Exchange and exclusion in the non-abelian anyon gas
EEXCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS
DOUGLAS LUNDHOLM AND VIKTOR QVARFORDT
Abstract.
We review and develop the many-body spectral theory of ideal anyons, i.e.identical quantum particles in the plane whose exchange rules are governed by unitaryrepresentations of the braid group on N strands. Allowing for arbitrary rank (dependent on N ) and non-abelian representations, and letting N → ∞ , this defines the ideal non-abelianmany-anyon gas. We compute exchange operators and phases for a common and wide classof representations defined by fusion algebras, including the Fibonacci and Ising anyonmodels. Furthermore, we extend methods of statistical repulsion (Poincar´e and Hardyinequalities) and a local exclusion principle (also implying a Lieb–Thirring inequality)developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequencesof unitary representations of the braid group, for which two-anyon exchange is nontrivial. Contents
1. Introduction 21.1. Anyons — abelian vs. non-abelian 31.2. Some mathematically rigorous results for abelian anyons 51.3. Main new results 82. Algebraic anyon models 112.1. Fusion 112.2. Braiding 142.3. Pentagon and hexagon equations 152.4. Vacuum, bosons and fermions 172.5. Abelian anyons 182.6. Fibonacci anyons 182.7. Ising anyons 203. Exchange operators and phases 213.1. Braid group representations 223.2. Exchange operators 253.3. Abelian anyons 263.4. Fibonacci anyons 273.5. Ising anyons 293.6. Clifford anyons 323.7. Burau anyons 324. Geometric and magnetic anyon models 334.1. Classical configuration space 334.2. Hamiltonian and Hilbert space 364.3. Statistics transmutation 404.4. Abelian anyons 44
Mathematics Subject Classification. a r X i v : . [ m a t h - ph ] S e p D. LUNDHOLM AND V. QVARFORDT
Introduction
Quantum statistics is fundamental to our understanding of the physical world. In threespatial dimensions this refers to the division of particles into either bosons, such as Higgs,photons, gluons and other force carriers as well as certain atoms such as He, or fermions,such as electrons, quarks and other ordinary matter particles including He atoms. Theircollective behavior underlies the explanation of everyday phenomena such as conductionvs. insulation, lasing, as well as stability of large systems such as planets and stars. Whenquantum systems of many particles are confined to two dimensions, however, other possibil-ities emerge. Such intermediate quantum statistics of exchange phases [LM77, GMS81] (asopposed to the exclusion principle [Gen40, Gen42, Hal91]), with associated particles knownas anyons [Wil82, Wu84], first appeared in theory in the 70’s and 80’s and filled a gap inthe logical argument used since the 1920’s to derive the boson/fermion dichotomy [Gir65,Kla68, Sou70, SW70, LD71, Fr¨o88]. They were later found to have an application withinthe fractional quantum Hall effect (FQHE), both in their abelian form [Lau99, ASW84]as well as in their non-abelian form [MR91, GN97]. The latter eventually spawned pos-sible applications to quantum computing and complexity theory [Kit03, Llo02, FKLW03].Rooted in configuration spaces and braid groups, these concepts naturally interact withpure mathematics and are e.g. entwined to knot theory, quantum groups, Chern-Simonstheory, and conformal field theory (CFT) [Wit89, MS89]. In a quantum gravity contextboth abelian [tH88, DJ88, Car90, KS91] and non-abelian [PRE15] anyons appear, the for-mer as a toy model for point masses in 2+1 dimensions or cosmic strings in 3+1 dimen-sions, and the latter proposed for the degrees of freedom on the spherical event horizonof ‘normal’ black holes. Reviews focusing on the physics of anyons are numerous, seee.g. [DMV03, For92, Fr¨o90, IL92, Jac90, Kha05, Ler92, LR16, Myr99, NSS +
08, Ouv07,Ste08, Wil90], while for introductions to the mathematical techniques involved we refer to[DRW16, DˇST01, FG90, LSSY05, Lun19, MS19, MS95, Rou16, RW18].The concrete emergence of anyons from underlying systems of bosons and fermions hasbeen studied recently [LR16, YGL +
19, BLLY20]. On the experimental side one of thedifficulties involved is in measuring phase interference for individual particles (see [NLGM20]
XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 3 for a recent proposal) while another, more robust, route is to observe indirect effects ofstatistics in density distributions [CDLR19]. A fundamental problem [CJ94, MS94, NW94]in this regard concerns the relationship between exchange phases and exclusion (which in thefermionic case was attributed to Pauli [Pau47]), and this is only beginning to be addressedin a mathematically precise manner for anyons.1.1.
Anyons — abelian vs. non-abelian.
The most direct route to quantum statis-tics is to start from an N -body Schr¨odinger wave function Ψ : ( R ) N → C , where x =( x , . . . , x N ) ∈ ( R ) N are the positions of N point particles in the plane. The square of itsamplitude | Ψ(x) | is normalized (cid:82) R N | Ψ | = 1 and carries the interpretation of a probabilitydensity to find the particle labeled j at the position x j ∈ R , at the same instant in timefor j = 1 , . . . , N . In the case that the particles are all identical (indistinguishable), thisprobability must be symmetric, | Ψ( x , . . . , x j , . . . , x k , . . . , x N ) | = | Ψ( x , . . . , x k , . . . , x j , . . . , x N ) | , j (cid:54) = k. This leaves the possibility for an exchange phase:Ψ( x , . . . , x j , . . . , x k , . . . , x N ) = e iαπ Ψ( x , . . . , x k , . . . , x j , . . . , x N ) , j (cid:54) = k. (1.1)By arguments of logical consistency, it turns out that this phase must be independent ofwhich pair of particles are exchanged. Thus, if α = 0 (or α ∈ Z ) then Ψ is symmetricand this defines bosons , subject to Bose-Einstein statistics [Bos24, Ein24], while if α = 1(or α ∈ Z + 1) then Ψ is antisymmetric and this defines fermions , subject to Fermi-Dirac statistics [Fer26, Dir26]. In particular, bosons admit product states (Bose-Einsteincondensate) Ψ( x , . . . , x N ) = N (cid:89) j =1 u ( x j ) , (1.2)having independent identical distribution in a one-body state u ∈ L ( R ), while fermionsare necessarily correlated subject to Pauli’s exclusion principle [Pau25, Pau47]Ψ(x) = 0 if x j = x k for j (cid:54) = k, (1.3)and are spanned by Slater determinants (exterior products) of one-body statesΨ(x) = u ∧ . . . ∧ u N (x) = ( N !) − / det[ u k ( x j )] . (1.4)If α / ∈ Z then (1.1) defines anyons (as in ‘any phase’ [Wil82]) with statistics parameter α , and requires that Ψ is a multivalued function, since a double exchange is not the identity.The ideal N -partice Hamiltonian operator, which we take to be the non-relativistic kineticenergy, ˆ T = − ∆ x = − N (cid:88) j =1 ∆ x j is defined to act on these different functions Ψ, with sufficient regularity and suitable bound-ary conditions on the unit square Q = [0 , , for example, to make the system finite. Theground states of the form (1.2) respectively (1.4) immediately produce calculable groundstate energies, E N = inf spec ˆ T ; for bosons (i.e. the ideal Bose gas ) simply E ( α =0) N = N λ ( − ∆ N Q ) = 0 , ¯ E ( α =0) N = N λ ( − ∆ D Q ) = 2 π N, (1.5) D. LUNDHOLM AND V. QVARFORDT respectively for fermions (i.e. the ideal Fermi gas ) the semiclassical
Weyl’s law [Wey12] E ( α =1) N = N − (cid:88) k =0 λ k ( − ∆ N Q ) = 2 πN + o ( N ) , ¯ E ( α =1) N = N − (cid:88) k =0 λ k ( − ∆ D Q ) = 2 πN + o ( N ) , (1.6)as N → ∞ , where λ ≤ λ ≤ . . . denote the eigenvalues of − ∆ N / D Q , the Neumann/DirichletLaplacian on Q , ordered according to their multiplicity. On the other hand, the anyoniccase presents a real difficulty as it is not directly reducible to simple product states butactually turns out to be equivalent to a system of interacting bosons or fermions withcomplicated many-body correlations. In fact, the operator ˆ T acting on multivalued Ψsubject to (1.1) is equivalent to the magnetic operatorˆ T α = N (cid:88) j =1 − i ∇ x j + α (cid:88) k (cid:54) = j ( x j − x k ) −⊥ , x −⊥ := ( − y, x ) x + y for x = ( x, y ) ∈ R \ { } , acting on bosonic Ψ. Thus mathematical techniques for interacting Bose gases are essential.If Ψ takes values in some Hilbert space F , then the condition (1.1) may be replaced byΨ( x , . . . , x j , . . . , x k , . . . , x N ) = U Ψ( x , . . . , x k , . . . , x j , . . . , x N ) , j (cid:54) = k, (1.7)where U ∈ U( F ), the group of unitary operators on F . In general there will be topologicalconsistency conditions on the possibilities for exchange (considered properly as continuous loops in the configuration space of particle positions) and one must consider a representationof the corresponding braid group, i.e. a homomorphism ρ : B N → U( F ) . (1.8)We will later define precisely what we mean by this. In our context, where we considerideal anyons, the most general model we may consider (referred to as a geometric anyonmodel ) will be in one-to-one correspondence with such a representation ρ . The case (1.1)is a special case where each generator σ j of B N is represented as the phase ρ ( σ j ) = e iπα .Whereas phases are abelian, a representation (1.8) such that ρ ( σ j ) do not all commute isnon-abelian (non-abelian anyons are also known as nonabelions or plektons in the liter-ature). We will focus attention on a particular family of non-abelian representations whicharise naturally from the perspective of quantum field theory [FRS89, FG90] and whichinclude a number of anyon models which have been proposed to be relevant in condensedmatter contexts such as the FQHE. Some of these, such as one known as the Fibonacci anyonmodel, have even been proposed as good candidates for topologically protected quantumcomputing [FKLW03, Kit06, NSS +
08, RW18]. We refer to this general class of representa-tions as algebraic anyon models .Our main aim in this work is to connect the relatively well-developed general algebraictheory of non-abelian anyons to the hitherto relatively undeveloped many-body spectraltheory and analysis of corresponding geometrically defined Laplace operators ˆ T ρ , in orderto eventually be able to compute or estimate a physically essential property such as theground state energy in the many-body (thermodynamic) limit N → ∞ . XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 5 e i pαπ e i (2 p +1) απ p p Figure 1.1.
One- respectively two-particle loops of abelian anyons in theplane, with their respective phases and braid diagrams obtained by project-ing and ordering the particles on the horizontal axis and with time runningupwards on the vertical axis. In each loop p other particles are enclosed, andthe total obtained phase is απ times the number of simple braids appearingin the diagram.1.2. Some mathematically rigorous results for abelian anyons.
Apart from a fewspecial systems such as for 2 ≤ N ≤ N [Wu84, Cho91b, Cho91a], the spectrum of the many-anyon Hamiltonian ˆ T α remains unknown for α / ∈ Z . It has been noted however that ina rotationally symmetric situation such as the harmonic oscillator potential the groundstate (g.s.) in the bosonic representation must have a very specific total angular momentum[CS92] L ∼ − α (cid:18) N (cid:19) corresponding to a relative angular momentum − α for each pair of particles. This impliesthat the g.s. energy E N as a function of α will have to have level crossings between differentsuitable L and is only likely to be smooth on intervals of lengths tending to zero as N → ∞ . Statistical repulsion , generalizing the Pauli principle (1.3) for fermions, manifests con-cretely in three ways: primarily as an effective scalar repulsion between pairs of particles,making | Ψ | smaller along the diagonals of the configuration space, secondarily as a non-trivial growth of the local Neumann energy with the number of particles, and tertiarily as adegeneracy pressure in the density (e.g. a Thomas–Fermi profile compared to a condensedone-body profile). The first effect has been observed already for N = 2 anyons in earlyworks, manifested as a centrifugal-barrier repulsion due to the fractional relative angularmomentum. In the abelian many-anyon gas it was brought forward by D. L. and Solovej in[LS13a, LS13b] and quantified by the ‘fractionality’ of αα N = min p ∈{ , ,...,N − } min q ∈ Z | (2 p + 1) α − q | , which entered as a coupling constant in a many-anyon Hardy inequality (as quadratic forms)ˆ T α ≥ α N N (cid:88) ≤ j This type of inequality had been proven for fermions ( α N = 1 ∀ N ) by Hoffmann-Ostenhoffet al. [HOHOLT08, Theorem 2.8] (improvements in higher dimensions were discussed in[FHOLS06]). They had in fact proven a similar bound also for anyons [HOHOLT08, Theo-rem 2.7] although in this case with a much weaker constant, replacing 4 α N /N by C α,N = min p ∈{ ,...,N − } (cid:18) p min q ∈ Z | pα − q | (cid:19) , which vanishes e.g. for fermions. For this result they used a many-body version of a magneticHardy inequality for an Aharonov-Bohm singular field due to Laptev and Weidl [LW99] andgeneralized by Balinsky [Bal03] to the case of many singularities, in a sense considering asingle particle in the field of the others fixed (monodromy; cf. Figure 1.1, left). On the otherhand, the validity of (1.9) for fermions boils down to the Poincar´e inequality in pairwiserelative coordinates, namely that the energy for an antipodal-antisymmetric and nonzerofunction on the unit circle S (or unit sphere S d − ) cannot be zero, (cid:90) π | u (cid:48) ( ϕ ) | dϕ ≥ (cid:90) π | u ( ϕ ) | dϕ, if u ( ϕ + π ) = − u ( ϕ ) . Such a relative Poincar´e inequality was generalized to the anyonic setting in [LS13a] inthe form of a symmetry adaptation of the magnetic inequality of Laptev and Weidl. Formultivalued functions this amounts to (cid:90) π | u (cid:48) ( ϕ ) | dϕ ≥ min q ∈ Z | α − q | (cid:90) π | u ( ϕ ) | dϕ, if u ( ϕ + π ) = e iαπ u ( ϕ ) . This provides then the first manifestation of statistical repulsion (rooted in simple exchangeor half-monodromy; cf. Figure 1.1, right).Also importantly for the subsequent development, a more powerful local version of theHardy inequality was introduced (it will be given below in its improved form). In the casethat α ∗ := inf N ≥ α N = lim N →∞ α N is positive, which is true iff α is an odd-numerator rational number, this local version ofthe inequality (1.9) was used to derive local energy estimates for ˆ T α (with Neumann b.c.on Q N ) E N ≥ Cα N ( N − + ≥ Cα ∗ ( N − + (1.10)with an explicit numerical constant C > 0. This provides a local exclusion principle — a secondary manifestation of statistical repulsion — and it may be applied iterativelyin a way which in the case of fermions goes back already to Dyson and Lenard [DL67]in their ingenious proof of the thermodynamic stability of fermionic matter with Coulombinteraction (see e.g. [LS10, Lun19]). Namely, the corresponding bound for fermions, whichfollows immediately from (1.6) with λ k ≥ π for k ≥ 1, is E N ≥ π ( N − + . It was shown in [LS13a, LS14] that such a linear bound in N is sufficient to estimate theabelian anyon gas energy E N ≥ Cα ∗ N + o ( N )as N → ∞ . The function α (cid:55)→ α ∗ appearing in these bounds and supported on odd-numerator rationals is a variant of the Thomae or ‘popcorn’ function ; cf. Figure 1.2. XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 7 αα ∗ αf ( j (cid:48) α ∗ ) Figure 1.2. Left: The popcorn function α (cid:55)→ α ∗ . Right: A numerical lowerbound to the function α (cid:55)→ f ( j (cid:48) α ∗ ) from [LL18, Fig. 6]. The continuousorange curve indicates α (cid:55)→ c ( α ) for comparison (see text).These bounds were subsequently improved in works of D. L. with Larson respectivelySeiringer; first in [LL18] considering a refinement of the many-anyon Hardy inequality toyield E N ≥ f (cid:0) j (cid:48) α N (cid:1) ( N − + where the function f (cid:0) j (cid:48) α (cid:1) ∼ πα as α → α (itsdefinition and uniform bounds will be given below), and then with a scale-covariant methodin [LS18] to E N ≥ c ( α ) N, N ≥ , where c ( α ) = min (cid:8) f (cid:0) j (cid:48) α (cid:1) , . (cid:9) . The latter bound thus removes the dependence on α ∗ , although with a weaker constant; cf. Figure 1.2. In fact it was shown in [LS18] that thehomogeneous ideal abelian anyon gas is subject to uniform linear bounds in α ∈ [0 , 1] (byperiodicity and a conjugation symmetry Ψ (cid:55)→ Ψ, α (cid:55)→ − α , it suffices to study this range): Theorem 1.1 ( Uniform bounds for the ideal abelian anyon gas [LS18]) . For any sequence of abelian N -anyon models ρ N : B N → U(1) , ρ N ( σ j ) = e iαπ , with statisticsparameters α = α ( N ) ∈ [0 , and n-anyon exchange parameters α n = α n ( N ) ∈ [0 , , wehave the following uniform bounds for the ground-state energy, inf spec ˆ T α : C ( ρ N ) N (cid:0) − O ( N − ) (cid:1) ≤ E N ≤ ¯ E N ≤ π N (cid:0) O ( N − / ) (cid:1) , where C ( ρ N ) := max (cid:8) C ( ρ N ) , f (cid:0) j (cid:48) α N (cid:1)(cid:9) , ν/ ≤ f ( j (cid:48) ν ) ≤ πν (1 + ν ) ∀ ν ∈ [0 , , and C ( ρ N ) := 14 min { E , E , E } ≥ c ( α ) := 14 min (cid:8) f ( j (cid:48) α ) , . (cid:9) . Further, there exist universal constants C ≥ C > such that, if α is independent of N , C α ≤ lim inf N →∞ E N /N ≤ lim sup N →∞ ¯ E N /N ≤ C α. D. LUNDHOLM AND V. QVARFORDT The latter limits are the energy per particle and density, thus, the abelian anyon gas with α ∈ (0 , 1] exhibits extensivity in the energy like the Fermi gas (1.6) (and unlike the Bosegas (1.5)), as a consequence of the statistical repulsion between pairs of particles.Another development in [LS13a, LL18, LS18] concerns the Lieb-Thirring inequality (cid:104) Ψ , ˆ T α Ψ (cid:105) ≥ Cα (cid:90) R (cid:37) Ψ ( x ) d x , α ∈ [0 , , (1.11)where the one-body density (cid:37) Ψ is the marginal of the probability distribution (cid:37) Ψ ( x ) := N (cid:88) j =1 (cid:90) R N − | Ψ( x , . . . , x j − , x , x j +1 , . . . , x N ) | (cid:89) k (cid:54) = j d x k . Such an inequality was first proved for fermions by Lieb and Thirring [LT75, LT76] withthe goal of simplifying Dyson and Lenard’s proof of stability of matter. It is a powerfulcombination of both the uncertainty principle and the exclusion principle, and manifeststhe tertiary form of statistical repulsion as a degeneracy pressure in the density. Namelythe r.h.s. of (1.11) is on the form of the Thomas–Fermi approximation [Tho27, Fer27]inf spec ˆ T α = inf bosonic Ψ: (cid:82) R N | Ψ | =1 (cid:104) Ψ , ˆ T α Ψ (cid:105) ≈ inf (cid:37) ≥ (cid:82) R (cid:37) = N (cid:90) R πα(cid:37) ( x ) d x , (1.12)which employs the extensive ideal Fermi gas energy locally at x ∈ R (with a mean-fieldmotivated guess for intermediate α ; cf. [Sen91, CS92, LBM92, CDLR19]). In [LS13a] it wasobserved that the local exclusion principle (1.10) may be used to prove the Lieb-Thirringinequality, with a constant involving α ∗ , and the successive improvements in [LL18, LS18] forthe local energy led to corresponding improvements of the constant in (1.11). Furthermore,we should mention that a possibility of minimizing the energy due to statistical repulsionusing certain clustering states (in a sense generalizing (1.2) and (1.4) for particular α ) wasdiscussed in [LS13b, Lun17], although this still remains an open problem.Another line of approach to understanding the many-anyon spectrum is to considerthe limit α → 0, known as almost-bosonic anyons . In the corresponding mean-field(or ‘average-field’ ) theory (see e.g. [CWWH89, IL92]) of weakly magnetically interactingbosons, the Thomas–Fermi approximation (1.12) has been rigorously justified in a partic-ular limit via regularized (extended, i.e. non-ideal) anyons [LR15, CLR17, CLR18, Gir20],although with a few surprises, such as a