A Quantum Impurity Model for Anyons
Enderalp Yakaboylu, Areg Ghazaryan, Douglas Lundholm, Nicolas Rougerie, Mikhail Lemeshko, Robert Seiringer
AA Quantum Impurity Model for Anyons
E. Yakaboylu, ∗ A. Ghazaryan, D. Lundholm, N. Rougerie, M. Lemeshko, and R. Seiringer IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400 Klosterneuburg, Austria Uppsala University, Department of Mathematics - Box 480, SE-751 06 Uppsala, Sweden Universit´e Grenoble- Alpes & CNRS, LPMMC (UMR 5493) - B.P. 166, F-38042 Grenoble, France (Dated: October 29, 2020)One of the hallmarks of quantum statistics, tightly entwined with the concept of topological phases of matter,is the prediction of anyons. Although anyons are predicted to be realized in certain fractional quantum Hallsystems, they have not yet been unambiguously detected in experiment. Here we introduce a simple quantumimpurity model, where bosonic or fermionic impurities turn into anyons as a consequence of their interaction withthe surrounding many-particle bath. A cloud of phonons dresses each impurity in such a way that it e ff ectivelyattaches fluxes / vortices to it and thereby converts it into an Abelian anyon. The corresponding quantum impuritymodel, first, provides a new approach to the numerical solution of the many-anyon problem, along with a newconcrete perspective of anyons as emergent quasiparticles built from composite bosons or fermions. Moreimportantly, the model paves the way towards realizing anyons using impurities in crystal lattices as well asultracold gases. In particular, we consider two heavy electrons interacting with a two-dimensional lattice crystalin a magnetic field, and show that when the impurity-bath system is rotated at the cyclotron frequency, impuritiesbehave as anyons as a consequence of the angular momentum exchange between the impurities and the bath. Apossible experimental realization is proposed by identifying the statistics parameter in terms of the mean squaredistance of the impurities and the magnetization of the impurity-bath system, both of which are accessible toexperiment. Another proposed application are impurities immersed in a two-dimensional weakly interactingBose gas. I. INTRODUCTION
A topological classification of interacting quantum statesis crucial in the context of current research on topologicalstates of matter [1–4]. The discovery of such states in thefractional quantum Hall e ff ect (FQHE) [5] has revolutionizedour understanding of the quantum properties of matter, andhence they are becoming landmarks for the current as well asfuture research directions in physics. One of the most importantcharacterizations of topological states of matter is in termsof the underlying fractionalized excitations. As proposed byLaughlin [6], the excitations in the FQHE are fractionally-charged quasiparticles, which were later demonstrated to beanyons with fractional statistics [7]. Since then, anyons havereceived a significant amount of attention, also because of theirpotential role in quantum computation [8–11]. Experimentalevidence of anyons, on the other hand, is not yet conclusiveand currently contested, despite that two recent works havereported encouraging results in that direction [12, 13].Anyons are a type of quasiparticle whose quantum statisticsinterpolates between bosons and fermions. They occur only inlower-dimensional systems, i.e. mainly in two dimensions andto some extent in one dimension, although the latter will notbe our focus here. The possibility of anyons in two dimensionsmay be traced to the algebraic triviality of the rotation groupSO(2) and the topological non-triviality of a 2D configurationspace with a point removed. Indeed, the symmetrization pos-tulate which had been taken for granted during the first halfof a century of quantum mechanics was called into questionfor such geometries in the 1960’s-80’s [14–23]. As elaboratedfor the first time by Leinaas and Myrheim [18], this leads to ∗ [email protected] the possibility that when two identical particles are exchangedin two dimensions, the statistics parameter α can assume anyintermediate value between 0 (bosons) and 1 (fermions): ψ A ( r , ϕ + π ) = e i πα ψ A ( r , ϕ ) , (1)where ψ A ( r , ϕ ) is the two-body wave function in relative co-ordinates, ( r , ϕ ). As a consequence, anyons have a peculiarproperty: when they are interchanged twice in the same way,the wave function does not return to the original. Namely,under a 2 π rotation, the relative wave function is not single-valued, ψ A ( r , ϕ + π ) = e i πα ψ A ( r , ϕ ). Nevertheless, if ψ A ( r , ϕ )is an eigenstate of some Hamiltonian ˆ H , one may introducea single-valued wave function, ψ ( r , ϕ ) = exp[ − i αϕ ] ψ A ( r , ϕ ),which is governed by the Hamiltonian e − i αϕ ˆ H ( ∂/∂ϕ ) e i αϕ = ˆ H ( ∂/∂ϕ + i α ) , (2)where the statistics parameter now emerges as a gauge field.This establishes a connection between the statistics and gaugefields, and further implies that the orbital angular momentumof two particles in relative coordinates, which is given by − i ∂/∂ϕ + α , is nonintegral [18, 20]. Such a configuration canbe obtained with a magnetic field which substitutes the role ofthe statistics gauge field. This concept of anyons was discussedby Wilczek [20, 21], who realized them as a flux-tube-charged-particle composite – a charged particle ‘orbiting around’ amagnetic flux απ .The picture of flux-tube-charged-particle composites pro-vides a formal description of anyons. From the practical pointof view, however, it does not give much insight concerning aphysical realization. Indeed there has been a recent upsurgein interest concerning the realization of anyons as emergentquasiparticles in experimentally feasible systems, in particularfrom the perspective of deriving robust, testable predictions a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t such as density signatures [24–30]. Also the emergence ofanyons in a FQHE setting by means of attachment of flux viaLaughlin quasiholes was recently revisited and elaborated onin Ref. [31]. Here our main motivation is to define a physicalHamiltonian for a bipartite system such that the statistics gaugefield emerges as a consequence of the interaction between thetwo subsystems [32, 33]. In particular, due to its experimentalfeasibility, we consider a quantum impurity model.The presence of individual quantum particles, called impuri-ties, is almost inevitable in many quantum settings. In severalsituations, ranging from crystals to helium nanodroplets toneutron stars, impurities are coupled to a complex many-bodyenvironment [34–38]. Their interaction with a surroundingquantum-mechanical medium is the focus of quantum impu-rity problems. Impurities do not only appear to be good de-scriptions of experimental reality, but also provide intricateexamples of quantum critical phenomena [39, 40]. In general,quasiparticles formed by impurities are considered as an el-ementary building block of complex many-body systems. Awell-known example is the polaron, which has been introducedas a quasiparticle consisting of an electron dressed by latticeexcitations in a crystal [34–36]. Over the years, with the helpof recent advances in ultracold atomic physics, which enablea high degree of control over experimental parameters suchas interactions and impurity concentration, quantum impuri-ties have been investigated in several di ff erent experimentaland theoretical studies [37, 41–50]. In recent works [29, 33],it has been demonstrated that the many-particle environmentmanifests itself as an external gauge field with respect to theimpurities. It has been also shown in Ref. [29] that the angularmomentum of a quantum planar rotor becomes fractional whenit interacts with a 2D many-particle environment. This config-uration resembles a two-anyon problem in relative coordinateswhen the relative distance between impurities is fixed.In this manuscript, we consider identical impurities im-mersed in a many-particle bath and show that they turn intoanyons in the introduced model as a consequence of theirinteraction with the surrounding bath. Particularly, we treatthe impurities as a slow / heavy and the surrounding bath asa fast / light system, and demonstrate that the latter manifestsitself as a statistics gauge field with respect to the impurities.Excitations of the surrounding bath – a cloud of phonons – at-tach vortices to each impurity so that the quasiparticle formedfrom the impurity dressed by phonons becomes an anyon. Themain contributions of the manuscript can be summarized asfollows:(1) The introduced impurity model provides a new perspec-tive of anyons as emergent quasiparticles built from compositebosons or fermions and a coherent state of quantized vortices.Such a perspective is not only conceptually helpful but alsoopens up new and simple numerical approaches to investigatethe spectra of many anyons.(2) The model further paves the way towards realizing anyonsin terms of impurities in standard condensed matter systemssuch as crystal lattices and ultracold gases. This o ff ers a sig-nificant practical advantage over the strongly correlated mate-rials which to date typically have been investigated to realizeanyons. (3) The statistics parameter of the emerging anyons is identifiedas the phonon angular momentum, which allows us to measurethe statistics parameter in terms of the mean square distanceof the impurities and the magnetization of the impurity-bathsystem.The paper is organized as follows. In Sec. II we presentthe basic machinery for anyons and give a brief review of theregular ideal anyon Hamiltonian. Afterwards, in Sec. III, basedon the emergent gauge picture, we derive the Hamiltonian of aquantum impurity model whose adiabatic limit corresponds toanyons. We present a transparent model where the impuritiescouple only to a single phonon mode. Then, we exemplifythe model by investigating two- and three-impurity problemsand their exact numerical spectra. In Sec. IV we present anew approach to the numerical solution of the many-anyonproblem and provide a new perspective of anyons in termsof composite bosons or fermions. We discuss a possible ex-perimental realization of the model in Sec. V by consideringheavy impurities interacting with collective excitations of abath within the Fr¨ohlich-Bogoliubov model. We conclude thepaper in Sec. VI with a discussion of our results and questionsfor future work. The Appendix provides some further techni-cal details. Throughout the manuscript we use natural units( (cid:126) ≡ M ≡ II. ANYON HAMILTONIAN
