Model wavefunctions for an interface between lattice Laughlin and Moore-Read states
MModel wavefunctions for an interface between lattice Laughlin and Moore-Read states
B(cid:32)la˙zej Jaworowski ∗ and Anne E. B. Nielsen † Max-Planck-Institut f¨ur Physik komplexer Systeme, D-01187 Dresden, Germany
We use conformal field theory to construct model wavefunctions for an interface between latticeversions of a bosonic ν = 1 / ν = 5 / I. INTRODUCTION
One of the characteristic features of the topologicalorders is the existence of nontrivial physical phenomenaat the edges or interfaces with an another topologicalphase. While the former can be used to characterize asingle topological phase [1, 2], the latter can tell us howtwo different topological phases are related to each other(e.g. if one of them can be transformed into the other byanyon condensation [3–10]). In experiments, the inter-faces can be potentially useful for example for isolatinga certain edge mode to prove its existence [11, 12]. Fromthe perspective of applications, even when both sides areAbelian, the interfaces can have non-Abelian zero-energymodes similar to the Majorana zero modes [13–15], whichcan encode quantum logic gates [14–17].The interfaces involving a non-Abelian state at least onone side are more complicated than their Abelian coun-terparts and our knowledge of them is less complete. Anattempt at a systematic understanding of such interfacesis the topological symmetry breaking formalism (anyoncondensation) [3–10]. This approach can tell us whichanyons can cross the interface, as well as whether theinterface is gapped or gapless and the nature of the in-terface modes. However, there is a number of thingsone cannot determine in this way, such as the nonuniver-sal, microscopic details of the interface and the anyons.Moreover, while being very general, it is also abstract.Thus, there is a need of concrete examples embodyingthese mathematical concepts and making them more ac-cessible. There are some analytically solvable models de-scribing interfaces, although they are so far limited onlyto the cases with a nonchiral topological order on bothsides [4, 18, 19].However, the study of interfaces between chiral topo-logical orders is also important. In particular, the frac-tional quantum Hall states, which have been thoroughly ∗ [email protected] † On leave from Department of Physics and Astronomy, AarhusUniversity, DK-8000 Aarhus C, Denmark studied experimentally [20], are chiral. The interfacesbetween Abelian quantum Hall states were already cre-ated, e.g. in Camino et al.’s interferometric experimentswhich aimed at demonstrating the fractional statistics[21]. There are also some proposals for creating interfaceswith a non-Abelian state on at least one side [11, 22, 23].There is also a considerable effort directed at creatinglattice versions of FQH states. Numerical works predictthe appearance of discretized FQH states in a range ofsimple lattice models [24–27]. Experiments have shownthe creation of such topological orders in a moir´e super-lattice in graphene subjected to a strong magnetic field[28]. A number of alternative experimental designs wereproposed, involving e.g. optical lattices [29–37]. In thisway, one can also create discretized versions of bosonicFQH states, so far not seen in experiments. Therefore,the study of interfaces between lattice FQH states – bothbosonic and fermionic – has experimental relevance.Although, to the best of our knowledge, exactly solv-able models of FQH interfaces (neither Abelian nor non-Abelian, neither lattice nor continuum) have not beenfound so far, such systems were studied both numer-ically and analytically. For Abelian states, there aresystematic field theory approaches based on K matri-ces (related to topological symmetry breaking) [38, 39]or Chern-Simons theory [40, 41]. A more detailed, mi-croscopic treatment was achieved by constructing modelwavefunctions from conformal field theory, either in thecontinuum, by forming matrix product states from CFTvertex operators and combining matrices belonging todifferent FQH states [22, 23], or in the lattice by combin-ing the vertex operators directly [42]. Numerical calcu-lations were also performed using exact diagonalization,although this method has limitations as it can be usedfor small system sizes [22, 23, 43].In the case of non-Abelian states, apart from thetopological symmetry breaking studies [5–10], there havebeen some works which tried to determine the prop-erties of the interface modes at some gapless inter-faces (Halperin/Pfaffian, Pfaffian/anti-Pfaffian) and/orthe behaviour of anyons in the presence of these inter-faces by using the “giant hole approach” [44] or by cou- a r X i v : . [ c ond - m a t . s t r- e l ] D ec pling the effective edge theories [11, 45–49]. Moreover,the latter approach was used for a Pfaffian/Pfaffian inter-face to determine the gapping conditions and the scalingof entanglement entropy for the gapped case [50]. Mi-croscopic studies are very rare. For a Halperin/Pfaffianinterface, a model wavefunction was constructed frominfinite-dimensional MPS based on conformal field the-ory [12]. For a Pfaffian/anti-Pfaffian interface, DMRGstudies was performed [51, 52]. In both cases, continuumsystems were investigated.Here, we perform a microscopic study of an anotherexample: an interface between a fermionic Pfaffian stateand an Abelian bosonic Laughlin state. In contrast tothe former works on non-Abelian interfaces, we focuson the lattice case. We construct model wavefunctionsfrom conformal field theory and study their propertiesusing Monte Carlo methods. These wavefunctions de-scribe both the ground state and the localized anyonicexcitations (both quasiholes and quasielectrons). Fromthe ground state wavefunction (Section II), we obtainquantities such as particle density, correlation functionor entanglement entropy. We note that the latter wasvery rarely studied for non-Abelian interfaces. Using theanyon wavefunctions (Section III) we can investigate thedensity profile, charge and statistics of such excitations.Such a microscopic study of localized non-Abelian anyonsof a chiral topological order in the presence of an inter-face was, up to our knowledge, not performed before.Moreover, while the previous microscopic works on FQHinterfaces concentrated on a single interface, here we alsodiscuss the case of multiple Laughlin islands within aMoore-Read system, arguing that for local interactionsacross the interfaces, the parity of the particle numberat each island cannot be changed locally, leading to theemergence of topological degeneracy resembling that ofMajorana zero modes. II. THE GROUND STATEA. The construction of the wavefunction