slightly bigger constant than 2 π due to self-interactions and the emergence of a vortex lattice [CDLR19]. Note that this approachis typically not possible for non-abelian models, at least not the algebraic anyon modelsstudied here whose associated exchange phases are distributed at discrete positions on theunit circle, and therefore it will not be our focus in this work.1.3. Main new results. In this work we give, to our knowledge, a first mathematically rig-orous study of the ground-state properties of the ideal non -abelian many-anyon gas, taking N → ∞ . Two-particle energies, second virial coefficients and other pairwise statistics-dependent properties have been investigated in the past for certain non-abelian models, no-tably [Ver91, LO94] as well as [MTM13b, MTM13a] for non-abelian Chern-Simons (NACS)particles, and these could be argued to be dominant for the dilute gas.After dealing with the basic tasks of defining an N -anyon model in sufficient generality,its kinetic energy operator and ground-state energy, the main aim is to find a non-abelian XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 9 equivalent of the statistical repulsion to generalize the above results for abelian anyons.Again we choose to take the route via a pairwise Poincar´e inequality and a many-anyonHardy inequality, however this requires first to define and compute for an arbitrary sequenceof anyon models ρ N : B N → U( F N ) two-anyon exchange operators U p , which are theunitary matrices in (1.7) given that p other anyons are encircled in the exchange of twoanyons. It is a consequence of the uncertainty principle that U ∼ ρ N ( σ j ) does not suffice.Our first main observation, Theorem 3.9, concerns the reduction of U p in any algebraicanyon model to an exchange involving only three objects, thus giving a recipe on how tocompute it in general. We illustrate this by computing explicit exchange operators U p andphases (eigenvalues e iβπ ) for a few models of special interest, including Fibonacci and Isingmodels as well as the simplest non-trivial Burau model (which is not of the same algebraictype). The resulting data of interest concerning the exchange properties of the model aresummarized by the exchange parameters for p enclosed particles β p := min { β ∈ [0 , 1] : e iβπ or e − iβπ is an eigenvalue of U p } , p ∈ { , . . . , N − } , and for n ∈ { , , . . . , N } involved particles (worst case of up to n − α n := min p ∈{ , , ,...,n − } β p . In Section 4.2 we define the kinetic energy operator ˆ T ρ for an arbitrary representation ρ (1.8). We then extend methods of local statistical repulsion — in all three of its manifesta-tions — to general anyon models. In particular, we prove a many-anyon Hardy inequalityˆ T ρ ≥ α N N (cid:88) ≤ j 14 min { α / , . } . In particular: • For Fibonacci anyons the exchange parameters are α = β = 3 / and α N = β =1 / for N ≥ , and hence (1.14) holds with C ( ρ N ) ≥ / (numerically C ( ρ N ) (cid:38) . ) for N ≥ . • For Ising anyons the exchange parameters are α N = β = 1 / for all N ≥ , andhence (1.14) holds with C ( ρ N ) ≥ / (numerically C ( ρ N ) (cid:38) . ) for N ≥ . algebraic geometric magnetic ρ : B N → U( D ) most general most realisticmost computational statistics transmutation Figure 1.3. Anyon models.The inhomogeneous gas is covered by the Lieb–Thirring inequality, Theorem 6.10, whichgeneralizes (1.11) to (cid:104) Ψ , ˆ T ρ Ψ (cid:105) ≥ Cα (cid:90) R (cid:37) Ψ ( x ) d x , for a universal constant C > 0. An immediate application is the thermodynamic stabilityof any Coulomb-interacting (2D in 3D, and otherwise ideal) anyon gas for which α ( N ) > N ; cf. [LS14, Theorem 21] and [Lun19, Sec. 7.3.3].Due to the necessity in this field of employing tools from all corners of mathematics,we take the style of a review and will introduce the necessary concepts as they arise. Wewill go from algebraic to geometric and magnetic models and, via functional inequalities ofPoincar´e, Hardy and Lieb–Thirring, finally to the ‘physical’/thermodynamic application.In Section 2 we recall the essentials of algebraic anyon models, which include models con-sidered in FQHE contexts and of relevance to topological quantum computing. Section 3 isdevoted to defining and computing general exchange operators and phases for these modelsof interest. Previous work for abelian anyons has relied on the existence of a magneticrepresentation, however as we will discuss in Section 4.3, it is not obvious that such a rep-resentation should exist for all non-abelian models. Therefore we take a different approachvia covering spaces, as outlined in [FM89, MS95]. The technical machinery for this is re-called in Sections 4.1-4.2 and 4.8. In Section 5 statistical repulsion is discussed, based onour generalization (1.13) of the many-anyon Hardy inequality for an arbitrary geometricanyon model. Finally, in Section 6 these tools are applied to prove bounds for the homo-geneous anyon gas energy as well as the potentially nonhomogeneous gas by means of theLieb–Thirring inequality.We use boldface to help the reader see whenever a new key concept arises. Althoughwe have tried to uniformize and streamline the various notations found in the literature,beware e.g. that the popular symbol σ has multiple meanings (spectrum, anyon, braid, . . . ).The results in this work are based in part on the MSc thesis of V. Q. [Qva17]. Acknowledgments. D. L. gratefully acknowledges financial support from the SwedishResearch Council (grant no. 2013-4734) and the G¨oran Gustafsson Foundation (grant no.1804). D. L. thanks Gustav Brage, Erik S¨onnerlind, Oskar Weinberger and Joel Wiklundfor fruitful discussions on braid group representations in their BSc thesis projects at KTHEngineering Physics [Wei15, BS18, Wik18]. We have also benefited significantly from dis-cussions with John Andersson, Eddy Ardonne, Tilman Bauer, Alexander Berglund, GeraldGoldin, Hans Hansson, Simon Larson, Jon Magne Leinaas, Tomasz Maciazek, Dan Petersen,Adam Sawicki, Robert Seiringer, Jan Philip Solovej, Yoran Tournois, and Susanne Viefers. XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 11 Algebraic anyon models A potential confusion for a newcomer to the vast literature on anyons is that, while achoice of an ideal N -anyon model is mathematically equivalent to a choice of a braid grouprepresentation (1.8), as we shall make precise in Section 4, for any physical realization itmakes all the difference in the world whether the corresponding particles are treated asclassical or quantum mechanical. That is, one could think of localizing the positions ofthe particles to a specific set of points and then perform braiding operations on them as ifthey were classical objects. This is convenient from the practical perspective of quantumcomputation and is sometimes the way that anyons are portrayed. We stress however thatan anyonic object is a consequence of identity at a fundamental quantum/mathematicallevel and can never be classical, just like bosons and fermions cannot. The uncertaintyprinciple forbids us to talk about precise positions unless we have access to infinite energy,and demands that we consider all possibilities that are not logically excluded. One wayto clarify the differences in perspectives is on one hand to consider a realization of a braidgroup representation (such as an algebraic anyon model or a FQHE quasihole ansatz) asa kinematical framework, and on the other to require an associated geometric or magneticanyon model in which also the dynamics (actual quantum Hamiltonian, Hilbert space etc.)of the particles — actual anyons — is specified (see [For91, LR16] for a related discussion).For an algebraic anyon model we specify the following information:1. Particle types / labels / topological charges.2. Rules for fusion / splitting.3. Rules for braiding.Intuitively, fusion takes into account how any subsystem of particles looks from afar,and is convenient when there is a flexible/large number of particles. The fully consistentmachinery of fusion and braiding, which was initially developed in the context of CFT andChern-Simons [Wit89, MS89] as well as algebraic QFT [FRS89, FG90], led eventually tothe notion of a unitary braided fusion category / modular tensor category [Kit06,Appendix E] [RSW09]. This is a highly technical subject and, luckily, over the years ithas been simplified and destilled through a number of excellent lecture notes, reviews andtheses into a form which is accessible to an average humanoid mathematical physicist. Wehave attempted to continue this effort and give here only a very brief summary of the partsthat are necessary for our purposes. Thus we follow the notations of [Pre04, Kit06, Bon07,NSS + 08, TTWL08, FdG10, Wan10, Pac12, DRW16, Sim16, Ton16, MG17, FS18] (althoughnote convention differences!).2.1. Fusion. Fusion rules. We start with a finite set of labels L = { a, b, c, . . . } , whose elements areinterpreted as the different types of anyons (or the topological charges ) present in themodel. It must always contain a special element 1 ∈ L called the vacuum . The elementsof L are assumed to generate a commutative, associative fusion algebra with unit 1. Thismeans that we may take formal linear combinations of labels and have a binary composition × such that for a, b ∈ L , a × b = (cid:88) c ∈L N cab c, (2.1) where the nonnegative integers N cab ∈ N are called fusion multiplicities . If N cab (cid:54) = 0 wewrite c ∈ a × b and say that c is a valid result of fusion, and N cab then counts the numberof distinguishable ways that this type of fusion may occur, with each one of them called a fusion channel . The model is formally (1) called non-abelian if there exists some a and b in L such that (cid:88) c ∈L N cab ≥ , i.e. if there are several possible ways to fuse a and b , and otherwise, i.e. if both the productand process of fusion are always unique, it is abelian . The typical models considered inthe literature are actually multiplicity free , i.e. N cab ∈ { , } for all a, b, c ∈ L , but insteadthere may be several possible fusion products c ∈ a × b , each defining a unique fusionchannel.To each anyon type a ∈ L is associated a quantum dimension d a ≥ N a = [ N cab ] b,c ∈L . By the Perron-Frobenius theorem, thesecan also be found by replacing (2.1) by the corresponding system of polynomial equations: d a d b = (cid:88) c ∈L N cab d c . Furthermore, the fusion algebra (2.1) has the property that to each a ∈ L there is a uniqueinverse element, or charge conjugate , ¯ a ∈ L such that 1 ∈ a × ¯ a . In particular, ¯1 = 1.Also note that 1 × a = a = a × N c a = N ca = δ ca .2.1.2. Fusion diagrams and spaces. Given a possible fusion c ∈ a × b , i.e. N cab (cid:54) = 0, weconstruct a Hilbert space V cab by assigning an element (cid:104) a, b ; c, µ | of an abstract orthonormalbasis to each fusion channel, enumerated by an index µ ∈ { , , . . . , N cab } : V cab := Span C {(cid:104) a, b ; c, µ |} µ . The dual space to this fusion space is called a splitting space : V abc := Span C {| a, b ; c, µ (cid:105)} µ . We picture each fusion resp. splitting channel basis state diagrammatically, with time run-ning upwards, as: a bµc respectively a bµc or simply ba c . In the last diagram we have suppressed the channel index µ (anticipating multiplicity-freemodels) and the arrows and allow time to flow from down/right to up/left (this allows fortremendous typographical simplifications upon composing diagrams). If one prefers, onemay turn the diagrams (or time) around and replace fusion spaces by splitting spaces, andvice versa. Beware that the conventions of the literature are mixed. Also note that by theproperties of the inverse/conjugate/unit we have the simple diagrams a a , a a , ¯ aa , a ¯ a . (1) This terminology is justified in [RW16]. XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 13 Fusions/splittings may be diagrammatically composed into more complicated processes;a typical one is shown in Figure 2.1, left. In case a single charge d ∈ L splits consecutivelyinto charges a, b, c , there are two distinct ways to represent this by diagrams: b ca e d respectively b ca de , (2.2)with corresponding composed splitting spaces defined by V abcd := (cid:77) e V abe ⊗ V ecd respectively ˜ V abcd := (cid:77) e V aed ⊗ V bce . (2.3)Note that we must sum over all possible intermediate charges e ∈ a × b resp. e ∈ b × c , andthat there could also be a multiplicity N cab = dim V abc etc. in each splitting process. The di-agrams (2.2) (possibly with additional suppressed labels µ ∈ { , . . . , N eab } , ν ∈ { , . . . , N dec } ,etc.) represent the basis elements of the corresponding space. All bases are naturally orderedaccording to a choice of order of the elements in L .In the case of further successive splitting we define the standard splitting space V a a ··· a n c := (cid:77) b ,b ,...,b n − V a a b ⊗ V b a b ⊗ V b a b ⊗ . . . ⊗ V b n − a n c = a a a a n − a n b b · · · b n − b n − c (cid:12)(cid:12)(cid:12)(cid:12) for all possible intermediatecharges b , b , . . . , b n − . Again, in the case of nontrivial multiplicities one must also consider all intermediate labels µ , . . . , µ n − of the respective spaces. Also note that by fusion with the vacuum we mayalways shift the diagrammatic representation: a a a b b . . . = a a a a b b . . . F-symbols: Associativity of fusion. The requirement of associativity of the fusionalgebra, ( a × b ) × c = a × ( b × c ) , enforces natural isomorphisms on the fusion spaces V abcd and ˜ V abcd . Definition 2.1 (F operator) . The F operator (or F matrix ) F abcd : V abcd = (cid:76) e V abe ⊗ V ecd → ˜ V abcd = (cid:76) e V aed ⊗ V bce is an isomorphism, diagrammatically represented by F : b ca e d (cid:55)→ b ca de = (cid:88) f F abcd ; fe b ca f d . In the case of nontrivial fusion multiplicities one may write | a, e ; d, α (cid:105) ⊗ | b, c ; e, β (cid:105) = (cid:88) f,µ,ν (cid:104) F abcd (cid:105) ( f,µ,ν ) , ( e,α,β ) | a, b ; f, µ (cid:105) ⊗ | f, c ; d, ν (cid:105) , where the components F abcd ; fe = [ F abcd ] f,e , expressed in the standard bases, are known as the F-symbols . The following lemma will be useful when computing the F -symbols: Lemma 2.2. Consider the splitting space V abcd in a multiplicity-free model. When one ofthe particle types is trivial, i.e. a, b, c or d equals , then dim V abcd = 1 , and furthermore if a, b or c equals , then the corresponding F -matrix F abcd is trivial. Explicitly that is F bcd = F bcd ; bd = 1 ,F a cd = F a cd ; ac = 1 ,F ab d = F ab d ; db = 1 . Proof. By the defining properties of the fusion algebra, b c e d = b c b d = b c dd , and so on. (cid:3) Braiding. Now that we have defined the fusion spaces, we may consider braidingoperators on these spaces. The simplest cases define the R and B symbols.2.2.1. R-symbols: Commutativity of fusion. One requires that the result of fusing a with b must be the same as fusing b with a . That is, fusion is commutative, a × b = b × a. This gives rise to a natural isomorphism between the corresponding fusion/splitting spacesfor each possible fusion channel. Definition 2.3 (R operator) . The R operator R ab is an isomorphism on each fusionchannel c ∈ a × b R ab : V bac → V abc , or in terms of the basis states, unitary matrices R abc ∈ U( N cab ), R ab : | b, a ; c, µ (cid:105) (cid:55)→ (cid:88) ν [ R abc ] νµ | a, b ; c, ν (cid:105) , diagrammatically represented by R ab : a bµc (cid:55)→ a bµc = (cid:88) ν (cid:104) R abc (cid:105) νµ a bνc . In the case of trivial fusion multiplicities we thus see that the R operator is diagonal, R ab : a bc (cid:55)→ a bc = R abc a bc , with R abc ∈ U (1) the R-symbols . That is, there is no mixing of the c label (and mixing of a and b cannot occur since these are fixed in the definition).We require trivial braiding with the vacuum: R bc = 1 and R a c = 1, for all a, b, c ∈ L . XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 15 B-symbols: Braiding of standard fusion states. We note that b ca e d = (cid:88) f (cid:0) F − (cid:1) acbd ; fe b ca df = (cid:88) f R bcf (cid:0) F − (cid:1) acbd ; fe b ca df = (cid:88) g (cid:88) f F abcd ; gf R bcf (cid:0) F − (cid:1) acbd ; fe b ca g d , where the ordering of the symbols is natural for matrix multiplication (also with regard toany suppressed multiplicities). The above operation on splitting states in V abcd defines theB operator. Definition 2.4 (B operator) . The B operator B abcd is an isomorphism on the splittingspace V abcd , given by b ca e d = (cid:80) g B abcd ; ge b ca g d , with the B-symbols B abcd ; ge := (cid:88) f F abcd ; gf R bcf (cid:0) F − (cid:1) acbd ; fe . Symbolically we write this relationship as B = F RF − . Lemma 2.5. Consider the fusion space V abcd in a multiplicity-free model. When one of theparticle types is trivial, i.e. a, b, c or d equals , then dim V abcd = 1 and the corresponding B -matrix B abcd is one-dimensional, B bcd = B bcd ; bc = R bcd ,B a cd = B a cd ; ad = R cc = 1 ,B ab d = B ab d ; da = R b b = 1 ,B abc = B abc c ¯ b = R bc ¯ a . Proof. This follows directly from the diagrams b c c d = b cd = R bcd b cd = R bcd b c b d , b ca b = b ca a = R bca b ca a = R bca b ca c , and so on. (cid:3) Pentagon and hexagon equations. There are certain consistency conditions to bemet upon combining fusions and braidings. Apart from the geometrically more obviousrelations that have been coded into the diagrammatic framework, such as e.g. Figures 2.1and 3.1 (for their formal origins, see e.g. [MS89, Eq. (4.3)], [FG90, Eq. (4.89)-(4.91)] or[Kit06, Eq. (206)]), there are also a few less obvious ones. a a . . . a n b b . . . b n c = a b dd c a b dd c Figure 2.1. Some typical fusion/splitting/braiding diagrams. Figure 2.2. Diagram corresponding to the pentagon equation (2.4).One such family of constraints are the Pentagon equations (Figure 2.2) F pzwu ; tq F xytu ; sp = (cid:88) r ∈L F xyzq ; rp F xrwu ; sq F yzws ; tr , (2.4)and another the Hexagon equations (Figure 2.3) R xzp F xzyu ; qp R yzq = (cid:88) r ∈L F zxyu ; rp R rzu F xyzu ; qr , (2.5) (cid:0) R xzp (cid:1) − F xzyu ; qp (cid:0) R yzq (cid:1) − = (cid:88) r ∈L F zxyu ; rp ( R rzu ) − F xyzu ; qr . (2.6)Furthermore, there is a remaining gauge freedom in the space of solutions to theseequations; see e.g. [Bon07, Section 2.5] and [Kit06, Appendix E.6]. Namely, for states | µ (cid:105) ∈ V abc one may consider a change of basis (cid:103) | µ (cid:48) (cid:105) = (cid:88) µ (cid:2) u abc (cid:3) µµ (cid:48) | µ (cid:105) XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 17 Figure 2.3. Diagrams corresponding to the hexagon equations (2.5)-(2.6).with u abc ∈ U( N cab ). In the multiplicity-free case these are simply phases, and amount tothe transformations ˜ F abcd ; ef = u afd u bcf u abe u ecd F abcd ; ef , ˜ R abc = u bac u abc R abc . With the totality of constraints it has been shown that there are only finitely many gaugeequivalence classes of solutions with a given fusion algebra (2.1) (Ocneanu rigidity; see[Kit06, ENO05]).We now consider a few central examples of algebraic anyon models which solve the aboveconstraints.2.4. Vacuum, bosons and fermions. A simplest example of an algebraic anyon modelis to take L = { } , so that all symbols are trivial and there is only the vacuum state. Thisis the trivial/vacuum model .The next-to-simplest case is two symbols L = { , ψ } , where ¯ ψ = ψ and ψ × ψ = 1. Notethat together with the default fusion relations 1 × × ψ = ψ , these make up therelations of the group Z . Here F = 1 up to a gauge phase, the pentagon equations aretrivial while the hexagon equations reduce to R aca + c R aba + b = R a, ( b + c ) a + b + c , (2.7)where addition is that in Z , with the only non-trivial