In general, one can derive the statistics gauge field withinChern-Simons theory [51–54]. The action of a system ofnon-relativistic charged particles with mass M coupled to theAbelian Chern-Simons gauge field A µ is given by S = M (cid:90) dt N (cid:88) q = ˙ x q + (cid:90) d y A µ j µ + κ (cid:90) d y (cid:15) µνρ A µ ∂ ν A ρ . (3)Here j µ ( y ) = (cid:80) Nq = ˙ y µ δ (2) ( y − x q ) is a point-like source, κ = / (2 πα ) the level parameter, which assumes any number [55],and (cid:15) µνρ the 2 + µ, ν, ρ indicate the 2 + µ = µ = , δ S /δ A = A = α A , one obtains thestatistics gauge field: A iq = ∂∂ x iq N (cid:88) p (cid:48) > p Θ p (cid:48) p = N (cid:88) p ( (cid:44) q ) = (cid:15) i j (cid:16) x jp − x jq (cid:17) | x p − x q | , (4)where (cid:80) q > p denotes the summation over both of the particleindices q and p with the condition q > p , Θ qp = tan − (cid:104) x qp / x qp (cid:105) is the relative polar angle between particles q and p , x iqp = x iq − x ip , and we define (cid:15) i j = (cid:15) i j with (cid:15) =
1. The Hamiltonianfor N ideal non-interacting anyons with statistics parameter α can be written asˆ H N -anyon = − N (cid:88) q = (cid:104) ∇ q + i α A q (cid:105) . (5)Without loss of generality, we consider the Hamiltonianˆ H N -anyon to act on bosonic states, i.e. the Hilbert space H = L ( R N ) of square-integrable functions on R N which aresymmetric w.r.t. exchange. Later we also consider fermionicstates by changing α to α − L ( R N ). A. Regular anyon Hamiltonian
The eigenvalue problem for the Hamiltonian (5) has beensolved analytically only for the two-anyon case [18, 21, 56](see also Refs. [57, 58] for a system of two anyons in the pres-ence of the Coulomb potential), while for three or more parti-cles only part of the spectrum is known exactly. The three- andfour-anyon spectra have been investigated by means of numeri-cal diagonalization techniques [59–62], and a subspace of exacteigenstates is also known analytically for arbitrary N [63–67].Rigorous upper and lower bounds on the exact ground-stateenergy were established in Refs. [68–73]. Another approachhas been to first regularize the Hamiltonian (5) by making thefluxes extended [74–77], and in this situation an exact average-field theory and a corresponding Thomas-Fermi theory may bederived in the almost-bosonic limit α ∼ N − → H = − r (cid:32) ∂∂ϕ + i α (cid:33) − r ∂∂ r (cid:32) r ∂∂ r (cid:33) + r . (6)We observe that this form of the Hamiltonian allows neither aperturbative treatment of the problem nor the use of diagonal-ization techniques with respect to the free operator. Namely,the matrix element (cid:104) l , m | r − | l , m (cid:105) , where the state | l , m (cid:105) is theeigenstate of the harmonic oscillator in polar coordinates withthe principal and magnetic quantum numbers l and m , respec-tively, is logarithmic divergent for the m = H = r − α ˆ H r α , which is a self-adjoint Hamil-tonian in the L space weighted by the measure r α + dr . Inthe transformed Hamiltonian the divergent term vanishes [54].In fact, this transformation corresponds to a ‘real gauge trans-formation’ leading to an imaginary vector potential. In other words, it leads to the replacement of ∂/∂ r → ∂/∂ r + α/ r inthe Hamiltonian (6). If we further combine this transformationwith the one given in Eq. (2), the regular (singular-free) two-anyon Hamiltonian can be obtained directly from the bosonic( α =
0) Hamiltonian via the transformation:ˆ˜ H = exp (cid:2) − i α ( ϕ − i ln r ) (cid:3) ˆ H exp (cid:2) i α ( ϕ − i ln r ) (cid:3) . (7)We can generalize this and obtain the regular N -anyon Hamil-tonian asˆ˜ H N -anyon = N (cid:89) q > p z − α qp ˆ H N -boson N (cid:89) q > p z α qp = − N (cid:88) q = (cid:104) ∇ q + i α ˜ A q (cid:105) (8)with the complex gauge field˜ A q = ∇ q N (cid:88) p (cid:48) > p (cid:16) Θ p (cid:48) p − i ln r p (cid:48) p (cid:17) , (9)where we define ˆ H N -boson = − (cid:80) Nq = ∇ q / z qp = x qp + ix qp = e i ( Θ qp − i ln r qp ) , and r qp = | x q − x p | . In the gauge field (9) thesecond term identifies the imaginary vector potential that elim-inates the singularities arising due to the first term. This can beobserved with the disappearance of the ˜ A q term in the Hamilto-nian, i.e., ˜ A q = A q − A q = N -anyon Hamiltonian in this gauge can bewritten asˆ˜ H N -anyon = − N (cid:88) q = ∇ q + i α N (cid:88) p ( (cid:44) q ) = (cid:15) i j x jpq + ix ipq r pq ∂∂ x iq . (10)The above Hamiltonian is again self-adjoint in a weightedspace (with the weight (cid:81) q > p r α qp multiplying the usual mea-sure), and it may serve as a model Hamiltonian for the calcu-lation of the corresponding anyon spectra. Therefore, in therest of the paper we will also consider this computationallyconvenient regular Hamiltonian ˆ˜ H N -anyon , even though the mainemphasis will be placed on the physical one ˆ H N -anyon . III. IMPURITY MODEL
We start by considering impurities immersed in a weaklyinteracting bath. Within the Fr¨ohlich-Bogoliubov theory [36,92, 93], a general Hamiltonian of an impurity problem is givenby ˆ H qim = − N (cid:88) q = ∇ q + Λ (cid:88) l = ω l ˆ b † l ˆ b l (11) + Λ (cid:88) l = λ l ( x ) (cid:16) e − i β l ( x ) ˆ b † l + e i β l ( x ) ˆ b l (cid:17) . Here the first term corresponds to the kinetic energy of theidentical impurities, which are considered to be either bosonsor fermions. The second term is the kinetic energy of themany-particle bath, whose collective excitations are given witha gapped dispersion relation ω l . In what follows, we will referthese excitations simply as “phonons” regardless of its actualmeaning. Λ is the total number of phonon modes. The creationand annihilation operators associated to each phonon mode,ˆ b † l and ˆ b l , obey the commutation relation (cid:104) ˆ b l , ˆ b † l (cid:48) (cid:105) = δ ll (cid:48) . Thefinal term describes the interaction between the impurities andthe many-particle bath. While we assume that λ l ( x ) is a realfunction, β l ( x ) will at this stage be allowed to be complex for alater purpose. Both functions may depend on all the variables x = { x , . . . , x N } .As we discuss in more detail in Sec. V on specific appli-cations, we consider heavy impurities. This, together with agapped dispersion relation, allows us to treat the bosonic orfermionic impurities as the slow system and the rest of theHamiltonian as the fast one:ˆ H fast ( x ) = (cid:88) l ω l ˆ b † l ˆ b l + (cid:88) l λ l ( x ) (cid:16) e − i β l ( x ) ˆ b † l + e i β l ( x ) ˆ b l (cid:17) , (12)where the coordinates of impurities, x , are regarded as parame-ters. We note that even if β l is complex, ˆ H fast ( x ) is self-adjointin the ˆ w = exp (cid:104) − (cid:80) l Im[ β l ]ˆ b † l ˆ b l (cid:105) -weighted Fock space. Theeigenstates and eigenvalues of the Hamiltonian (12) can befound by applying the following two transformations:ˆ S = exp − i (cid:88) l β l ˆ b † l ˆ b l , ˆ U = exp − (cid:88) l λ l ω l (cid:16) ˆ b † l − ˆ b l (cid:17) , (13)where the transformation ˆ S is, in general, a similarity transfor-mation, as β l might be complex, whereas ˆ U is unitary. There-fore, the eigenstates can be written as ˆ S | ψ n (cid:105) with | ψ n (cid:105) = ˆ U | n (cid:105) .Here the states | n (cid:105) symbolically represent normalized phononstates with the collective index n . Namely, | (cid:105) is the vac-uum state of the bath, | (cid:105) ≡ ˆ b † l | (cid:105) a one-phonon state, and soon. (Later, in simplified models, where we consider a singlephonon mode, the states | n (cid:105) correspond to the usual harmonic-oscillator eigenstates). The eigenvalues, on the other hand,are given by ε n = (cid:80) ni = ω l i + ε with the ground state energy ε = − (cid:80) l λ l /ω l .Let us assume that there exists a large energy gap betweenthe vacuum state | (cid:105) and the excited states ˆ b † l | (cid:105) , i.e., we con-sider the limit ω l → ∞ , which corresponds to a physical real-ization of heavy impurities interacting with gapped excitationsof a bath, see Sec. V. In this adiabatic limit the lowest energyspectrum of the Hamiltonian (11) is given by the Schr¨odingerequation − N (cid:88) q = (cid:104) ∇ q + i G q (cid:105) + W ( x ) χ E ( x ) = E χ E ( x ) , (14)where G q = − i (cid:104) ψ | ˆ S − ∇ q ˆ S | ψ (cid:105) = − (cid:88) l ( λ l ( x ) /ω l ) ∇ q β l ( x ) (15)is the emergent gauge field and W ( x ) the emergent scalar po-tential. The corresponding lowest energy eigenstates of the Hamiltonian (11) are, then, given by | Ψ E ( x ) (cid:105) = χ E ˆ S ˆ U | (cid:105) ; (16)see Appendix A for details. If we are able to match the emer-gent gauge field (15) with the statistics gauge field, the afore-mentioned adiabatic solution of the full problem described bythe Hamiltonian (11) corresponds to the problem of N anyonsinteracting with the potential W ( x ). Although our essentialfocus is the statistics gauge field given by Eq. (4), i.e., thematching of G q = α A q , we also consider the other choice (9)within a toy model for computational purposes. We emphasizethat the condition ω l → ∞ does not necessarily cancel outthe emergent gauge field G q , as we demonstrate in particularexamples below.At first sight, it looks like we have made the N -anyon prob-lem more complicated, as we consider in Eq. (11) bosonic orfermionic particles in a many-particle bath, instead of particlesinteracting with the statistics gauge field. However, the corre-sponding quantum impurity setup, first, lays the groundworkfor realizing anyons in experiment (an experimental realizationwill be proposed in Sec. V). Furthermore, as will be discussednext, this formulation reveals new insights into the structureof the anyon Hamiltonians (5) and (10), and on how anyonsmay emerge from composite bosons (fermions) – the topo-logical bound state of a boson (fermion) and an even numberof quantized vortices (cf. e.g. [94]). Finally, it introducesnew techniques for numerical investigation of the N -anyonproblem. A. Emergent interacting anyons
By using the definition of the statistics gauge field (4) as wellas imposing adiabaticity in the problem leading to Eqs. (14)and (15) we can identify the parameters of the many-particlebath that turn the impurities into anyons. In order to present fea-tures of the introduced model in a transparent way, we considerfor simplicity the following l -independent expressions: β l ( x ) = β ( x ) = − s N (cid:88) q > p Θ qp , (17)constant λ l = λ , and ω l = ω . Here s is an integer such that β ( x q ) has the correct periodicity under the continuous exchangeof particles and the Hamiltonian (11) commutes with the per-mutation operators. Without loss of generality it can be set toits lowest possible non-trivial value s =
