψ in terms of that of the one-body states ϕ k of the Slater determinants in (9) yields the tentative bound 2mE 0 (N ) ν 2πK 2 /L 2 ∼ 2πα * ¯ N, which again matches the available lower bounds. Note also that the repulsive pair potential αW R that emerged above matches in the dilute limit the point interaction conventionally introduced to regularize ideal anyons [84].

Conclusions
With the ansatz given by the discussed trial states, we have reduced the difficult problem of bounding the ground-state energy of a system of N abelian anyons with even-numerator rational statistics parameter to the study of the N -dependence of the quantity which is essentially the energy of a repulsive 2D Bose gas described by Φ but weighted by |ψ α | 2 . One could try to estimate this using the techniques of Dingle, Jastrow and Dyson (see [60,64] and references therein). Alternatively, Monte Carlo methods could prove useful in this formulation. In any case, since the weight |ψ α | 2 is designed so as to cancel any long-range correlations by means of its clustering properties, and since the energy of a dilute 2D Bose gas is logarithmically small [61,63], the discussed approach indeed looks very promising. Also, if the anyons are not completely free but an additional attraction is added then it seems rather clear from the above expression with suitable Φ that they would prefer to cluster in this way. Finally, let us remark that if these are indeed the correct (approximative) ground states for a many-body system of abelian anyons, then they could possibly also explain from a more fundamental perspective the occurrence of such clustering states in the FQHE (cf. [15, pp. 239-240] and note that the usual Read-Rezayi states are supposed to be built of k-clusters of α = 2/k anyons in a zero magnetic field). Furthermore, the elementary excitations of such an abelian anyon condensed ground state may, according to well-known properties of clustering states, in turn be non-abelian anyons.