equation( R ψψ ) = 1 . There are thus two such models: R ψψ = ± e inπ , n = 0 , 1, denoted Z ( n )2 and correspond-ing to bosons and fermions respectively. Note that there is also an accompanying gradingin the fusion whether we have even or odd numbers of particles, ψ N = (cid:40) , N even, ψ, N odd. Abelian anyons. Recall that an abelian model is characterized by the fact that thereis a unique result of fusion, a × b = c = b × a .As a generalization of the above Z abelian model, and in order to still only have a finite collection of charges L , we consider an elementary charge α ∈ [0 , 2) such that q = e iπα is a finite root of unity. Either α = µ/ν is a reduced fraction with µ ∈ { , , . . . , ν − } and ν ≥ q ν = 1 an odd root of unity, or α = µ/ν is a reduced fraction with µ ∈ { , , . . . , ν − } and ν ≥ 1, i.e. q ν = − ν and define the fusion rules [ α ] × [ β ] = [ α + β mod 2]or equivalently, with α = µ/ν , β = λ/ν , we may use the notation[ µ ] ν × [ λ ] ν = [ µ + λ mod 2 ν ] ν , the relations of Z ν . However in the case that ν is odd and µ, λ even we obtain the subgroup Z ν of even integers. The conjugate charge is ¯ α = − µ/ν mod 2 i.e. [ µ ] ν = [ − µ mod 2 ν ] ν .Since [ µ ] × νν = [2 νµ ] ν = [0] ν = 1, iteration of the hexagon equation (2.7) yields( R [ µ ] ν , [ µ ] ν [2 µ ] ν ) ν = R [ µ ] ν , [ νµ ] ν [( ν +1) µ ] ν R [ µ ] ν , [ νµ ] ν [( ν +1) µ ] ν = R [ µ ] ν , [2 νµ ] ν [ µ ] ν = 1 , and similarly to the previous case we have a family of models satisfying these equations —all the 2 ν :th roots of unity. Let L = { , α , α , . . . , α ν − } denote the one with R ααα = e iαπ ,i.e. the elementary anyon α has the statistics parameter α . In the case of even-numerator α we have α ν = 1 and a reduction to a group of ν roots of unity L = { , α , α , . . . , α ν − } .Note that the quantum dimension is d a = 1 for all a ∈ L , since d a d ¯ a = d = 1 and (cid:89) a ∈L d a = d [ (cid:80) a ∈L a ] , and which would otherwise produce a contradiction.2.6. Fibonacci anyons. The Fibonacci anyon model is determined by a single nontriv-ial anyon type L = { , τ } (with τ = τ ) and the fusion rules1 × , × τ = τ × τ, τ × τ = 1 + τ. The quantum dimension d τ is the largest solution to d τ = 1 + d τ . Hence, d τ = φ > 1, where φ := (1 + √ / τ are of interest,and the corresponding standard splitting spaces V τ n ∗ = V τ n ⊕ V τ n τ are spanned by elementsof the form τ τ ; τ τ τ , τ τ τ τ ; τ τ τ τ τ , τ τ τ τ τ , τ τ τ τ τ τ ; τ τ τ τ τ τ , τ τ τ τ τ τ τ , τ τ τ τ τ τ τ , τ τ τ τ τ τ τ , τ τ τ τ τ τ τ τ ; . . . ;with dimensions growing according to the Fibonacci sequence. Namely, the total chargecan be either 1 or τ and if it is τ then the next-to-last intermediate charge can be either1 or τ , enumerating the states of V τ n − ∗ recursively, while if the total charge is 1 then thenext-to-last must be τ , while the second next-to-last can then be either 1 or τ , enumeratingthe states of V τ n − ∗ , and so on. XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 19 Definition 2.6. The n :th Fibonacci number Fib( n ) is defined by the recurrence relationFib(0) = 0 , Fib(1) = 1 , Fib( n ) = Fib( n − 1) + Fib( n − . For example: n − − − n ) 2 − n ) = φ n − ( − φ ) − n √ . From the above observations we havedim V τ n = Fib( n − , dim V τ n τ = Fib( n ) , dim V τ n ∗ = Fib( n + 1) . See e.g. [TTWL08] and [MG17, Examples 3.9 and 3.12] for explicit solutions of theconstraints. The nontrivial pentagon equations read F ττcτ ; da F aτττ ; cb = F τττd ; ce F τeττ ; db F τττb ; ea , for all allowed a, b, c, d, e internal edges of fusion trees, and the hexagon equations R ττc F ττττ ; ca R ττa = (cid:88) b F ττττ ; cb R τbτ F ττττ ; ba , for a, b, c internal edges of fusion trees.One can show that the only potentially nontrivial F-symbols are F τττ and F ττττ (allothers allowed by fusion equal 1), and that the pentagon equations and unitarity imply F τττ = 1 and F ττττ = (cid:20) φ − ηφ − / ¯ ηφ − / − φ − (cid:21) =: F, (with a convenient abuse of notation), where the phase | η | = 1 is a remaining gauge param-eter which we will put to η = 1 for simplicity (cf. [MG17]). Note that with this conventiondet F = − F = , i.e. F − = F = ¯ F , as matrices. Hence, τ ττ τ = τ ττ τ , τ ττ τ = φ − τ ττ τ + φ − / τ ττ τ τ , τ ττ ττ = φ − / τ ττ τ − φ − τ ττ τ τ , and the inverse transformations are found by a direct exchange of diagrams.There are two sets of solutions for the nontrivial R-symbols (with continued abuse ofnotation): R := R ττ ⊕ R τττ = diag( R ττ , R τττ ) , with R ττ = e πi/ , R τττ = e − πi/ , (2.8) respectively their complex conjugates R ττ = e − πi/ , R τττ = e πi/ . (2.9)Hence for the nontrivial B-symbols, B = F RF : B ττ ∗ = R (i.e. B ττa = R ττa , a ∈ { , τ } ) , B τττ = R τττ , (2.10)and (yet another abuse of notation) B := B ττττ = (cid:20) B ττττ ;11 B ττττ ;1 τ B ττττ ; τ B ττττ ; ττ (cid:21) = (cid:20) φ − e − πi/ φ − / e πi/ φ − / e πi/ − φ − (cid:21) , (2.11)i.e. diagrammatically τ τ τ = e πi/ τ τ τ , τ τ τ τ = e − πi/ τ τ τ τ , τ ττ τ = e − πi/ τ ττ τ , τ ττ τ = φ − e − πi/ τ ττ τ + φ − / e πi/ τ ττ τ τ , τ ττ τ τ = φ − / e πi/ τ ττ τ − φ − τ ττ τ τ . One could also take all symbols complex conjugated, corresponding to the choice (2.9).We will stick to the first choice (2.8) throughout, while the second (2.9) amounts to timereversal of all diagrams. Note further that B − = B † = ¯ B and B = F R F = .2.7. Ising anyons. The Ising anyon model is described by the set of charges L = { , σ, ψ } (with σ = σ and ψ = ψ ), i.e. two nontrivial anyon types, with the nontrivialfusion rules σ × σ = 1 + ψ,σ × ψ = ψ × σ = σ,ψ × ψ = 1 . The quantum dimensions are thus d ψ = 1 and d σ = √ 2. The σ and ψ are called Ising anyonand Majorana fermion, respectively. Even though there is the additional particle type inthis model, the possible fusion/splitting basis states involving the Ising anyon σ are actuallyless numerous than those of the Fibonacci anyon ( d σ < d τ ), the first few being σ σ ; σ σ σ , σ σ σ ψ ; σ σ σ σ σ , σ σ σ σ ψ σ ; σ σ σ σ σ σ , σ σ σ σ σ ψ σ , σ σ σ σ σ σ ψ , σ σ σ σ σ ψ σ ψ ; . . . ;There are two (see e.g. [MG17, Example 3.10]) solutions for the F-symbols: F σσσσ = ± √ (cid:20) − (cid:21) =: F,F σψσψ ; σσ = F ψσψσ ; σσ = − , XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 21 all other symbols being trivial. Note that also in this case F − = F = ¯ F . There are actuallyeight different anyon models with the above fusion algebra, characterized by the value ofthe “ topological spin ” (cf. [Kit06, Sec. 10.5-10.6], [BGN11]) θ σ := 1 √ (cid:0) R σσ + R σσψ (cid:1) = e i π (2 j +1) / , j = 0 , , . . . , . We consider in this work one set of solutions for the R -symbols, with θ σ = e πi/ : R σσ = e − πi/ , R σσψ = e πi/ , R σψσ = R ψσσ = e − πi/ = − i, R ψψ = − , and another is found by taking complex conjugates. Corresponding diagrams are given in[Kit06, Table 1]. Hence for the nontrivial B-symbols, we have directly from Lemma 2.5 B ψψ = B ψψ ψψ = R ψψ = − ,B σψσ = B σψσ ; σψ = R σψσ = − i,B ψσσ = B ψσσ ; ψσ = R ψσσ = − i,B σσ ∗ = R σσ ∗ ,B ψσσ = B ψσσ σσ = R σσψ = e πi/ ,B σψσ = B σψσ σψ = R ψσσ = − i,B σσψ = B σσψ ψσ = R σψσ = − i, (2.12)and by simple computation one finds also B ψψψψ = B ψψψψ ;11 = R ψψ = − ,B σψψσ = B σψψσ ; σσ = R ψψ = − ,B ψσψσ = B ψσψσ ; σ = − R σψσ = i,B ψψσσ = B ψψσσ ;1 σ = − R ψσσ = i,B ψσσψ = B ψσσψ ; σσ = R σσ = e − πi/ ,B σψσψ = B σψσψ ; σ = − R ψσσ = i,B σσψψ = B σσψψ ;1 σ = − R σψσ = i, as well as B = F RF = B σσσσ = 12 (cid:20) R σσ + R σσψ R σσ − R σσψ R σσ − R σσψ R σσ + R σσψ (cid:21) = 1 √ (cid:20) e πi/ e − πi/ e − πi/ e πi/ (cid:21) . (2.13)Note again that B − = B † = ¯ B , while B = F R F = − i .3. Exchange operators and phases Here we show how to compute the operators and corresponding phases associated withan exchange of two anyons in an arbitrary algebraic anyon model, and illustrate the proce-dure for abelian, Fibonacci and Ising anyons. We also consider a couple of other common . . . j . . . N = = Figure 3.1. Braid diagrams corresponding to the generator σ j of B N , i.e. acounterclockwise exchange of particles/strands j and j +1 with time runningupwards, and the relations σ σ σ = σ σ σ respectively σ σ = σ σ of B .representations that arise in a different way. For an introduction to the braid group and itsrepresentations we refer to [Bir74, KT08].3.1. Braid group representations.Definition 3.1 (Braid/permutation group) . Recall that the braid group on N strands, B N , may be defined as the group with generators σ , . . . , σ N − subject to the relations σ j σ k = σ k σ j , if | j − k | ≥ ,σ j σ j +1 σ j = σ j +1 σ j σ j +1 , j = 1 , , . . . , N − , and similarly for their inverses σ − j . The symmetric/permutation group S N may bedefined by the same generators and relations as the braid group, with the additional relations σ j = 1 for j = 1 , , . . . , N − 1, hence σ − j = σ j in this case.See Figure 3.1 for corresponding diagrams. An important example that we will returnto frequently in this work is the exchange of two anyons, while enclosing exactly p otheranyons. This is represented by the group element and diagram (Figure 1.1, right):Σ p := σ σ . . . σ p σ p +1 σ p . . . σ σ ↔ t t .. t p t . (3.2)In fact, the number of generators of the braid group may be reduced to only two: Lemma 3.2. For any N ≥ , B N is generated by the two elements σ and Θ N := σ σ . . . σ N − , and furthermore σ k = Θ k − N σ Θ − kN , (3.3) for any k ∈ { , , . . . , N − } . The proof is straightforward by induction; see e.g. [Wei15, Theorem 3.5]. One shouldnote however that the group relations expressed in terms of { σ , Θ N } are more complicated.As a consequence of this alternative presentation we have also the following similarityproperty. We write as usual ˜ U ∼ U if ˜ U = S − U S for some isometry S . XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 23 Lemma 3.3. For any N ≥ , any representation ρ : B N → GL( V ) and any pair j, k ∈{ , , . . . , N − } of generators, we have similarity ρ ( σ j ) ∼ ρ ( σ k ) . Furthermore, any unitary representation ρ : B N → U( V ) is completely reducible into or-thogonal blocks of irreducible unitary representations ρ = ⊕ n ρ n , and if ρ n ( σ j ) is scalar (ascalar multiple of the identity) for some j then so is ρ n .Proof. This is a trivial consequence of the presentation (3.3): ρ ( σ k ) = ρ (Θ k − N ) ρ ( σ ) ρ (Θ k − N ) − , hence ρ ( σ ) ∼ ρ ( σ ) ∼ . . . ∼ ρ ( σ N − ). For the second statement see e.g. [Wei15, Proposi-tions 4.2 and 4.7]. (cid:3) Definition 3.4 (Abelian vs. non-abelian representations) . Naturally, we call a represen-tation ρ : B N → GL( V ) abelian if [ ρ ( σ j ) , ρ ( σ k )] = 0 for all j, k ∈ { , . . . , N − } , and non-abelian otherwise.Now look for concrete representations ρ ( σ j ) on the spaces of our algebraic anyon models.We will use the following notation for the standard fusion/splitting spaces: V a,t n c = Span (cid:40) t ta b b · · · t tb n − b n − c : for all possible b j (cid:41) V a,t n ∗ = Span (cid:40) t ta b b · · · t tb n − b n − b n : for all possible b j (cid:41) V ,t n ∗ = Span (cid:40) t t t b · · · t tb n − b n − b n − : for all possible b j (cid:41) = V t,t n − ∗ V ∗ ,t n ∗ = Span (cid:40) t tb b b · · · t tb n − b n b n +1 : for all possible b j (cid:41) Note how the only thing that differs are the fixing of left and right charge sectors. Alsonote the chain of inclusions V ,t n c ⊆ V ,t n ∗ ⊆ V ∗ ,t n ∗ , and V ∗ ,t n ∗ = (cid:77) a ∈L c ∈ a × t n V a,t n c . (3.4) Definition 3.5 ( Splitting basis representation ) . Denote ρ n ( σ j ) the representation ofthe braid generator σ j on the standard splitting space V ∗ ,t n ∗ , mapping a basis state to t t . . . t t . . . tb b . . . b j +1 . . . b n +1 , for all possible intermediate b k ∈ L , k ∈ { , . . . , n + 1 } . Hence the j th anyon goes in front of the j + 1th anyon, while in ρ n ( σ − j ) the j + 1th anyongoes in front of the j th anyon. The action for an arbitrary braid b = σ n k k . . . σ n km k m is defined similarly by composition upwards (multiplication on the left corresponding to braiding laterin time), thus obtaining a D n -dimensional unitary representation ρ n : B n → U( V ∗ ,t n ∗ ), with D n = dim V ∗ ,t n ∗ = (cid:88) b ,...,b n − N b b t N b b t . . . N b n +1 b n t (thus typically D n ∼ d nt , as n → ∞ ).Subrepresentations may then also be obtained by restricting to V a,t n c for allowed valuesof a = b and c = b n +1 , ρ n = ⊕ ρ a,t n c . (Note that braiding does not change these values, cf.Lemma 3.7 below.) Theorem 3.6. We have for the splitting basis representation ρ n ρ n ( σ j ) ∼ (cid:77) R tt , where the sum is over the labels a, c ∈ L of subrepresentations V a,t n c in the decomposition (3.4) and the intermediate possibilities d of splitting spaces V attd .Proof. On each V a,t n c in (3.4) use Lemma 3.3 and the definition of the B-operator, B ∼ R , ρ a,t n c ( σ ) = (cid:77) d ∈ a × t × t B attd ⊗ d,t n − c ∼ (cid:77) d ∈ a × t × t R tt ∗ ⊗ d,t n − c . (cid:3) In the case that consecutive braids are considered, the following diagram, allowing for anarbitrary central element c , will be particularly important. Lemma 3.7. Useful diagram: t c t a b d e = (cid:88) f,g,h B act d ; fb B ft t e ; gd B at cg ; hf t c t a h g e . (3.5) Proof. We can replace c t a b d = (cid:80) f B act d ; fb c t a f d and continue: t c t a b d e = (cid:88) f B act d ; fb t c t a f d e = (cid:88) f B act d ; fb (cid:88) g B ft t e ; gd t c t a f g e = (cid:88) f B act d ; fb (cid:88) g B ft t e ; gd (cid:88) h B at cg ; hf t c t a h g e . (cid:3) XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 25 Exchange operators. We may now generalize the simple exchange of two anyons,allowing for other anyons to be topologically enclosed in the exchange loop. Given a repre-sentation ρ : B n → U( F ), with n ≥ p + 2, one may thus consider the two-anyon exchangeoperator (2) corresponding to the braid (3.2), U p := ρ (Σ p ) = ρ ( σ ) ρ ( σ ) . . . ρ ( σ p ) ρ ( σ p +1 ) ρ ( σ p ) . . . ρ ( σ ) ρ ( σ ) . (3.6)However we will also define a more general class of exchange operators acting on the standardsplitting states. Definition 3.8 (Exchange operator) . Let U t,c,t denote counterclockwise exchange of a pairof anyons of type t around one anyon of type c acting on any corresponding standardsplitting state, that is, according to (3.5), U t,c,t : t c ta b d e (cid:55)→ t c ta b d e = (cid:88) f,g,h B actd ; fb B ftte ; gd B atcg ; hf t c ta h g e . (3.7)More generally, if the exchange is done in such a way as to enclose p anyons of type t , . . . t p ,we write U t, { t ,...,t p } ,t , U t, { t ,...,t p } ,t : t t t a a a a . . . t p ta p +1 a p +2 a p +3 (cid:55)→ t t .. t p ta a . . . a p +3 . As usual, if an anyon t is repeated p times we write t p , and if the anyon type t is understoodwe denote for brevity U p := U t,t p ,t . This is the representation of (3.6) obtained from thesplitting basis representation ρ n = p +2 in Definition 3.5.We shall only need to compute U p up to isometries — in fact we only need to know itseigenvalues. Therefore, as it turns out, the order of the inner anyons is not important in U t, { t ,...,t p } ,t , only their fusion products. Theorem 3.9 (Reduction of exchange operator [Qva17]) . The exchange operator is givenup to similarity by U t, { t ,...,t p } ,t ∼ (cid:77) c ∈ t × ... × t p U t,c,t where the direct sum is taken over all fusion channels of the fusion t × t × . . . × t p . Thatis, c is a possible result of the fusion t × . . . × t p , counted with multiplicity. (2) [FdG10] call the matrices ρ ( σ j ) ‘elementary braiding operations’ (EBO), and we may refer to these as elementary exchange operators, however we have not seen a name convention given to U p in the literature. Proof. We make a change of basis at the base of the braid and use the diagrammatic rules.Namely, using a sequence of F-moves, where t moves to the same tree as t , t to the sameas t and t , etc: a a a p +3 t t · · · t p t F − (cid:55)−−→ t t t t · · · t p t F − p (cid:55)−−−→ c t p t p − t p − ... t t The states are then enumerated by the intermediate total charge c ∈ t × t × . . . × t p . Inother words we have decomposed the space V a tt ...t p ta p +3 ∼ = (cid:77) c ∈ t × t × ... × t p (cid:101) V t ...t p c ⊗ V a tcta p +3 , where (cid:101) V is the space of states formed of successive splitting of the intermediate charge c (cf. (2.3)) into the anyons t p , t p − , . . . , t , forming a standard splitting tree (in this case alsoknown as a ‘staircase configuration’).We may then use the diagrammatic identities of Figure 2.1 (which actually translate tothe hexagon equations [MS89, p.196-197], [Kit06, p.81-82]) to place the braid at the basestate, thereby reducing the operation to U t,c,t acting only on the factor V a ,tcta p +3 . Finally wemay use the inverse isomorphism F-moves to transform the resulting unbraided state intothe original standard form. Hence, in our shorthand notation, U t, { t ,...,t p } ,t = ( F ) p − (cid:0) ⊕ c U t,c,t (cid:1) ( F − ) p − . The procedure is similar to the one depicted in [MS89, Figure 33] corresponding to thefirst p simple braids or first B-move of (3.7). (cid:3) We illustrate this theorem with some important examples of anyon models.3.3. Abelian anyons. In an arbitrary abelian anyon model with elementary exchangephase ρ ( σ j ) = e iαπ , we have the one-dimensional exchange operator (complex phase factor) U p = U α , α p , α = ρ ( σ ) . . . ρ ( σ p ) ρ ( σ p +1 ) ρ ( σ p ) . . . ρ ( σ ) = e i (1+2 p ) απ . An alternate derivation is using B abca + b + c = R bcb + c : U p = U α , α p , α = B α p αα p +1 ; α p α B α p ααα p +2 ; α p +1 α p +1 B αα p α p +1 ; αα p = R α p αα p +1 R ααα R αα p α p +1 = e i (1+2 p ) απ . Note that in Theorem 3.9 we have always a unique result of fusion α p = α × p .More generally we could also have considered a reducible abelian model: ρ ( σ j ) = S − diag( e iα π , . . . , e iα D π ) S, j = 1 , . . . , N − , S ∈ U( D ) , i.e. ρ ∼ ρ α ⊕ · · · ⊕ ρ α D , for which U p = S − diag( e i (1+2 p ) α π , . . . , e i (1+2 p ) α D π ) S, p ≥ . Note that this formula also provides a way to test if a given model is non-abelian: XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 27 ReIm • e iπ/ ( p ≡ • e i π/ ( p ≡ • e iπ ( p ≡ • e i π/ ( p ≡ • e i π/ ( p ≡ Figure 3.2. Exchange phases U p , p modulo 5, on the complex unit circlefor abelian anyons with α = 3 / Proposition 3.10. Let ρ : B N → U( D ) be a representation with ρ ( σ j ) ∼ diag( e iα π , . . ., e iα D π ) (the similarity here is not necessarily uniform in j ). If, for some p ≥ , U p (cid:28) diag( e i (1+2 p ) α π , . . . , e i (1+2 p ) α D π ) then ρ is non-abelian, i.e. there exist j (cid:54) = k so that [ ρ ( σ j ) , ρ ( σ k )] (cid:54) = 0 . The proof is immediate from the above observation for arbitrary abelian ρ .3.4. Fibonacci anyons. In the case of Fibonacci anyons the exchange operator is U p = U τ,τ p ,τ . Since p anyons may fuse either to 1 or τ with multiplicities τ p = Fib( p − · p ) · τ, by Theorem 3.9 we have U τ,τ p ,τ ∼ U ⊕ Fib( p − τ, ,τ ⊕ U ⊕ Fib( p ) τ,τ,τ . We compute the respective blocks in this decomposition:3.4.1. U τ, ,τ . Begin by observing τ τ = τ τ . This exchange operator acts on the spacewith (ordered) basis (cid:40) τ τ τ , τ τ τ τ , τ ττ τ , τ ττ τ , τ ττ τ τ (cid:41) . By (2.10)–(2.11) we find (let us again drop some indices for simplicity) U = ρ ( σ ) = B ττ ττ B τττ ; ττ B τττ ττ B ττττ ;11 B ττττ ;1 τ B ττττ ; τ B ττττ ; ττ = R ττ R τττ R τττ B B τ B τ B ττ = R ττ ⊕ R τττ ⊕ R τττ ⊕ B. Taking the first conjugation convention (2.8), we observe by B ∼ R thatspec( B ) =: σ ( B ) = σ ( R ) = { R ττ , R τττ } = { e πi/ , e − πi/ } , and thus σ ( U ) = (cid:8) R ττ (mult. 2) , R τττ (mult. 3) (cid:9) = (cid:8) e πi/ (mult. 2) , e − πi/ (mult. 3) (cid:9) . U τ,τ,τ . This exchange operator acts on the space with (ordered) basis (cid:40) τ τ τ τ τ , τ τ τ τ τ , τ τ τ τ τ τ , τ τ ττ τ , τ τ ττ τ τ , τ τ ττ τ τ , τ τ ττ τ τ , τ τ ττ τ τ τ (cid:41) . Grouping the basis into different charge sectors, we obtain ρ ( σ ) = R τττ ⊕ R ⊕ B ⊕ B B τ R τττ B τ B ττ ρ ( σ ) = R τττ ⊕ B ⊕ R ⊕ R τττ B B τ B τ B ττ U = ρ ( σ ) ρ ( σ ) ρ ( σ ) = ( R τττ ) ⊕ ( RBR ) ⊕ ( BRB ) ⊕ M, where BRB ∼ RBR = RF RF R and M = R τττ ( B ) + B τ B τ B ττ B τ B τ R τττ B τ ( B ττ ) + B B τ R τττ B τ B τ R τττ B ( R τττ ) B τ B ττ R τττ B τ ( B ττ ) + B B τ R τττ B τ B ττ R τττ ( B ττ ) + B τ B τ R τττ . In this case we obtain (as can be verified simply by computer algebra)( R τττ ) = e πi/ ,σ ( RBR ) = σ ( BRB ) = { e πi/ , e − πi/ } ,σ ( M ) = { e πi/ , e πi/ , e − πi/ } , and hence σ ( U ) = (cid:8) e πi/ (mult. 3) , e πi/ (mult. 2) , e − πi/ (mult. 3) (cid:9) . Corollary 3.11 ([Qva17]) . The eigenvalues of U p for Fibonacci anyons are (cf. Figure 3.3) σ ( U p ) = e i π/ with multiplicity Fib( p + 3) ,e iπ/ with multiplicity 2 Fib( p ) ,e − iπ/ with multiplicity 3 Fib( p ) ,e − i π/ with multiplicity 3 Fib( p − . Reducing to U p | V τ p ∗ yields the same eigenvalues with multiplicity Fib( p +1) , Fib( p ) , Fib( p ) ,respectively Fib( p − . Note that the above expressions are valid for all p ≥ p ) is defined also fornegative p . XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 29 ReIm • e iπ/ ( p ≥ • e − iπ/ ( p ≥ • e i π/ ( p ≥ • e − i π/ ( p (cid:54) = 1) Figure 3.3. Eivenvalues of U p on the complex unit circle for Fibonaccianyons (in one convention, another is obtained by complex conjugation).3.5. Ising anyons. Let us compute the exchange matrices U σ, { σ p ,ψ q } ,σ resp. U ψ, { σ p ,ψ q } ,ψ .The fusion multiplicities are σ n +1 = 2 n · σ, σ n = 2 n − · (1 + ψ ) , and hence by Theorem 3.9 we have (note that these act on differently sized spaces) U σ,σ n +1 ,σ ∼ U ⊕ n σ,σ,σ ,U σ,σ n ,σ ∼ U ⊕ n − σ, ,σ ⊕ U ⊕ n − σ,ψ,σ ,U ψ,σ n +1 ,ψ ∼ U ⊕ n ψ,σ,ψ ,U ψ,σ n ,ψ ∼ U ⊕ n − ψ, ,ψ ⊕ U ⊕ n − ψ,ψ,ψ . The exchanges involving ψ are easiest to compute as they are all diagonal: U ψ, ,ψ = R ψψ = − ,U ψ,ψ,ψ = R ψψ = − ,U ψ,σ,ψ = R σψσ R ψψ R σψσ = , i.e. for all valid labels a, b, c, d ∈ { , σ, ψ } , ψ ψa b c = − ψ ψa b c , ψ ψ ψa b c d = − ψ ψ ψa b c d , ψ σ ψa b c d = + ψ σ ψa b c d . (3.8)For the ones involving braiding of σ ’s we obtain off-diagonal matrices however:3.5.1. U σ, ,σ . This exchange operator acts on the space σ σ = σ σ with (ordered)basis (cid:40) σ σ σ , σ σ σ ψ , σ σσ σ , σ σσ ψ σ , σ σψ σ , σ σψ σ ψ (cid:41) . By (2.12)–(2.13) we have immediately U σ, ,σ = ρ ( σ ) = R σσ R σσψ B σσσσ ;11 B σσσσ ;1 ψ B σσσσ ; ψ B σσσσ ; ψψ R σσψ R σσ = R ⊕ B ⊕ ˜ R, where ˜ R := R ψ ⊕ R . Since σ ( B ) = σ ( R ) = σ ( ˜ R ) = { R , R ψ } , we find σ ( U σ, ,σ ) = (cid:8) R (mult. 3) , R ψ (mult. 3) (cid:9) = (cid:8) e − πi/ (mult. 3) , e πi/ (mult. 3) (cid:9) . U σ,σ,σ . This exchange operator acts on the space with (ordered) basis (cid:40) σ σ σ σ σ , σ σ σ σ ψ σ , σ σ σσ σ , σ σ σσ ψ σ , σ σ σσ σ ψ , σ σ σσ ψ σ ψ , σ σ σψ σ σ , σ σ σψ σ ψ σ (cid:41) . Again, by grouping into charge sectors, ρ ( σ ) = R σσ R σσψ B σσσσ ;11 B σσσσ ;1 ψ B σσσσ ; ψ B σσσσ ; ψψ B σσσσ ;11 B σσσσ ;1 ψ B σσσσ ; ψ B σσσσ ; ψψ R σσψ R σσ = R ⊕ B ⊕ B ⊕ ˜ R,ρ ( σ ) = B σσσσ ;11 B σσσσ ;1 ψ B σσσσ ; ψ B σσσσ ; ψψ R σσ R σσψ R σσψ R σσ B σσσσ ;11 B σσσσ ;1 ψ B σσσσ ; ψ B σσσσ ; ψψ = B ⊕ R ⊕ ˜ R ⊕ B. Hence U σ,σ,σ = ρ ( σ ) ρ ( σ ) ρ ( σ ) = RBR ⊕ BRB ⊕ B ˜ RB ⊕ ˜ RB ˜ R, and using that BRB ∼ RBR = RF RF R and that (by computation) σ ( BRB ) = σ ( B ˜ RB ) = σ ( ˜ RB ˜ R ) = { e − πi/ , e πi/ } , we obtain σ ( U σ,σ,σ ) = (cid:8) e − πi/ (mult. 4) , e πi/ (mult. 4) (cid:9) . XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 31 ReIm • e − i π/ (even p ≥ • e − iπ/ (all p ) • e i π/ (even p ≥ • e i π/ (all p ≥ Figure 3.4. Eivenvalues of U p on the complex unit circle for Ising anyons.3.5.3. U σ,ψ,σ . This exchange operator acts on the space with (ordered) basis σ ψ σ σ σ , σ ψ σ σ σ ψ , σ ψ σψ σ σ , σ ψ σψ σ σ ψ , σ ψ σσ ψ σ , σ ψ σσ ψ σ . Here we use Lemma 3.7 on each of these basis states, and (2.12)-(2.13), to find U σ,ψ,σ = diag (cid:0) B ψσσ ; ψσ B ψσσ σσ B σψσ ; σψ , B ψσσ ; ψσ B ψσσψ ; σσ B σψσ ; σψ , B ψψσσ ;1 σ B σσ σσ B ψσψσ ; σ , B ψψσσ ;1 σ B σσψ ; σσ B ψσψσ ; σ (cid:1) ⊕ ˜ B = diag (cid:0) e − πi/ , e πi/ , e πi/ , e − πi/ (cid:1) ⊕ ˜ B, where the last two states yield the off-diagonal part˜ B := (cid:34) B σψσψ ; σ B σσσσ ; ψψ B σσψψ ;1 σ B σψσψ ; σ B σσσσ ;1 ψ B σσψ ψσ B σψσ σψ B σσσσ ; ψ B σσψψ ;1 σ B σψσ σψ B σσσσ ;11 B σσψ ψσ (cid:35) = 1 √ (cid:20) e − πi/ e − πi/ e − πi/ e − πi/ (cid:21) , with σ ( ˜ B ) = { e − πi/ , e πi/ } . Hence σ ( U σ,ψ,σ ) = (cid:8) e − πi/ (mult. 3) , e πi/ (mult. 3) (cid:9) . In summary, Corollary 3.12. The eigenvalues of U p = U σ,σ p ,σ for Ising anyons are (cf. Figure 3.4) σ ( U σ, ,σ ) = (cid:8) e − πi/ , e πi/ (cid:9) , each with multiplicity , and for n ≥ σ ( U σ,σ n ,σ ) = (cid:8) e − πi/ , e − πi/ , e πi/ , e πi/ (cid:9) , each with multiplicity · n − , respectively, for n ≥ σ ( U σ,σ n +1 ,σ ) = (cid:8) e − πi/ , e πi/ (cid:9) , each with multiplicity n +2 . Reducing to U p | V σ p ∗ yields the same eigenvalues with multiplicity , n − , respectively n . Clifford anyons. Another anyonic model that has been considered in the literatureis obtained by taking a composition of Ising anyons with an abelian factor of α = − / ρ Clifford ( σ j ) := ρ α = − / ( σ j ) ρ Ising ( σ j ) . Because the resulting representation may alternatively be defined using Clifford algebrasand spinors [NW96, Iva01], these anyons may be referred to as Clifford anyons (or “half-quantum vortices from Majorana fermions”, see e.g. [Ton16, Ch. 4.2.2], and [DRW16, The-orem 4.8]).Using U p = U Clifford p = e − (2 p +1) πi/ U Ising p , we find σ ( U ) = (cid:8) e − πi/ , e πi/ (cid:9) ,σ ( U ) = (cid:8) e − πi/ , e πi/ (cid:9) = {± i } ,σ ( U ) = (cid:8) e − πi/ , e − πi/ , e πi/ , e πi/ (cid:9) ,σ ( U ) = (cid:8) ± (cid:9) ,σ ( U ) = (cid:8) e − πi/ , e − πi/ , e πi/ , e πi/ (cid:9) ,σ ( U ) = (cid:8) ± i (cid:9) , and so on, with σ ( U k + l ) = σ ( U l ), k ≥ l = 1 , , , 4. The first few of these U p have alsobeen computed in [BS18, Wik18] using representations of Clifford algebras.3.7. Burau anyons. The standard (unreduced) Burau representation [Bur35] is sim-ply a deformation of the (defining) permutation representation on C N with deformationparameter z ∈ C \ { } : ρ : B N → GL( C N ) ρ ( σ j ) = j − ⊕ (cid:20) − z z (cid:21) ⊕ N − j − . (3.9)Thus it has rank N , but may, depending on z , be reduced either once or twice to anirreducible representation of rank N − N − 2. At the outset these are not unitary butfor z ∈ U(1) close enough to 1 they may be unitarized [Squ84]. For more details we referto e.g. [Wei15], [DRW16, Sec. 2.3]. Note that these are different than the algebraic anyonmodels considered above since their dimension grows linearly with N .As a simplest example we may consider N = 3, reduced and unitarized: ρ : B → U(2) ρ ( σ ) = 12 (cid:20) − w + w + 1 − w √ w + w − − √ w + w − + 1 − w √ w + w − − √ w + w − + 1 − w − w + 1 (cid:21) ,ρ ( σ ) = 12 (cid:20) − w + w + 1 + w √ w + w − − √ w + w − + 1+ w √ w + w − − √ w + w − + 1 − w − w + 1 (cid:21) , where w = √ z = e iπα and | α | < / 3; see [Wei15]. We find that σ ( U ) = σ ( ρ ( σ )) = σ ( ρ ( σ )) = { , − w } XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 33 while U = ρ ( σ σ σ ) = ρ ( σ σ σ ) = diag( w , − w ) . Proposition 3.10 then verifies that ρ is necessarily non-abelian for these w (except possiblyat w = 1), but it may also be verified directly (also at w = 1) by the identity ρ ( σ σ ) − ρ ( σ σ ) = (cid:20) w √ w + w − − √ w + w − + 1 − w √ w + w − − √ w + w − + 1 0 (cid:21) . In fact, a central result in the theory of braid group representations is that the rank ofany non-abelian representation must grow at least linearly with N , and that the Buraurepresentations are of the smallest rank: Theorem 3.13 (Formanek [For96]) . Let ρ : B N → U( D ) be a unitary braid group repre-sentation, and N > . If D < N − , then ρ is abelian. Furthermore, if D = N − or D = N − then ρ is either abelian or of Burau type (i.e. equivalent to the product of anabelian and a reduced Burau representation). The proof is given in [For96, Theorem 23] for irreducible representations ρ : B N → GL( C D ), and the above formulation follows from the complete reducibility of any uni-tary representation (see Lemma 3.3). Also the case D ≤ N for arbitrary N ≥ Geometric and magnetic anyon models The general definition of the Hilbert space of anyon models was discussed in some detailin [MS95], and the choice of a self-adjoint Hamiltonian in [BCMS93, DFT97, LS14]. See also[Sou70, FM89, MD93], [KLS99, Appendix A], [DGH99, DˇST01, GM04], [Lun19, Sec. 3.7],[MS19] for further details concerning the motivations for these definitions. For backgroundon fiber bundles and connections we refer to [Nak03, Tau11].4.1. Classical configuration space. Let us temporarily consider particles as points in R d for general d ∈ N , before fixing d = 2. The configuration space for N distinguishable particles in R d is simply the set ( R d ) N of N -tuples of coordinates. Let (cid:52) (cid:52) N := { ( x , . . . , x N ) ∈ ( R d ) N : ∃ j (cid:54) = k s.t. x j = x k } denote the fat diagonal of this space, i.e. the coincidence set of at least two particles. Theconfiguration space for N distinct particles in R d is then the set ( R d ) N \ (cid:52) (cid:52) N of N -tuplesof distinct coordinates. Consider the natural (3) actions of permutations σ ∈ S N on ( R d ) N ,( σ, x) (cid:55)→ σ. x = ( x σ − (1) , . . . , x σ − ( N ) )(x , σ ) (cid:55)→ x .σ = ( x σ (1) , . . . , x σ ( N ) ) . Identifying the orbits of S N under this action on distinct particles, one obtains the config-uration space for N identical particles in R d , C N := (cid:16) ( R d ) N \ (cid:52) (cid:52) N (cid:17) (cid:46) S N = P N ( R d ) = (cid:110) X = { x , . . . , x N } ⊂ R d : | X | = N (cid:111) , (3) We may regard a tuple x = ( x j ) Nj =1 as a map j (cid:55)→ x j , so that x .σ = x ◦ σ indeed acts on the right. the space of N -point subsets of R d . For reference we fix a particular point in C N ,x = (1 e , e , . . . , N e ) (cid:55)→ X = { e , e , . . . , N e } , where e is the first unit basis vector in R d (for d = 2 this fixes a real axis in C ∼ = R ).This choice of base point will provide us with a reference ordering of the particles, andcorresponds naturally to the arrangement of strands in Figure 3.1.Consider now the continous motion of identical particles in R d . Given a continuous path γ : [0 , → C N from a point X = γ (0) to a point Y = γ (1), there is an induced action on C N : ( γ, X ) (cid:55)→ γ.X = Y In the same way we obtain a (trivial) action of continuous loops γ s.t. γ.X = X . Compo-sitions ( γ, η ) (cid:55)→ γη of paths and loops are associative under these actions. A continuousexchange of identical particles with initial configuration X ∈ C N is a loop γ in C N basedat X . In the case that we are interested in exchanges modulo homotopy equivalence,which is the case for free particles, the relevant group of loops is the fundamental group π ( C N ) = π ( C N , X ). Theorem 4.1 (Configuration space topology) . The space C N is path-connected and itsfundamental group is π ( C N ) = , d = 1 ,B N , d = 2 ,S N , d ≥ . The proof amounts to Artin’s correspondence between geometric and algebraic braidgroups [Art25, Art47], and the simple-connectedness of the configuration space of distinctparticles for d ≥ 3. For details we refer to e.g. [Bir74, Theorem 1.8], [KT08], or [Wei15].The covering space of C N is a fiber bundle˜ C N → C N with fiber π ( C N )and may be defined as the space of paths in C N based at X up to homotopy equivalence, γ ∼ γ (cid:48) iff ∃ F : [0 , × [0 , → C N cont. s.t. F (0 , · ) = γ, F (1 , · ) = γ (cid:48) ,F ( s, 0) = γ (0) , F ( s, 1) = γ (1) ∀ s ∈ [0 , . Thus, with the usual composition of paths, the action of paths and loops γ in C N lifts to afaithful action on ˜ C N , ( γ, ˜ X ) (cid:55)→ γ. ˜ X , with γ. ˜ X = γ (cid:48) . ˜ X for homotopic paths γ ∼ γ (cid:48) in C N (with the same endpoints), where the image of the action of loops γ at X ∈ C N is exactlythe induced group π ( C N , X ) of equivalence classes [ γ ] of homotopic loops based at X . Thecanonical projection to the endpoint of a path is denoted ˜pr : ˜ C N → C N . By definition ˜ C N is simply connected, π ( ˜ C N , ˜ X ) = 1.Hence, taking the reference points X (cid:55)→ ˜ X ↔ ( X , X ∈ ˜ C N maybe expressed ˜ X = ˜ γ . ˜ X = γ . ˜ X = [ γ ] . ˜ X for some path ˜ γ in ˜ C N based at ˜ X , or equivalently, using its projection γ = ˜pr ◦ ˜ γ in C N based at X = ˜pr( ˜ X ) and ending at X = ˜pr( ˜ X ), or using simply the equivalence class [ γ ]of homotopic such paths. XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 35 Any two points ˜ X, ˜ X (cid:48) ∈ ˜ C N with the same projection X = ˜pr( ˜ X ) = ˜pr( ˜ X (cid:48) ) differ onlyby an element of π ( C N ) = π ( C N , X ), since˜ X = [ γ ] . ˜ X = [ γ (cid:48) γ (cid:48)− γ ] . ˜ X = ([ γ (cid:48) ][ γ ]) . ˜ X = ˜ X (cid:48) . [ γ ] , [ γ ] = [ γ (cid:48)− γ ] ∈ π ( C N , X ), where we have defined an action of equivalence classes ofhomotopic loops based at X on the right,( ˜ X, [ γ ]) (cid:55)→ ˜ X. [ γ ] := [ γ γ ] . ˜ X . In other words we first do γ at X and then the path γ taking us to X = ˜pr( ˜ X ). Thecorresponding representation L : π ( C N ) → Diff( ˜ C N ) on the left, s.t. L [ γ ][ η ] = L [ γ ] L [ η ] , is L [ γ ] ( ˜ X ) := ˜ X. [ γ − ] = [ γ γ − ] . ˜ X . The group π ( C N ) then acts in this way as a group of diffeomorphisms of ˜ C N , and C N =˜ C N /π ( C N ). By writing each point ˜ X ∈ ˜ C N using a simple path γ from X to X and a loop γ at X , we have a way of representing ˜ X = ([ γ ] . ˜ X ) . [ γ ] as a pair ( X, [ γ ]) ∈ C N × π ( C N )(this is not a canonical idenfication but rather a choice of local trivialization of the bundle).The corresponding set of points ( X, ↔ [ γ ] . ˜ X , with γ a simple path from X to X ,constitutes a fundamental domain in ˜ C N in bijection to C N .Further, one may interchange the above actions via the conjugation of pathsAd η [ γ ] := [ ηγη − ] , for compatible paths η, γ in C N . Namely, given ˜ X = [ γ ] . ˜ X and [ γ ] ∈ π ( C N , X ),[ γ ] . ˜ X = [ γγ ] . ˜ X = [ γ ( γ − γγ )] . ˜ X = ˜ X. Ad γ − [ γ ] , with Ad γ − [ γ ] ∈ π ( C N , X ). Conversely, if [ γ ] ∈ π ( C N , X ) then˜ X. [ γ ] = Ad γ [ γ ] . ˜ X with Ad γ [ γ ] ∈ π ( C N , X ).For d = 1, trivially ˜ C N ∼ = C N since no continuous exchange is possible and the particlesmay be ordered canonically along the real line according to the choice X ↔ x ↔ ˜ X .For d ≥ C N ∼ = C N × S N , i.e. each point X in the configuration space has N !different representatives corresponding to each permutation σ ∈ S N , and we may thereforereconstruct the space of distinct particles as precisely the covering space( R d ) N \ (cid:52) (cid:52) N ∼ = ˜ C N ∼ = { ( X, σ ) : X ∈ C N , σ ∈ S N } , with the reference point ˜ X ↔ ( X , ↔ x , the ordered N -tuple of particles.For concreteness, we will from now on fix d = 2 so that the relevant group is the braidgroup B N . In this case we deal with an infinite set of representatives for each point of C N :˜ C N (cid:51) ˜ X ↔ ( X, b ) , X ∈ C N , b ∈ B N . It is now convenient to consider two projections:˜ C N ˜pr −→ C N pr ←− ( R ) N \ (cid:52) (cid:52) N ( X, b ) (cid:55)→ X = { x , . . . , x N } (cid:55)→ x = ( x , . . . , x N ) In fact there is also a canonical projection ˆpr : B N → S N defined by quoting with theadditional relations { σ j = 1 } of S N and thus we may in similarity to the higher-dimensionalcase consider˜ C N ˆpr −→ C N × S N ∼ = ( R ) N \ (cid:52) (cid:52) N pr −→ C N ( X, b ) (cid:55)→ ( X, ˆpr( b )) ↔ x = ( x , . . . , x N ) (cid:55)→ X = { x , . . . , x N } ( X , b ) (cid:55)→ ( X , ˆpr( b )) ↔ ˆpr( b − ) . x = x . ˆpr( b ) (cid:55)→ X so that the composition is ˜pr = pr ◦ ˆpr. Inverses may then be defined,pr − ( X ) := ( X, , ˜pr − ( X ) := ( X, , mapping C N to fundamental domains in ( R ) N respectively ˜ C N . Example . In the case N = 2 we may identify C ∼ = C × R + × R , where the configuration space of the relative angle C = S / Z = [0 , π ) per. is a circle,pr − ( C ) = [0 , π ) is a single-cover of C and a half-circle, pr − ( C ) = [0 , π ) per. is a double-cover and a circle, while the full covering space ˜ C = R is the real line containing thefundamental domain ˜pr − ( C ) = [0 , π ). Points on ˜ C are indexed by C and π ( C ) = Z , thewinding number of the loop relative to a base point.4.2. Hamiltonian and Hilbert space. One means of quantizing a classical system is tofind a formulation in terms of a Poisson algebra of observables and then look for repre-sentations as operators on a Hilbert space (see e.g. [Thi07, Lun19] for introductions). Weconsider quantizations of the classical non-relativistic kinetic energy, i.e. the Hamiltonianfunction (4) T = N (cid:88) j =1 p j , (4.1)and thus seek representations of the momenta p = ( p j ) Nj =1 = ( p jk ) j =1 ,...,N,k =1 , as differ-entiation operators. The important point here is that the momenta are conjugate to theconfiguration variables X = { x , . . . , x N } of identical particles, which take values on themanifold C N . Locally, i.e. on any topologically trivial open subset Ω ⊆ C N the particlesremain distinguishable, and may thus be given representatives x = ( x , . . . , x N ) on coordi-nate charts pr − (Ω) ⊆ R N , by the correspondence above. We may then use that, locallyw.r.t x in these charts, the classical canonical Poisson algebra { x jk , p j (cid:48) k (cid:48) } = δ jj (cid:48) δ kk (cid:48) , j, j (cid:48) ∈ { , . . . , N } , k, k (cid:48) ∈ { , } , can be given a Schr¨odinger representation as the quantum CCR algebra i [ˆ x jk , ˆ p j (cid:48) k (cid:48) ] = δ jj (cid:48) δ kk (cid:48) , j, j (cid:48) ∈ { , . . . , N } , k, k (cid:48) ∈ { , } , represented on a Hilbert space H Ω = L (Ω; F ), where F is a representation Hilbert spacefor any internal degrees of freedom (observables other than these x ’s and p ’s), and then thedifferent charts Ω are to be patched together to a representation on all of the configurationspace manifold C N . The natural geometric framework [MD93, DGH99, DˇST01], [Lun19,Remark 3.29] for this problem is to consider a hermitian fiber bundle E → C N over the (4) For simplicity we put the mass equal to 1 / (cid:126) = 1. XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 37 configuration space, with fiber F , endowed with a connection A and thus a covariant deriv-ative ∇ A . We are in this work, as a first step, interested in ideal, free anyon models so thatthe pure effects of statistics may be isolated. Physically, this means that upon restrictingthe particles to topologically trivial subsets Ω ⊆ C N they should behave as ideal, free anddistinguishable particles with the usual momenta ˆ p j = − i ∇ x j and the usual free kinetic en-ergy ˆ T = (cid:80) Nj =1 ( − ∆ x j ) and no interactions. Geometrically, this means that the connection A should be (locally) flat. Furthermore, for simplicity we only consider finite-dimensionalspaces F . We have the following convenient classification of such bundles: Theorem 4.3 (Classification theorem of flat bundles) . There is a 1-to-1 correspondence be-tween flat hermitian vector bundles E ρ → C N with fiber F = C D and unitary representationsof B N on F , ρ : B N → U( F ) , up to conjugation (similarity) in U( F ) . In other words, given N and fiber F , the moduli space of flat connections is exactlyHom( B N , U( F )) / U( F ) (4.2)where the action of U( F ) on representations ρ ∈ Hom is the adjoint one,( S, ρ ) (cid:55)→ Ad S ρ = SρS − : b (cid:55)→ Sρ ( b ) S − . A detailed proof of this well-known classification theorem is given in [Tau11, Theorem 13.2],and a further discussion of the correspondence in [Mic13, Chapter 5] as well as [MS19]. Thiscorrespondence motivates the following definition of an arbitrary model of ideal, free anyons. Definition 4.4 (Geometric anyon model) . By a geometric N -anyon model we meanhere a flat hermitian vector bundle E ρ → C N , or equivalently, a unitary representation ρ : B N → U( F ). Similarly, a geometric many-anyon model is a family of geometricanyon models ρ N : B N → U( F N ) for some sequence of N ≥ N are allowed; multiples, odd/even, etc.), where B is the trivial group.Further, the Hilbert space of a geometric N -anyon model is defined as the space of L global sections on the bundle E ρ → C N , i.e. the L -closure of smooth global sections, H Nρ := (cid:26) Ψ ∈ Γ( C N , E ρ ) : (cid:90) C N | Ψ | F < ∞ (cid:27) , where Γ( M, E ) denotes the set of smooth sections on the bundle E → M . Example . Using the map ˆpr : B N → S N and the canonical representation ˆ ρ : S N → U( N )of σ j acting as permutation matrices ( z = 1 in (3.9)), we may construct a bundle (the canonical permutation bundle ) with fiber F = C N and the canonical open covering { Ω σ } σ ∈ S N of R N \ (cid:52) (cid:52) N → C N with locally constant transition functions t σσ (cid:48) : Ω σ ∩ Ω σ (cid:48) → U( N ) , X (cid:55)→ ˆ ρ ( σσ (cid:48)− ) , σ, σ (cid:48) ∈ S N . The connection is locally trivial but its holonomy nontrivial due to the transition functions.The space Γ( C N , E ρ ) of smooth global sections can equivalently be characterized as thespace C ∞ ρ ( ˜ C N ; F ) of smooth ρ -equivariant functions on the covering space ˜ C N , i.e. func-tions Ψ ρ : ˜ C N → F such that, given any two points ˜ X = [ γ ] . ˜ X and ˜ Y = ˜ γ. ˜ X in ˜ C N connected by a path ˜ γ in ˜ C N and such that its projection γ = pr ◦ ˜ γ in C N , based at thepoint X = γ (0) ∈ C N , is a closed loop, γ.X = γ (1) = X , we haveΨ ρ ( ˜ Y ) = ρ ( b − )Ψ ρ ( ˜ X ) , (4.3) where b = Ad γ − [ γ ] ∈ B N is the equivalence class at X to which the loop γ belongs,i.e. ˜ Y = ˜ X.b = L b − ( ˜ X ). Thus we have an isomorphism H Nρ ∼ = L ρ ( ˜ C N ; F ) (see [MS95,Lemma 2.4]), where the inner product of the latter space is defined (cid:104) Φ ρ , Ψ ρ (cid:105) L ρ = (cid:90) C N (cid:104) Φ ρ ( ˜ X ) , Ψ ρ ( ˜ X ) (cid:105) F dX. In the integrand ˜ X denotes any pre-image of X ∈ C N under the projection ˜pr : ˜ C N → C N .This integrand is indeed well defined thanks to the equivariance condition (4.3), because if˜ X (cid:48) = ˜ X.b , b ∈ B N , then (cid:104) Φ ρ ( ˜ X (cid:48) ) , Ψ ρ ( ˜ X (cid:48) ) (cid:105) F = (cid:104) ρ ( b − )Φ ρ ( ˜ X ) , ρ ( b − )Ψ ρ ( ˜ X ) (cid:105) F = (cid:104) Φ ρ ( ˜ X ) , Ψ ρ ( ˜ X ) (cid:105) F , by the unitarity of the representation ρ . The measure dX on C N is derived from the usualLebesgue measure d x on R N via the projection pr, such that (cid:82) Ω dX = N ! (cid:82) pr − (Ω) d x.The condition (4.3) is in fact used in the proof of Theorem 4.3 to explicitly constructthe flat bundle E ρ associated to ρ ; cf. [Tau11, Chapter 13.9.3]. A ρ -equivariant functionmay be considered a relation Ψ ρ ⊆ ˜ C N × F with a unique v ∈ F to each ˜ X ∈ ˜ C N and suchthat if ( ˜ X, v ) ∈ Ψ ρ then ( ˜ X.b, ρ ( b − ) v ) ∈ Ψ ρ for all b ∈ B N . Then the bundle E ρ → C N isprecisely the set of orbits in ˜ C N × F of the right action( ˜ X, v ) .b := ( ˜ X.b, ρ ( b − ) v ) , ˜ X ∈ ˜ C N , v ∈ F , b ∈ B N , with projection [( ˜ X, v )] (cid:55)→ ˜pr( ˜ X ). Conversely, given a smooth global section Ψ ∈ Γ( C N , E ρ ),there is to each ˜ X a unique v ∈ F such that Ψ( ˜pr( ˜ X )) = [( ˜ X, v )], which defines a ρ -equivariant function Ψ ρ ( ˜ X ) := v (again, see [MS95, Lemma 2.4]).Given an N -anyon model ρ and locally flat bundle E ρ → C N we now have a meansof defining a covariant derivative ∇ ρ Ψ on smooth sections Ψ ∈ Γ( C N , E ρ ) which on anytopologically trivial subset Ω ⊆ C N reduces to the ordinary derivative on Ψ : Ω → F∇ Ψ( X ) = ( ∇ j Ψ) j =1 ,...,N = ( ∂ Ψ /∂x jp ) p =1 , j =1 ,...,N and thus a kinetic energy operator ˆ T ρ = ( − i ∇ ρ ) ∗ ( − i ∇ ρ ) which on Ω reduces to the ordinaryone for distinguishable particlesˆ T ρ Ψ( X ) = N (cid:88) j =1 ( − i ∇ j ) Ψ = − N (cid:88) j =1 (cid:88) p =1 , ∂ Ψ /∂x jp . The corresponding quadratic form T ρ [Ψ] := (cid:90) C N |∇ ρ Ψ | F N dX (4.4)may be extended to the L -sections for which the integral is finite, by taking the closure ofsmooth sections with compact support on C N (see below).In the ρ -equivariant setting, a function Ψ ρ ∈ C ∞ ρ ( ˜ C N ; F ) has a derivative at any point˜ X ∈ ˜ C N , defined by ∇ Ψ ρ = (cid:18) ∂ Ψ ρ ∂x j , ∂ Ψ ρ ∂x j (cid:19) j =1 ,...,N , ∂ Ψ ρ ∂x jp = lim t → t (Ψ ρ ( γ tjp . ˜ X ) − Ψ ρ ( ˜ X )) (4.5) XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 39 where γ tjp is the path [0 , (cid:51) s (cid:55)→ { x , . . . , x j + st e p , . . . , x N } in C N defining the corre-sponding tangent vector. The index j as it stands here is artificial and depends on whichordering of particles we choose, and we may use the projection ˆpr : ˜ X (cid:55)→ x before applyingthe path on the j th entry of the tuple x and then project again to C N with pr. Hence thevector in (4.5) depends in fact on the pre-image of X in B N . In any case, this ambiguity isremoved in the permutation-invariant sum |∇ Ψ ρ | F N = N (cid:88) j =1 (cid:88) p =1 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ Ψ ρ ∂x jp (cid:12)(cid:12)(cid:12)(cid:12) F . (4.6)The kinetic energy T ρ [Ψ ρ ] := (cid:90) C N |∇ Ψ ρ ( ˜ X ) | F N dX (4.7)is then again well defined because of equivariance, ∂ Ψ ρ ∂x jp ( ˜ X.b ) = lim t → t (Ψ( γ tjp . ( ˜ X.b )) − Ψ( ˜ X.b )) = ρ ( b − ) ∂ Ψ ρ ∂x j (cid:48) p ( ˜ X ) , j (cid:48) = ˆpr( b )[ j ] , (4.8) (cid:104)∇ Φ( ˜ X (cid:48) ) , ∇ Ψ( ˜ X (cid:48) ) (cid:105) F N = N (cid:88) j =1 (cid:88) p =1 , (cid:28) ρ ( b − ) ∂ Φ ρ ∂x j (cid:48) p ( ˜ X ) , ρ ( b − ) ∂ Ψ ρ ∂x j (cid:48) p ( ˜ X ) (cid:29) F = (cid:104) ρ ( b − ) ∇ Φ( ˜ X ) , ρ ( b − ) ∇ Ψ( ˜ X ) (cid:105) F N = (cid:104)∇ Φ( ˜ X ) , ∇ Ψ( ˜ X ) (cid:105) F N , and is identical to (4.4) by the above correspondence. We consider the closure of this form(4.7) on Ψ ρ ∈ C ∞ ρ,c , the ρ -equivariant smooth functions on ˜ C N such that ˜pr(supp Ψ ρ ) ⊆ C N is compact, which then defines a Sobolev subspace H ρ ⊆ L ρ of anyon wave functions withfinite expected kinetic energy. This is also a Hilbert space with the inner product (cid:104) Φ , Ψ (cid:105) H ρ := (cid:90) C N (cid:16) (cid:104) Φ( ˜ X ) , Ψ( ˜ X ) (cid:105) F + (cid:104)∇ Φ( ˜ X ) , ∇ Ψ( ˜ X ) (cid:105) F N (cid:17) dX. Definition 4.6 (Anyon Hamiltonian) . We define the kinetic energy operator ˆ T ρ on the full N -anyon Hilbert space H Nρ as the unique self-adjoint operator corresponding to the closednon-negative quadratic form T ρ in (4.7), with operator domain D ( ˆ T ρ ) ⊆ H ρ ⊆ H Nρ (see e.g.[Tes14, Theorem 2.14]). Considering ˆ T ρ as initially defined on C ∞ ρ,c , with supports awayfrom diagonals (cid:52) (cid:52) N , this is the Friedrichs extension . Remark . One may consider other extensions, such as the Krein extension (5) , howevercomparing to the two-anyon case where the extension theory has been carried throughin detail [GHKL91, MT91, BS92, ACB95, CO18, CF20] these other extensions would allcorrespond to introducing additional attractive interactions between the anyons. Thustaking the Friedrichs extension is in alignment with our wish in this work to isolate the (5) This is the smallest (in terms of energy) non-negative extension while Friedrichs is the largest. Theextension theory for the 2-anyon problem is actually somewhat analogous to that of two bosons in 3D,admitting a circle of extensions, with the functions of the Friedrichs extension vanishing as r α and those ofKrein behaving like r − α as the relative distance r → 0. Indeed, just as in 3D, we anticipate that there couldbe good physical reasons to consider these other extensions whenever anyons arise as emergent particlesalong with other interactions. Non-Friedrichs extensions are sometimes referred to as soft-core anyons. effects of statistics and study purely ideal anyons. See [DFT94] for a possible approachto the other extensions in the many-anyon case, and [LS14] for some further mathematicaldetails in the abelian Friedrichs case. Note that the issue here concerns point interactions supported at (cid:52) (cid:52) N while we could also have introduced ordinary scalar interactions by addingpair (or higher order) potentials to the Hamiltonian (4.1). Both these types of modificationswould alter the behavior for distinguishable particles as well.In the many-anyon context one may consider the group B ∞ defined as the direct limitof the N -strand groups with respect to the inclusion maps B N (cid:44) → B N +1 sending σ j (cid:55)→ σ j .It is also natural to consider sequences of representations ρ N : B N → U( F N ) such that therespective inclusions B N (cid:44) → B N +1 and C ρ N ( B N ) (cid:44) → C ρ N +1 ( B N +1 ) (the group algebras)commute [RW12, Definition 2.1], although not all representations are of this type, such asthe Burau representations [DRW16, Sec. 6.1].For geometric many-anyon models we may also consider the Fock space H ∞ := ∞ (cid:77) N =0 H Nρ N , where H ρ := F = C is the vacuum sector. Recall that in order to have a non-abelianmodel it is necessary that dim F N ≥ N − F as we take N → ∞ , as one may do for bosons and fermions.4.3. Statistics transmutation. Given N and fiber F , let the trivial or bosonic bundle be the bundle with trivial geometry, taking the trivial representation ρ + ( σ j ) := + for all j , thus E ρ + = C N × F . We note that any function Ψ + ∈ H N + of the standard bosonic N -body Hilbert space H N + := L (( R ) N ; F ) := (cid:8) Ψ ∈ L ( R N ; F ) : Ψ(x .σ ) = Ψ(x) ∀ σ ∈ S N (cid:9) , may be considered an equivariant function Ψ ρ + ∈ H Nρ + of the bosonic bundle. Namely,by the canonical map ˆpr : B N → S N , we have in ˜ C N pre-images of C N labeled by permu-tations ˆpr( b ). A symmetric function Ψ + ∈ H N + thus extends to a function Ψ ρ + ( ˜ X.b ) :=Ψ + (x . ˆpr( b )) = Ψ + (x), x = ˆpr( ˜ X ). Conversely, any function Ψ ρ + : ˜ C N → F with symmetryΨ ρ + ( ˜ X ) = Ψ ρ + ( ˜ X.b ) = Ψ ρ + ( X ) , b ∈ B N , where ˜ X is any pre-image in ˜ C N of a point X ∈ C N , may be identified with a function in H N + by defining Ψ + (x) := Ψ ρ + (cid:0) pr(x) (cid:1) on R N \(cid:52) (cid:52) N and using that L ( R N \(cid:52) (cid:52) N ) = L ( R N )by the vanishing measure of (cid:52) (cid:52) N in R N . Note however that to make it a probabilitydistribution in L ( R N ) (i.e. on the distinguishable configuration space as is conventional)it is necessary to normalize Ψ + := Ψ ρ + / √ N !. We note for example that, given any ρ -equivariant Ψ ρ , the scalar function | Ψ ρ | F (amplitude) is ρ + -equivariant, | Ψ ρ ( ˜ X.b ) | F = | ρ ( b − )Ψ ρ ( ˜ X ) | F = | Ψ ρ ( ˜ X ) | F , which is to say that | Ψ ρ | F defines a probability distribution on C N for L ρ -normalized Ψ ρ .Furthermore, we have that H ( R N \ (cid:52) (cid:52) N ) = H ( R N ) (see e.g. [LS14, Lemma 3] and, more XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 41 generally, [LLN19, Appendix B]) so that we may as well identify the bosonic Sobolevspaces H ρ + ∼ = H := C ∞ c ( R N \ (cid:52) (cid:52) N ) ∩ L H ( R N ) . Thus, after taking the closure, the functions do not have to vanish at (cid:52) (cid:52) N , and indeed forexample the Gaussian Ψ(x) = e −| x | ∈ H .There is also an analogous version for fermions, where the fermionic bundle is definedby ρ − ( σ j ) := − for all j , i.e. ρ − = sign ◦ ˆpr ⊗ . Instead of H N + we then consider the fermionic N -body Hilbert space H N − := L (( R ) N ; F ) := (cid:8) Ψ ∈ L ( R N ; F ) : Ψ(x .σ ) = (sign σ )Ψ(x) ∀ σ ∈ S N (cid:9) , Taking Ψ ρ − ( ˜ X.b ) := Ψ − (x . ˆpr( b )) = sign ˆpr( b − )Ψ − (x), we obtain a section of the fermionicbundle, and vice versa Ψ − (x) := Ψ ρ − ( ˜ X ) (modulo normalizations) with any lift ˜ X (cid:55)→ x (cid:55)→ X defines an antisymmetric function. We define analogously the fermionic Sobolev space H ρ − ∼ = H := C ∞ c ( R N \ (cid:52) (cid:52) N ) ∩ L H ( R N ) , and in this case we will see (by means of the Hardy inequality in Section 5) that the functionswith finite kinetic energy necessarily vanish on (cid:52) (cid:52) N , as had been expected from (1.3).Note that to any flat hermitian vector bundle E → C N with fiber F , or representation ρ : B N → U( F ), there is also the associated flat principal bundle P → C N with fiber U( F )(and vice versa). Again the correspondence to ρ -equivariant functions carries over to P ,which may be defined as the set of orbits in ˜ C N × U( F ) of the right action( ˜ X, g ) .b := ( ˜ X.b, ρ ( b − ) g ) , ˜ X ∈ ˜ C N , g ∈ U( F ) , b ∈ b N , with the projection [( ˜ X, g )] (cid:55)→ ˜pr ˜ X = X . Using a similar identification as for Γ( C N , E ) then,the set of smooth global sections Γ( C N , P ) is the set of smooth functions u ρ : ˜ C N → U( F )that are ρ -equivariant, u ρ ( ˜ X.b ) = ρ ( b − ) u ρ ( ˜ X ) . (4.9)Recall that a hermitian vector bundle of rank n is trivial iff there exist n global sec-tions which are pointwise orthonormal thus defining a frame { u ( X ) , . . . , u n ( X ) } in F and,equivalently, iff there exists a global section of the principal bundle, so that P ∼ = C N × U( F ).Thus, triviality of E and P is equivalent to the existence of u ρ satisfying (4.9).Now, given an anyon model ρ : B N → U( F ), assume that there exists a smooth function u ρ : ˜ C N → U( F ) satisfying (4.9), i.e. a global section of P . Then, given any symmetricfunction Ψ + ∈ H N + one obtains a ρ -equivariant function Ψ ρ ∈ H Nρ by takingΨ ρ ( ˜ X ) := u ρ ( ˜ X )Ψ + ( ˜ X ) . (4.10)Conversely, given a ρ -equivariant function Ψ ρ ∈ H Nρ , the functionΨ + ( ˜ X ) := u ρ ( ˜ X ) − Ψ ρ ( ˜ X ) (4.11)satisfies Ψ + ( ˜ X.b ) = Ψ + ( ˜ X ), where ˜ X is any pre-image in ˜ C N of X ∈ C N , and may thereforebe identified with a function in H N + . Considering the kinetic energy, we may write using (4.10) T ρ [Ψ ρ ] = (cid:90) C N |∇ ( u ρ Ψ + )( ˜ X ) | F N dX = (cid:90) C N |∇ A Ψ + ( ˜ X ) | F N dX = (cid:90) C N |∇ A Ψ + ( X ) | F N dX, (4.12)where the covariant derivative ∇ A = ∇ + A , is given in terms of the connection A : ˜ C N → u ( F ) N ( u ( F ) is the Lie algebra of U( F )), A ( ˜ X ) := u ρ ( ˜ X ) − ∇ u ρ ( ˜ X ) . This expression descends to ˆpr( ˜ C N ) = R N \ (cid:52) (cid:52) N thanks to the symmetry A j ( ˜ X.b ) = u ρ ( ˜ X ) − ρ ( b ) ∇ j (cid:48) [ ρ ( b − ) u ρ ]( ˜ X ) = A j (cid:48) ( ˜ X )and thus the integrand to C N , thanks to the permutation invariance of the sum, as in (4.6).We call the above operation statistics transmutation as it takes us from one model ofanyons to another, to the cost of introducing the non-trivial connection or gauge potential A . In fact in the abelian case F = C , u ( F ) = i R , one has A j = i A j for magnetic vectorpotentials A j : R N \ (cid:52) (cid:52) N → R with magnetic fieldcurl A j = ∇ u † ρ ∧ ∇ u ρ + u † ρ ∇ ∧ ∇ u ρ = , as follows due to differentiating u † ρ u ρ = 1. We have thus represented the geometric anyonmodel ρ in this way as a magnetic anyon model using bosons and magnetic potentials.Also in the non-abelian case one could talk about non-abelian “magnetic” gauge fields withzero curvature, Dω A = dω A + ω A ∧ ω A = 0 , ω A = (cid:88) j,k A jk dx jk = u † ρ du ρ . One may do the corresponding transformations (4.10)-(4.11)-(4.12) also with a fermionicfunction as reference (and indeed there are good physical reasons for doing so), althoughfor simplicity we will stick to the bosonic case in all that follows as it will be seen not to bea loss in generality. Definition 4.8 (Transmutable anyon model) . A geometric N -anyon model, determined bya braid group representation ρ : B N → U( F ), will be called transmutable if its associatedflat principal bundle P → C N is trivial, i.e. if it admits a smooth global section in Γ( C N , P )or a ρ -equivariant function u ρ : ˜ C N → U( F ). The corresponding transmuted model withbosons and the gauge potential A = u † ρ ∇ u ρ in the kinetic energy (4.12) will be called a magnetic anyon model corresponding to ρ . The geometric model formulation is alsoreferred to as the anyon gauge and the magnetic model as the magnetic gauge .Bosons are then obviously transmutable with the global section u ρ = . Naturally thisleads to the following question which we have not seen duly discussed in the literature(among the exceptions we find [Blo80, Dow85, SISI88, HMS89, FM89, ISIS90, DGH99,MS95], as well as very recently [MS19] which take an alternative approach via graph con-figuration spaces): Question 4.9. Which (geometric) anyon models are transmutable? Only in the abelian case do we find a complete answer: XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 43 Theorem 4.10. All abelian anyon models are transmutable. A proof was given by [Dow85] and [MS95, Theorem 2.22] (see also e.g. [MS19, Example 5])and involves an observation that torsion in homology H ( B N , Z ) is the same as torsionin cohomology H ( B N , Z ), with an explicit global section given below, which was alsoimplicitly used in the earlier works [Wu84, FM91].Since a fermionic model is abelian it is thus transmutable into a bosonic model. Fur-thermore, two-anyon models are trivially transmutable since they are also abelian (thoughpossibly of higher rank and reducible). Also recall that for rank D = dim F < N − N > 6, any representation of B N is necessarily abelian (Theorem 3.13) and therefore thebundle trivializes to a sum of line bundles. Geometrically, the moduli space (4.2) of possibleanyon models or 2D quantum statistics splits into components of non-transmutable models(i.e. isomorphism classes of topologically non-trivial bundles) and a component of mutuallytransmutable models (which in turn may a priori consist of several connected components ofdifferent flat connections on a topologically trivial bundle; see however [MS19, Section 3.3])including the circle ρ α ⊗ F , α ∈ [0 , ρ α =0 = ρ + of the bosonic/trivial bundle.The classification of isomorphism classes of higher-rank flat hermitian vector bundles on C N for N > Definition 4.11 (Whitney sum) . Given two geometric N -anyon models, determined byrepresentations ρ : B N → U( F ) resp. ρ : B N → U( F ), their Whitney sum is thegeometric N -anyon model determined by ρ = ρ ⊕ ρ : B N → U( F ⊕ F ). This is thebundle E ρ = E ρ ⊕ E ρ .Namely, as noted in [MS95, p.10], the bundles trivialize under the taking of their k -foldWhitney sum ⊕ k ρ for large enough k , because of the following finiteness theorem for thecohomology ring H ∗ ( C N , Z ) = H ∗ ( B N , Z ) [Arn70, Arn14, Vai78]: Theorem 4.12 (Arnold’s Cohomology Theorems [Arn14, p.201]) . Finiteness: With the exception of H ∼ = H ∼ = Z , the cohomology groups H j ( B N , Z ) are allfinite. Furthermore, H j ( B N , Z ) = 0 for j = 2 and for j ≥ N .Repetition: H j ( B N +1 , Z ) = H j ( B N , Z ) for all j and N .Stability: H j ( B N , Z ) = H j ( B j − , Z ) for all N ≥ j − . Theorem 4.13. Given any flat vector bundle E → C N , there exists a k ∈ N such that the k -fold Whitney sum (cid:76) k E is trivial.Proof. Hermitian vector bundles are classified up to stable equivalence by their Chernclasses, c j ∈ H j , which trivialize in ⊕ k E for large enough k if they are only torsion.Furthermore, a hermitian vector bundle of sufficiently high rank is trivial iff it is stablytrivial [AE12]. (cid:3) Remark . As a relevant comparison, note that with the analogous definitions, a pair of spinless fermions in dimension d = 3 is not transmutable to bosons. Namely, in relativecoordinates C ∼ = C × R + × R , where the relative angles are on C ∼ = S / ∼ , the sphere with antipodal points identified.Here ˜ C ∼ = S and hence a global section u ρ : ˜ C → U(1) is a smooth function on the spherewith the equivariance relation u ρ ( − x ) = ρ ( τ ) u ρ ( x ) , x ∈ S , where τ is the generator of π ( C ) = Z and ρ the representation. For bosons ( ρ + = 1)we can take u ρ + = 1 while for fermions ( ρ − ( τ ) = − 1) there is no such function by theBorsuk-Ulam theorem (any continuous and odd complex-valued function on the sphere hasa zero somewhere). However, taking the Whitney sum, ρ = ρ − ⊕ ρ − , there is e.g. theglobal section u ρ : S → U(2) , u ρ ( x, y, z ) = x (cid:20) (cid:21) + y (cid:20) − ii (cid:21) + z (cid:20) − (cid:21) , x + y + z = 1 , that trivializes the corresponding bundle. (6) Abelian anyons. The fiber for an irreducible abelian model ρ α ( σ − j ) := e iαπ , α ∈ ( − , F = V α N c = C , where the total charge c = α × N corresponds to the totalnormalized magnetic flux of all particles (mod 2). The convention we use here on thesign of α (conjugation symmetry) will help to recover the familiar conventions with theequivariance condition (4.3) actually being an action of B N on the right.A global section is given by the analytic continuation to ˜ C N of u ρ α (x) := U (x) α , wherewe use z j := x j + ix j ∈ C and the boson-fermion transmutation U : L / asym → L / sym , ( U Ψ)(x) := (cid:89) ≤ j The corresponding kinetic energy for bosonic Ψ + = u − ρ α Ψ ρ α isˆ T α := N (cid:88) j =1 ( − i ∇ j + A α,j ) , T α [Ψ + ] = 1 N ! (cid:90) R N | ( − i ∇ j + A α,j )Ψ + | . In this formulation there is the possibility to extend the statistics parameter α to all of R and consider the gauge-equivalent magnetic models − i ∇ + A α +2 n = U − n ( − i ∇ + A α ) U n , D ( ˆ T α +2 n ) = U − n D ( ˆ T α ) , n ∈ Z . For α = 0 the domain of the Friedrichs extension is D ( ˆ T ρ + ) = H = L ∩ H ( R N )which defines free bosons, while for α = 1 it is D ( ˆ T ρ − ) = U − H = U − L ∩ H ( R N )which defines free fermions in a bosonic representation (see [LS14, Section 2.2]).In the reducible case F = C D , we take a basis of joint eigenvectors s.t. ρ ( σ − j ) = S diag( e iγ π , . . . , e iγ D π ) S − ,S ∈ U( D ), and the corresponding global section is the analytic continuation to ˜ C N of u ρ (x) = S diag ( U (x) γ , . . . , U (x) γ D ) S − . Thus, if { v n } is a basis in F of joint unit eigenvectors such that ρ ( σ − j ) v n = e iγ n π v n , thenwe may consider the subspace in H ρ consisting of functionsΨ( ˜ X ) = D (cid:88) n =1 U ( ˜ X ) γ n Φ n ( X ) v n , (4.13)where Φ n ∈ C ∞ c, sym ( R N \ (cid:52) (cid:52) N ), for which (cid:90) C N | Ψ | = D (cid:88) n =1 N ! (cid:90) R N | Φ n | , T ρ [Ψ] = D (cid:88) n =1 T γ n [Φ n ] , i.e. (Φ n ) ∈ ( H N + ) D models a collection of N -anyon wave functions in the magnetic repre-sentation with statistics parameters α = γ n , n ∈ { , . . . , D } .For a glimpse of how such abelian models may arise in a FQHE context, see e.g. the briefintroduction [Tou20, Chapter 2], and [LR16] for a more precise application (accountingfor certain ambiguities [For91, For92] in the original derivation [ASW84]). Another recentrealization is via quantum impurity problems, polarons and angulons [YGL + 19, BLLY20].4.5. Fibonacci anyons. We consider the representation ρ N : B N → U( F N ) on the split-ting space given in Definition 3.5. Imposing that we should have exactly N anyons, as wellas irreducibility, we may thus take F Nc := V τ N c , where there is a remaining choice to bemade for the total charge c ∈ { , τ } : F N ∼ = C Fib( N − and F Nτ ∼ = C Fib( N ) .Presently we do not know if this model is transmutable or not. However, there is acorrespondence to FQHE and CFT (Read-Rezayi states [RR99, CGT01, AKS05, Lun17])which is similar to that for the Ising model discussed below, and which seems to suggesttransmutability [AS07, TA20]. Ising anyons. Here we could work with two types of particles (the Ising anyon σ and the Majorana fermion ψ ) explicitly present in the system and having dynamics, but tosimplify matters slightly we consider N anyons of only the more interesting type σ , withfiber F N = V σ N σ ∼ = C ( N − / for N odd and F Nc = V σ N c ∼ = C N/ − for N even, where either c = σ or c = ψ .The corresponding model, also including the fermions ψ , arises in the FQHE context, sup-posedly as an effective model for quasiparticles modeled by a family of electron (fermion)wave functions known as Moore-Read or Pfaffian states [MR91, NW96]. Another represen-tation for these states are as correlators of a CFT related to the Ising model. This storyis beyond the scope of the present article (see e.g. [HHSV17] and [Tou20, Chapter 3.4] forrecent reviews), but it suffices to note that, given positions x j ∈ R , j = 1 , . . . , N lifting to˜ X ∈ ˜ C N there exists a family of states { Ψ n ( ˜ X ) } Dn =1 with values in an M -fermion Hilbertspace C D ∼ = F N ⊆ L ( R M ) and some constant C > M → ∞(cid:104) Ψ n ( ˜ X ) , Ψ m ( ˜ X ) (cid:105) F N = Cδ nm + O ( e −| x j − x k | )if | x j − x k | → ∞ [GN97, BGN11]. Thus the model appears to be transmutable by theexistence of this trivialization, at least in some limit and for N = 3 or 4, and D = 2.However the connection in this family of models is the Berry connection induced from theembedding F N (cid:44) → L ( R M ) and which is not necessarily locally flat.Another way to obtain its transmutability would be to use that is expected (as part of theCFT correspondence) to arise as subrepresentations of NACS with gauge group G = SU (2).4.7. NACS. There is a certain family of representations which arise in the context of Non-Abelian Chern-Simons (NACS) theory [Koh87, GMM90, Ver91, Lo93, LO94, BJP94,MTM13b, MTM13a], These may be explicitly defined as magnetic models on the trivialbundle [Koh87], and are hence transmutable. Typically one formulates the models in aholomorphic gauge for convenience.Take a compact Lie group G (for example G = SU(2)) and a unitary representation ρ : G → U( F ) (for example F = C ), such that the generators L a of the Lie algebra g of G are mapped to operators ˆ L a ∈ L ( F ). Using path-ordered exponentials, we may write g = exp P (cid:90) γ (cid:88) a L a dg a , ρ ( g ) = exp P (cid:90) γ (cid:88) a ˆ L a dg a . Let F N = (cid:78) N F , with ρ acting canonically on each of the factors ρ ( j )1 ( g ) = exp P (cid:90) γ (cid:88) a ˆ L ( j ) a dg a = ⊗ ρ ( g ) ⊗ , ˆ L ( j ) a = ⊗ ˆ L a ⊗ . Defining U ρ ( ˜ X ) := exp P (cid:90) Γ πκ (cid:88) j (cid:54) = k (cid:88) a ˆ L ( j ) a ˆ L ( k ) a d log( z j − z k ) , ˜ X = [Γ] . ˜ X , then F N carries a representation ρ : B N → U( F N ) with a globally defined section u ρ ( ˜ X ) := U ρ ( ˜ X ) / | U ρ ( ˜ X ) | F N . The parameter 4 πκ − ∈ Z is known as the level of the representation, and U ρ formallysolves the Knizhnik-Zamolodchikov equations [KZ84], [Koh87], [LO94, Eq. (39)]. XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 47 The representation ρ will typically be reducible into irreducible blocks (‘conformal blocks’) ρ n and one could say that, while ρ n may not be transmutable individually, their sum ρ is.4.8. Local kinetic energy. Given a subset Ω ⊆ R we define the local configurationspace of identical particles on Ω C N (Ω) := (cid:0) Ω N \ (cid:52) (cid:52) N (cid:1) (cid:46) S N = P N (Ω) , the set of N -point subsets of Ω. Let now Ω be open and simply connected (homotopic to R ). Similarly to the full space we may again define the covering space˜ C N (Ω) → C N (Ω)as the space of continuous paths in C N (Ω) modulo homotopy equivalence. We choose asreference base point for these paths a point X (Ω) ∈ C N (Ω), which is exactly the ordered setof points X ∈ C N ( R ) as before, only translated and scaled to fit the domain Ω. Denote apath in C N ( R ) which performs this scaling and translation by γ Ω , so that X (Ω) = [ γ Ω ] .X and ˜ X (Ω) := [ γ Ω ] . ˜ X . With the corresponding scaling and translation [ γ Ω γ (cid:48)− ] to matchthe reference points X (Ω) and X (Ω (cid:48) ) there is then an inclusion˜ C N (Ω (cid:48) ) (cid:44) → ˜ C N (Ω) if Ω (cid:48) ⊆ Ω . By the homotopy equivalence Ω ∼ R we have π ( C N (Ω) , X (Ω)) ∼ = π ( C N ( R ) , X ) = B N , [ γ ] . ˜ X (Ω) = [ γγ Ω ] . ˜ X = [ γ Ω γ − γγ Ω ] . ˜ X = ˜ X . Ad γ − [ γ ] , [ γ ] ∈ π ( C N (Ω) , X (Ω)) , ˜ X (Ω) . [ η ] = [ γ Ω η ] . ˜ X = [ γ Ω ηγ − γ Ω ] . ˜ X = Ad γ Ω [ η ] . ˜ X (Ω) , [ η ] ∈ π ( C N ( R ) , X ) . The actions of paths and loops in ˜ C N thus carry over to ˜ C N (Ω) straightforwardly, consideringthere only paths contained in C N (Ω). Definition 4.15 (Function space) . Given an N -anyon model ρ : B N → U( F ) and a simplyconnected open domain Ω ⊆ R , the space L ρ ( ˜ C N (Ω); F ) of ideal N -anyon wave func-tions on Ω is the closure of the space of ρ -equivariant smooth functions Ψ ∈ C ∞ ρ,c ( ˜ C N ( R ); F )with ˜pr(supp Ψ) ⊆ C N ( R ) compact,Ψ( ˜ X.b − ) = ρ ( b )Ψ( ˜ X ) , b ∈ B N , (4.14)w.r.t. the norm (cid:107) · (cid:107) L ρ induced by the inner product on C N (Ω) (cid:104) Φ , Ψ (cid:105) L ρ := (cid:90) C N (Ω) (cid:104) Φ( ˜ X ) , Ψ( ˜ X ) (cid:105) F dX = 1 N ! (cid:90) Ω N (cid:104) Φ( ˜ X ) , Ψ( ˜ X ) (cid:105) F d x . (4.15) Definition 4.16 (Kinetic energy) . Given an N -anyon model ρ : B N → U( F ) and a simplyconnected domain Ω ⊆ R , we define a Sobolev space H ρ ( ˜ C N (Ω); F ) of wave functions Ψwith finite expected kinetic energy on Ω T Ω ρ [Ψ] := (cid:90) C N (Ω) |∇ Ψ( ˜ X ) | F N dX. (4.16)This is the Hilbert space with inner product (cid:104) Φ , Ψ (cid:105) H ρ := (cid:90) C N (Ω) (cid:16) (cid:104) Φ( ˜ X ) , Ψ( ˜ X ) (cid:105) F + (cid:104)∇ Φ( ˜ X ) , ∇ Ψ( ˜ X ) (cid:105) F N (cid:17) dX (4.17) obtained by taking the closure of smooth ρ -equivariant functions on ˜ C N ( R ) w.r.t. thecorresponding norm, H ρ ( ˜ C N (Ω); F ) := C ∞ ρ,c ( ˜ C N ; F ) H ρ . We may then define the N -anyon ground-state energy on Ω, N ≥ E N (Ω) := inf (cid:40)(cid:90) C N (Ω) |∇ Ψ( ˜ X ) | F N dX : Ψ ∈ H ρ ( ˜ C N (Ω); F ) , (cid:90) C N (Ω) | Ψ | F dX = 1 (cid:41) . For convenience we also define E (Ω) := 0 and the one-particle energy E (Ω) := inf (cid:26)(cid:90) Ω |∇ Ψ | F : Ψ ∈ H (Ω; F ) , (cid:90) Ω | Ψ | F = 1 (cid:27) = 0 . If we need to indicate the anyon model ρ we write E ρN (Ω).Again, the ρ -equivariance (4.14) of functions assures that the integrands in (4.15), (4.16)and (4.17) are well defined. We will also need the corresponding one-body density: Definition 4.17 (One-body density) . Let ρ : B N → U( F ) be an N -anyon model andΩ ⊆ R a simply connected open subset. For Ψ ∈ L ρ ( ˜ C N (Ω); F ) a normalized N -body wavefunction on Ω we define its one-body density on Ω as the function (cid:37) Ψ ∈ L (Ω), where (cid:37) Ψ ( x ) at x ∈ Ω is the marginal of the probability distribution | Ψ | F at X = { x } ∪ X (cid:48) , (cid:37) Ψ ( x ) := (cid:90) C N − (Ω \{ x } ) | Ψ( { x } ∪ X (cid:48) ) | F dX (cid:48) = 1( N − (cid:90) Ω N − | Ψ( x , x (cid:48) ) | F d x (cid:48) . It may equivalently be defined via its integral over a subset Ω (cid:48) ⊆ Ω (cid:90) Ω (cid:48) (cid:37) Ψ = (cid:90) C N (Ω) N (cid:88) j =1 { x j ∈ Ω (cid:48) } | Ψ | F dX, (the expected number of particles on Ω (cid:48) ) and the Lebesgue differentiation theorem, takingΩ (cid:48) = B ε ( x ), ε → Definition 4.18 (Energy density) . Similarly, we define for Ψ ∈ H ρ ( ˜ C N (Ω); F ) a localexpected kinetic energy on Ω (cid:48) ⊆ Ω T Ω (cid:48) ⊆ Ω ρ [Ψ] := (cid:90) Ω (cid:48) T ρ, Ψ ( x ) d x = (cid:90) C N (Ω) N (cid:88) j =1 { x j ∈ Ω (cid:48) } |∇ j Ψ | F dX, and its corresponding kinetic energy density T ρ, Ψ ∈ L (Ω), T ρ, Ψ ( x ) := (cid:90) C N − (Ω \{ x } ) |∇ x Ψ( { x } ∪ X (cid:48) ) | F dX (cid:48) . Occasionally we shall suppress the F on the norms to make the notation less heavy. XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 49 Statistical repulsion Given an N -anyon model ρ : B N → U( F ) with associated exchange operators { U p } N − p =0 ,we may define an exchange parameter for p enclosed particles β p := min { β ∈ [0 , 1] : e iβπ or e − iβπ is an eigenvalue of U p } , and for n ∈ { , , . . . , N } involved particles (i.e. worst case of up to n − α n := min p ∈{ , , ,...,n − } β p . To a simple two-particle exchange U = ρ ( σ ) is then associated the parameter α = β . Inthe case of irreducible abelian anyons, with exchange phases U p = e i (2 p +1) απ , we have α n = min p ∈{ , ,...,n − } min q ∈ Z | (2 p + 1) α − q | . The goal of this section is to derive explicit lower bounds for the local kinetic energy interms of the above model-dependent parameters which quantify the strength of repulsionamong the particles due to their statistics (exchange phases).5.1. Relative configuration space. Let Ω ⊆ R be a convex open set. Consider thesymmetric exchange of two particles inside Ω with the remaining N − x , x ): r = x − x , R = ( x + x ) ∈ Ω , x (cid:48) = ( x , . . . , x N ) ∈ Ω N − , x( r ; R ; x (cid:48) ) := (cid:0) R + r , R − r , x , . . . , x N (cid:1) ∈ R N To keep all points in Ω we must restrict the values of r ∈ R to the subsetΩ rel ( R ; x (cid:48) ) = (cid:8) r ∈ R : x( r ; R ; x (cid:48) ) ∈ Ω N \ (cid:52) (cid:52) N (cid:9) . For these we have a surjection X ( r ; R ; x (cid:48) ) := pr(x( r ; R ; x (cid:48) )) = (cid:8) R + r , R − r , x , . . . , x N (cid:9) ∈ C N (Ω) . However, Ω rel is not simply connected. For example, in the case Ω = R we haveΩ rel = R \ { , ± x − R ) , . . . , ± x N − R ) } and X ( ± e ; e ; 3 e , . . . , N e ) = X = X ( R ) , while if ¯Ω = Q = [0 , is the unit square then Ω rel ⊆ [ − , is a rectangle centered at with up to 2 N + 1 points removed symmetrically around the origin (one point being theorigin).Note that several configurations ( r ; R ; x (cid:48) ) map to the same point X ( r ; R ; x (cid:48) ), and we havefor example the antipodal symmetry X ( − r ; R ; x (cid:48) ) = X ( r ; R ; x (cid:48) ) , r ∈ Ω rel ( R ; x (cid:48) ) . Upon fixing R and x (cid:48) and a base point r ∈ Ω rel we may extend r (cid:55)→ X ( r ; R ; x (cid:48) ) to a smoothmap on the covering space ˜Ω rel ( R ; x (cid:48) ) → Ω rel ( R ; x (cid:48) )˜ r (cid:55)→ ˜ X (˜ r ; R ; x (cid:48) ) (cid:55)→ X ( r ; R ; x (cid:48) ) such that ˜ X (˜ r ; R ; x (cid:48) .σ ) = ˜ X (˜ r ; R ; x (cid:48) ) .b for σ ∈ S N − and some b ∈ B N . By composing withΨ ∈ C ∞ ρ,c we then have a smooth map ψ : ˜Ω rel → F , ψ (˜ r ) := Ψ (cid:0) ˜ X (˜ r , R , x (cid:48) ) (cid:1) (5.1)such that ∂ k ψ (˜ r ) = (cid:88) j,q ∂ Ψ ∂x jq ∂x jq ∂r k (˜ r ) = ± (cid:18) ∂ Ψ ∂x k − ∂ Ψ ∂x k (cid:19) ( ˜ X (˜ r )) , where the sign depends on the order of exchange of particles 1 and 2 done in ˜ r . Namely,the action of any β ∈ π (Ω rel , r ) carries over to a corresponding one on ˜ X ,˜ X (˜ r .β ; R ; x (cid:48) ) = ˜ X (˜ r ; R ; x (cid:48) ) .b, some b = b ( β ) ∈ B N , so that, with the transformation rule (4.8), ∂ k ψ (˜ r .β ) = ± ρ ( b − ) (cid:18) ∂ Ψ ∂x (cid:48) k − ∂ Ψ ∂x (cid:48) k (cid:19) ( ˜ X (˜ r )) , j (cid:48) = ˆpr( b )[ j ] , and hence |∇ ψ (˜ r .β ) | = 14 |∇ Ψ − ∇ Ψ | for any β (cid:55)→ b (cid:55)→ ˆpr( b ) wich leaves invariant the partition { , } , { , , . . . , N } .Furthermore, given any r = | r | > rS is contained in Ω rel , wemay consider the corresponding lift of this circle to a subset in ˜Ω rel isomorphic to R . Bythe convexity of Ω, this will be possible on any of the annuli A p ⊆ Ω rel defined by r ∈ I p , I = ( r := 0 , r ) , I = ( r , r ) , . . . , I M − = ( r M − , r M ) , I M = ( r M , r max ) , ≤ M ≤ N − , with r max := dist( , ∂ Ω rel ) = 2dist( R , ∂ Ω), and having written the relative distances to theother particles in increasing order0 ≤ r = 2 | x σ (1) − R | ≤ r = 2 | x σ (2) − R | ≤ . . . ≤ r N − = 2 | x σ ( N − − R | for a suitable permutation σ ∈ S N − .Comparing now to the two-particle relative configuration space ˜ C → C in Example 4.2,we have the curve R → S → S / ∼ = C (with x ∼ − x ) of relative angles ϕ (cid:55)→ e ( ϕ ) := (cos ϕ, sin ϕ ) (cid:55)→ {± (cos ϕ, sin ϕ ) } , which after fixing a base point ˜ e (0) ∈ ˜ C lifts to a unique curve R → ˜ C ϕ (cid:55)→ ˜ e ( ϕ ) := [ γ ϕ ] . ˜ e (0) , γ ϕ ( t ) := {± e ( ϕt ) } , t ∈ [0 , . In the same way there is for r ∈ I p a corresponding smooth curve R → ˜ C N (Ω) → C N (Ω)obtained by taking ˜ r = r ˜ e ( ϕ ) with the remaining coordinates fixed, i.e. ϕ (cid:55)→ ˜ X ( r ˜ e ( ϕ ); R ; x (cid:48) ) (cid:55)→ X ( r e ( ϕ ); R ; x (cid:48) ) . Given a base point on the curve, for example˜ X ( r ˜ e (0); R ; x (cid:48) ) = [ η ] . ˜ X (Ω) = [ ηγ Ω ] . ˜ X for some fixed path η from X (Ω) to X ( r e (0); R ; x (cid:48) ) (possibly via X ( r ; R ; x (cid:48) )), this curvecan also be represented as ϕ (cid:55)→ ˜ X ( r ˜ e ( ϕ ); R ; x (cid:48) ) = [Γ ϕ ] . ˜ X ( r ˜ e (0); R ; x (cid:48) ) XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 51 using the path in C N (Ω)Γ ϕ ( t ) := (cid:8) R + rγ ϕ ( t ) , R − rγ ϕ ( t ) , x , . . . , x N (cid:9) , t ∈ [0 , . We may then define a smooth function v : R → F as the restriction of Ψ to this particularcurve, v ( ϕ ) := ψ ( r ˜ e ( ϕ )) = Ψ( ˜ X ( r ˜ e ( ϕ ); R ; x (cid:48) )) , (5.2)with v (cid:48) ( ϕ ) = ∇ ψ · r ˜ e (cid:48) ( ϕ ) = ± (cid:20) (cid:18) ∂ Ψ ∂x − ∂ Ψ ∂x (cid:19) ( − r sin ϕ ) + 12 (cid:18) ∂ Ψ ∂x − ∂ Ψ ∂x (cid:19) ( r cos ϕ ) (cid:21) . We also compute ∂ r ψ ( r ˜ e ( ϕ )) = ∇ ψ · e ( ϕ ) = ± (cid:20) (cid:18) ∂ Ψ ∂x − ∂ Ψ ∂x (cid:19) cos ϕ + 12 (cid:18) ∂ Ψ ∂x − ∂ Ψ ∂x (cid:19) sin ϕ (cid:21) , which combines with the previous expression to yield |∇ ψ | F = | ∂ r ψ ( r ˜ e ( ϕ )) | F + 1 r | v (cid:48) ( ϕ ) | F = 14 |∇ Ψ − ∇ Ψ | F . (5.3)A two-particle exchange on C with winding number n ∈ Z yields e ( nπ ) = ( − n e (0) and˜ e ( nπ ) = [ γ nπ ] . ˜ e (0) = [ γ π ] n . ˜ e (0) = ˜ e (0) . [ γ π ] n , with [ γ π ] the generator of the respective braid group π ( C ) = Z (base point {± e (0) } ).With our corresponding curve on C N (Ω)˜ X ( r ˜ e ( nπ ); R ; x (cid:48) ) = [Γ nπ ] . ˜ X ( r ˜ e (0); R ; x (cid:48) ) = ˜ X ( r ˜ e (0); R ; x (cid:48) ) .b n , with b = Ad [ ηγ Ω ] − ([Γ π ]) ∈ B N the generator of exchange. In this case we perform con-tinuous two-particle exchanges which are not necessarily simple but may, depending on r ,involve other particles. Topologically we have a situation similar to (3.2), i.e. [Γ π ] ∼ b ∼ Σ p where there are p enclosed particles if r ∈ I p . Thus the ρ -equivariance of Ψ ∈ C ∞ ρ implies v ( nπ ) = Ψ( ˜ X ( r ˜ e ( nπ ); R ; x (cid:48) )) = Ψ( ˜ X ( r ˜ e (0); R ; x (cid:48) ) .b n ) = ρ ( b − n ) v (0)where ρ ( b ) ∼ ρ (Σ p ) = U p , the corresponding two-anyon exchange operator.5.2. Poincar´e inequality. The statistical repulsion between a pair of anyons boils downto the following simple version of the Poincar´e inequality. Lemma 5.1 (Poincar´e inequality for semi-periodic functions) . Let ψ ∈ H ([0 , π ]; F ) satisfythe semi-periodic boundary condition ψ ( π ) = U ψ (0) , (5.4) for some U ∈ U( F ) , then (cid:90) π | ψ (cid:48) ( ϕ ) | F dϕ ≥ λ ( U ) (cid:90) π | ψ ( ϕ ) | F dϕ, (5.5) where λ ( U ) := inf { λ ∈ [0 , 1] : e iλπ or e − iλπ is an eigenvalue of U } . (5.6) Proof. We consider the operator D on the space L ([0 , π ]; F ) with Dψ ( ϕ ) = − iψ (cid:48) ( ϕ ) anddomain given by functions ψ ∈ H ([0 , π ]; F ) satisfying the b.c. (5.4). This is a self-adjointoperator and its spectrum is given explicitly by