2. The minus sign in(17) is chosen simply to eventually make our emergent anyonspositively oriented.We now introduce a unitary transformation ˆ a † l = (cid:80) l (cid:48) v ll (cid:48) ˆ b † l (cid:48) ,with (cid:80) l (cid:48)(cid:48) v l (cid:48) l (cid:48)(cid:48) v ∗ ll (cid:48)(cid:48) = δ l (cid:48) l , such that ˆ a † = (cid:80) Λ l = ˆ b † l / √ Λ and con-sider the choice of ω = √ Λ , which simplifies the Hamiltonian.Then, the impurity Hamiltonian (11) can be written asˆ H qim = − N (cid:88) q = ∇ q + ω (cid:16) ˆ a † ˆ a + λ (cid:17) + λω (cid:16) F ˆ a † + F − ˆ a (cid:17) (18)with F = N (cid:89) q > p (cid:32) z qp | z qp | (cid:33) . (19)For later convenience we use the notation F − , instead of F ∗ ,even though they are equivalent in this particular case. InEq. (18) we have neglected the term (cid:80) Λ l ≥ ω ˆ a † l ˆ a l as the im-purities couple only to a single phonon mode, and omit thesubindex, ˆ a † → ˆ a † . We further subtracted the ground stateenergy of the phonon part of the Hamiltonian, − ωλ . Then, defining α = λ , (20)and keeping the coupling λ fixed in the limit of ω → ∞ , thelowest energy spectrum of the quantum impurity problem (18)is governed by Eq. (14) with the gauge field G q = α A q andscalar potential W = α (cid:80) Nq = A q .The emergence of the anyon Hamiltonian can also be ob-tained without going to the gauge picture, by direct diago-nalization of the Hamiltonian (18). Namely, if we apply thecorresponding ˆ S and ˆ U transformations,ˆ S = F ˆ a † ˆ a , ˆ U = exp (cid:104) − (cid:112) α/ (cid:16) ˆ a † − ˆ a (cid:17)(cid:105) , (21)the transformed Hamiltonian can be written asˆ H (cid:48) qim = ˆ U − ˆ S − ˆ H qim ˆ S ˆ U = ω ˆ a † ˆ a (22) − N (cid:88) q = (cid:34) ∇ q + i A q ( x ) (cid:32) ˆ a † − (cid:114) α (cid:33) (cid:32) ˆ a − (cid:114) α (cid:33)(cid:35) . If we diagonalize the Hamiltonian in the basis of the eigenstatesof the free operators, the transition between di ff erent phononstates for any bosonic (respectively fermionic) eigenstate | Φ (cid:105) of the free N -particle Hamiltonian is quantified by the matrixelement (cid:68) (cid:104) n | ˆ H (cid:48) qim | m (cid:105) (cid:69) Φ / (cid:68) (cid:104) m | ˆ H (cid:48) qim | m (cid:105) − (cid:104) n | ˆ H (cid:48) qim | n (cid:105) (cid:69) Φ with n (cid:44) m . In particular, the transition matrix element between thevacuum state and the one-phonon state is (cid:112) α/ (cid:68)(cid:80) q (cid:16) i [ ∇ q , A q ] + − (1 + α ) A q (cid:17)(cid:69) Φ ω − (cid:68)(cid:80) q (cid:16) i [ ∇ q , A q ] + − (1 + α ) A q (cid:17)(cid:69) Φ , (23)where [ , ] + is the anti-commutator. Then, in the limit of ω → ∞ the transition matrix element becomes negligible (we refer toAppendix C for more details), and hence, the vacuum expecta-tion value (cid:104) | ˆ H (cid:48) qim | (cid:105) = − N (cid:88) q = (cid:104) ∇ q + i α A q ( x ) (cid:105) + W ( x ) , (24)decouples from the rest of the spectrum. This is basicallythe statement of the adiabatic theorem. As a result, theenergy levels belonging to the vacuum sector governed byEq. (24) describes interacting anyons in the potential W ( x ) = α (cid:80) Nq = A q ( x ). We note, however, that for ideal point-like anyons the expectation values (cid:104) A q (cid:105) Φ in Eq. (23) are diver-gent for certain bosonic eigenstates of the free Hamiltonian,as we discussed in Sec. II A, and therefore, the adiabatictheorem breaks down. In order to avoid these singularitiesthe corresponding spectrum can be calculated instead fromthe fermionic end by changing α to α − L ( R N ) to L ( R N ) for the impurities.Let us investigate the form of the Hamiltonian (18) and theemergence of anyons in more detail. In this simplified modelthe interaction between the impurities and phonon field is de-scribed by the factor F . The ˆ S transformation attaches twoflux / vortex units to bosonic (fermionic) impurities to convertthem into composite bosons (fermions). (In fact, in this par-ticular case (19) we call it a flux rather than a vortex, as it isjust a phase factor.) Then, a cloud of phonons, which mani-fests itself through a coherent state, dresses each impurity andforms a quasiparticle governed in the limit of ω → ∞ by theanyon Hamiltonian (24). The corresponding coherent state ofphonons is given by | ψ (cid:105) = ˆ U | (cid:105) = e − α/ ∞ (cid:88) n = (cid:16) − √ α/ (cid:17) n √ n ! | n (cid:105) , (25)which involves infinitely many phonons, weighted accordingto α . In this adiabatic limit, the states in the Fock space de-couple from each other, and each resulting energy level isfilled by anyons with the statistics parameter 2 n + α , where n corresponds to the energy levels of the Fock space sector.Specifically, if the impurities are initially bosons, in the vacuumstate, n =
0, they turn into interacting anyons with statisticsparameter α and spectrum described by Eq. (24). In the ex-cited states, n (cid:44)
0, on the other hand, the impurities becomecomposite bosons with 2 n units of flux and interacting withthe statistics gauge field α A q as well as the scalar potential α (1 + n ) (cid:80) q A q . See Appendix C for more details concerningthe diagonalization of Eq. (22).Below we focus only on the vacuum state and present twosimple examples. For an easier comparison with the resultsexisting in the literature, we investigate impurities confinedadditionally in a harmonic-oscillator potential and allow forfermionic impurities with the coupling α − = s λ in orderto deal with the singular interaction. Furthermore, since theinteraction term depends only on relative coordinates, we factorout the center-of-mass problem, and make a transformationto relative coordinates. In general, the relative coordinatesfor an N -body problem are given by Jacobi coordinates: R = (cid:80) Ni = z i / √ N and u m = ( (cid:80) mi = z i − mz m + ) / √ m ( m + m runs from 1 to N −
1. Two anyons
As a first example, we consider the simplest case – the two-impurity problem. The Hamiltonian in Jacobi coordinates withthe notation u = r exp( i ϕ ), which is simply usual relativecoordinates for a 2-body problem, becomesˆ H = − ∇ + r + ω (cid:32) ˆ a † ˆ a + α − (cid:33) + ω (cid:114) α − (cid:104) e i ϕ ˆ a † + e − i ϕ ˆ a (cid:105) , (26)where α ≥
1. This two-impurity problem can be solved numer-ically by diagonalizing the Hamiltonian with the eigenstatesof the free Hamiltonian – the first line of Eq. (26). Theseeigenstates are the phonon states | n (cid:105) times the antisymmetricimpurity states. The latter are the usual harmonic oscillatorwave functions: Φ lm = (cid:115) l !)( l + | m | + e − r / r m L ml ( r ) exp( im ϕ ) / √ π , (27)where L ml ( r ) are associated Laguerre polynomials. The corre-sponding eigenvalues are given by 2 l + | m | +
1. The antisymmet-ric impurity states in Jacobi coordinates follow from the parityof the angular quantum number of the impurities m : whileeven m refers to bosons, the odd ones correspond to fermions.Due to the finite number of impurity states considered in nu-merics, there is an intricate relation between the number ofimpurity wave functions, maximum number of phonons, andactual value of ω in order to achieve convergence. The con-verged result of the anyonic spectra for the Hamiltonian (26)is obtained with ω (cid:38)
20, up to 5 number of phonons, andseveral hundred impurity states in the fermionic basis, whichis shown in In Fig.1 (Top). In fact, this result can also be foundanalytically. Namely, it follows from Eq. (24) that the lowestlevels are described by the Hamiltonian (cid:104) | ˆ H (cid:48) | (cid:105) = − r (cid:32) ∂∂ϕ + i ( α − (cid:33) − α − − r ∂∂ r (cid:32) r ∂∂ r (cid:33) + r . (28)Then, the corresponding eigenvalues directly fol-low from the harmonic oscillator ones by replac-ing | m | with (cid:112) ( m + ( α − + α −
1) to yield2 l + (cid:112) ( m + ( α − + α − +
1, which agrees withthe numerical result shown in Fig.1 (Top).
2. Three anyons
As a next example, we study the three-impurity problem. InJacobi coordinates, with the notation u = η = η x + i η y and u = ξ = ξ x + i ξ y , the Hamiltonian is given byˆ˜ H = − (cid:16) ∇ η + ∇ ξ (cid:17) + (cid:16) η + ξ (cid:17) + ω (cid:32) ˆ a † ˆ a + α − (cid:33) + ω (cid:114) α − (cid:32) η − ηξ | η − ηξ | (cid:33) ˆ a † + H.c. (29)
FIG. 1. Calculations of the two-anyon (Top) and three-anyon (Bot-tom) spectra for the interacting anyon model in an external harmonic-oscillator potential (18). The energies are given in units of the har-monic frequency. The spectra have been calculated from the fermionicend, i.e., the coupling is chosen as α − = s λ such that α = l max = m max =
21, and ω =
23 with up to 5 phonons for the two-anyon case.For the three-anyon case we consider all the antisymmetric impuritywave functions (30) restricted by the condition E n max m max ≤
26 andlimit the maximum number of phonons to 10 with ω =
54. For clarity,we do not display all the curves in the second plot. with α ≥ η = η x + η y , and ξ = ξ x + ξ y . In contrast tothe two-impurity problem, the implementation of the permu-tation symmetry for the impurity states in Jacobi coordinatesis not trivial due to the relations P η = ( η + √ ξ ) / P ξ = ( √ η − ξ ) /
2. Accordingly, by following the conventionintroduced by Kilpatrick and Larsen [95], we use hyperspheri-cal coordinates, ( ρ, θ, φ, ψ ), η = ρ e − i ψ (cos θ cos φ + i sin θ sin φ ) ,ξ = ρ e − i ψ (cos θ sin φ − i sin θ cos φ ) , where the symmetrization of the wave function is straightfor-ward. Namely, the bosonic ( + ) or fermionic (-) wave functionsare given by Φ ± nm νµ = (cid:115) n !)( m + n + e − ρ / ρ m L m + n ( ρ ) Y ± m νµ ( θ, φ, ψ ) , (30)where Y ± m νµ ( θ, φ, ψ ) are the hyperspherical harmonics; seeAppendix B. The wave functions are normalized over thevolume dV = ρ d ρ cos(2 θ ) d θ d φ d ψ for ρ ∈ [0 , ∞ ) , θ ∈ [ − π/ , π/ , φ ∈ [ − π/ , π/ , ψ ∈ [0 , π ]. Here, n = , , , · · · is the radial quantum number and m = , , , · · · is one of the angular momentum numbers such that the cor-responding spectrum reads E nm = (2 n + m + ν, µ ) have the same parity as m and they are restricted by: | ν | , | µ | ≤ m and ν = q with anon-negative integer q ; see also Ref. [96]. In Fig. 1 (Bottom),we show the corresponding spectra. In the diagonalizationprocedure over one thousand fermionic impurity states are con-sidered and the maximum number of phonons is limited to10 with ω ≈
50 resulting in converged spectra in the regimeconsidered.