Our approach follows the CFT construction for lat-tice quantum Hall wavefunctions from Refs. [42, 53, 54],which are based on the idea devised by Moore and Readfor continuum quantum Hall wavefunctions [55]. We con-sider a system of N sites on a plane, each one at a com-plex position z j = x j + iy j . The sites can be occupiedby at most one particle, with n j ∈ { , } denoting theoccupation number of site j . That is, each site hosts afermionic or hardcore bosonic degree of freedom (a sys-tem can contain sites of both types). We set the chargeof each particle to unity. In contrast to the continuumcase, we do not assume a constant magnetic field, butrather set it to zero everywhere except at the positionsof the lattice sites. That is, we attach an infinitely thinsolenoid, containing η i ∈ R + flux quanta, to each site.The total number of flux quanta is N φ = (cid:80) i η i and can be different from N , which means that we can define twokinds of filling factors. In the simplest case of a single MRor Laughlin state, the particle number M is conserved, sowe can define a “lattice filling” ν lat = M/N and a “topo-logical filling” ν = M/N φ = 1 /q , q ∈ N + . The formerdescribes the degree of discretization (the lower ν lat , thedenser are the lattice points, i.e. the closer we are to thecontinuum), while the latter tells us which state do wediscretize (i.e. it is equal to the Landau level filling factorof the corresponding continuum state).Any wavefunction in our system can be written in theoccupation number basis, | Ψ (cid:105) = 1 C (cid:88) n Ψ( n ) | n (cid:105) (1)where n is a vector of occupation numbers of all sites, | n (cid:105) is a corresponding basis state (to define it unambigu-ously we fix the order of fermionic creation operators inthe definition of | n (cid:105) to be the same as the order of siteindices), Ψ( n ) are the unnormalized wavefunction coeffi-cients and C is the normalization constant (which we fixto be real without loss of generality).To construct a model CFT wavefunction for a singlelattice quantum Hall state, a vertex operator V i ( z i , n i )of a certain conformal field theory (depending on whichkind of state we want to create) is assigned to each site.The wavefunction is then given by the correlator of thevertex operators for all sites,Ψ( n ) = (cid:104) | (cid:89) i V i ( z i , n i ) | (cid:105) . (2)This method can be generalized to interfaces [42]. Ingeneral, for two given quantum Hall states, there can bemany different types of interfaces, depending on the inter-actions between the particles on the two sides. A wave-function for a particular type can be created by forminga correlator of the form (2), but made from the vertexoperators of two different CFTs. Such a quantity is well-defined when the two CFTs can be embedded in a thirdone, which puts a restriction on the states for which thismethod can be applied. We note that in Refs. [12, 22, 23],model wavefunctions for interfaces (Laughin/Halperin,Pfaffian/Halperin) in continuous systems were createdby patching together infinite-dimensional matrix productstates, derived from conformal field theory, representingthe two different fractional quantum Hall states. Ourapproach is similar, but it uses the CFT vertex opera-tors directly, without the need of matrix product staterepresentation.So far, we employed this method only for AbelianLaughlin states [42]. Here, we use it to study an inter-face between a bosonic Laughlin state and a non-Abelianfermionic Moore-Read state, both at topological filling ν = 1 /
2. They are described by U (1) and U (1) × IsingCFTs, respectively. The embedding condition is satis-fied, as the U (1) part is the same for both states.We assume that the system consists of two parts. Theleft one with, which consists of the first N L sites with η i = η L each, is described by the MR state. In the rightone, consisting of next N R sites up to N = N L + N R ,the particles are in the Laughlin state. The number offlux quanta per site is set to a constant within each part, η i = η R for i (cid:54) = N L and η i = η R for i > N L , but it candiffer between the parts, i.e. we can have η L (cid:54) = η R . Wenote that in general the two parts of the system can beof any shape and can be split into disconnected regions,but we will use the L and R labels for simplicity, as thisis the geometry that we will study numerically in thiswork.More specifically, the planar systems considered in thiswork consist of sites arranged in a square lattice of size( N xL + N xR ) × N y . The interface is parallel to the y direction, as shown in Fig. 1 (a). Without the loss ofgenerality, we set the lattice constant to unity and theposition of the interface to x = 0.The vertex operators are given by V i ( z i , n i ) = (cid:40) V Ising ,i ( z i , n i ) V Laughlin ,i ( z i , n i ) for i ≤ N L V Laughlin ,i ( z i , n i ) for i > N L . (3)Here, the V Laughlin ,i ( z i , n i ) and V Ising ,i ( z i , n i ) are theLaughlin-like and Ising-like parts of the vertex operator,respectively. V Laughlin ,i == (cid:40) e iπ ( j − η L n i : e qni − ηL √ q φ ( z i ) : for i ≤ N L e iπ ( j − η R n i : e qni − ηR √ q φ ( z i ) : for i > N L , (4)with φ ( z i ) being a free chiral bosonic field and q = 2 inour case, ensuring the topological filling ν = 1 / q = 1 (an interface between a bosonic MRstate and a fermionic integer quantum Hall state, bothat ν = 1), but then double occupancy of the L sites hasto be allowed [54]. In this work we restrict the study onlyto the q = 2 case.The Laughlin-like part is the same for all sites, exceptfrom the different values of η i on the two sides. The Isingpart is V Ising ,i ( z i , n i ) = ψ ( z i ) n i , (5)where ψ ( z i ) is a chiral Majorana field. In contrast to theLaughlin term, the Ising one is assigned only to the L sites.By evaluating the correlator (2), we obtain the unnor-malized wavefunction coefficientsΨ( n ) = δ n Pf (cid:32) z (cid:48) i − z (cid:48) j (cid:33) (cid:89) i 6, differing by the way the chargeis distributed among L and R parts within each config-uration | n (cid:105) . Each such a configuration has well-defined M L and thus it corresponds to the charge M L − N L ,as a background charge − η/q = − is associated withevery L site, and each particle has unit charge. Thecharge is not well-defined for the entire | Ψ (cid:105) state, as itis a linear combination of different configurations withdifferent values of M L , because pairs of particles can betransferred across the interface. However, a charge mod-ulo 2, ∆ Q = (cid:0)(cid:0) M L − N L (cid:1) mod 2 (cid:1) is well-defined, andequal to ∆ Q = 0 , . , , . N L mod 8) = 0 , , , η , it ispossible to derive a parent Hamiltonian for single Moore-Read [54, 56] or Laughlin [57] lattice quantum Hall states.However, it is not straightforward to extend these calcu-lations to the case when the system is described by twodifferent CFTs. Another way to connect our wavefunc-tion to a Hamiltonian is to find a short-range Hamilto-nian, whose ground state is approximated by our wave-function, following the approach from Ref. [54]: diago-nalizing the Hamiltonian numerically and optimizing itscoefficients to maximize the overlap between the groundstate and our wavefunction. However, this would requireextensive exact-diagonalization or DMRG calculations,and we leave it for future works.Often the wavefunction itself, which may, but does nothave to, be related to a Hamiltonian, can provide in-sights into the inner structure of given topological or-der. For example, the Laughlin wavefunction revealedthe physical mechanism of FQHE and the nature of thefractionalized excitations [58]. The Kalmeyer-Laughlinwavefunction provided a vital example of a chiral spinliquid [59, 60]. The concepts of Haldane hierarchy [61] orcomposite fermions [62] were also embodied in wavefunc-tions. Speaking of the interfaces, Regnault et al. used amodel wavefuncion to study the nature of the interfacemodes [12, 22, 23], while we employed our model state todetermine the properties of localized anyons in presenceof the interface [42]. Therefore, we believe that the studyof the wavefunction itself is important and thus, in thiswork we focus solely on | Ψ (cid:105) . 