Edge Mode Combinations in the Entanglement Spectra of Non-Abelian Fractional Quantum Hall States on the Torus
EEdge Mode Combinations in the Entanglement Spectra ofNon-Abelian Fractional Quantum Hall States on the Torus
Zhao Liu,
1, 2
Emil J. Bergholtz,
3, 4
Heng Fan, and Andreas M. L¨auchli
3, 5 Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann Straße 1, D-85748 Garching, Germany Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38, D-01187 Dresden, Germany Dahlem Center for Complex Quantum Systems and Institut f¨ur Theoretische Physik,Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany Institut f¨ur Theoretische Physik, Universit¨at Innsbruck, A-6020 Innsbruck, Austria (Dated: October 29, 2018)We present a detailed analysis of bi-partite entanglement in the non-Abelian Moore-Read fractional quantumHall state of bosons and fermions on the torus. In particular, we show that the entanglement spectra can bedecomposed into intricate combinations of different sectors of the conformal field theory describing the edgephysics, and that the edge level counting and tower structure can be microscopically understood by consideringthe vicinity of the thin-torus limit. We also find that the boundary entropy density of the Moore-Read state ismarkedly higher than in the Laughlin states investigated so far. Despite the torus geometry being somewhat moreinvolved than in the sphere geometry, our analysis and insights may prove useful when adopting entanglementprobes to other systems that are more easily studied with periodic boundary conditions, such as fractional Cherninsulators and lattice problems in general.
PACS numbers: 73.43.Cd, 71.10.Pm, 03.67.-a
I. INTRODUCTION
Quantum correlations give rise to many exotic phases ofmatter that cannot be characterized in terms of traditionalconcepts, such as local order parameters and symmetry. Re-cently, tools from the field of quantum information (QI) havebeen used to quantify such correlations . Of special inter-est among the applications are systems in which more tradi-tional condensed-matter methods are of limited use, for exam-ple topologically ordered matter . Fractional quantum Hall(FQH) states stand out as experimentally verified topologi-cally ordered phases driven by interactions, and their possi-ble applications in the context of quantum computation areof great current interest . The microscopic understandingof these phases is mainly based on ad hoc , albeit brilliant,guesswork and numerical wave-function overlap calcula-tions in small systems. A fundamental problem with usingwave-function overlaps as a probe is, however, that it nec-essarily vanishes in the thermodynamic limit (for any real-istic interaction). Recently, it has been realized that (bi-partite) entanglement measures, most saliently the von Neu-mann entropy and the entanglement spectrum can pro-vide valuable insights into these states—in principle even inthe thermodynamic limit.In this work, we focus our attention on entanglement in thearchetypical non-Abelian FQH state, namely the Moore-Readstate , which has received a tremendous amount of attentionrecently as a potential platform for topological quantum com-putation. Previous theoretical studies have accumulated ev-idence that the ground state of the two-dimensional electrongas at the Landau level filling fraction ν = 5 / is well de-scribed by the Moore-Read state, which may be thought ofas paired composite fermions and has quasiparticles possess-ing fractional charge ± e/ and obeying non-Abelian braidstatistics . In recent experiments, both the fractional charge x y A B A (a) (b) B FIG. 1. (Color online) The torus setup (a) compared with the orbitalpartition on the sphere (b). The dark lines indicate the centers ofthe single-particle states and the differently shaded regions denotethe approximate spatial partitioning corresponding to the half-blockorbital partitioning. Red arrows represent the artificial edge statesinduced by splitting the system into A and B . and non-Abelian braid statistics have been claimed , butthe interpretations of the experiments is still under debate .Another possible host of the Moore-Read state is the Bose-Einstein condensate under rapid rotation, in which the bosonicstate at ν = 1 is particularly promising . However, the ex-perimental realization of the bosonic FQHE is extremely chal-lenging, although some strategies to overcome the difficultieshave been proposed .To study bi-partite entanglement, we (artificially) divide asystem into two parts A and B (Fig. 1). In a tensor productHilbert space, H = H A ⊗ H B , any pure state | Ψ (cid:105) AB can bedecomposed using the Schmidt decomposition , | Ψ (cid:105) AB = (cid:88) i e − ξ i / | ψ Ai (cid:105) ⊗ | ψ Bi (cid:105) , (1)where