B. A toy model for free anyons
Eq. (24), which describes the lowest levels of the impurityproblem (18) in the limit of ω → ∞ , can also serve as a newplatform for studying the original N -anyon problem withoutthe presence of the interaction potential W , i.e., free anyons.Namely, instead of Eq. (17), if we define β l ( x ) = β ( x ) = − s N (cid:88) q > p (cid:16) Θ qp − i ln r qp (cid:17) (31)with the same ω and λ as in the previous case, the emergentgauge field is given by Eq. (9). In this case the correspondingimpurity Hamiltonian analogous to Eq. (18) can be written asˆ˜ H qim = − N (cid:88) q = ∇ q + ω (cid:16) ˆ a † ˆ a + λ (cid:17) + λω (cid:16) ˜ F ˆ a † + ˜ F − ˆ a (cid:17) (32)with ˜ F = N (cid:89) q > p z qp . (33)We note that as ˜ F − (cid:44) ˜ F ∗ , the Hamiltonian (32) is not Hermi-tian in this case. Nevertheless, its non-Hermiticity is harmlessfor our purposes, and indeed the fast part of the Hamiltonian isself-adjoint in a Fock space weighted by | ˜ F | a † ˆ a . This allowsus to rewrite the Hamiltonian in the gauge picture so that thelowest energy levels of the Hamiltonian (32) become real in thelimit of ω → ∞ . Namely, if we diagonalize the Hamiltonian,the energy levels belonging to the vacuum state in this limit aregiven by Eq. (24) with the replacement A q → ˜ A q . Moreover,as ˜ A q =
0, the emergent scalar potential W ( x ) vanishes and hence Eq. (24) for this case simply corresponds to the free N -anyon Hamiltonian defined in the regular gauge (10): (cid:104) | ˆ˜ H (cid:48) qim | (cid:105) = − N (cid:88) q = (cid:104) ∇ q + i α ˜ A q ( x ) (cid:105) . (34)Therefore, the Hamiltonian (32) describes free anyons in thelimit of ω → ∞ . The mechanism of how anyons emerge outof the impurity-bath coupling in this particular model is verysimilar to the previous interacting case. The only di ff erenceis the form of composite bosons / fermions. In contrast to thefactor F , in the Hamiltonian (32) the attachment of flux / vortexis performed by ˜ F . As the latter includes also a length, we willcall the multiplication by ˜ F vortex attachment. Therefore, inthis toy model, impurities are dressed by vortices by means ofphonons.Similar to the previous interacting anyon case, the corre-sponding two- and three-impurity problems in this toy modelcan be solved by diagonalizing the Hamiltonian (32) with theeigenstates of the free Hamiltonian by considering the impu-rities confined additionally in a harmonic-oscillator potential.Instead of following this approach, below we present new com-putational techniques for the numerical solution, along with anew concrete perspective of anyons in relation to compositebosons or fermions. IV. A NEW PERSPECTIVE ON ANYONS
The simplified impurity model given in Eq. (32) or (18)also provides some useful analytical insights for the N -anyonproblem. Here we focus solely on the Hamiltonian (32) cor-responding to free anyons. Nevertheless the interacting case,Eq. (18), follows straightforwardly. The Hamiltonian (32) canbe diagonalized in the Fock space with the displacement oper-ator ˆ T = exp (cid:104) − √ α (cid:16) ˜ F ˆ a † − ˜ F − ˆ a (cid:17) / √ (cid:105) . Then, the N -anyonspectrum, which emerges in the limit of ω → ∞ , is given bythe Hamiltonian˜ H N -anyon = (cid:104) | ˆ T − ˆ˜ H qim ˆ T | (cid:105) = ∞ (cid:88) n = e − α/ n ! (cid:18) α (cid:19) n ˆ˜ H (2 n ) N -comp , (35)where we made use of the coherent state (25) (see Appendix Cfor details). Here the Hamiltonianˆ˜ H (2 n ) N -comp = ˜ F − n ˆ H N -boson ˜ F n = − N (cid:88) q = (cid:104) ∇ q + i n ˜ A q (cid:105) (36)describes N composite bosons (fermions) – the topologicalbound state of a boson (fermion) and an even number of quan-tized vortices. This formula shows concretely how anyonsemerge from composite bosons or, put it di ff erently, it depictshow a fractional vortex manifests itself through integer vortices.A similar expression can also be obtained for the interactinganyon model (18) by removing tildes from ˜ F and ˜ F − in the dis-placement operator. In this case the expression (35) describesthe interacting anyon model.First of all, Eq.(35) naturally simplifies to H N -boson for α →
0, and, in general, it can be given by a power seriesˆ˜ H N -anyon = ∞ (cid:88) n = n ! (cid:18) α (cid:19) n K n (37)with K n = n (cid:88) j = (cid:32) nj (cid:33) ( − n − j ˆ˜ H (2 j ) N -comp . (38)One sees from Eq. (24) or Eq. (34) that K n = n ≥ K also vanishes as aresult of the fact that ˜ A q =
0. Therefore, the power seriesexpansion (37) simply yieldsˆ˜ H N -anyon = ˆ˜ H (0) N -comp + α (cid:16) ˆ˜ H (2) N -comp − ˆ˜ H (0) N -comp (cid:17) . (39)It is straightforward to show that the above simple expressionis equivalent to the N -anyon Hamiltonian (10). In this form weare quite justified to term this process as “statistics transmuta-tion” from bosons at α = α = / fractionalvalues of α . As we discuss below, Eq. (39) admits new numeri-cal techniques for studying various many-anyon problems. A. New numerical techniques for free anyons
Although the resulting formula (39) looks almost trivial, wewould like to emphasize that it is derived within the introducedquantum impurity model by the non-trivial algebraic propertiesof the coherent state (25). This, first, shows the consistencyof our approach, and further provides a new approach to thenumerical solution of the N -anyon problem. Specifically, theHamiltonian (39) also readsˆ˜ H N -anyon = H N -boson + α (cid:16) ˜ F − H N -boson ˜ F − H N -boson (cid:17) . (40)The above Hamiltonian can be further written in terms of theimpurity basis states, | Φ m (cid:105) , which are the (anti-)symmetriceigenstates of the free N -particle Hamiltonian ˆ H N -boson , as E N -anyon = E N -boson + α (cid:16) ˜ Z − E N -boson ˜ Z − E N -boson (cid:17) , (41)where we define the elements of the matrices (cid:16) E N -boson (cid:17) nm = (cid:104) Φ n | ˆ H N -boson | Φ m (cid:105) , and (cid:16) ˜ Z (cid:17) nm = (cid:104) Φ n | ˜ F | Φ m (cid:105) , (cid:16) ˜ Z − (cid:17) nm = (cid:104) Φ n | ˜ F − | Φ m (cid:105) . (42)We note that the inverse matrix ˜ Z − is deduced from the rela-tion ( ˜ Z ˜ Z − ) nm = (cid:80) k (cid:104) Φ n | ˜ F | Φ k (cid:105)(cid:104) Φ k | ˜ F − | Φ m (cid:105) = δ nm , where weuse the fact that the (anti-)symmetrizer, (cid:80) k | Φ k (cid:105)(cid:104) Φ k | = S , com-mutes with the interaction term ˜ F , and S| Φ k (cid:105) = | Φ k (cid:105) for all(anti-)symmetric states.We underline that all the diagonalization techniques that weare aware of in the literature are based on the diagonalization of the matrix whose entries are given by the interaction term (cid:104) Φ n | [ ∇ q , ˜ A q ] + | Φ m (cid:105) . However, Eq. (41) is based on the matrix Z ,which is much easier to construct in comparison to the former.Thus, the matrix equation (41) could be of use as a powerfultechnique in calculating the N -anyon spectrum. This will beillustrated below in particular examples.
1. Examples
We again consider our impurities confined additionally ina harmonic-oscillator potential. Moreover, since in this toymodel there are no singular terms in the corresponding tran-sition matrix element (23), i.e., ˜ A q =
0, we consider bosonicimpurities. We first consider the two-impurity problem. TheHamiltonian in relative coordinates is given byˆ˜ H = − ∇ + r + ω (cid:18) ˆ a † ˆ a + α (cid:19) + ω (cid:114) α (cid:104) ( √ re i ϕ ) ˆ a † + ( √ re i ϕ ) − ˆ a (cid:105) . (43)The corresponding matrix ˜ Z can be constructed from the matrixelements (cid:104) Φ lm | ( √ re i ϕ ) | Φ l (cid:48) m (cid:48) (cid:105) and the matrix ˜ Z − from theelements (cid:104) Φ lm | ( √ re i ϕ ) − | Φ l (cid:48) m (cid:48) (cid:105) , where Φ lm are the harmonicoscillator wave functions (27). We note that as the phase factorexcludes the m = m (cid:48) = l + | m + α | + ≈ H = (cid:16) P η + P ξ (cid:17) + (cid:16) η + ξ (cid:17) + ω (cid:18) ˆ a † ˆ a + α (cid:19) (44) + ω (cid:114) α (cid:32)
12 ( η − ηξ ) ˆ a † + η − ηξ ) − ˆ a (cid:33) . In this particular example, instead of constructing the matrices˜ Z and ˜ Z − separately, we first constructed the matrix ˜ Z − byusing the symmetric impurity wave functions (30). Afterwardswe take its pseudoinverse and define ˜ Z . Although the otherway around is also possible, we find the former way moreconvenient for numerical reasons as ˜ Z − is a more stable matrix.Similar to the two-impurity case, all the entries of ˜ Z − are finitedue to the presence of the phase factor in the interaction term;see Appendix B for the calculation of the matrix elementsin hyperspherical coordinates. In Fig. 2 (Bottom), we showthe corresponding spectra, which agree with the known three-anyon spectra [59, 60]. We found that one can access thespectrum for the interval 0 ≤ α ≤ Z − is quite straightforward; see FIG. 2. Calculations of the two-anyon (Top) and three-anyon (Bottom)spectra for the original harmonic-oscillator anyon problem withoutthe scalar potential W (32). The energies are given in units of theharmonic frequency. The spectra are here calculated from the bosonicend by using Eq. (41). The applied parameters are l max = m max =
20 for the two-anyon case. For the three-anyon case we consider allthe symmetric impurity wave functions (30) restricted by the condition E n max m max ≤ Appendix B. This appears to us as a very promising way fornumerical investigation of various many-anyon problems.