10 0 10 x n ( x ) (a) 10 5 0 5 x (b)10 0 10 x n ( x ) (c) 10 5 0 5 x (d)(e) (f) 5101520 N y n i n i FIG. 2. The particle density plots for the ground state ofsystems with an interface. (a)-(d) The superimposed resultsof density as a function of x for various cylindrical systems.The results are divided into four groups, (a), (b), (c), (d),corresponding to ( N L mod 8) = 0 , , , 6. (e), (f) Exampleplots of particle density for planar systems of size (4 + 4) × × 6, respectively. B. Numerical results - particle density Once we have the wavefunction (6), we can study itsproperties numerically using Monte Carlo methods. Inparticular, it is straightforward to obtain the average par-ticle density (cid:104) n i (cid:105) . On a cylinder, the density is constantin the y direction, so we define the density as a functionof x (cid:104) n ( x ) (cid:105) = 1 N y (cid:88) i (cid:104) n i (cid:105) δ ( x − x i ) (9)We investigate this quantity for a number of systemswith different sizes, which are shown in 2 (a)-(d). Whenthe cylinder is thin, the states display large oscillationsin density within the L part. In particular, at N y = 1, wehave either (cid:104) n i (cid:105) = 0 or (cid:104) n i (cid:105) = 1, with no fractional values,reminiscent of the thin torus limut of the continuum FQHstates. As the cylinder gets wider, the density in the L bulk becomes close to 3 / 4, as expected for a η = 3 / ν = 1 / N y increases. In the R part, independently of N y , the density is close to 1 / η = 1, ν = 1 / N L mod 8) groups display four qual-itatively different patterns of particle density. This ismost striking when comparing ( N L mod 8) = 4, whichhas an additional “step” (i.e. a local maximum of density N y Q L / N y N xL =4, N xR =4 N xL =6, N xR =5 N xL =6, N xR =6 N xL =8, N xR =8 N xL =10, N xR =10 N xL =12, N xR =12 N xL =2, N xR =3 N xL =10, N xR =9 N xL =14, N xR =5 N xL =16, N xR =4 N L mod 8=0 N L mod 8=2 N L mod 8=4 N L mod 8=6 10 12 14 16 18 20 N y Q L / N y FIG. 3. The total charge Q L in part L per unit of interfacelength, as a function of interface length N y . The differentcolors correspond to different system sizes in the x direction.The marker shapes denote the values of ( N L mod 8). Theinset shows the magnification of the right part of the plot. near the interface in part R ) to ( N L mod 8) = 0, wheresuch a feature is absent. For wider cylinders, the densityprofiles in all four groups become similar.In the case of planar systems, the density patterns aremore complicated, as the translational invariance is lost,and the density inhomogenities exist near the interfaceand all three edges of the L part. Two examples areshown in Fig. 2 (e),(f).The background charge − η i /q changes abruptly from − / − / x as Q ( x ) = (cid:88) i ( (cid:104) n i (cid:105) − η i /q ) δ ( x − x i ) , (10)and a total charge accumulated in part I as Q I = (cid:88) i ∈ I ( (cid:104) n i (cid:105) − η I /q ) . (11)Apart from thin cylinders, the excess charge is con-centrated mostly near the interface x = ± . 5, with Q ( − . ≈ − Q (0 . Q ( − . ≈ Q L . Therefore, let us study the lat-ter quantity as a function of system size. To be precise,instead of Q L itself, we investigate the charge in part L per unit of interface length, i.e. Q L /N y . If there is afixed density pattern in the thermodynamic limit, thenthis quantity should converge to a fixed value.The results are shown in Fig. 3. The different col-ors correspond to different system sizes in the x direc- y j y i | C ij | L edgeL bulkL, near interfaceR, near interfaceR bulkR edge FIG. 4. The correlation for x i = x j function as a function of y j − y i at different locations of x i = x j : in the bulk of eachpart ( x = − , x = 4 . x = − . x = 10 . x = − . x = 0 . × tion, while the marker shapes refer to the four classes( N L mod 8). It can be seen that at N y = 1 (the right-most points from every class) the charge modulo 2 equals∆ Q . As we increase N y , Q L /N y in all four classes seemsto display convergence towards a fixed, negative value of Q L /N y , lying between − . 03 and − . Q ( − . 5) behaves very similarly to Fig. 3 forwide cylinders. On the other hand, qualitative differencesarise in the thin cylinder limit. At N y = 1 limit, there areonly two possible values: Q ( − . 5) = 1 / x = − . 5) or Q ( − . 5) = − / x = − . ± / N L . Moreover, in Ref. [51] theauthors observed that the dipole moment with respect tothe interface is constant also outside the thin-torus limit.This dipole moment was of topological origin, as it arosefrom the difference of the quantized Hall viscosities onboth sides of the interface. In contrast, in our work thedipole moment is not constant. This can be seen easily bynoting that for large enough N y , Q ( − . /N y is roughlyconstant, therefore Q ( − . 5) should grow more or less lin-early with N y . Since most of the charge is concentratedat x = ± . 5, this means that the dipole moment mustgrow with N y . To verify this, we evaluated the dipolemoment explicitly, confirming that it is indeed not con-served. N y S ( ) Middle of the L regionNext to the interface (L) InterfaceMiddle of the R region N y N y FIG. 5. The second R´enyi entropy as a function of cylindercircumference for three series of systems: (a) (8 + 8) × N y , (b)(10 + 4) × N y and (c) (12 + 4) × N y . C. Numerical results – correlation function We expect that our interface is gapless. This is becausethe edges of the Laughlin state are described by a chiralLuttinger liquid [1], while the MR state has also a singleMajorana fermion edge mode [63, 64]. In the effective in-terface theories considered so far for various non-Abelianinterfaces [11, 45–50], the Majorana mode can be gappedonly when paired with a second Majorana mode. Sincethere is just a single Majorana mode in the system, weexpect that it cannot be gapped.It is expected that gapless systems generated by short-range interactions have exponentially decaying correla-tion functions. In our case, we do not have the parentHamiltonian, so we cannot ensure that our wavefunc-tion indeed can be generated by a short-range interac-tion. But assuming it is the case, the correlation functionwould give us some indication of the gaplessness of theinterface.The correlation function is given by C ij = (cid:104) n i n j (cid:105) − (cid:104) n