the states | ψ Ai (cid:105) ( | ψ Bi (cid:105) ) form an orthonormal basis forthe subsystem A ( B ) and the entanglement “energies” ξ i ≥ a r X i v : . [ c ond - m a t . s t r- e l ] J a n are related to the eigenvalues, λ i , of the reduced densitymatrix, ρ A = tr B | Ψ (cid:105) AB AB (cid:104) Ψ | , of A as λ i = e − ξ i .For topologically ordered states in two dimensions, the en-tanglement entropy contains topological information aboutthe state: S A = − tr[ ρ A ln ρ A ] = − (cid:80) i λ i ln λ i = (cid:80) i ξ i e − ξ i , is expected to scale as S A ≈ αL − nγ + O (1 /L ) , where L is the (total) block boundary length, n is the numberof disconnected boundaries, and γ characterizes the topologi-cal field theory describing the state .Li and Haldane realized that the full so-called entangle-ment spectrum (ES), { ξ i } , contains much more informationthan entanglement entropy. In particular, when plotted againstthe natural quantum numbers of the system, it shows a remark-able similarity with the conformal field theory (CFT) describ-ing the chiral edge states of the FQH states.To make practical use of the entanglement concepts, it isinstrumental to find a protocol with which the theoreticalideas can be (numerically) tested in realistic circumstances.The most widely used concept of partitioning the system interms of the single-particle orbitals was introduced by Haque,Zozulya, and Schoutens in their study of the topological en-tanglement entropy in Laughlin states on the sphere. A numer-ical determination of γ (and α ) in realistic circumstances re-quires information about S A for a number of different bound-ary lengths, L . Because of its technical simplicity, early at-tempts to obtain the entropy scaling in FQH states focusedon the sphere geometry . However, as recently demonstratedfor Abelian FQH states, a substantially better finite size scal-ing can be obtained on the torus where the boundary lengthcan be varied continuously by varying the aspect ratio [cf.Fig. 1(a) and 1 (b)]. (The idea of obtaining entanglemententropy scaling through varying discrete torus circumferenceswas also used in Ref. 22 for the dimer model on the triangularlattice.) Importantly, this extra degree of freedom available onthe torus also provides a handle on when the extrapolationsneeded to extract γ can be trusted (and when they cannot).With a few very recent exceptions , the efforts made inthe study of the ES in FQH states are also numerical .In addition to these works, there has been a large number ofrecent studies extending the range of applicability of the ESto an increasing number of physical systems . The studiesof the ES in FQH states have focused predominantly on thesphere geometry. In this case, there is a genuine benefit withthis choice since it amounts to probing the physics of a singleFQH edge while the natural partition on the torus correspondsto two oppositely oriented edges [cf. the red arrows in Fig.1(a) and 1 (b), respectively]. A benefit with the torus setupis, however, that one can continuously connect to the exactlysolvable thin-torus limit from which many of the propertiesof the ES can be understood microscopically .On the sphere one finds that the ES has a chiral structure that is intimately related to the squeezing rule of model FQHstates that holds on genus-0 manifolds. The structure of thesqueezed configurations also provides physical insight similarto what is possible in the thin-torus limit. While the squeezingrule does not hold on the torus (genus-1), the ES can never- theless be described by combining two edge spectra, as wasshown in Ref. 28 for the Laughlin state.In spite of the technical difficulties involving two separateedges, these issues are worth dealing with, in particular sincethere are many physical systems of great interest that are onlyapproachable using periodic boundary conditions. Specifi-cally, regular two-dimensional lattices do not admit generallya defect-free embedding onto the sphere (because of their dif-ferent Euler characteristics). In particular, the recently pro-posed fractional Chern insulators appear to belong to thiscategory.The two-edge picture on the torus is reportedly difficultto extend to non-Abelian FQH states due to their non-trivialground-state degeneracies, which do not result from simplecenter-of-mass translations as in the Abelian case. Thus, it isnot a priori clear how to choose the ground state, | Ψ (cid:105) AB , in(1) (or alternatively, how to define the density matrix of thefull system A ∪ B ) out of this degenerate set. Note that theissue of degenerate ground states does not occur in the spherecase in which the model states are unique maximal densityzero modes of their respective parent Hamiltonians.Here, we