B. Emergence of anyons from composite bosons
In addition to the new numerical approach discussed above,Eq.(35) provides a natural geometric interpretation of thisstatistics transmutation of impurities in terms of vector bundles.Note that in general we may consider the N -anyon Hamilto-nian (5) resp. (10) and its domain of functions as defining acomplex line bundle (i.e. a rank one hermitian vector bundle)over the configuration space of N identical particles in theplane. In the cases considered here there are no further mag-netic interactions than the statistical one, and this then definesa locally flat line bundle which is characterized solely by thestatistics parameter α ∈ R . Using the trivial bosonic bundle α = e i πα . We note that α and α + ff erent bundles. Afamily of such bundles α = n , n ∈ Z , may be characterizedgeometrically by the minimal winding number n of the phaseas two particles are simply exchanged, or the winding number2 n as one particle continuously encircles another one. This isthe same as the number of unit fluxes attached to each particle,and the permutation symmetry enforces the same number toeach particle. We may thus talk about an even-integer familyof bosonic bundles having the physical interpretation as com-posite bosons with 2 n quanta of flux attached to each boson,cf. Eq. (36). Note that multiplication by F of Eq. (19) resp. F ∗ changes these winding numbers, and indeed these are thegauge transformations that unitarily transform one such bundleinto the other [97].Thus, our starting point in the statistics transmutation wasthe geometrically trivial bundle H of regular bosonic states onthe plane R on which the free Hamiltonian − (cid:80) q ∇ q / ff ectively considering a semi-infiniteladder {H ˜ F n | n (cid:105)} , n = , , , . . . , of even-integer bundles ofcomposite bosons. The factor ˜ F n ensures that the windingnumber of the phase under simple exchange increases by n ,and equivalently that the vorticity attached to each particle is2 n . In Eq. (32) we have introduced the possibility of hoppingfrom one such bundle, winding number or vorticity, to the nexthigher one by means of the interaction term ˜ F ˆ a † and hence thesymmetric (thus staying within the family of bosonic bundles)attachment of a minimal number of vortices to each particle (cf.the arguments leading to Eq. (17)), as well as the correspondinghopping to the next lower bundle using the term ˜ F − ˆ a and thusthe detachment of a minimal number of vortices. We then havethe interpretation of Eq. (32) that we are introducing an energygap ω between each level of the bundles (which in this contextmay be interpreted or defined as the energy cost of creatingthe corresponding number N ( N −
1) of vortices), and enablingthe hopping between consecutive bundles on the ladder by anon-zero amplitude λω . In the simultaneous limit of both largeenergy gap ω and large hopping amplitude, while keeping theirratio λ fixed, what then emerges according to Eq. (34) is thefractional bundle labeled by the fraction of vorticity per particle α = λ . The phonon state (25) attaches a Poisson-distributedsequence of weights on each integer bundle on the ladder,resulting in the superposition (35) of bundles. Furthermore,according to the alternative form (39), it may be equivalentlyunderstood as a linear (modulo weights) deformation betweenthe two lowest bosonic bundles at α = α =
2. When λ = α = ff ectively achieves a complete transmutationinto the next integer level by means of the unitary equivalenceof ( | ˜ F | / ˜ F ) ˆ H ( ˜ F / | ˜ F | ) = F ∗ ˆ HF to ˆ H . We anticipate that thisgeometric perspective on the statistics transmutation could0be extended, for instance to the setting of higher-rank vectorbundles and non-Abelian anyons. V. EXPERIMENTAL REALIZATION
We now investigate a possible experimental configuration ofthe quantum impurity model (11). Let us consider N identicalimpurities, say bosons, with mass M , immersed into a weaklyinteracting many-particle bath, whose collective excitationsare given by phonons with a gapped dispersion ω ( k ). Thiscreates the necessary energy gap between the phonon states inthe limit of M → ∞ , which allows us to consider the problemin the adiabatic limit. Furthermore, we consider impuritiesconfined to two dimensions and leave out the direct interactionbetween impurities, as the latter is irrelevant for our discussion.In analogy to the Fr¨ohlich-Bogoliubov theory, the Hamiltonianof such a model is given byˆ H exp = − M N (cid:88) q = ∇ q + (cid:88) k ω ( k ) ˆ b † k ˆ b k (45) + (cid:88) k (cid:16) V ( k , x )ˆ b † k + V ∗ ( k , x )ˆ b k (cid:17) with (cid:80) k = (cid:82) d k / (2 π ) . Here ˆ b † k and ˆ b k are the creationand annihilation operators for a phonon with the wave vec-tor k and frequency ω ( k ). They obey the commutation re-lation [ˆ b k , ˆ b † k (cid:48) ] = (2 π ) δ ( k − k (cid:48) ). The last term in Eq. (45)describes the impurity-phonon interaction with the coupling V ( k , x ), which depends on the coordinates of impurities x = { x , . . . , x N } .As a first step, we decompose the creation and annihilationoperators in polar coordinates,ˆ b † k = (cid:114) π k ∞ (cid:88) µ = −∞ i µ e − i µϕ k ˆ b † k µ , (46)with (cid:104) ˆ b k µ , ˆ b † k (cid:48) µ (cid:48) (cid:105) = δ ( k − k (cid:48) ) δ µµ (cid:48) . The Hamiltonian (45) is givenby ˆ H exp = − M N (cid:88) q = ∇ q + (cid:88) k ,µ ω ( k ) ˆ b † k µ ˆ b k µ (47) + (cid:88) k ,µ λ µ ( k , x ) (cid:104) e − i β µ ( k , x ) ˆ b † k µ + e i β µ ( k , x ) ˆ b k µ (cid:105) where (cid:80) k = (cid:82) ∞ dk , and V µ ( k , x ) = (cid:115) k (2 π ) (cid:90) d ϕ k V ( k , x ) i µ exp( − i µϕ k ) , (48)which has been further decomposed into V µ ( k , x ) = λ µ ( k , x ) exp( − i β µ ( k , x )). The Hamiltonian (47) is of the formof the general model Hamiltonian (11). As a result, accordingto Eq. (15), the emergent gauge field can be written as G q = − (cid:88) k ,µ (cid:16) λ µ ( k , x ) /ω ( k ) (cid:17) ∇ q β µ ( k , x ) . (49) FIG. 3. Experimental proposal where the Fr¨ohlich polarons turn intoanyons. Heavy electrons with mass M immersed in a 2D materialare subjected to the magnetic field B . If the electron-bath system isrotated at the cyclotron frequency Ω = B / (2 M ), the polarons becomeanyons. The setup can also be extended to a 2D Bose gas. If the bosonic impurities interact with the phonons in sucha way that the emergent gauge field (49) matches with thestatistics gauge field (4), they turn into anyons in the limitof M → ∞ . The eventual experimental realization of themodel, therefore, reduces to the feasibility of such an impurity-phonon interaction. In general, the gauge field (49) is non-zero when the integral (cid:80) k (cid:16) λ µ ( k , x ) /ω ( k ) (cid:17) ∇ q β µ ( k , x ) is not anodd function under µ , which is a manifestation of breakingtime reversal symmetry. This can be achieved by applying amagnetic field or rotation to the system. In principle, suchan interaction is feasible with the state-of-art techniques inultracold atomic physics, for instance in a rotating Bose gas.Below we present a simple and intuitive realization within awell-known problem – the Fr¨ohlich polaron. A. Fr¨ohlich Polarons as Anyons
Let us consider two electrons confined in a plane interactingwith longitudinal optical phonons, ω ( k ) = ω . The corre-sponding Fr¨ohlich Hamiltonian for two impurities is given byEq. (45) with the following coupling [36] V ( k , x , x ) = V ( k ) (cid:16) e − i k · x + e − i k · x (cid:17) , (50)where V ( k ) is the Fourier component of the impurity-phononinteraction in real space. We further apply a magnetic fieldto the impurities along the z -direction. Then, the Fr¨ohlichHamiltonian is given byˆ H F = − ( ∇ − i a ) M − ( ∇ − i a ) M + (cid:88) k ω ˆ b † k ˆ b k + (cid:88) k V ( k ) (cid:16) e − i k · x + e − i k · x (cid:17) ˆ b † k + H.c. (51)where a i = B ( − y i , x i ) / x − y plane at the cyclotron frequency Ω = B / (2 M ). The1experimental setup is depicted in Fig. 3. The rotation of thesystem allows us to factor out the center-of-mass coordinatesin the limit of M → ∞ . The Hamiltonian (51) can be writtenin the rotating coordinate system asˆ H FR = e − it Ω ˆ J z (cid:32) ˆ H F − i ∂∂ t (cid:33) e it Ω ˆ J z + i ∂∂ t (52) = M (cid:16) − ∇ − ∇ + M Ω ( x + x ) (cid:17) + Ω ˆ Λ z + (cid:88) k ω ˆ b † k ˆ b k + (cid:88) k V ( k ) (cid:16) e − i k · x + e − i k · x (cid:17) ˆ b † k + H.c.Here ˆ J z = ˆ L z + ˆ L z + ˆ Λ z is the total angular momentumof the impurity-bath system along the z -direction, with ˆ L i z being the angular momentum of the i -th impurity and ˆ Λ z thecollective angular momentum operator of the bath. Next weintroduce relative and center-of-mass coordinates, r and R ,respectively, and then apply the unitary Lee-Low-Pines (LLP)transformation [98], ˆ T LLP = exp (cid:104) − i ˆ R · (cid:80) k k ˆ b † k ˆ b k / √ (cid:105) . Wedecompose the creation and annihilation operators in polarcoordinates, where the angular momentum operator simplyreads ˆ Λ z = (cid:80) k ,µ µ ˆ b † k µ ˆ b k µ . The transformed Hamiltonian can berewritten asˆ H (cid:48) FR = − M ∇ r + M Ω r + (cid:88) k ,µ ω µ ˆ b † k µ ˆ b k µ (53) + (cid:88) k ,µ λ µ ( k , r ) (cid:104) e − i µϕ ˆ b † k µ + e i µϕ ˆ b k µ (cid:105) + ˆ h R . Here ω µ = ω + µ Ω is the e ff ective phonon dispersion relation,where the second term arises as a consequence of the rotation.The impurity-bath coupling strength is given by λ µ ( k , r ) = (cid:112) k / (2 π ) V ( k ) J µ ( kr / √ (cid:2) + ( − µ (cid:3) , (54)which follows from the Jacobi-Anger expansion, exp[ i k · x ] = (cid:80) µ i µ J µ ( kr ) exp[ i µ ( ϕ − ϕ k )], with J µ ( kr ) being the Bessel func-tion of the first kind. The last term in Eq. (53),ˆ h R = M ∇ R − i (cid:88) k k ˆ b † k ˆ b k / √ + M Ω R , (55)is the Hamiltonian for the center-of-mass motion that couplesto the many-particle bath. We note that the coupling term, (cid:80) k ∇ R · k ˆ b † k ˆ b k / M , is negligible in the limit of M → ∞ , as themomentum operator scales as √ M . Therefore, the center-of-mass coordinate decouples in the transformed Hamiltonian. Ina similar way, the contribution of the term ( (cid:80) k k ˆ b † k ˆ b k ) / M tothe fast Hamiltonian is also negligible in the limit of M → ∞ .Consequently, ˆ h R will be omitted hereafter.In realistic situations the maximum number of phonons n max interacting with impurities is finite. Furthermore, because ofthe finite size of the first Brillouin zone, we consider a naturalcut-o ff for the phonon wave vector k max . This puts an upperlimit for the µ -summation as well as for the k -integral. Thelimitation on the k -integral can a ff ect the small distance behav-ior of the impurities. Nevertheless, as the repulsive Coulomb interaction between two electrons prevents us from consideringsmall distances, we will ignore this cut-o ff . The cut-o ff for the µ -summation, on the other hand, is essential in order to havea spectrum bounded from below. Namely, the ground stateenergy of the fast Hamiltonian can be written as ε gs = min { , n max ( ω − Ω µ max ) } − (cid:90) ∞ dk µ max (cid:88) µ = − µ max λ µ ( k , r ) ω µ , (56)and we consider the case ( ω − Ω µ max ) >