i (cid:105)(cid:104) n j (cid:105) , (12)and can be easily computed using Monte Carlo. For theease of presentation, we choose sites with x i = x j andinvestigate C ij as a function of y j − y i . The results forthe different values of x i = x j are shown in Fig. 4.In the bulk, the correlation function seems to decayroughly exponentially. However, near the edges its decayseems to be slower. Near the interface, the values of thecorrelation function halfway across the cylinder seem tobe located roughly between the results for the bulk andthe edge, still showing the lack of exponential decay. Thissuggests that the interface is indeed gapless (providedthat it is generated by a short-range interaction). D. Entanglement entropy The topological properties of the interface can mani-fest themselves in the entanglement entropy when the cutcoincides with the interface. While this issue was studiedusing field theory for Laughlin/Laughlin [38, 39] or Pfaf-fian/Pfaffian interfaces [50], for the Pfaffian/Laughlininterface, up to our knowledge, there were no predic-tions how the entropy should scale with the interfacelength. Thus, we are going to study the entropy numeri-cally, using the Monte Carlo method outlined in [65, 66](see also our previous work where we used this method[42, 53, 54]). Within this approach, the second R´enyientropy can be obtained by sampling two independentcopies of the system.In Fig. 5, we show the entanglement entropy scaling forthree series of systems, of size: (8 + 8) × N y , (10 + 4) × N y and (12 + 4) × N y . The cut is parallel to the interface,as shown in Fig. 1. We are interested especially in thefour positions of the cut: in the bulks of the two sides,precisely at the interface and right next to the interfaceon the left (i.e. x = − x = − N xL + (cid:98) N xL / (cid:99) + 1 for the L side and x = (cid:98) N xR / (cid:99) for the R side, with (cid:98)(cid:99) denoting the floorfunction. If the interface wavefunction indeed describesthe expected topological orders, then, by applying a lin-ear fit, S (2) ( N y ) = AN y − γ (13)we should recover the expected value of γ : γ L = ln(8) / γ R = ln(2) / 2, corresponding to the topological en-tanglement entropies of the MR and Laughlin state, re-spectively. These values are indicated by blue and redticks, respectively, on the y axes of the plots. The redand blue lines denote the fits. Because for thin cylindersthe linear scaling is distorted by finite-size effects, we dis-card these systems from the calculation. That is, in thefit we include only the data points denoted by filled sym-bols. The fits seem to cross the N y = 0 line relativelyclose to the predicted values. For γ R , the agreement isgood: we obtain 0 . ± . . ± . . ± . × N y , (10 + 4) × N y and (12 + 4) × N y , re-spectively, compared to ln(2) / ≈ . R part has a ν = 1 / γ L , we obtain − . ± . − . ± . − . ± . 15, respectively, compared to ln(8) / ≈ . 03. That is,the agreement is worse, and the error bars are much big-ger. In addition, the result for part L seems to dependstrongly on the position of the cut and on which datapoints we take into account on the fit. Also, while thefits in Fig. 5 were performed without the inclusion of MCerror bars in the weights, including them makes the re-sult even more dependent on the number of included datapoints. The detailed analysis is contained in Appendix A.Nevertheless, the fitted values oscillate around the pre-dicted value and are clearly nonzero. Thus, we concludethat the L part is also topologically ordered, and theresults are consistent with the Moore-Read topologicalorder, although not indicating it clearly.What happens with the entropy when the cut coincideswith the interface (black markers in Fig. 5)? For almostall the investigated systems, the entropy at the interfaceis lower than in the bulks of both sides (excluding somethin cylinders). However, as N y increases, the interfaceentropy increases faster than the R bulk entropy, thuswe can expect that the former will finally dominate overthe latter. For large enough N y , the scaling seems to belinear. Because we do not have compelling theoreticalarguments that in this case such a scaling is expected inthe thermodynamic limit, we do not rule out the pos-sibility that the percieved linear dependence is in factnonlinear, and the nonlinearity would show up for larger N y . Nevertheless, assuming that it is linear, we performthe fit. The obtained values are close to ln(8) / 2, i.e. thetopological entanglement entropy of the left part. If thisis indeed the case, this is similar to the case of Laughlinstates at fillings 1 /q L , 1 /q R such that q R = a q L , a ∈ N + [38, 39, 42]. However, the fitted parameters are subjectedto the same distortions and uncertainties as γ R , thus wecannot conclude that it is indeed the case.How far does the influence of the interface extendsinto the L and R parts? Next to the interface on theright ( x = 1), the entanglement entropy values for largeenough N y are similar to the ones in the bulk R part.However, on the left, the influence of the interface is ap-parent in the first column of sites next to it ( x = − N y systems) are lower than in the L bulk ofthe same system (see the violet markers in Fig. 5). Thefit for large N y also yields values roughly close to thetheoretical value of γ L , although again the results areuncertain due to the dependence of the fitted value onthe data points included. Thus, we do not rule out thepossibility that near the interface there might be somevariation of γ , e.g. a similar increase as in the Laughlin-Laughlin interfaces [42]. III. THE SYSTEMS WITH ANYONS Having determined the ground state wavefunctions, wenow wonder, what are the properties of the localized any-onic excitations above the ground state. A. The construction of the wavefunction The wavefunctions including anyons can be obtainedby inserting further vertex operators into the correlator(2). These operators depend on parameters w i , the com- plex coordinates of the anyons. The state is given by | Ψ (cid:105) α = 1 C (cid:88) n Ψ α ( n , w ) | n (cid:105) . (14)There are two differences between (14) and (1). First,now the wavefunction coefficients, as well as the normal-ization constant, depend on the external parameters, theanyon coordinates w . Secondly, there can be more thanone degenerate state, hence we introduced the index α .Moreover, while for the ground state the fermion parityconservation was enforced by the Pfaffian factor, in thepresence of anyons the correlators are nonzero both foreven and odd M L . To restore the fermion parity con-servation, we assume that the interaction generating ourwavefunction allows to exchange particles through theinterface only in pairs. Then, the Hilbert space dividesinto two disconnected parts, with even and odd M L . Wefocus on the case of even M L .While the particle coordinates are restricted to the lat-tice sites, the quasihole coordinates can be located any-where on the plane/cylinder. In such a way, we will beable to move them smoothly, which will be importantwhen