adopt a very simple and natural choice for the setof | Ψ (cid:105) AB and show that a similar, but significantly richer, two-edge picture also holds true for the ES of non-Abelian FQHstates on the torus. Specifically, we disentangle the physics ofthe edge modes appearing in the entanglement spectra in eachof the topologically distinct sectors of the Moore-Read stateof both fermions and bosons. We find that, even for a givencut in one of the ground states, the resulting towers are gener-ated from combinations of different sectors of the underlyingconformal field theory.We also carefully analyze the scaling of the von Neumannentropy in the various sectors of the Moore-Read state. Wefind that the total entropy as well as the area-law entropydensity, α , can be estimated (in particular quite accurately inthe case of bosons) while the extrapolation is too sensitive tofaithfully determine the topological part, γ .The remainder of this article is organized as follows. InSection II, we introduce the physical model and the methodwe use to obtain the ground states and calculate the ES. InSection III, we analyze the ES from two distinct perspectives.On the one hand, we explain the ES as the combination ofedge modes and discuss the quantitative relation in this com-bination. On the other hand, we use the thin-torus limit anda perturbation theory to illuminate the microscopic origin ofthe observed ES, including the counting rules in different edgesectors. Finally, we discuss the entanglement entropy in Sec-tion IV. II. MODEL AND METHOD
We study a two-dimensional N -boson (fermion) systemsubject to a perpendicular magnetic field on a torus with peri-ods L and L in the x and y directions. The full symmetryanalysis of this system was first provided by Haldane —herewe use a convenient representation thereof. Periodic boundaryconditions require that L L = 2 πN s (in units of the mag-netic length) where N s is the (integer) number of magneticflux quanta (the number of vortices for rotating Bose-Einsteincondensates). We choose a basis of normalized single-particlelowest Landau level (LLL) wave functions as ψ j = 1 (cid:112) L π / ∞ (cid:88) n = −∞ e [ i ( πjL + nL ) x − ( y + nL + πjL ) / , (2)where j = 0 , , , ..., N s − can be understood as thesingle-particle momentum in units of π/L . Because ψ j iscentered along the line y = − πj/L , the whole system canbe divided into N s orbitals that are spatially localized in the y − direction (but delocalized in the x − direction). There aretwo translation operators, T α , α = 1 , , that commute withthe Hamiltonian H (and any translational invariant operator);they obey T T = e πiN/N s T T , and operators have eigen-values e πiK α /N s , K α = 0 , ..., N s − . T corresponds to x -translations and K = (cid:80) Ni =1 j i (mod N s ) is the total x -momentum in units of π/L . T translates a many-body stateone lattice constant L /N s = 2 π/L in the y -direction andincreases K by N . At filling factor ν = p/q (with p and q co-prime), T q commutes with T , and T k ( k = 0 , , . . . q − )generate q degenerate orthogonal states, which have different K . This is the q -fold center of mass degeneracy common toall eigenstates of a translational invariant operator in a Lan-dau level. Thus, the energy eigenstates are naturally labeledby a two-dimensional vector K α = 0 , ..., N s /q − , where e πiK q/N s is the T q -eigenvalue.We use exact diagonalization to obtain the Moore-Readstates, which are zero-energy ground states of certain three-body Hamiltonians (see Appendix A), in the orbital basis. TheMoore-Read states are non-Abelian states, for which the de-generacy on the torus is enhanced (in this case by a factor )compared to the q -fold degeneracy discussed above. It is read-ily seen from the thin-torus configurations (the ground statesas L → ) that they are not simply the translations of eachother (see below). To extract the ES, we choose the groundstates as eigenstates of T and T q and bipartition the systeminto blocks A and B , which consist of l A consecutive orbitalsand the remaining N s − l A orbitals, respectively. We labelevery ES level by the particle number N A = (cid:80) j ∈ A n j andthe total momentum K A = (cid:80) j ∈ A jn j (mod N s ) in block A ,where n j is the particle number on the orbital j . (In this work,we present data only for the case in which l A = N s / .)To understand the ES, it is essential to understand what thepartitioning of the state looks like in the thin-torus limit. Forthe bosonic case, there are three different thin-torus patternsleading to the following partitions (for N = N s = 16 ): | | | | ± | | . (3)For the fermionic case, there are six different thin torus pat-terns and the following partitions (for N = 16 , N s = 32 ): | | | | | | ± | | | | ± | | . (4)The bold block is our subsystem A . For bosons in (3),we have two qualitatively different cuts: | · · · | and | · · · | ( | · · · | gives a mirror image of this).For fermions in (4), we have four qualitatively different cuts: | · · · | ( | · · · | ), | · · · | , | · · · | , and | · · · | ( | · · · | ).We stress that, as long as the edges are sufficiently wellseparated, one can understand the entanglement in terms oftwo non-interacting edges whose details depend on the localenvironment around the cuts . This holds true also for thestates that are connected to a thin-torus configuration which isa linear superposition of two individual terms—in these casesthe ES is composed of two shifted and superimposed mirrorimages corresponding to the ES of a single term respectively.Our procedure is different from that in Refs. 33 and 42where the authors calculate the ES via a mixed state den-sity matrix of the form ρ = d (cid:80) di =1 | Ψ i (cid:105) AB AB (cid:104) Ψ i | where {| Ψ i (cid:105) AB } denote d degenerate ground states. With this recipeone finds that the ES corresponds to the superimposed ES of all the d thin-torus patterns. For the entanglement entropy,such a mixed state prescription essentially shifts S A ( L ) bya constant and would thus result in a shifted prediction forthe topological contribution, γ . In the case of Abelian states,it turns out that averaging the entropies (rather than the den-sity matrices) over the different sectors, or equivalently overthe possible translations of the region A , significantly reducesfinite-size corrections and yields results in excellent agree-ment with theory . We note that the mixed-state prescriptionshifts the entropies of Abelian states by a constant value, ln d ,and would thus lead to a topological entropy different from thetheoretical predictions for the spatial (as opposed to orbital)cut—in fact, it would lead to γ = 0 . For non-Abelian states,it is not yet settled which orbital basis prescription would leadto the same topological entropy as for the spatial cut. FIG. 2. (color online) The edge modes of the environment in the leftpanel (black dots) and the environment in the middle panel (greendots) can combine to form a tower in the right panel (blue crosses).The relation in Eq. (5) is shown by the parallelogram. The edgemode at ∆ k = − pointed by the solid black arrow and the edgemode at ∆ k = 1 pointed by the solid green arrow can generate thelevel at ∆ k = 0 pointed by the solid blue arrow. This data comesfrom the ES of bosons in the 11 sector (see Fig. 5). III. ENTANGLEMENT SPECTRA: TWO-EDGE PICTUREAND THIN-TORUS ANALYSIS
The most prominent N A sectors of the ES of the Moore-Read state for N = 16 are displayed in Fig. 5 ( ν = 1 bosons)and Fig. 6 ( ν = 1 / fermions). The gross features of theMoore-Read ES on the torus are very similar to that of theLaughlin state—in both cases, multiple towers are formed .In this section, we analyze the ES from two different perspec- tives: We explain the tower structure in terms of combinationsof edge modes and highlight intriguing relations between theES levels within the towers as well as between the levels indifferent particle number sectors. Moreover, we use the ex-actly solvable thin-torus limit and perturbation theory to un-derstand the formation of various edge environments and tow-ers.The observed towers in the numerical ES can be reproducedby first assigning the edge modes of individual edge environ-ments and then combining them appropriately. The numberof independent edge modes at momentum ∆ k in an edge en-vironment is determined by the underlying edge theory. Theedge theory of the Moore-Read state is richer than that of theLaughlin state and contains a free boson branch as well as aMajorana fermion branch . The details are recapitulated inAppendix B for completeness. It is important to note thatthere are different sectors of the edge theory and that theycome with different predictions for the counting of states asa function of momentum. This is reflected in our numericallyobtained ES, where we observe the edge environments withdifferent counting rules. It is interesting to see that two edgeenvironments with different counting rules can also combineto form a tower.There are intriguing quantitative relations in the combina-tion of edge modes as first pointed out for the Laughlin statein Ref. 28. An explicit example of how two edges, with dif-ferent dispersion, add up to a tower is given in Fig. 2. Moregenerally, each edge mode can be labeled by three parameters:the edge environment X to which it belongs, its momentumshift ∆ k i compared with the bottom mode of the environment X , and the change of the subsystem particle number ∆ N X A inthe environment X compared with the thin-torus state. Twoedge modes with entanglement energy ξ ( X , ∆ k i , ∆ N X A ) and ξ ( Y , ∆ k j , ∆ N Y A ) , respectively (here we assume ∆ k i ≤ and ∆ k j ≥ ), combine to form a level in the X Y tower withentanglement energy ξ ( X Y , ∆ k i + ∆ k j , ∆ N X A + ∆ N Y A ) = ξ ( X , ∆ k i , ∆ N X A ) + ξ ( Y , ∆ k j , ∆ N Y A ) −