0, where the groundstate is given by the vacuum state ˆ S ˆ U | (cid:105) .It follows from Eq. (49) that the corresponding emergentgauge field for the ground state of the Hamiltonian (53) isgiven by G = α ( r ) r e ϕ (57)with α ( r ) = − µ max (cid:88) µ = − µ max (1 + ( − µ ) µ ( ω + µ Ω ) (cid:90) ∞ dk k π (cid:16) V ( k ) J µ ( kr / √ (cid:17) . (58)In general, the emergent gauge field does not necessarily cor-respond to a statistics gauge field, as the r -dependence of α ( r ) relies on the form of V ( k ). Now we will have a closerlook at the derivation of the Fr¨ohlich Hamiltonian, where V ( k )emerges as the Fourier component of the interaction betweenthe impurity and surrounding many-particle bath in real space.In the regular Fr¨ohlich polaron, impurities are consideredto be confined to two dimensions, but interact with a threedimensional bath, and V ( k ) emerges as a consequence of theinteraction between the electron and the polarization field [99],which leads to V ( k ) = (cid:113) √ πγ F / k with γ F being the Fr¨ohlichcoupling constant for the electron confined in two dimensions.This form of the coupling describes surface polarons [99–102].In this case, the statistics parameter (58) scales as 1 / r . Nev-ertheless, if we apply a strong magnetic field, of the order of Ω ∼ ω , the relative wave function of the impurities is local-ized in r -space, and hence the relative motion of two impuritiesis described by only the relative angle. In this case, we canomit the r -dependency of the statistics parameter, or consider alimited range, where α ( r ) is approximately constant. We note,however, that for such a strong magnetic field ( ω − Ω µ max ) < S and ˆ U transformations. The emergence ofanyons in such a scenario is similar to the model investigatedin Ref. [29], where the relative distance between impuritieswas assumed to be constant.
1. 2D phonon bath
Instead of a three-dimensional bath, we now consider aquasi-2D bath. Namely, we consider impurities confined to twodimensions, and also interacting with a 2D bath. The latter canbe achieved by assuming that the confinement of a 3D bath inthe z direction is so strong that we can ignore the excitations in2 × - 6 × - 6 × - 6 × - 6 FIG. 4. The relative statistics parameter (59) as a function of thedimensionless cyclotron frequency Ω /ω for the impurities immersedin a 2D ionic crystal. We note that the e ff ective scalar potentialdepends also on the statistics parameter. The applied parameters are γ = ω and µ max =
50. For typical parameters of optical phonons( ω ≈ Hz) the cyclotron frequency is at the order of MHz. that direction, or we can consider a bath in the form of a singlelayer of atoms in a two-dimensional lattice, such as graphene.In this case, the polarization field behaves like a 1 / r field,instead of the 1 / r behavior of a 3D bath. This follows fromthe fact that in two-dimensional materials the Coulomb lawscales as ln r , which has already been observed and investigatedin several experimental works [103–105]. Then, the Fouriercomponent is given by V ( k ) = (cid:112) πγ/ k with some constant γ ,which we call the 2D Fr¨ohlich coupling constant. As a resultof this, the emergent gauge field yields G = e ϕ α/ r with thestatistics parameter α = γω Ω µ max (cid:88) µ = , even µ ( ω − µ Ω ) − , (59)where we use the relation (cid:82) ∞ dkJ µ ( kr ) / k = / (2 | µ | ) for µ (cid:44) M → ∞ , when the impurity-bath system is rotated at thecyclotron frequency. The adiabaticity condition (23) can bewritten as (cid:68) [ ∇ q , A q ] + (cid:69) /ω ∝ Ω /ω (cid:28) (cid:68) [ ∇ q , A q ] + (cid:69) ∝ (cid:68) ∂ ˆ H /∂α (cid:69) = ∂ E /∂α ∝ Ω . Therefore, the limit of M → ∞ reads as M (cid:29) B /ω in an experimental configuration. The absolutestatistics parameter ((59) minus one in the fermionic case)emerges as a function of the dispersion, cyclotron frequency,and the 2D Fr¨ohlich coupling constant. In Fig. 4 we show thestatistics parameter as a function of the dimensionless cyclotronfrequency Ω /ω .The origin of the emergent anyons can be intuitively under-stood in terms of the relative angular momentum of impuritiesimmersed in a bath. By using Eq. (15) and bearing in mindthat the ˆ S transformation (13) can be written asˆ S = exp − i ϕ (cid:88) k ,µ µ ˆ b † k µ ˆ b k µ = exp (cid:16) − i ϕ ˆ Λ z (cid:17) , (60) the emergent gauge field is given by G = −(cid:104) | ˆ U − ˆ Λ z ˆ U | (cid:105) e ϕ / r .In other words, the statistics parameter is simply given bythe expectation value of the angular momentum of the many-particle bath in the coherent state α = −(cid:104) ˆ Λ z (cid:105) coherent state , (61)which can assume any number, as the coherent state is not aneigenstate of the angular momentum operator. Therefore, therelative angular momentum of the impurities is shifted by α and hence becomes nonintegral.This result is consistent with the conservation of the angularmomentum of the impurity-bath system. The total angularmomentum of the impurity-bath system, which consists ofthe relative angular momentum of the impurities and collec-tive angular momentum of the bath, assumes an integer value.However, this does not necessarily imply that each of them,separately, assumes an integer value. On the contrary, as weshow in Eq. (61), the relative angular momentum of the impu-rities is possibly nonintegral, and so is the collective angularmomentum of the bath, in such a way that their sum is integer.Namely, the expectation value of the total angular momentumcan be written as (cid:104) ˆ J z (cid:105) Ψ E = (cid:104) ˆ L z (cid:105) Ψ E + (cid:104) ˆ Λ z (cid:105) Ψ E , (62)where | Ψ E (cid:105) is the total eigenstate of the impurity-bath systemand ˆ L z the relative angular momentum of the impurities. Then,it follows from Eq. (16) that in the limit of M → ∞ the firstterm is given by (cid:104) ˆ L z (cid:105) Ψ E = m − (cid:104) ˆ Λ z (cid:105) coherent state , with m being aninteger. The second term in Eq. (62), on the other hand, reads (cid:104) ˆ Λ z (cid:105) Ψ E = (cid:104) ˆ Λ z (cid:105) coherent state so that the total angular momentumis integer at the end. This is the manifestation of anyons anal-ogous to the picture of Wilczek’s flux-tube-charged-particlecomposite. In this picture fractional values of the angular mo-mentum stems from the fact that the photon field manifestsitself as a classical field via the magnetic flux, even thoughthe total angular momentum of the electron-photon systemassumes a half-integer value, when the angular momentum ofthe magnetic flux is taken into account [106, 107]. Thereby,anyons can here be interpreted as impurities ‘orbiting around’a ‘magnetic flux’ created by the many-particle bath throughthe coherent state.Moreover, the manifestation of the statistics parameter interms of the angular momentum of the many-particle bath,i.e., Eq. (61), implies that the statistics parameter can be mea-sured in experiment by detecting the phonon angular momen-tum. Recent works show that this is feasible; see for instanceRefs. [108, 109]. The relation (61) allows us also to propose anovel method to measure the statistics parameter. Namely, ifwe take the derivative of the Hamiltonian (53) with respect tothe cyclotron frequency Ω , we obtain ∂ ˆ H (cid:48) FR /∂ Ω = B r / + ˆ Λ z .If we further use the Hellman-Feynman theorem in the limit of M → ∞ , where the total state, which is given by Eq. (16), isseparable, the statistics parameter is given by α = B (cid:104) r (cid:105) − ∂ E ∂ Ω . (63)Here we take the expectation value of r with respect to theground state of the impurities such that E is the anyonic3ground state energy. We further note that there arises alsothe term (cid:104) ∂ ˆ h R /∂ Ω (cid:105) = B (cid:104) R (cid:105) / H (cid:48) FR , the expectation value simply assumes an integer number: (cid:104) ∂ ˆ h R /∂ Ω (cid:105) = n R x + n R y + n R x ( y ) being the energy levelin the center-of-mass dimension x ( y ). Consequently, we ne-glect its contribution to the statistics parameter. In Eq. (63) thesecond term defines the magnetization of the system M , i.e., ∂ E (cid:48) FR /∂ Ω = − M M , which is routinely measured in torquemagnetometry setups to probe the 3D and 2D Fermi surfacesof di ff erent types of materials [110–114]. The first term, onthe other hand, can be measured with a standard time-of-flightmeasurement [115]. We note that the relation (63) is remi-niscent of the recently proposed method to observe anyonicstatistics in the FQHE [28], where the statistics parameter isdefined in terms of the mean square radius of the density dis-tribution of atoms. Here, anyons can be observed in a muchsimpler condensed matter system by measuring the magnetiza-tion of the impurity-bath system and mean square distance ofthe impurities.