evaluating their statistics.We study two classes of anyons of the Moore-Readstate. The basic non-Abelian excitations are constructedusing the following vertex operator [56, 67] V NA ,i,m ( w i ) = σ ( w i ): exp (cid:18) p i √ q φ ( z i ) (cid:19) : , (15)where σ ( w i ) is the holomorphic spin operator of the chiralIsing CFT, and p i = 1 / p i = − / 2, correspond to aquasihole and a quasielectron, respectively. We note thatthe latter are difficult to construct in the continuum [68],whereas for the lattice their construction is simple – itrequires only flipping the sign of the p i .The other group of excitations consists of Laughlin-likeAbelian anyons, described by the vertex operator [69] V A ,i ( w i ) = : exp (cid:18) p i √ q φ ( z i ) (cid:19) : , (16)where p i is now integer. This vertex operator describesalso the excitations of the Laughlin state. Thus, theseanyons are valid topological excitations of the entire sys-tem. In contrast, the ones generated by the vertex oper-ator (15) are valid topological excitations only within the L part. Nevertheless, technically we can also attempt toput such an anyon in the R part and see what happens.We will refer to these two groups as “Abelian” and“non-Abelian” for brevity, although the reader shouldbear in mind that the anyonic content of the Moore-Readstate is richer than the considered cases. We denote thenumbers of non-Abelian and Abelian anyons as R NA and R A , and their total number as R = R NA + R A . For con-venience, we will also assume that the anyons are indexedin such a way that the first R NA are non-Abelian, andthe rest are Abelian.The wavefunction coefficients for even M L are nowgiven by the following correlatorΨ α ( w , n ) == (cid:104) | R NA (cid:89) i =1 V NA ,i ( w i ) R (cid:89) i = R NA +1 V A ,i ( w i ) M (cid:89) i =1 V i ( z i , n i ) | (cid:105) α == I α ( w , n ) J ( w , n ) , (17)where the index α means that we take only the conformalblock where the Ising fields fuse to α , and I α and J are theIsing and Jastrow parts of the wavefunction, respectively.The latter is given by J ( w , n ) = (cid:104) | R (cid:89) i =1 : exp (cid:18) p i √ q φ ( z i ) (cid:19) : M (cid:89) i =1 V i ( z i , n i ) | (cid:105) == δ n (cid:89) i 12 with twoAbelian anyons and two non-Abelian anyons (a quasihole anda quasiparticle of each type). The plots (a)-(e) correspond tothe case when the non-Abelian anyons are both located in the L part, while in (f)-(j) one of them is located in the Laughlinpart. The radial density plots are arranged in the same wayas the anyons in (a),(f). The “ × ” symbols in (a), (f) denotethe anyon coordinates. andΦ ( kl )( mn ) == Pf (cid:32) ( w k − z (cid:48) i )( w l − z (cid:48) i )( w m − z (cid:48) j )( w n − z (cid:48) j ) + ( i ↔ j ) z (cid:48) i − z (cid:48) j (cid:33) (25) B. Anyon charge and density distribution Let us now verify that the anyons are well localizedand that their charges agree with the theoretical predic-tion − p i /q . We define the excess charge at site i in thepresence of anyons as the difference˜ Q i = (cid:104) n i (cid:105) an − (cid:104) n i (cid:105) GS , (26)where the index “an” means that the density is evaluatedin the presence of anyons, and “GS” means the densityevaluated in the ground state (i.e. without anyons). Notethe difference from the definition used in Sec. II B - nowwe do not care for the background charge, but only forthe difference of the particle density distributions withand without anyons (otherwise we would always obtainsome excess charge near the interface). The charge of thelocalized anyon is studied by investigating the charge ac-cumulated within some radius around the anyon position w k , ˜ Q k ( r ) = (cid:88) i θ ( r − | z i − w k | ) ˜ Q i . (27)If the anyons are localized, ˜ Q k ( r ) should converge to afixed value quickly as we increase r .Fig. 6 (a) shows the distribution of charge ˜ Q i in thecase of two Abelian and two non-Abelian anyons on acylinder. Each of the anyons is located in a part where itis a valid topological excitation. It can be seen that theyare indeed well localized, with most of the charge con-centrated near their positions. The calculation of ˜ Q k ( r ),displayed in Fig. 6 (b)-(e), shows that they indeed seemto converge to a value close to − p i /q as r increases.As noted in Sec. III A, the definition of our wavefunc-tion does not forbid us to put the non-Abelian anyonswithin the R part. The result of exchanging one Abelianand one non-Abelian anyon positions from Fig. 6 (a) isshown in Fig. 6 (f). It can be seen that still the charge isconcentrated mostly in their vicinity, and approximatelyhas the expected value − p i /q .We note that in some cases, even with Abelian anyonsonly, there is some additional charge modulation at theinterface. This is a finite-size effect, whose strength de-creases with N y . The detailed analysis of this effect canbe found in Appendix B. We note that a similar phe-nomenon was encountered for Laughlin-Laughlin inter-faces [42]. C. Anyon statistics To check whether the “anyons” we investigate are trueanyons, we have to evaluate their statistics. We will con-sider the processes in which a single mobile anyon l en-circles other, static anyons. The effect of anyon braidingis given by Ψ = γ M γ B Ψ (28)where Ψ is a vector of degenerate wavefunctions | Ψ α (cid:105) for all possible values of α , while γ M and γ B are themonodromy and Berry matrices. The monodromy matrixcan be evaluated from the analytical continuation of thewavefunctions, while the Berry matrix can be written as γ B = exp (cid:16) i θ B (cid:17) , where the elements of θ B are given by θ Bαβ = i (cid:73) P (cid:104) ψ α | ∂∂w l ψ β (cid:105) d w l + c . c ., (29)where P is the path of the l th anyon.To proceed further, we need to show that the conformalblocks are orthogonal if there is more than one. The over-laps can be computed using Monte Carlo, as explainede.g. in Ref. [56]. In Fig. 7 (a) and (b) we plot the over-lap between the two conformal blocks for two cases offour non-Abelian anyons and two Abelian ones in foursystems, depicted in Fig. 7 (c)-(f). A general trend of 100 200 300 N || I | (a) 100 200 300 N (b) 12 34 56 (c) (d) 12 34 56 (e) 12 34 56 (f) FIG. 7. The overlap between conformal blocks as a functionof the number of sites for the planar systems of size (2 k + k ) × k , for k = 4 , , , 7. Both subplots correspond to 4 non-Abelian anyons in the L part and two Abelian ones in the R part: (a) p = p = − . p = p = 0 . p = 1 = − p ,(b) p = p = p = p = 0 . p = p = − 1. The rest ofthe plots, (c)-(f), show the systems taken into account in thecalculation. The blue and orange points denote the sites andthe anyons, respectively. The numbers denote the ordering ofthe anyons, which fixes the basis for the degenerate states via(23). overlap decreasing with N is seen, with (cid:104) Ψ ψ | Ψ i (cid:105) of theorder 10 − for the largest system. This shows that inlarge systems that can be studied using Monte Carlo theconformal blocks are already close to orthogonality, andwe can expect that in the thermodynamic