λ n,k = µ n + 2 k ∈ R , n ∈ { , . . . , dim F } , k ∈ Z , with corresponding orthonormal eigenfunctions u n,k ( ϕ ) = e iλ n,k ϕ v n , where U v n = e iµ n π v n , µ n ∈ ( − , F of eigenvectors of U . By the spectral theorem, the l.h.s.of (5.5) is then (cid:107) Dψ (cid:107) = (cid:10) ψ, D ψ (cid:11) ≥ inf n,k λ n,k (cid:107) ψ (cid:107) = λ ( U ) (cid:107) ψ (cid:107) , according to the definition (5.6). (cid:3) For completeness we also state a version suitable for magnetic anyon models modeledusing bosons: Lemma 5.2 (Gauge-transformed Poincar´e inequality) . Let ψ ∈ H ([0 , π ]; F ) satisfy theperiodic boundary condition ψ ( π ) = ψ (0) , and assume A ∈ C ([0 , π ]; u ( F )) , where u ( F ) are the anti-hermitian operators on F . Then (cid:90) π (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ddϕ + A (cid:19) ψ ( ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) F dϕ ≥ λ ( U ) (cid:90) π | ψ ( ϕ ) | F dϕ, (5.7) where U = exp P ( (cid:82) π A ) ∈ U( F ) is the path-ordered exponential of A .Proof. Define the function v ( ϕ ) = U ( ϕ ) ψ ( ϕ ), where U ( ϕ ) := exp P ( (cid:82) ϕ A ( t ) dt ), i.e. (bydefinition) U (0) = and U (cid:48) ( ϕ ) = A ( ϕ ) U ( ϕ ) for all ϕ ∈ [0 , π ]. Then v (cid:48) ( ϕ ) = ψ (cid:48) ( ϕ ) + A ( ϕ ) ψ ( ϕ ), so the l.h.s. of (5.7) is (cid:82) π | v (cid:48) | , and furthermore v satisfies the b.c. v ( π ) = U ( π ) ψ ( π ) = U ψ (0) = U v (0) . Now we may apply Lemma 5.1 to v and use | v | = | ψ | to conclude the lemma. (cid:3) Diamagnetic inequality. Consider estimates due to a polar decomposition of F : Lemma 5.3 (Angular diamagnetic inequality) . Let ψ ∈ H ([0 , π ]; F ) , | ψ | F = (cid:0)(cid:80) k | ψ k | (cid:1) / where ψ k are the components in an ON basis of F . Then | ψ | F ∈ H ([0 , π ]; R + ) , | ψ (cid:48) ( ϕ ) | F ≥ (cid:12)(cid:12) | ψ | (cid:48)F ( ϕ ) (cid:12)(cid:12) (5.8) pointwise a.e. on [0 , π ] , and thus (cid:90) π | ψ (cid:48) ( ϕ ) | F dϕ ≥ (cid:90) π (cid:12)(cid:12) | ψ | (cid:48)F ( ϕ ) (cid:12)(cid:12) dϕ. (5.9) Proof. This is a direct consequence of the usual diamagnetic inequality for functions withvalues in C ∼ = R (see e.g. [LL01, Theorem 6.17]), i.e. pointwise a.e. | ( u , u ) (cid:48) | = | u (cid:48) | + | u (cid:48) | ≥ | (cid:113) u + u (cid:48) | , where the r.h.s. is zero if u + u = 0. Applied inductively, | u (cid:48) | + . . . + | u (cid:48) n − | + | u (cid:48) n | ≥ | (cid:113) u + . . . + u n − (cid:48) | + | u (cid:48) n | ≥ | (cid:113) u + . . . + u n (cid:48) | , XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 53 etc., extends to F ∼ = C D ∼ = R D . Thus, pointwise a.e. | ψ (cid:48) | F = (cid:88) k | ψ (cid:48) k | ≥ (cid:12)(cid:12) | ψ | (cid:48)F (cid:12)(cid:12) , and since | ψ | F ∈ L and ψ (cid:48) ∈ L we also have | ψ | F ∈ H and the bound (5.9). (cid:3) Lemma 5.4 (Spatial diamagnetic inequality) . Let Ω ⊆ R be a simply connected Lipschitzdomain. Let Ψ ∈ H ρ ( ˜ C N (Ω); F ) , | Ψ | F = (cid:0)(cid:80) k | ψ k | (cid:1) / , then | Ψ | F ∈ H ( ˜ C N (Ω); R + ) extends to a function | Ψ | F ∈ H (Ω N ; R + ) , |∇ Ψ( ˜ X ) | F N ≥ (cid:12)(cid:12) ∇| Ψ | F ( ˜ X ) (cid:12)(cid:12) R N pointwise a.e. on C N (Ω) , and (cid:90) C N (Ω) |∇ Ψ( ˜ X ) | F N dX ≥ (cid:90) C N (Ω) (cid:12)(cid:12) ∇| Ψ | F ( ˜ X ) (cid:12)(cid:12) R N dX = 1 N ! (cid:90) Ω N (cid:12)(cid:12) ∇| Ψ | F (x) (cid:12)(cid:12) R N d x . (5.10) Proof. The inequality (5.8) may be replaced by a derivative in any direction, | ∂ j,k Ψ | F ≥ (cid:12)(cid:12) ∂ j,k | Ψ | F (cid:12)(cid:12) . We therefore have a pointwise a.e. inequality on ˜ C N (Ω) as above, |∇ Ψ | F N = (cid:88) j,k | ∂ j,k Ψ | F ≥ (cid:88) j,k (cid:12)(cid:12) ∂ j,k | Ψ | F (cid:12)(cid:12) = (cid:12)(cid:12) ∇| Ψ | F (cid:12)(cid:12) R N . Furthermore both sides of this inequality are invariant under the action of B N due tothe ρ -equivariance of Ψ, and thus descend to C N (Ω). Hence we have a function | Ψ | F ∈ H ρ + ( C N (Ω); R + ) satisfying the bound (5.10). This extends to a function | Ψ | F ∈ H (Ω N \(cid:52) (cid:52) N ; R + ). We may also consider | Ψ | F as a function in L (Ω N ) = L (Ω N \ (cid:52) (cid:52) N ) andits distributional derivative ∇| Ψ | F ∈ H − (Ω N ). Since H ( R N \ (cid:52) (cid:52) N ) = H ( R N ) thisfunction extends to H (Ω N ); cf. [LS14, Lemmas 3 and 4] and more generally [LLN19,Appendix B]. In fact Ψ ∈ H ρ is the limit of Ψ n ∈ C ∞ ρ,c and the inequality (5.10) for suchfunctions implies for the limit | Ψ | F ∈ H (Ω N ) ∩ H ( R N \ (cid:52) (cid:52) N ) = H (Ω N ). (cid:3) The diamagnetic inequality shows that inf spec ˆ T ρ ≥ inf spec ˆ T ρ + , i.e. bosons alwaysprovide a lower bound to the energy, since given any Ψ ∈ H ρ ( ˜ C N (Ω); F ) or sequence Ψ n ∈ C ∞ ρ converging to Ψ we obtain T Ω ρ [Ψ n ] ≥ T Ω ρ + (cid:2) | Ψ n | F (cid:3) ≥ E ρ + N (Ω) , and thus E ρN (Ω) ≥ E ρ + N (Ω). Furthermore, we obtain by finiteness of the integral T Ω ρ + (cid:2) | Ψ | F (cid:3) = T [Φ] = 1( N − (cid:90) Ω N − (cid:90) Ω |∇ x Φ( x , x (cid:48) ) | d x d x (cid:48) , Φ(x) = | Ψ( X ) | F , the Sobolev embedding | Ψ( · , x (cid:48) ) | F ∈ L p (Ω), 2 ≤ p < ∞ , for a.e. x (cid:48) ∈ Ω N − [LL01, Theo-rems 8.5 and 8.8]. Hardy inequality. We are now ready to prove our first main result concerning thedynamics of the anyon gas, which extends the many-body Hardy inequality [LL18, Theo-rem 1.3] for irreducible abelian anyons in the magnetic representation to arbitrary geometricanyon models. Theorem 5.5 ( Hardy inequality for non-abelian anyons ) . Let ρ : B N → U( F ) be an N -anyon model with exchange parameters ( β p ) p ∈{ ,...,N − } and α N = min p ∈{ ,...,N − } β p , and let Ω ⊆ R be open and convex. Then, for any Ψ ∈ H ρ ( ˜ C N (Ω); F ) , (cid:90) C N (Ω) |∇ Ψ | F N dX ≥ N (cid:90) C N (Ω) (cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) j =1 ∇ j Ψ (cid:12)(cid:12)(cid:12)(cid:12) F dX + 4 N (cid:90) C N (Ω) (cid:88) j We adapt the strategy of the proof of [LL18, Theorem 1.3] to functions living on thecovering space. By definition of the space H ρ , w.l.o.g. Ψ ∈ C ∞ ρ,c where ˜pr(supp Ψ) is compactin C N , i.e. Ψ is zero close to (cid:52) (cid:52) N . We may also use invariance of |∇ Ψ | = (cid:80) j |∇ j Ψ | underthe action of B N and thus of S N to replace N ! copies of the integral (cid:82) C N (Ω) by (cid:82) Ω N .Now use the parallelogram identity n (cid:88) j =1 | z j | = 1 n (cid:88) ≤ j We consider the term (cid:90) Ω N − (cid:90) Ω |∇ Ψ − ∇ Ψ | d x d x d x (cid:48) = (cid:90) Ω N − (cid:90) Ω (cid:90) Ω rel ( R ;x (cid:48) ) |∇ Ψ − ∇ Ψ | d r d R d x (cid:48) where, following the strategy of Section 5.1, we make the 1-to-1 coordinate transformationto relative coordinates R ∈ Ω, r ∈ Ω rel with measure d x d x = d r d R . Given a configurationx (cid:48) ∈ Ω N and R ∈ Ω, by (5.3) it remains then to study the integral (cid:90) Ω rel ( R ;x (cid:48) ) |∇ Ψ −∇ Ψ | d r = 4 (cid:90) Ω rel ( R ;x (cid:48) ) |∇ ψ (˜ r ) | d r = 4 M (cid:88) p =0 (cid:90) A p |∇ ψ (˜ r ) | d r +4 (cid:90) A c |∇ ψ (˜ r ) | d r with the smooth function ψ in (5.1) defined on the covering space ˜Ω rel and A c := Ω rel \∪ p A p .On each annulus A p we write in terms of polar coordinates (cid:90) A p |∇ ψ (˜ r ) | d r = (cid:90) r p +1 r = r p (cid:90) πϕ =0 (cid:18) | ∂ r ψ | + 1 r | ∂ ϕ ψ | (cid:19) dϕ rdr. For the first term we use the pointwise diamagnetic inequality (5.8) | ∂ r ψ | ≥ (cid:12)(cid:12) ∂ r | ψ | (cid:12)(cid:12) while in the second we may at fixed r ∈ I p identify | ∂ ϕ ψ | = | v (cid:48) ( ϕ ) | for the smooth function(5.2) of the relative angle ϕ ∈ R which obeys the semi-periodicity v ( nπ ) = ρ ( b − ) n v (0) , n ∈ Z , with ρ ( b − ) ∼ U − p . We stress that the choice of a base point ˜ X ( r ˜ e (0); R ; x (cid:48) ) = [ η ] . ˜ X (Ω) =[ ηγ Ω ] . ˜ X for the curve used in the definition of v (respectively ψ ) is arbitrary since if we in-stead took [ η (cid:48) ] . ˜ X (Ω) = ([ η ] . ˜ X (Ω)) . [ γ − η − η (cid:48) γ Ω ] they differ by the braid B = [ γ − η − η (cid:48) γ Ω ]and thus, as long as it preserves our division of the particles, we have ∂ ϕ [Ψ( ˜ X ( r ˜ e ( ϕ ); R ; x (cid:48) ) .B )] = ρ ( B − ) ∂ ϕ [Ψ( ˜ X ( r ˜ e ( ϕ ); R ; x (cid:48) ))] , leaving | v (cid:48) ( ϕ ) | unchanged. In the same way we have v ( ϕ + π ) = ρ ( b − ) v ( ϕ ) and thus (cid:90) π | v (cid:48) | = 2 (cid:90) π | v (cid:48) | ≥ λ ( U − p ) (cid:90) π | v | = λ ( U p ) (cid:90) π | v | , by the Poincar´e inequality, Lemma 5.1. Therefore (cid:90) A p |∇ u (˜ r ) | d r ≥ (cid:90) r p +1 r = r p (cid:90) π (cid:18)(cid:12)(cid:12) ∂ r | u | (cid:12)(cid:12) + λ ( U p ) r | u | (cid:19) dϕ rdr The remaining term can be bounded using the diamagnetic inequality, (cid:90) A c |∇ ψ | ≥ (cid:90) A c | ∂ r ψ | ≥ (cid:90) A c (cid:12)(cid:12) ∂ r | ψ | (cid:12)(cid:12) . Summarizing, (cid:90) Ω N |∇ Ψ − ∇ Ψ | d x ≥ (cid:90) Ω N (cid:12)(cid:12) ∂ r | Ψ | (cid:12)(cid:12) + M (cid:88) p =0 A p ( r ) β p | Ψ | | r | d x , which, after bounding uniformly λ ( U p ) = β p ≥ α N in terms of the exchange param-eters of the anyon model, simplifies further with the remaining support (cid:80) Mp =0 A p ( r ) = B r max (0) ( r ) = Ω ◦ Ω ( x , x ).The other terms in (5.12) involving the pairs ( x j , x k ) are exactly analogous in termsof corresponding relative coordinates ( r jk , R jk ). After collecting these we may finally useagain the B N -invariance of the collection of all the terms to write the integrals on C N (Ω).This proves the first bound of the theorem.The second bound of the theorem follows exactly as in [LL18], passing again to polarcoordinates on Ω rel and considering u ( r ) = | Ψ( X ) | F in (cid:90) r max (cid:18) | u (cid:48) | + α N r | u | (cid:19) rdr ≥ λ (cid:90) r max | u | rdr. The minimizer of the Rayleigh quotient satisfies the Bessel equation − u (cid:48)(cid:48) ( r ) − u (cid:48) ( r ) /r + ν u ( r ) /r = λu ( r ) , u (0) = 0 , u (cid:48) ( r max ) = 0 , i.e. u ( r ) = J ν ( j (cid:48) ν r/r max ), with the eigenvalue λ = ( j (cid:48) ν ) /r , ν = α N ∈ [0 , j (cid:48) ν are given in [LL18, Appendix A]. (cid:3) Note that Theorem 5.5 implies the simpler inequality on Ω = R , Ψ ∈ H ρ , T ρ [Ψ] ≥ C N (cid:88) j Proposition 5.6 (Counterexamples for α = 0, global case) . A Hardy inequality of the type (5.13) cannot hold with any positive constant C N in the case that ρ : B N → U( D ) is abelianwith α = 0 or ρ : B → U(2) is the non-abelian Burau representation given in Section 3.7with w = 1 ( α = α = 0 ).Proof. For ρ : B N → U( D ) abelian with α = 0 we may take a joint unit eigenvector v ∈ C D such that ρ ( σ j ) v = v ∀ j and consider the state Ψ = Φ ε v ∈ H , where Φ ε ∈ H ∩ C ∞ c ( R N \ (cid:52) (cid:52) N ) approximates in H the product state Φ (x) = e −| x | = (cid:81) Nj =1 e −| x j | as ε → 0. Thus Ψ is ρ -equivariant,Ψ( ˜ X.σ − j ) = Φ ε ( X ) v = ρ ( σ j )Ψ( ˜ X ) , (cid:82) C N | Ψ | = ( N !) − (cid:82) R N | Φ ε | → C , and T ρ [Ψ] = (cid:90) C N |∇ Ψ( ˜ X ) | dX = 1 N ! (cid:90) R N |∇ Φ ε | → N ! (cid:90) R N |∇ Φ | < ∞ , while, due to the non-integrability of the inverse-square potential, (cid:90) R N | Ψ | | x − x | d x = (cid:90) R N | Φ ε | | x − x | d x → ∞ as ε → . For N = 2 any representation is abelian and thus the above counterexample applies ifand only if α = β = 0.For N = 3 we consider the reduced unitarized Burau representation with F = C and w = 1: ρ ( σ ) = 12 (cid:20) −√ −√ − (cid:21) , ρ ( σ ) = 12 (cid:20) √ √ − (cid:21) , ρ ( σ σ σ ) = (cid:20) − (cid:21) . XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 57 Note that ρ ( σ j ) = and thus it descends to a non-abelian representation of S = { , σ , σ , σ σ , σ σ , σ σ σ } . The eigenvectors corresponding to eigenvalue 1 are v := 12 (cid:20) √ − (cid:21) , v := 12 (cid:20) √ (cid:21) , resp. v − v = (cid:20) (cid:21) , and therefore we cannot simply take a constant eigenvector as in the abelian case. Instead,take two disjoint domains Ω and Ω in R and define Ψ( ˜ X ) = Ψ( ˆpr( ˜ X )) byΨ( x , x , x ) := (cid:88) σ ∈ S f ( x σ (1) , x σ (2) ) g ( x σ (3) ) ρ ( σ ) v = 2 f ( x , x ) g ( x ) v + 2 f ( x , x ) g ( x )( − v ) + 2 f ( x , x ) g ( x )( v − v ) , where f ∈ H ( R ; C ) supported in Ω and g ∈ H ( R ; C ) supported in Ω . We also usedthat ρ ( σ ) v = v − v and ρ ( σ ) v = v − v . It then follows that the three terms aboveare pairwise orthogonal in L ρ and (cid:90) C | Ψ | = 2 (cid:90) Ω | f ( x , x ) | d x d x (cid:90) Ω | g ( x ) | d x . Furthermore, (cid:90) C |∇ Ψ | =2 (cid:90) Ω (cid:0) |∇ f ( x , x ) | + |∇ f ( x , x ) | (cid:1) d x d x (cid:90) Ω | g ( x ) | d x + 2 (cid:90) Ω | f ( x , x ) | (cid:90) Ω |∇ g ( x ) | d x . and Ψ is ρ -equivariant by definition,Ψ( ˜ X.σ − j ) = (cid:88) σ ∈ S f ( x σ j σ (1) , x σ j σ (2) ) g ( x σ j σ (3) ) ρ ( σ j σ ) v = ρ ( σ j )Ψ( ˜ X ) . Again, by fixing g and taking f ∈ C ∞ c (Ω \ (cid:52) (cid:52) ) to approximate a product state u ( x ) u ( x )where, e.g., u is the g.s. of the Dirichlet laplacian on Ω , we find that T ρ [Ψ] stays uniformlybounded while (cid:90) R | Ψ | | x − x | ≥ (cid:90) Ω | f ( x , x ) | | x − x | d x d x (cid:90) Ω | g | grows unboundedly. This contradicts the validity of the Hardy inequality (5.13) for anypositive value of the constant C N . (cid:3) Proposition 5.7 (Counterexamples for α = 0, local case) . In the case that ρ : B N → U( D ) is abelian with ρ ( σ − j ) ∼ diag( e iγ n π ) Dn =1 we have E ρN (Ω) ≤ min n ∈{ ,...,D } E ρ γn N (Ω) , with E ρ α N ([0 , ) ≤ πN ( N − α π ( N − α (1 − παN ) , if ≤ α < (2 πN ) − , (5.14) and thus for Ω = [0 , (and in fact for any bounded convex Ω ) the Hardy inequality ofTheorem 5.5 necessarily trivializes for α = β = 0 . Proof. Use the abelian magnetic representation (4.13) to estimate for any 1 ≤ n ≤ DE ρN (Ω) ≤ inf (cid:40) T Ω γ n [Φ n ] : Φ n ∈ C ∞ c, sym ( R N \ (cid:52) (cid:52) N ) , (cid:90) C N (Ω) | Φ n | = 1 (cid:41) = E ρ γn N (Ω) . The bound (5.14) for the abelian energy on Ω = [0 , was proved in [LS18, Lemma 3.2]using a Dyson-type ansatz [Dys57]. In fact, if γ n = 0 and Ω ⊆ R bounded we maytake a sequence of Φ n ∈ C ∞ c, sym ( R N \ (cid:52) (cid:52) N ) approximating the constant function Φ =( | Ω | N /N !) − / , such that (cid:90) C N (Ω) | Φ n | = 1 , T Ω0 [Φ n ] = 1 N ! (cid:90) Ω N |∇ Φ n | → , yielding E ρN (Ω) = 0. (cid:3) Remark . The question concerning the behavior of the optimal constant in (5.13) for N → ∞ and α ∗ = 0 is an interesting but difficult open problem (in fact open even forfermions with α ∗ = 1 [FHOLS06]). It is a consequence of the uncertainty principle that inthe ground state we must also consider exchanges U p with p > A p can have a higher weight if β p < β . This has been discussed to some extentin [LS13b, Lun17, LL18, Lun19], and it would be even more relevant for point-attractiveanyons (cf. Remark 4.7). Actually only the latter form of the Hardy inequality relies onthe regularity assumptions on Ψ at (cid:52) (cid:52) N implied by the Friedrichs extension. See also theremarks following Lemma 5.12 below.5.5. Scale-covariant energy bounds. The goal of this subsection is to use the positivityof the anyonic energy due to the repulsion from the Hardy inequality for just a few particlesto derive positivity and in fact a quadratic growth for large numbers of particles. Weare here guided by the scale-covariant method introduced in [LS18] and formulated quitegenerally in [LLN19] (see also [Lun19, Sec. 5.5]): Lemma 5.9 ( Covariant energy bound ; [LLN19, Lemma 4.1]) . Assume that to any n ∈ N and any cube Q ⊂ R d there is associated a non-negative number (‘energy’) e n ( Q ) satisfying the following properties, for some constant s > : • (scale-covariance) e n ( λQ ) = λ − s e n ( Q ) for all λ > ; • (translation-invariance) e n ( Q + x ) = e n ( Q ) for all x ∈ R d ; • (superadditivity) For any collection of disjoint cubes { Q j } Jj =1 such that their unionis a cube, e n (cid:16) J (cid:91) j =1 Q j (cid:17) ≥ min { n j }∈ N J s.t. (cid:80) j n j = n J (cid:88) j =1 e n j ( Q j ); • (a priori positivity) There exists q ≥ such that e n ( Q ) > for all n ≥ q .There then exists a constant C > independent of n and Q such that e n ( Q ) ≥ C | Q | − s/d n s/d , ∀ n ≥ q. XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 59 Note that scale-covariance with s = 1 as well as translation-invariance follows directlyfrom the definition of the anyonic energy E N (Ω): Lemma 5.10 (Covariance) . Given Ω ⊆ R open and simply connected, x ∈ R and λ > ,we have E N ( λ Ω + x ) = λ − E N (Ω) . (5.15) Proof. Denote Ω λ := λ Ω + x . For X = { x , . . . , x N } ∈ C N (Ω) we have X λ := λX + x ∈C N (Ω λ ), and for ˜ X ∈ ˜ C N (Ω) an equivalence class of paths from X (Ω) to X = ˜pr( ˜ X ) wehave ˜ X λ := λ ˜ X + x ∈ ˜ C N (Ω λ ) a corresponding translated and scaled equivalence class ofpaths from X (Ω λ ) to X λ = ˜pr( ˜ X λ ). Given Ψ λ ∈ H ρ (Ω λ ) we define Ψ( ˜ X ) := λ N Ψ λ ( ˜ X λ ).Then, using dX λ = λ N dX , we have an L ρ -isomorphism (cid:90) C N (Ω) | Ψ( ˜ X ) | F dX = (cid:90) C N (Ω λ ) | Ψ λ ( ˜ X λ ) | F dX λ , and, with the corresponding rescaling in (4.5), (cid:90) C N (Ω) |∇ Ψ( ˜ X ) | F N dX = λ (cid:90) C N (Ω λ ) |∇ Ψ λ ( ˜ X λ ) | F N dX λ , and vice versa by inversion ( λ, x ) (cid:55)→ ( λ − , − x ). Applied to a sequence of minimizers yields(5.15). (cid:3) Lemma 5.11 (Superadditivity) . For K ≥ , let { Ω k } Kk =1 be a collection of disjoint, convexsubsets of R . For any (cid:126)n ∈ N K with (cid:80) k n k = N , let (cid:126)n denote the characteristic functionof the subset of C N ( R ) where exactly n k of the points X = { x , . . . , x N } are in Ω k , for all ≤ k ≤ K . Let W ( X ) := (cid:88) (cid:126)n K (cid:88) k =1 E n k (Ω k ) (cid:126)n ( X ) , and assume Ω := ∪ k Ω k ◦ is also convex. We then have (cid:90) C N (Ω) |∇ Ψ | ≥ (cid:90) C N (Ω) W | Ψ | (5.16) for any Ψ ∈ H ρ ( ˜ C N (Ω)) , and in particular E N (Ω) ≥ min (cid:126)n K (cid:88) k =1 E n k (Ω k ) . Proof. Using that = (cid:80) (cid:126)n (cid:126)n on C N (Ω), we have for any Ψ ∈ C ∞ ρ,c ( ˜ C N ) (cid:90) C N (Ω) |∇ Ψ | dX = (cid:88) (cid:126)n (cid:90) C N (Ω) |∇ Ψ | (cid:126)n dX, (5.17)where for any ˜ X = [ γ ] .X (Ω) ∈ ˜ C N (Ω) |∇ Ψ( ˜ X ) | (cid:126)n = K (cid:88) k =1 (cid:88) j : x j ∈ Ω k |∇ j Ψ( ˜ X ) | (cid:126)n . Hence for each (cid:126)n , the integral to be considered is K (cid:88) k =1 (cid:90) C N − nk (Ω \ Ω k ) (cid:90) C nk (Ω k ) (cid:88) j : x j ∈ Ω k |∇ j Ψ( ˜ X ) | dX k (cid:126)n dX ck . (5.18)Fixing the set of points X ck ∈ C N − n k (Ω \ Ω k ) and considering the set X k ∈ C n k (Ω k ) s.t. X = X k ∪ X ck , we have by ˜ X = [ γ ] . ˜ X (Ω) a path γ in Ω where n k of the points at X N (Ω)move into Ω k and N − n k into Ω \ Ω k . Any such path can equivalently be taken via firstacting on ˜ X N (Ω) with a braid b ∈ B N , then selecting a partition of particles at X N (Ω),say { , . . . , n k } , { n k + 1 , . . . , N } , then moving the former set of points via X n k (Ω k ) to X k and the latter via X N − n k (Ω k (cid:48) ) to X ck for some k (cid:48) (cid:54) = k . Note that any additional encirclingof the points in X k around the fixed set X ck in this process may be taken care of by writingthe corresponding action of [ γ k ] ∈ π ( C n k (Ω \ X ck ) , X k ) on ˜ X k (cid:55)→ X k as ˜ X N (Ω) .b for some b ∈ B N (actually in the subgroup of the pure braid group which keeps the indices fixed).Furthermore, any ˜ X k ∈ ˜ C n k (Ω k ) can be represented as ˜ X k = [ γ k ] . ˜ X n k (Ω k ) for a path γ k of n k points in Ω k and using ˜ X n k (Ω k ) .β if it is a loop β ∈ B n k (cid:44) → B N . In this way weconstruct a surjective map˜ C n k (Ω k ) × ˜ C N − n k (Ω \ Ω K ) × B N (cid:51) ( ˜ X k , ˜ X ck , b ) (cid:55)→ ˜ X ∈ ˜pr − (supp n ) ⊆ ˜ C N (Ω) . Hence, if we define for a fixed X n k (Ω k ) and X ck with fixed lift to ˜ X ∈ ˜ C N (Ω) the functionΨ k ( ˜ X k ) := Ψ( ˜ X ) we have Ψ k ( ˜ X k .