2. 2D weakly interacting Bose gas
The proposed setup can be extended to di ff erent many-particle environments such as a two-dimensional weakly in-teracting Bose gas or a film of liquid helium. The Fr¨ohlich-Bogoliubov regime of these impurity problems is governedby the Hamiltonian (45); see Refs. [116–119] for the detailson the validity of the Fr¨ohlich-Bogoliubov theory. In suchenvironments, however, the dispersion of the correspondingexcitations ω ( k ) is, in general, gapless. Nevertheless, realisticcircumstances, such as finite size of the BEC, impose a naturallow-momentum cut-o ff k min for the dispersion. This allows usto investigate these impurity problems within the adiabatic theo-rem as well. Furthermore, if the condition ω ( k min ) − µ max Ω > S and ˆ U transformations. This condition is satisfied for small values ofthe cyclotron frequency. Moreover, the cyclotron frequencyshould be less than the transverse trapping frequency. Other-wise, the atom density in the gas drops down and the healinglength can become arbitrarily large, and the problem cannotbe described within the Fr¨ohlich-Bogoliubov theory. Underthese conditions Eq. (58) remains valid by replacing ω withthe dispersion ω ( k ): α ( r ) = − µ max (cid:88) µ = − µ max (1 + ( − µ ) µ (cid:90) ∞ k min dk k π (cid:16) V ( k ) J µ ( kr / √ (cid:17) ( ω ( k ) + µ Ω ) . (64)For instance, let us consider the impurities inside a two-dimensional Bose gas. The latter can be considered a weaklyinteracting gas if the condition n a B (cid:28) n is the density of the Bose gas and a B the boson-boson scatteringlength parameterizing the contact boson-boson interaction. The Bogoliubov dispersion is given by ω ( k ) = ck (cid:113) + k ξ / . (65)Here c = (cid:112) g BB n / m B and ξ = (2 m B g BB n ) − / are the speed ofsound and the healing length of a weakly interacting Bose gas,respectively. m B is the boson mass and g BB is the boson-bosoncoupling constant given by g BB = π/ ( m B ln(1 / n a B )). Thecoupling, on the other hand, is V ( k ) = √ n (2 π ) − g IB (cid:32) ξ k + ξ k (cid:33) / (66)with g IB being the impurity-boson coupling constant. Wefurther define two dimensionless numbers β = g IB n / ( π c )and β = m B / M , which characterize the Bogoliubov-Fr¨ohlichHamiltonian in 2D; see Ref. [116] for details. In Fig. 5 wepresent α ( r ) as a function of the dimensionless distance r √ n in a parameter regime where the Fr¨ohlich-Bogoliubov theory isapplicable. Apart from small distances, α ( r ) is approximatelyconstant, and hence, the emergent gauge field behaves as thestatistics gauge field. We finally note that the same formal-ism can also be extended beyond the Fr¨ohlich model, wheretwo-phonon scattering processes as well as additional phonon-phonon interactions should be included [117]. In this case, onecan also investigate impurities strongly interacting with thebath.Regarding the preparation of the system in order to observethe anyonic behavior of impurities in a 2D Bose gas, we firstnote that the impurity-bath interaction for a 2D Bose gas hasnot been achieved yet. Nevertheless, its realization is not outof reach as the sound propagation was already measured ex-perimentally in a 2D Bose fluid [120]. Rotation of the system,on the other hand, was already demonstrated for a cold atomicBEC [121]. Furthermore, the adiabaticity condition Ω (cid:28) ω allows us to measure the statistics parameter by a time-of-flightmeasurement after sudden release of trapping potential and arf spectroscopic measurement of impurity state, respectively,which have been already performed (see Ref. [115] for thetime-of-flight measurement and Refs. [42, 122] for the rf spec-troscopy of impurity states in 3D BEC systems). Finally, theanyonic behavior of impurities can also potentially be observedby direct interference measurements [123]. This is also fea-sible by imaging of the interference pattern of the impurityatoms after sudden release of the trapping potential. VI. DISCUSSIONS
In this paper, we have introduced a quantum impurity modelwhere the surrounding many-particle bath manifests itself asthe statistics gauge field with respect to the impurities, and thelowest energy spectra correspond to the anyonic spectra. Interms of the quasiparticle picture, anyons can here be identifiedas impurities which are first converted into composite bosons(fermions) and then dressed by a coherent state of phononsweighted according to the statistics parameter. Its magnitudein turn depends on the ratio between the phonon energy gapand the hopping amplitude in the adiabatic limit.4
FIG. 5. The relative statistics parameter of Eq. (64) as a function ofthe dimensionless distance r √ n for impurities immersed in a 2DBose gas. The long distance behavior of the emergent gauge fieldcorresponds to the statistics gauge field. The applied Bogoliubovparameters are β = β = / ξ = . B = µ max =
10, and k min = k . The latter is found by solving the equation ω ( k ) − µ max Ω =
0. For densities n >
100 the parameter √ n ξ >
1, which is accessiblein the current experiments, see for instance Ref. [120].
The introduced model reveals new numerical techniques forstudying the N -anyon problem. Specifically, the analyticalform of Eq. (39) or Eq. (41) provides new routes to obtainingthe N -anyon spectrum. The direct approach is to calculate thematrices ˜ Z and ˜ Z − separately by explicitly evaluating interac-tion integrals in the impurity basis using Eq. (42). One can alsocalculate ˜ Z algebraically by writing the harmonic-oscillatorHamiltonian in terms of the ladder operators. Afterwards theinverse matrix ˜ Z − can be evaluated from the pseudoinverseof Z . The general procedure of the evaluation of ˜ Z and ˜ Z − numerically for more than three impurities will be the subjectof future work. Moreover, some common techniques used inquantum impurity problems, such as the Diagrammatic MonteCarlo (DiagMC) and Density Matrix Renormalization Group(DMRG) technique, can be applied to the introduced impuritymodel for numerical studies of the N-anyon problem.As an experimental proposal, we considered heavy electronsinteracting with the excitations of a two-dimensional ioniccrystal subject to a magnetic field. If the impurity-bath systemis rotated at the cyclotron frequency, the impurities behaveas anyons. We showed that the statistics parameter manifestsitself through the expectation value of the angular momentumof the many-particle bath in the coherent state. This makes itpossible to measure the statistics parameter in experiment interms of the mean square distance of the impurities and themagnetization of the impurity-bath system. Furthermore, ithas been shown that the proposed setup is applicable to otherbosonic baths, such as a two-dimensional Bose gas. In thiscase the long distance behavior of the emergent gauge fieldresembles the statistics gauge field. A possible experimentalmeasurement of the statistics parameter might be more feasiblein such an environment due to recent advances in ultracoldatomic physics.To summarize, we have undertaken the first step towardsrealizing anyons by using quantum impurities. This new de-scription of anyons promises to shed new light on the field of fractional statistics and related branches of physics, suchas the FQHE. The formalism developed in this manuscript isbased on the consideration of a non-degenerate ground stateof the fast Hamiltonian, which corresponds to a U(1) gaugefield in the adiabatic limit, and hence Abelian anyons. If wefurther consider an impurity problem that exhibits some degen-eracy in the ground state, then the emergent gauge field in theadiabatic limit could very well correspond to a non-Abeliangauge field. This would allow us to extend the model to realizenon-Abelian anyons in terms of quantum impurities. In thecontext of impurity problems, this can potentially be achievedby considering internal degrees of freedom of phonons. Suchan approach could then allow us to use quantum impurities asa platform for topological quantum computation. ACKNOWLEDGMENTS
We are grateful to M. Correggi, A. Deuchert, and P.Schmelcher for valuable discussions. We also thank the anony-mous referees for helping to clarify a few important points inthe experimental realization. A.G. acknowledges support bythe European Unions Horizon 2020 research and innovationprogram under the Marie Skłodowska-Curie grant agreementNo 754411. D.L. acknowledges financial support from theG¨oran Gustafsson Foundation (grant no. 1804) and LMU Mu-nich. R.S., M.L., and N.R. gratefully acknowledge financialsupport by the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation pro-gramme (grant agreements No 694227, No 801770, and No758620, respectively).