limit the or-thogonality will be achieved.It can be shown [56] that, assuming the conformalblocks are orthogonal or there is just one, we can ex-press the Berry phase (29) solely using the normalizationconstant θ Bαβ = iδ αβ (cid:73) P C α ∂∂w l C α d w l + c . c . (30) 1. Abelian anyons In the case when the l th anyon is Abelian, the Berryphase can be computed analytically, under the assump-tion that the anyons are well-localized (which is sup-ported by the numerical results from Sec. III B). Thepartial derivative ∂∂w l in such a case does not act on theIsing part of the wavefunction. Thus, we can easily gen-0eralize the reasoning from Refs. [70]. Knowing the wave-function coefficients (17), we can evaluate the derivative ∂C α ∂w l = C α (cid:88) k p l (cid:104) n k (cid:105) w l − z k − C α (cid:88) k p l η k q ( w l − z k ) ++ C α (cid:88) k ( (cid:54) = l ) p l p k q ( w l − w k ) . (31)Thus, for an anticlockwise path winding at most oncearound each site and each anyon, the Berry phase is θ Bαβ = δ αβ (cid:34) i (cid:73) P (cid:88) k p l (cid:104) n k (cid:105) w l − z k d w l + c . c . (cid:35) ++ 2 πδ αβ (cid:88) k : z k ∈ S p i η k q − πδ αβ (cid:88) k : w k ∈ S,k (cid:54) = i p i p k q (32)where S is the region of space encircled by the path P .To deal with the first term of Eq. (32), we note thatwe are in fact not interested in the phase θ Bαβ itself, be-cause it contains both the statistical contribution and theAharonov-Bohm contribution, arising from the encircledsites. To get rid of the latter, we compute the differ-ence of phases with and without encircled anyons. Forsimplicity, let us assume that we compute the mutualstatistics of anyons l and m , and all anyons other than m lie outside the region S . We consider two cases: θ B , in αβ ,when anyon m is inside S , and θ B , out αβ , when it is outside.We have θ B , in αβ − θ B , out αβ == δ αβ (cid:34) i (cid:73) P (cid:88) k p l ( (cid:104) n k (cid:105) in − (cid:104) n k (cid:105) out ) w l − z k d w l + c . c . (cid:35) + − π p l p m q (33)where (cid:104) n k (cid:105) in −(cid:104) n k (cid:105) out are the particle densities in the twocases. Now, the assumption of localized anyons comesinto play. If the anyons are localized and far from eachother, the density difference is nonzero only in the vicin-ity of the two locations of anyon m and is w l -independent.Thus, it can be taken out of the integral. Then, applyingthe residue theorem, we obtain θ B , in αβ − θ B , out αβ = − πp l (cid:88) k ( z k ∈ S ) ( (cid:104) n k (cid:105) in −(cid:104) n k (cid:105) out ) − π p l p m q (34)We note that the sum of density differences within region S is equal to the charge of the anyon m , i.e. − p m /q . Andthus, the Berry phase vanishes.Hence, the effect of the braiding is given by the mon-odromy matrix, which is equal to γ Mαβ = δ αβ exp (2 πip l p m /q ) (35) (a) 0.00 0.25 0.50 0.75 1.00Normalized position in the path0.900.920.940.960.981.001.02 C ( w ) / C (b) FIG. 8. (a), The path of the quasielectron motion for a planar(12 + 6) × 12 system with two non-Abelian quasiholes locatedin the L part and one Abelian quasielectron located in the R part. The orange dots mark the initial positions of the anyons,while the path is denoted by green lines. The quasihole movesanticlockwise along the path. (b) The corresponding ratio ofsquared normalization constants as a function of the quasi-electron position on the path. This recovers the Laughlin anyon statistics. The expres-sion is valid in the entire system, i.e. in the parts L and R and for paths crossing the interface, which is consistentwith the fact that the Abelian anyons are valid topo-logical excitations of both parts. We also note that theabove reasoning is valid even when p m is fractional, i.e.it yields also the mutual statistics of Abelian and non-Abelian anyons, but only if the latter is static. As faras this condition is fulfilled, there is no problem withputting a non-Abelian anyon in part R . The problemsarise when it moves, as we will see in the next subsection. 2. Non-Abelian anyons In the case where a non-Abelian anyon is mobile,we verify the vanishing of the Berry phase numerically.Following Refs. [56, 67], we rely on the fact that theBerry phase vanishes if the normalization constant C (and hence the integrand of (29)) is lattice-periodic in w l as long as anyon l is far away from other anyons.To see this is the case, let us consider a planar systemin which the anyon l moves along a rectangular pathconsisting of four segments P , P , P , P . We con-sider P and P being parallel to the x direction, with x increasing in the former and decreasing in the lat-ter, and located at y = y and y = y . Similarly P and P are parallel to the y direction, with y increas-ing in the former and decreasing in the latter, and arelocated at x = x and x = x . Moreover, we demandthat the rectangle has integer dimension in the units oflattice constants, i.e. x − x ∈ Z and y − y ∈ Z .Then, we note that (cid:82) P f ( w l )d w l = (cid:82) x x f ( x + iy )d x ,and (cid:82) P f ( w l )d w l = (cid:82) x x f ( x + iy )d x . If f ( w l ) is lattice-periodic, then f ( x + iy ) = f ( x + iy ), and the contri-butions of P and P cancel each other. Similarly, one1can show that the contribution of P cancels the contri-bution of P . Thus, on this special path the statistics aredetermined by the monodromy. And, since the statisticsare a topological property, we expect that they wouldnot change if the path is deformed. The above reasoningcan be regarded as a lattice generalization for the con-tinuum argument that the Berry phase vanishes when C is constant.The lattice periodicity can be demonstrated by cal-culating the ratio of squared normalization constants ineach point of some path, C ( w l ) /C , where C corre-sponds to the starting point of the path. The method ofcaluclating such ratios with Monte Carlo is described e.g.in [56]. Because the system size required for the simula-tion of a complete braiding process is too large even forthe Monte Carlo, we consider a square loop around a sin-gle lattice site (see Fig. 8 (a)), which will tell us how thenormalization constant changes as we move an anyon byone lattice constant in the x and y directions (or both).We focus on the case of two non-Abelian anyons, forwhich there is only one conformal block. We consider a(12+6) × 12 planar system and arrange the anyons in theway shown in Fig. 8 (a). Fig. 8 (b) shows the resultingratio of squared normalization constants while movingthe quasielectron around the small square of unit length.It can be seen that the dependence is nearly periodic,with ratio being close to 1 every time the anyon is at acorner of the square. The periodicity is not perfect - therestill are some discrepancies larger than the Monte Carloerror, which may be due to the insufficient separation ofthe quasielectron from the quasihole or from the systemedge.Therefore, we expect that the statistical phase will bedetermined by the monodromy. We focus on the statisti-cal contribution to the monodromy, i.e. the monodromyafter