β ) = Ψ( ˜ X.β ) = ρ ( β − )Ψ k ( ˜ X k ) . Note that it is also defined in a neighborhood of C n k (Ω k ). By making a smooth cut-off onΩ ck away from the points X ck we may consider this a smooth function on ˜ C n k ( R ) which is ρ -equivariant and with compact support ˜pr(supp Ψ k ). It is therefore in H ρ ( ˜ C n k (Ω k ); F ).Further, n k (cid:88) j =1 |∇ j Ψ k ( ˜ X k ) | = (cid:88) j : x j ∈ Ω k |∇ j Ψ( ˜ X ) | , for ˜ X ∈ supp (cid:126)n . Then (cid:90) C nk (Ω k ) n k (cid:88) j =1 |∇ j Ψ k ( ˜ X k ) | dX k ≥ E n k (Ω k ) (cid:90) C nk (Ω k ) | Ψ k | dX k and after putting this back into the original integral (5.18) and (5.17), and using that | Ψ k | = | Ψ | , we obtain the bound (5.16) in terms of the scalar and B N -symmetric potential W . (cid:3) In the sequel we denote simply by E N := E N ([0 , ) the N -anyon energy on the unitsquare with a given anyon model. Our starting point for positivitity is the following boundoriginally derived for abelian anyons in [LL18]: Lemma 5.12 (Bound for E N directly from Hardy; [LL18, Lemma 5.3]) . For ν > let j (cid:48) ν denote the first positive zero of the derivative of the Bessel function J ν , satisfying (5.11) .There exists a function f : [0 , ( j (cid:48) ) ] → R + satisfying t/ ≤ f ( t ) ≤ πt and f ( t ) = 2 πt (cid:0) − O ( t / ) (cid:1) as t → , XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 61 such that E N ≥ f (cid:0) ( j (cid:48) α N ) (cid:1) ( N − + . (5.19) Proof. After splitting the energy T ρ = κT ρ + (1 − κ ) T ρ and applying the diamagnetic in-equality Lemma 5.4 to the first part, and the Hardy inequality of Theorem 5.5 in the laterform to the second part, the proof is identical to that of [LL18, Lemma 5.3], where oneexpresses the bosonic N -particle energy as ( N − / t = j (cid:48) α N ) f ( t ) := 12 sup κ ∈ (0 , inf (cid:82) Q | ψ | =1 (cid:90) Q (cid:18) κ |∇ | ψ || + κ |∇ | ψ || + (1 − κ ) t Ω ◦ Ω ( x , x ) δ ( X ) | ψ | (cid:19) d x d x . The energy of the corresponding Schr¨odinger operator (with singular potential) on L ( Q )is then estimated in a standard way using projection on the constant function, yielding thebounds for f ( t ) stated in the lemma (see also Figure 1.2). (cid:3) Remark . We could replace the above bound by one which takes more of the particledistribution into account. Namely, after making a cut-off around the diagonals on theotherwise singular Hardy term we can take the average (i.e. expectation in the bosonic g.s.) (cid:88) j 2, the bound (5.19) will be trivial if α = 0, i.e. if E = 0. On the other hand, if α > E > E N > N ≥ Lemma 5.15 (A priori bounds in terms of E ; [LS18, Lemma 4.3]) . For any N ≥ wehave E N ≥ π (cid:0) N (cid:1) (cid:0) (cid:1) N − E (cid:0) π + 4 √ E (cid:1) + (cid:0) N (cid:1) (cid:0) (cid:1) N − E , and thus E N > if E > . The proof is exactly analogous to the abelian case, [Lun13, Proposition 2] and [LS18,Lemma 4.3], where we split the unit square Q in K = 4 smaller squares and use covariance(5.15) and superadditivity (5.16), with W ≥ W := 4 E (cid:88) k =1 (cid:88) (cid:126)n : n k =2 (cid:126)n , (for each k , we keep only the terms in W for which n k = 2), and (cid:90) Q N W = E (cid:18) N (cid:19) (cid:18) (cid:19) N − . Then we use the diamagnetic inequality of Lemma 5.4 to estimate the energy E N in termsof a standard bosonic or distinguishable particle problem, yielding the claimed bound (see[LS18, Lemma 4.3] for details). Lemma 5.16 ( Covariant energy bound ) . Given a sequence of anyon models ρ N : B N → U( F N ) with two-anyon exchange parameters α ( N ) ≥ α > (it could depend on the totalnumber N of particles but must then be uniformly bounded from below), the local N -anyonenergy E N ( Q ) on squares Q satisfies the criteria of Lemma 5.9 with s = 1 and q = 1 andtherefore there exists a constant C = C ( α ) > such that E N ≥ CN , N ≥ . For an explicit bound on this constant we will use the original method of [LS18, Lemma 4.8-4.9] and the local exclusion principle.5.6. Local exclusion principle. Given Ψ ∈ H ρ ( ˜ C N (Ω)), recall that it defines a corre-sponding one-body density (cid:37) Ψ ∈ L (Ω) and a kinetic energy density T ρ, Ψ ∈ L (Ω) (Defini-tions 4.17 and 4.18). Lemma 5.17 ( Local exclusion principle for non-abelian anyons ) . Given a sequenceof anyon models ρ N : B N → U( F N ) with exchange parameters α n = min p β p (they may alldepend on N ), the local n -anyon energy E n = E ρ N n ( Q ) on the unit square satisfies E n ≥ C ( ρ N ) n, ≤ n ≤ N, (5.20) where C ( ρ N ) := 14 min { E , E , E } ≥ 14 min { E , . } ≥ c ( α ) = 14 min (cid:8) f ( j (cid:48) α ) , . (cid:9) . Furthermore, for any simply connected open domain Ω ⊆ R and any square Q ⊆ Ω , any N ≥ and L -normalized Ψ ∈ H ρ N ( ˜ C N (Ω); F N ) with one-particle density (cid:37) Ψ on Ω , we have T Q ⊆ Ω ρ N [Ψ] ≥ C ( ρ N ) | Q | (cid:18)(cid:90) Q (cid:37) Ψ ( x ) d x − (cid:19) + , (5.21) where C ( ρ N ) := max (cid:8) C ( ρ N ) , f ( j (cid:48) α N ) (cid:9) ≥ max (cid:26) 14 min (cid:8) f ( j (cid:48) α ) , . (cid:9) , f ( j (cid:48) α N ) (cid:27) . Proof. By splitting the unit square Q into four equally large squares Q k , | Q k | = | Q | / E n ≥ m ∈{ n/ ,n/ ,...,n } E m . From this and a priori positivity follows the linear bound (5.20) (see [LS18, Lemma 4.8] fordetails). The bound on C ( ρ N ) in terms of E ≥ f ( j (cid:48) α ) follows from Lemma 5.15, wherethe positive root of ( π + 4 √ x ) + x = π is x ≥ . c ( α ) ≥ min { α / , . } using (5.11) and the bounds on f in Lemma 5.12. XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 63 For (5.21) we use the decomposition of the energy into particles in Q and in Q c = Ω \ Q , T Q ⊆ Ω ρ N [Ψ] = (cid:90) C N (Ω) N (cid:88) j =1 |∇ j Ψ( ˜ X ) | { x j ∈ Q } dX = (cid:88) (cid:126)n (cid:90) C N − n ( Q c ) (cid:90) C n ( Q ) (cid:88) j : x j ∈ Q |∇ j Ψ( ˜ X ) | dX (cid:126)n dX c and proceed similarly to the proof of superadditivity (Lemma 5.11), with Ω = Q andΩ k> ⊆ Q c (e.g. splitting Q c along the coordinate axes). Thus, T Q ⊆ Ω ρ N [Ψ] ≥ (cid:88) (cid:126)n (cid:90) C N − n ( Q c ) E n ( Q ) (cid:90) C n ( Q ) | Ψ( ˜ X ) | dX (cid:126)n dX c . Now use that from scale-covariance and (5.20) E n ( Q ) ≥ | Q | − C ( ρ N )( n − + for all 0 ≤ n ≤ N, while from our earlier Lemma 5.12 E n ( Q ) ≥ | Q | − f ( j (cid:48) α n )( n − + ≥ | Q | − f ( j (cid:48) α N )( n − + for all 0 ≤ n ≤ N. Defining the induced n -particle probability distribution on Q , n ∈ { , , . . . , N } , p n (Ψ; Q ) := (cid:88) (cid:126)n : n = n (cid:90) C N − n ( Q c ) (cid:90) C n ( Q ) | Ψ( ˜ X ) | dX (cid:126)n dX c , (5.22)with (cid:80) Nn =0 p n = 1 and (cid:80) Nn =0 np n = (cid:82) Q (cid:37) Ψ , we may use the convexity of x (cid:55)→ ( x − + toobtain T Q ⊆ Q ρ N [Ψ] ≥ N (cid:88) n =0 | Q | − C ( ρ N )( n − + p n ≥ | Q | − C ( ρ N )( np n − + , which proves (5.21). (cid:3) The ideal non-abelian anyon gas The homogeneous gas. The (zero-temperature) homogeneous anyon gas on adomain Ω is defined by taking a ground state Ψ of the kinetic energy T Ω ρ , either withDirichlet or Neumann boundary conditions. As N → ∞ and | Ω | → ∞ while keepingthe mean density ¯ (cid:37) := N/ | Ω | fixed ( the thermodynamic limit ) one expects on generalgrounds the one-body density (cid:37) Ψ to tend to the constant ¯ (cid:37) , regardless of the shape of(reasonable) Ω and the choice of b.c. (at least on average over large enough scales; cf. thealmost-bosonic limit [CDLR19]). Definition 6.1 (Dirichlet energy) . Given an N -anyon model ρ : B N → U( F ) and a simplyconnected domain Ω ⊆ R , we define the Dirichlet Sobolev space H ρ, ( ˜ C N (Ω); F ), ofwave functions Ψ ∈ L ρ vanishing on the boundary ∂ Ω and with finite expected local kineticenergy on Ω, as the closure of the space of functions Ψ ∈ C ∞ ρ,c ( ˜ C N (Ω)) (with ˜pr(supp Ψ) compactly supported in C N (Ω)) in the H ρ -norm. Taking the infimum of T Ω ρ [Ψ] of suchfunctions defines the Dirichlet ground-state energy on Ω¯ E N (Ω) := inf (cid:40)(cid:90) C N (Ω) |∇ Ψ( ˜ X ) | F N dX : Ψ ∈ H ρ, ( ˜ C N (Ω)) , (cid:90) C N (Ω) | Ψ | F dX = 1 (cid:41) , N ≥ , also ¯ E (Ω) := 0, and ¯ E (Ω) := inf (cid:8)(cid:82) Ω |∇ Ψ | : Ψ ∈ H (Ω; F ) , (cid:82) Ω | Ψ | = 1 (cid:9) . Since C ∞ ρ,c ( ˜ C N (Ω)) ⊆ C ∞ ρ,c ( ˜ C N ( R )) we trivially have E N (Ω) ≤ ¯ E N (Ω) , where, to distinguish the two, we refer to E N (Ω) as the Neumann energy on Ω. Wealso note that the one-body energy E with fiber F ∼ = C D is the same as E with F = C or F = R + . This follows by decomposing |∇ Ψ | = (cid:80) Dn =1 |∇ Ψ n | w.r.t. a basis in F , i.e. T | H (Ω; F ) = (cid:76) D T | H (Ω; C ) , and using the standard diamagnetic inequality, |∇ Ψ | ≥ |∇| Ψ || . Lemma 6.2 (Subadditivity) . Let Ω k , k = 1 , . . . , N be pairwise disjoint and simply con-nected subsets of R , and assume Ω := ∪ k Ω k ◦ is also simply connected. If ρ : B N → U( F ) is an arbitrary geometric N -anyon model then ¯ E N (Ω) ≤ N (cid:88) k =1 ¯ E (Ω k ) . Proof. Let u k ∈ C ∞ c (Ω k ; C ) and v ∈ F , | v | = 1, and consider the subset Ω = (cid:81) Nk =1 Ω k ⊆ R N where x j is localized on Ω j (thus making all particles distinguishable). Fix a representativex Ω ∈ Ω. This point is in 1-to-1 correspondence to X (and X (Ω)) by means of a simplepath γ Ω in C N , and any action of b ∈ B N on X induces a permutation Ω . ˆpr( b ) in R N anda lift ˜Ω .b in ˜ C N . Thus, by the disjointness of Ω k we can write the set of pre-images in ˜ C N ˜pr − pr(Ω) = (cid:71) b ∈ B N ˜Ω .b. Define for ˜ X ∈ ˜Ω ( b = 1) Ψ( ˜ X ) := u ( x ) u ( x ) . . . u N ( x N ) v and on ˜Ω .b for b (cid:54) = 1 Ψ( ˜ X.b ) := ρ ( b − )Ψ( ˜ X ) . Then we have a ρ -equivariant function such that for any ˜ X (cid:55)→ X = { x , . . . , x N } , x j ∈ Ω j , | Ψ( ˜ X ) | = | u ( x ) | . . . | u N ( x N ) | , and |∇ Ψ( ˜ X ) | = |∇ u ( x ) | | u ( x ) | . . . | u N ( x N ) | + . . . + | u ( x ) | . . . | u N − ( x N − ) | |∇ u N ( x N ) | . Thus, normalizing (cid:82) R | u k | = 1 and taking sequences of u k converging to the respectiveground states on Ω k realizing E (Ω k ), we obtain the upper bound of the lemma. (cid:3) XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 65 Theorem 6.3 (Uniform bounds for the homogeneous anyon gas) . For any sequence of N -anyon models ρ N : B N → U( F N ) with n-anyon exchange parameters α n = α n ( N ) ∈ [0 , we have the uniform bounds C ( ρ N ) N (cid:0) − O ( N − ) (cid:1) ≤ E N ( Q ) ≤ ¯ E N ( Q ) ≤ π N (cid:0) O ( N − / ) (cid:1) , (6.1) where C ( ρ N ) = max (cid:8) C ( ρ N ) , f ( j (cid:48) α N ) (cid:9) ≥ max (cid:26) 14 min (cid:8) f ( j (cid:48) α ) , . (cid:9) , f ( j (cid:48) α N ) (cid:27) . Remark . By scale-covariance, the g.s. energy per particle and unit density in the ther-modynamic limit N → ∞ , Ω = [ − L/ , L/ , L → ∞ at fixed density ¯ (cid:37) = N/ | Ω | is then e ( { ρ N } ) := lim inf N →∞ E N /N , max (cid:26) 14 min (cid:8) f ( j (cid:48) α ) , . (cid:9) , f ( j (cid:48) α ∗ ) (cid:27) ≤ e ( { ρ N } ) ≤ π , where α n are the lim inf of α n ( ρ N ) as N → ∞ . Remark . One can improve the upper bound in the case that the anyon model is trans-mutable and with a small statistics parameter, by treating the magnetic potential as a weakinteraction and using techniques for weakly interacting Bose gases [Dys57, LSSY05]. Thiswas done for almost-bosonic abelian anyons in [LS18, Lemma 3.2]. Proof. Given sub- and superadditivity and the local exclusion principle, the method toderive these bounds via splitting into smaller boxes is quite standard; we follow [LS18,Proposition 3.3 and Lemma 4.9].For the lower bound, let K ∈ N and split Q into K smaller squares Q k , k = 1 , . . . , K ,of equal size | Q k | = K − . By superadditivity Lemma 5.11 and scale-covariance Lemma 5.10we have for any state Ψ ∈ H ρ ( ˜ C N ( Q ); F N ) (cid:90) C N ( Q ) |∇ Ψ | ≥ (cid:88) (cid:126)n K (cid:88) k =1 K E n k (cid:90) C N ( Q ) (cid:126)n | Ψ | = K K (cid:88) k =1 N (cid:88) n =0 E n p n (Ψ; Q k ) , (6.2)where p n is the induced n -particle probability distribution on Q k , defined in (5.22). Definealso the average distribution of particle numbers γ n := K − K (cid:88) k =1 p n (Ψ; Q k ) , normalized N (cid:88) n =0 γ n = 1 , N (cid:88) n =0 nγ n = N/K =: ρ Q , which is the expected number of particles on any one of the smaller squares. Hence ther.h.s. of (6.2) is, by Lemma 5.17 and convexity, bounded from below by K N (cid:88) n =0 C ( ρ N )( n − + γ n ≥ C ( ρ N ) K (cid:32) N (cid:88) n =0 nγ n − (cid:33) + = C ( ρ N ) N ρ − Q ( ρ Q − + . Optimizing then ρ Q ∼ K , we take K := (cid:100) (cid:112) N/ (cid:101) and obtain 2(1 + (cid:112) /N ) − ≤ ρ Q ≤ 2, and thus E N ≥ C ( ρ N ) N (cid:16) (cid:112) /N ) − (1 + (cid:112) /N ) (cid:17) , which proves the claimed lower bound.For the upper bound let K be the smallest integer such that N ≤ K , and again splitthe unit square Q = [0 , into K smaller squares Q k of equal size. By subadditivityLemma 6.2 and the well-known one-body energy ¯ E ( Q ) = 2 π of the Dirichlet groundstate u ( x, y ) = sin( πx ) sin( πy ), we obtain¯ E N ( Q ) ≤ N (cid:88) k =1 E ( Q k ) + K (cid:88) k = N +1 E ( Q k ) = N K π ≤ N (1 + √ N ) π by scale-covariance. This proves the claimed upper bound. (cid:3) With our results of Corollaries 3.11-3.12 respectively Section 3.6, and bounds for f and j (cid:48) ν , we have then Corollary 6.6. For Fibonacci anyons the exchange parameters are α = β = 3 / and α N = β = 1 / for N ≥ , and hence (6.1) holds with C ( ρ N ) ≥ / for N ≥ . Corollary 6.7. For Ising anyons the exchange parameters are α N = β = 1 / for all N ≥ , and hence (6.1) holds with C ( ρ N ) ≥ / for N ≥ . Corollary 6.8. For Clifford anyons the exchange parameters are α N = β = 1 / for ≤ N ≤ and α N = β = 0 for N ≥ , and hence (6.1) holds with C ( ρ N ) ≥ / for N ≥ .Remark . Note that the numerical estimate of f ( j (cid:48) α ∗ ) in Figure 1.2 gives significantimprovements to these analytical lower bounds for C ( ρ N ): C ( ρ Fibonacci N ) (cid:38) . , C ( ρ Ising N ) (cid:38) . , respectively C ( ρ Clifford N ) (cid:38) . / . Let us make some additional remarks concerning the sharper first form of the Hardy in-equality of Theorem 5.5 and Remark 5.13. While Ising anyons exhibit a uniform statisticalrepulsion with β p independent of p , Fibonacci and in particular Clifford anyons could in prin-ciple prefer to cluster to minimize their repulsion, just like we consider this also a possiblepreferred behavior (balanced by the uncertainty principle) for abelian anyons with α ∗ < α [LS13b, Lun17]. We also note in a similar way that the statistical repulsion of Majoranafermions ψ in the Ising model is potentially weakened to some probability, U ψ,σ,ψ = + by(3.8), thus allowing for pairing to happen on larger scales.6.2. Lieb–Thirring inequality. In the case that the gas is not homogeneous, such as ifan additional external scalar potential is added to the Hamiltonian (4.1), a very powerfulbound is given by the Lieb–Thirring inequality, which estimates the kinetic energy of Ψlocally at x ∈ R in terms of the homogeneous gas energy at the corresponding density (cid:37) Ψ ( x ) (for fermions that would be the famous Thomas–Fermi appoximation (1.12)). Thebound combines the uncertainty principle and the exclusion principle into a uniform boundfor arbitrary N -anyon states Ψ ∈ H ρ . XCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS 67 Theorem 6.10 ( Lieb–Thirring inequality for ideal anyons ) . There exists a constant C > such that for any number of particles N ≥ , for any N -anyon model ρ : B N → U( F N ) with 2-particle exchange parameter α ∈ [0 , (which may depend on N ), and for any L -normalized N -anyon state Ψ ∈ H ρ ( ˜ C N ; F N ) , we have T ρ [Ψ] ≥ Cα (cid:90) R (cid:37) Ψ ( x ) d x . (6.3) Furthermore, given a one-body potential V : R → R and ˆ V ( X ) = (cid:80) Nj =1 V ( x j ) , inf spec ( ˆ T ρ + ˆ V ) ≥ Cα (cid:90) R V − ( x ) d x , (6.4) where V − := max {− V, } .Remark . For bosons, with α = 0, the inequality (6.3) cannot hold with any betterconstant since for product states (1.2) the l.h.s. is N (cid:82) |∇ u | while (cid:37) Ψ ( x ) = N | u ( x ) | in ther.h.s., so the ratio cannot be better than C GNS /N → 0, where C GNS := inf u ∈ H ( R ): (cid:82) R | u | =1 (cid:82) R |∇ u | (cid:82) R | u | is the optimal constant of the corresponding Gagliardo-Nierenberg-Sobolev inequality for N = 1. Remark . We expect that the optimal constant C in (6.3) is on the order of 2 π , whichis the Thomas-Fermi constant in 2D and is obtained in Weyl’s law (1.6) for the sum of theeigenvalues of the Laplacian on a bounded domain (although recall also our remarks after(1.12)). The standing conjecture for fermions [LT76] is that the optimal constant is slightlysmaller and equal to C GNS (which is only known numerically). However, proving this evenfor fermions is a difficult open problem; see e.g. [Fra20] for a recent review.We prove Theorem 6.10 as an application of the local exclusion principle, Lemma 5.17,combined with a local version of the uncertainty principle for bosons or distinguishableparticles. The method was introduced for abelian anyons in [LS13a] and goes back toDyson and Lenard’s approach to the proof of the stability of fermionic matter [DL67,Lemma 5]. We could simply replace [LS13a, Lemma 8] by our Lemma 5.17, however wegive for completeness a more immediate proof from [LNP16] formulated using a coveringlemma (see [Lun19] for a more detailed exposition of this local approach to Lieb-Thirringinequalities). Lemma 6.13 (Local uncertainty principle [LS13a, Lemma 9-10], [LS14, Lemma 14]) . For arbitrary d ≥ and N ≥ let Ψ be a wave function in H ( R dN ) , normalized (cid:82) R dN | Ψ | =1 , and let Q be an arbitrary cube in R d . Then there exists a constant C > dependingonly on d , such that T Q ⊆ R d [Ψ] = (cid:90) R dN N (cid:88) j =1 |∇ j Ψ | { x j ∈ Q } d x ≥ C (cid:82) Q (cid:37) /d Ψ (cid:16)(cid:82) Q (cid:37) Ψ (cid:17) /d − C | Q | /d (cid:90) Q (cid:37) Ψ , (6.5) with the one-body density (cid:37) Ψ ( x ) := N (cid:88) j =1 (cid:90) R d ( N − | Ψ( x , . . . , x j − , x , x j +1 , . . . , x N ) | (cid:89) k (cid:54) = j d x k . Lemma 6.14 (Covering lemma [LNP16]) . Let ≤ f ∈ L ( R d ) be a function with compactsupport such that (cid:82) R d f ≥ Λ > . Then the support of f can be covered by a collection ofdisjoint cubes { Q } in R d such that (cid:90) Q f ≤ Λ , ∀ Q and (cid:88) Q | Q | a (cid:32)(cid:20)(cid:90) Q f − q (cid:21) + − b (cid:90) Q f (cid:33) ≥ for all a > and ≤ q < Λ2 − d , where b := (cid:18) − d q Λ (cid:19) da − da + 2 d − > . Proof of Theorem 6.10. If α = 0 the theorem is trivially true, and otherwise we have E ( Q ) > Q given by Lemma 5.17with C ρ ≥ c ( α ) > 0. By definition of the space H ρ we may w.l.o.g. assume Ψ is smoothand the projection by ˜pr of its support to C N contained in the projection by pr of somelarge cube Q NL , Q L = [ − L, L ] .Let q = 1 and Λ = 5. If N ≤ Λ, then (6.3) follows immediately from (6.5) with Q = Q L , L → ∞ . If N > Λ, then we can apply Lemma 6.14 with f = (cid:37) Ψ and a = 1, and obtain acollection of disjoint cubes { Q } covering Q L . Using that T ρ [Ψ] = (cid:88) Q T Q ⊆ Q L ρ [Ψ]and the diamagnetic inequality of Lemma 5.4 T ρ [Ψ] ≥ T (cid:2) | Ψ | F (cid:3) , (cid:37) | Ψ | F = (cid:37) Ψ , and finally combining the bounds (6.5) and (5.21), we obtain( ε + 1) T ρ [Ψ] ≥ ε (cid:88) Q (cid:34) C (cid:82) Q (cid:37) (cid:82) Q (cid:37) Ψ − C | Q | (cid:90) Q (cid:37) Ψ (cid:35) + (cid:88) Q C ( ρ ) | Q | (cid:20)(cid:90) Q (cid:37) Ψ ( x ) d x − (cid:21) + ≥ εC (cid:82) R (cid:37) Λfor any fixed constant ε > ε ≤ C C ( ρ ) b . 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