APPENDIXA. Derivation of the emergent gauge field
In general, we consider a free N -boson system coupled toanother system. The corresponding total Hamiltonian can bewritten as ˆ H tot = − N (cid:88) q = ∇ q + ˆ H fast ( x ) , (67)with x = { x , . . . , x N } being the coordinates of bosons. Herethe first term is the kinetic energy of bosons. Since it is oftenmore convenient to study anyons in curvilinear coordinates,like polar coordinates as we discuss, we give the Laplacian incurvilinear coordinates in terms of the inverse metric tensor g i jq , with g q = det g i jq , ∇ q = √ g q (cid:88) i , j ∂∂ x iq √ g q g i jq ∂∂ x jq . (68)The second term in Eq. (67) describes the Hamiltonian ofthe system that couples to bosons, and we assume that ingeneral it is self-adjoint in a weighted space. The coordinates5of bosons, x , are regarded as parameters in the Hamiltonianˆ H fast ( x ), whose eigenvalue equation is given byˆ H fast ( x ) ˆ S | ψ n ( x ) (cid:105) = ε n ( x ) ˆ S | ψ n ( x ) (cid:105) , (69)where ˆ S = ˆ S ( x ) is a similarity transformation suchthat ˆ S − ˆ H fast ˆ S is Hermitian, and (cid:104) ψ n ( x ) | ψ m ( x ) (cid:105) = δ n , m .Note that the identity operator in the weighted Hilbertspace, where the Hamiltonian ˆ H fast is defined, is given byˆ S (cid:80) n | ψ n ( x ) (cid:105)(cid:104) ψ n ( x ) | ˆ S − = ˆ I .The total quantum state, which is defined via the eigenvalueequation ˆ H tot | Ψ E ( x ) (cid:105) = E | Ψ E ( x ) (cid:105) , can be expanded as | Ψ E ( x ) (cid:105) = (cid:88) n χ En ( x ) ˆ S | ψ n ( x ) (cid:105) , (70)where χ En ( x ) = (cid:104) ψ n ( x ) | ˆ S − | Ψ E ( x ) (cid:105) . Then, the eigenvalue equa-tion can be written as (cid:88) m H e ff nm χ Em ( x ) = E χ En ( x ) , (71)with the e ff ective Hamiltonian H e ff nm = − N (cid:88) q = (cid:88) l √ g q (cid:88) i , j (cid:34) δ nl ∂∂ x iq + (cid:104) ψ n | ˆ S − ∂∂ x iq ˆ S | ψ l (cid:105) (cid:35) × √ g q g i jq δ lm ∂∂ x jq + (cid:104) ψ l | ˆ S − ∂∂ x jq ˆ S | ψ m (cid:105) + ε n δ nm , (72)where (cid:104) ψ n | ˆ S − ∂/∂ x iq ˆ S | ψ m (cid:105) is the emergent gauge field. We notethat Eq. (71) with the Hamiltonian (72) is still exact. Now weassume that the spectrum of the Hamiltonian ˆ H fast ( x ) is discreteand non-degenerate at least in the n th level, and that the energysplittings between level n and the other levels, m (cid:44) n , are solarge that (cid:104) H e ff nm (cid:105)(cid:104) H e ff mm (cid:105) − (cid:104) H e ff nn (cid:105) (cid:28) . (73)Then, in this adiabatic limit the Schr¨odinger equation (71)simply reads − N (cid:88) q = √ g q (cid:34) ∂∂ x iq + (cid:104) ψ n | ˆ S − ∂∂ x iq ˆ S | ψ n (cid:105) (cid:35) √ g q g i jq × ∂∂ x jq + (cid:104) ψ n | ˆ S − ∂∂ x jq ˆ S | ψ n (cid:105) + W ( x ) χ En ( x ) = E χ En ( x ) , (74)where W ( x ) = ε n ( x ) (75) − N (cid:88) q = (cid:88) l (cid:44) n (cid:104) ψ n | ˆ S − ∂∂ x iq ˆ S | ψ l (cid:105) g i jq (cid:104) ψ l | ˆ S − ∂∂ x jq ˆ S | ψ n (cid:105) is the emergent scalar potential. B. The three-impurity matrix element in hypersphericalcoordinates
For the three-impurity problem the matrix elements of ˜ Z − in hyperspherical coordinates are given by (cid:104) Φ ± nm νµ | ρ − e i ψ (cid:2) A ( θ ) cos(3 φ ) + iB ( θ ) sin(3 φ ) (cid:3) − | Φ ± n (cid:48) m (cid:48) ν (cid:48) µ (cid:48) (cid:105) , (76)where A ( θ ) = cos( θ )(2 − cos(2 θ )), B ( θ ) = sin( θ )(2 + cos(2 θ ))with − π/ ≤ θ ≤ π/
4. The ρ - and ψ -integrals can be evaluatedanalytically. The latter integral, similar to the two-impurityproblem, excludes the diverging ρ -integrals. The φ -integral, onthe other hand, can be written as I w = (cid:90) π/ − π/ d φ e iw φ (cid:2) A ( θ ) cos(3 φ ) + iB ( θ ) sin(3 φ ) (cid:3) , (77)with w = × integer. Here, we used the hyperspherical har-monics [95, 96], which are given by Y ± m νµ ( θ, φ, ψ ) = (cid:32)(cid:32) √ − (cid:33) δ ν + (cid:33) (78) × π (cid:16) (cid:104) m , ν, µ (cid:105) ± ( − µ + ˜ m (cid:104) m , − ν, µ (cid:105) (cid:17) , where (cid:104) m , ν, µ (cid:105) = (cid:115) ˜ m !( ˜ m + α + β )!( m + α + β ( ˜ m + α )!( ˜ m + β )! Θ αβ ˜ m (sin(2 θ )) e i νφ e i µψ , (79) Θ αβ ˜ m ( x ) = (1 − x ) α/ (1 + x ) β/ P αβ ˜ m ( x ) , (80)with the Jacobi polynomials P αβ ˜ m ( x ) and the following numbers˜ m = m − max( | µ | , | ν | )2 , α = | ν + µ | , and β = | ν − µ | . (81)The integral (77) can be transformed into a rational functionof a complex variable by the substitution of z = exp(2 i φ ): I w = − i ( A + B ) (cid:73) C dz z + w / ( z + κ ) , (82)where the contour C is the unit circle, and κ = ( A − B ) / ( A + B ).This contour integral can be further written as I w = i ( A + B ) ∂∂κ (cid:73) C dz z + w / ( z + κ ) , (83)where the latter contour integral can be straightforwardly de-termined by using the residue theorem. For w ≥ I w = − π (cid:16) e i π w / + π w / (cid:17) A + B ) ∂∂κ (cid:16) H (1 − κ ) κ w / (cid:17) , (84)which yields I w = π e i π w / (cid:32) δ ( θ ) − wH ( θ )( A + B ) (cid:18) A − BA + B (cid:19) w − (cid:33) , (85)with H ( θ ) being the step function. The w < w → − w and θ → − θ in Eq. (85). Finally, the remaining θ -integral can becalculated numerically.6 C. Diagonalization of the impurity Hamiltonian
In this appendix we elaborate on why we expect to seethat the spectrum of the full Hamiltonian operator convergesto that of the lowest sector of the Fock space. Namely, wefind that any eigenstate with low energy will have vanishingcomponents outside the lowest sector in the adiabatic limit ω → ∞ . However, a few technical assumptions enter becausethe various components of the full operator depend on theparticle number.Starting with an impurity-bath coupled Hamiltonian of theform (18), H ω : = H + ω ˆ a † ˆ a + λω ( F ˆ a † + F − ˆ a ) + λ ω where ω ≥ λ ∈ R are parameters, and H : = N (cid:88) j = (cid:104) −∇ x j + V ( x ) (cid:105) acts in some N -body Hilbert space H , let us define for arbitrarycoupling γ ∈ R a deformed N -body operator H γ F : = N (cid:88) j = (cid:104) − ( ∇ x j + γ F j ) + V ( x ) (cid:105) . (86)For generality we allow for a potential V (which may dependon all the variables and thus also include interactions), and also for F to be any function of x , . . . , x N such that F j : = ∇ x j log F = F − ∇ x j F in (86) is well defined, say smooth, at least for non-coincident x j (we may then take an appropriate dense domain in H ).Using ˆ S = F ˆ n , ˆ n = ˆ a † ˆ a , and ˆ U = e − λ (ˆ a † − ˆ a ) in the expansion e X Ye − X = Y + [ X , Y ] +
12! [ X , [ X , Y ]] +
13! [ X , [ X , [ X , Y ]]] + . . . we obtain the transformationsˆ S − ∇ x j ˆ S = ∇ x j + F j ˆ n , ˆ S − ˆ a ( † ) ˆ S = F ( − ˆ a ( † ) , ˆ U − ˆ a ( † ) ˆ U = ˆ a ( † ) − λ, and thus H ω is similar to H (cid:48) ω : = ˆ U − ˆ S − H ω ˆ S ˆ U = H (cid:48) + ω ˆ n , H (cid:48) = N (cid:88) j = (cid:104) − ∇ x j + V ( x ) − (cid:0) ∇ x j · F j + F j · ∇ x j (cid:1) ˆ U − ˆ n ˆ U − F j ˆ U − ˆ n ˆ U (cid:105) . We compute for arbitrary n = , , , . . . ˆ U − ˆ n ˆ U | n (cid:105) = (cid:0) ˆ n − λ (ˆ a † + ˆ a ) + λ (cid:1) | n (cid:105) = ( n + λ ) | n (cid:105) − λ √ n + | n + (cid:105) − λ √ n | n − (cid:105) , ˆ U − ˆ n ˆ U | n (cid:105) = (cid:0) ˆ n − λ ˆ n (ˆ a † + ˆ a ) − λ (ˆ a † + ˆ a )ˆ n + λ ˆ n + λ ((ˆ a † ) + ˆ a ) − λ (ˆ a † + ˆ a ) + λ + λ (cid:1) | n (cid:105) = ( n + λ n + λ + λ ) | n (cid:105) − λ (2 n + + λ ) √ n + | n + (cid:105) + λ √ n + √ n + | n + (cid:105)− λ (2 n − + λ ) √ n | n − (cid:105) + µ √ n √ n − | n − (cid:105) , and thus obtain the non-trivial operator matrix elements (cid:104) n | H (cid:48) ω | n (cid:105) = N (cid:88) j = (cid:104) − ∇ x j + V ( x ) + ω n − (cid:0) ∇ x j · F j + F j · ∇ x j (cid:1) ( n + λ ) − F j (cid:0) ( n + λ ) + λ (1 + n ) (cid:1)(cid:105) = H ( n + λ ) F + ω n − λ (1 + n ) F , (cid:104) n + | H (cid:48) ω | n (cid:105) = N (cid:88) j = (cid:104) − (cid:0) ∇ x j · F j + F j · ∇ x j (cid:1) ( − λ √ n + − F j ( − λ (2 n + + λ ) √ n + (cid:105) = λ √ n + (cid:0) H − H F + n + λ ) F (cid:1) , (cid:104) n − | H (cid:48) ω | n (cid:105) = N (cid:88) j = (cid:104) − (cid:0) ∇ x j · F j + F j · ∇ x j (cid:1) ( − λ √ n ) − F j ( − λ (2 n − + λ ) √ n ) (cid:105) = λ √ n (cid:0) H − H F + n − + λ ) F (cid:1) , (cid:104) n + | H (cid:48) ω | n (cid:105) = N (cid:88) j = (cid:104) − F j λ √ n + √ n + (cid:105) = − λ √ n + √ n + F , (cid:104) n − | H (cid:48) ω | n (cid:105) = N (cid:88) j = (cid:104) − F j λ √ n √ n − (cid:105) = − λ √ n √ n − F . Hence, we have a symmetric pentadiagonal matrix of operators (cid:104) (cid:104) k | H (cid:48) ω | n (cid:105) (cid:105) = H λ F − λ F λ ( H − H F + λ F ) − λ √ F . . .λ ( H − H F + λ F ) H (1 + λ ) F + ω − λ F λ √ H − H F + + λ ) F ) − λ √ F . . . − λ √ F λ √ H − H F + + λ ) F ) H (2 + λ ) F + ω − λ F . . . − λ √ F λ √ H − H F + + λ ) F ) . . . − λ √ F . . . ... ... ... . . . Note that for the choice (33), F j = i ˜ A j , so F = F j = i A j , − F = A ≥
0, and H F = H i A = F ∗ H F is unitary equivalent to H . Furthermore, one has the unitary equivalenceof H i α A to H i ( α + n ) A for any integer n as well as the diamagnetic inequality (cid:104) H i α A (cid:105) Φ ≥ (cid:104) H (cid:105) | Φ | for any α ∈ R and Φ ∈ H [71],while H i γ ˜ A = | ˜ F | − γ/ H i γ A | ˜ F | γ/ , also implying isospectrality for H i ( α + n ) ˜ A . 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