removing the Aharonov-Bohm contribution (which,in case of one anyon moving on a closed loop, can be com-puted by subtracting the phase with and without the sec-ond anyon within the path). This statistical contributionin the case of a single Moore-Read state is well-known.For a single anticlockwise exchange, it is equal to γ M = e iπ ( p p /q − / . (36)In the case of an interface, there are additional termsinvolving R sites and non-Abelian anyons, but as long asthe braiding path is located in the L part, these terms donot contribute to the monodromy as no R site is encircledby the anyon, thus the result of an exchange is still givenby Eq. (36).We can also ask what happens if we put the non-Abelian anyons in the R part. In such a case, the mon-odromy indicates that the statistics become ill-defined.To see this, we note that now non-Abelian anyons encir-cle the R sites. The factors ( w l − z i ) p l n i for p l = ± / n i (cid:54) = 0, i.e. when a filled site is encircled. Thus,the monodromy depends on the path, i.e. it is not statis-tical. Moreover, since each configuration | n (cid:105) corresponds + − + − R RL ++ ++ ++ ++ FIG. 9. A schematic depiction of a system with two R islandson the L plane. The filled and empty circles denote parti-cles/holes and anyons, respectively. The positive and nega-tive charge is denoted by “+” and “ − ”, respectively. Thefollowing processes are depicted here: the exchange of par-ticles between L and R (the arrows at the bottom of eachisland), measurement of the charge (the arrow around theright island), and the change of parity (the arrows in the toppart of the picture). to different locations of the filled sites, each coefficient in(14) transforms in a different way and thus the effect ofa braiding is no longer a phase. Therefore, we concludethat it is not possible for non-Abelian anyons to cross theinterface. This conclusion does not depend on the Berryphase (for some consideration regarding the Berry phase,see Appendix C).We note that the nontrivial monodromy of an L par-ticle and an R anyon does not influence the boundarycondition for particles going around the cylinder (see Ap-pendix D). IV. MULTIPLE ISLANDS AND TOPOLOGICALDEGENERACY So far, we discussed the properties of a single interface.Now, let us consider two disconnected islands of the R type within an L plane, as in Fig. 9. Let us also assumethat the processes of exchange of particles through theinterfaces are local. That is, if the islands are sufficientlyfar apart from each other, a pair of particles annihilatedfrom part L should correspond to a creation of two par-ticles in island 1 or two particles in island 2, but notone particle in each island, as shown in Fig. 9. If we fixonly the total number of particles N , the Hilbert spacecontains configurations where the numbers of particles inthe first island N R ;1 is even and the ones where N R ;1 idodd. There is no local process connecting the two typesof configurations. Therefore, the Hilbert space fragmentsinto two disconnected subspaces. For each of them, we2can define a model ground state wavefunction using Eq.(6) or (17), but reducing the basis only to the given sub-space. If k islands are introduced, then the Hilbert spacefragments into 2 k − subspaces, which is reminiscent ofthe degeneracy of the Majorana modes. The appearanceof topological degeneracy in a similar setting of inter-faces forming several disconnected islands was alreadydiscussed in Ref. [4].The parity of N R ;1 can be measured by encircling anon-Abelian quasihole around it (see the arrow aroundthe right island in Fig. 9). Then, the term ( w l − z i ) p l n i .gives rise to a monodromy phase 0 if N R ;1 is even and π if it is odd. We also propose the following procedureto switch the parities at two islands. We create a local-ized Abelian quasihole-quasielectron pair with p i = ± L part(see the top part of Fig. 9). Then, we move the quasi-electron and a quasihole into the two different islands andrelease the pinning potentials. While in the L part theseexcitations were topological, in the R part they becomean ordinary particle and a hole, thus changing the parity.In such a way, the R islands can store quantum in-formation even though the interfaces are gapless. Wenote that essentially the same mechanism of creatingtopological degeneracy can be applied to the Laughlin-Laughlin interfaces from Ref. [42], thus connecting ourmodel wavefunctions to earlier results, predicting the ap-pearance of parafermion zero modes at some Laughlin-Laughlin interfaces [13]. V. CONCLUSIONS In this work, we have constructed model wavefunctionsfor lattice systems at filling ν = 1 / 2, in which part of thesystem is in the fermionic Moore-Read state, and the restis in a bosonic Laughlin state. We considered the cases inthe absence and presence of localized anyonic excitations.We have seen that the conditions of reflection and scal-ing invariance lead to different lattice filling factors ν lat ,i.e. different particle densities on the two sides of theinterface. For wide enough systems, these densities arenearly constant in the bulks of the two parts of the sys-tem, and their values are close to what is expected for therespective single quantum Hall states. Also, the constantterm γ of the entanglement entropy scaling in the bulks isconsistent with the values characterizing the topologicalorder of the respective quantum Hall states.As for the interface itself, we have found that somecharge accumulates in its vicinity, due to the fact thatthe particle density varies more smoothly than the back-ground charge. We observed a lack of exponential decayof the correlation function in its vicinity, consistent withthe prediction that the interface is gapless. We have alsoshown that the scaling of the entanglement entropy atthe interface is approximately linear, although the dataare too noisy to determine the coefficient exactly.We have studied the properties of the Laughlin anyons (which are valid topological excitations of the entire sys-tem) and the basic MR non-Abelian anyons. We havefound that the quasiparticles of both types are well-localized and have the expected charge irrespective oftheir location. However, the statistics become ill-definedif the path of a non-Abelian anyon passes through part R .Moreover, we argued that for multiple, disconnected is-lands of the R part within an L system, the particle num-ber parity at each island cannot be changed locally, i.e.it is topologically protected. However, it can be changedand measured by manipulating anyons within the L part.The presented construction can be modified and ex-tended in several ways. First, after allowing double oc-cupancy, one can consider an interface between a bosonicMR state and a fermionic integer quantum Hall effect.Secondly, one can also consider different fillings ν on bothsides, which would allow for all-bosonic or all-fermionicsystems, at the price of enforcing different charges of theparticles on the two sides. Finally, one can also use otherquantum Hall states – e.g. by forming a MR/Halperininterface, studied in [11, 12] for the continuous case. ACKNOWLEDGMENTS BJ was supported by START Fellowship by Founda-tion for Polish Science (FNP), no. 32.2019. Appendix A: More details on entanglement entropyresults Since the entanglement entropy scaling on the L sideand at the interface is noisy, here we provide additionalresults. In Fig. 10, we provide the fit parameters A, γ for all possible positions of the entanglement cut parallelto the interface. Moreover, we also study different setsof data points. Each color corresponds to a fit based ondatapoints N y, min , N y, min + 1 , . . . , N y, max .It can be seen that on the L side and at the interface,the results display large fluctuations, sometimes largerthan the result itself. Nevertheless, the obtained valuesseem to be consistent with the γ L = ln(8) / γ has a different value than on the left.On the other hand, in the R part, the results are closeto ln(2) / Appendix B: Density modulation at the interface inthe presence of anyons For some systems and some anyon configurations, weobserve that ˜ Q i is nonzero also far from the anyon posi-tions. Typically, the deviation is strongest near the in-terface, similarly to the Laughlin/Laughlin case [42]. InFig. 11, we show the charge density ˜ Q i for six examples3 A (a) 10 1517.520.0 (e) (i)024 A (b) (f) (j)0.20.40.6 (c) (g) (k)5 0 5 x 024 (d) 7.5 5.0 2.5 0.0 2.5 x (h) 10 5 0 x (l) FIG. 10. The constant term γ in the entanglement entropyscaling as a function of x position of the cut, for different setsof data points included. Columns 1,2,3, correspond to systemsof size (8 + 8) × N y , (10 + 4) × N y , (12 + 4) × N y , respectively.Rows 1 and 3 show the gradient A of the fit, while rows 2 and 4show γ , i.e. minus the intercept. The results in the first (last)two rows were obtained without (with) the inclusion of errorbars in the weights for the fit. The colors denote the differentsets of data points included, according to the colormap in theinset of (a) (the horizontal and vertical axes correspond to N y, min , N y, max , respectively). of anyon configurations for a (4 + 4) × N y and ( N L mod 8)). In Fig. 12(a) we plot the excess charge as a function of x ,˜ Q ( x ) = (cid:80) i ˜ Q i δ ( x i − x ) N y . (B1)for a series of (6 + 6) × N y systems, with two non-Abelianquasielectrons placed at x = − x = 4. It can be seen that the signof the density modulation depends on the parity of N y ,which corresponds to the two possibilities ( N L mod 8) =0 , 4. The magnitude of this density modulation decreaseswith N y . To quantify it, we define the total excess chargein part L, ˜ Q L = (cid:88) i 1, (d) an Abelian quasihole-quasielectronpair with p i = ± R particle-hole pair), (e)an Abelian and a non-Abelian quasihole-quasielectron pair( p i = ± . p i = ± 1) (f) an Abelian quasihole-quasielectronpair with p i = ± × ” symbol. In Fig. 12 (b) we plot the | ˜ Q L − Q L ;an | , i.e. the magni-tude of the excess charge in part L not associated withanyons. It decreases exponentially with N y , suggestingthat for infinitely wide cylinders the only excess charge isconcentrated in the vicinity of the anyon positions. Theexcess charge near the interface, ˜ Q ( x = − . x Q ( x ) (a) 6 8 10 12 14 16 18 20 N y | Q L Q L ; a n | (b) N y FIG. 12. The scaling of the density modulation at the inter-face for (6+6) × N y cylindrical systems. (a) The excess charge˜ Q ( x ) profile for different N y . (b) The absolute value of thetotal excess charge in the L part minus the anyon charge inthe L part as a function of N y . (a) 0.00 0.25 0.50 0.75 1.00Normalized position in the path0.940.960.981.00 C ( w ) / C (b) FIG. 13. (a) A path of the quasi-electron motion in a (2 +10) × 12 system. The orange points are the initial positionof the anyons, while the green lines denote the anticlockwisepath of the non-Abelian quasielectron motion. The secondanyon is a non-Abelian quasihole. (b) The ratio of squarednormalization constants in the given point on the path and inthe initial point. Appendix C: Berry phase for non-Abelian anyons inthe R part We have shown that if we put the non-Abelian anyonsin the R part, the monodromy in the braiding processbecomes ill-defined, and thus they lose their anyonic be-haviour. For completeness, here we consider the Berryphase contribution.Let us first consider a trivial case of N L = 0 and twonon-Abelian quasiholes ( p = p = 1 / w − w ) − / , is canceled by the ( w − w ) p p term from the Jastrow part. Thus, following thesame approach as in Sec. III C 1, we arrive at θ B , in − θ B , out = 2 π p p q = π/ N L = 2 is a little more complicated, butstill tractable analytically under the assumption of lo-calized anyons. Let us again focus on the case of twonon-Abelian quasiholes ( p = p = 1 / 2) and one Abelianquasielectron ( p = 0). The wavefunction is now givenbyΨ(w , n) = 2 − n n A n n ( w − w ) p p ( w − w ) p p ×× (cid:89) i =1 N (cid:89) j =3 ( w i − z j ) p i η j /q N (cid:89) i =1 ( w − z i ) p n i ×× (cid:89) i =1 N (cid:89) j =1 ( w i − z j ) p i η j /q (cid:89) i 2, we can attempt to prove the van-ishing of the Berry phase numerically, as in Sec. III C 2.However, we observe a lack of periodicity, as seen in Fig.13. Therefore, our current approach does not allow us toextract the value of the Berry phase. This does not ruleout an appearance of periodicity for systems too large tobe studied using our Monte Carlo software. Appendix D: Boundary conditions for a cylinderwith anyons The term ( w i − z j ) p i n j , creating nontrivial monodromywhen a non-Abelian anyon encircles a filled R site, hasa nontrivial effect also when an R particle encircles a non-Abelian anyon. This can occur on a cylinder. Thecylindrical system mapped to a complex plane looks asin Fig. 14 (a). Thus, if an R particle goes around acylinder, it encircles the entire L part, including the non-Abelian anyons located inside. But this can create anonzero phase, which seems to lead to a conclusion thatthe boundary conditions in the R part are determinedby the number of anyons in the L part. This would seemstrange, as we can consider another, equivalent mappingof a cylinder to the complex plane, where the R part isinside and no L anyons are encircled, as in 14 (b), andthus the boundary conditions for R particles should bedetermined only by quantities related to the R part.Let us consider a path K which encircles all N L L sites as well as k R , as well as anyons of total charge − P in /q sites. The only terms which generate nonzeromonodromy of a L particle on path K are ( z i − z j ) − n i η j and ( w l − z j ) p l n j . They give rise to a phase φ = 2 π ( P in − N L η L − kη R ) . (D1)However, we can write it also as φ = 2 π ( P − N L η L − N R η R ) −− π ( P out − ( N R − k ) η R ) , (D2)where P out is minus the charge of all anyons outside thepath in the units of 1 /q (i.e. their charge is − P out /q ),and P = P in + P out . Due to the charge neutrality (19),the first term vanishes. The second term depends onlyon the sites and anyons outside the path. 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