12 [ ξ ( X , , ∆ N X A ) + ξ ( Y , , ∆ N Y A )] . (5)The validity of the two-edge picture is insensitive to the cir-cumference L as long as the edges are sufficiently well sep-arated from each other, i.e. given that d ∼ L / πN s /L is large enough, which is equivalent to small enough L fora given system size. This is illustrated in Fig. 3, where thebreakdown of the two-edge picture is signaled for the larger L values, which is indeed a confirmation of the fact that thedecomposition of the entire ES into a combination of edgemodes is a highly non-trivial fact. Note that this breakdownoccurs despite the fact that the numerically exact Moore-Readstate is obtained for all L .The relative pseudo-energies of the assigned single-edgemodes depend smoothly on the torus thickness L , to somedegree even after the two-edge prediction breaks down. For a given L , the edge levels correspond well to the single-edge levels extracted from the ES on a sphere with a cor-responding length of the equator, as shown in Fig. 4. Forlarge boundary lengths, the dispersion of a single edge be-comes non-monotonic—at least in the case of fermions [cfFig. 4(b)], as can be inferred from the original data obtainedby Li and Haldane . This does not imply that the two-edgepicture will eventually break down, but does imply that theedge assignment becomes much more cumbersome at large L as the ∆ k = 0 levels no longer play the role of vacuumlevels of each tower. Also, for this reason it is very useful tofollow the evolution of the edge levels down to small L wherethe dispersion is monotonic in order to eventually understand FIG. 3. (color online) A plot of the main tower(s) of the ES in the0101 fermionic Moore-Read state for various L ( N A = 8 , N = N s / ). For small enough L (in this case L ≤ or so) theedges are well enough separated ( d ∼ L / πN s /L ) and thetwo-edge prediction (black crosses) reproduces the numerically ob-tained ES levels (blue squares). For larger L the edges are spatiallycloser and the two-edge prediction gradually breaks down. the ES also at large L .The adiabatic connection to the thin-torus ( L → ) limitalso enables us to understand more detailed features of theES by perturbing away from this solvable limit . The per-turbation theory is however hard to perform in a rigorousway, as was recently performed for the ES of one-dimensionalmodels . The reason for this is that the exponential behaviorof the matrix elements implies that higher-order contributionsfrom local terms come with amplitudes of the same order aslonger-range terms contribute at lower orders. Nevertheless,many insights can be gained from a perturbative perspective,as we discuss below.It is instructive to divide the perturbations, which are three-particle hopping processes, into three different classes. In thefirst class, three particles belong to the same subsystem andnone of them move across the edge. These processes do notqualitatively alter the entanglement between two subsystems.In the second class, two particles belong to one subsystem andone particle belongs to the other, but still none of them movesacross the edge. In the third class, some of the particles moveacross the edge. As we show below, the processes in the sec-ond class are responsible for generating new levels within atower, and those in the third class lead to levels in new tow-ers stemming from new edge environments. The entire ES ofthe Moore-Read state is built from successive combinations ofmany of the processes in each of these three classes.With the knowledge of the microscopic environment near acut in the thin-torus limit, the counting of each edge followsfrom the exclusion principle that no more than two bosons(fermions) occupy two (four) adjacent orbitals. Similar ex-clusion rules are, in addition to the thin-torus limit , also FIG. 4. (color online) The single-edge modes identified from the ESas a function of L . (a) The edge modes corresponding to the | cutin the bosonic Moore-Read state for N = N s = 12 . (b) The edgemodes corresponding to the | cut in the fermionic Moore-Readstate for N = 12 , N s = 24 . The rectangles contain the single-edge ES levels in the spherical geometry , here shifted: { ξ i } →{ ξ i + constant } , for the best comparison with the torus results at L = 14 for bosons and L = 16 for fermions. The number of fluxquanta on the sphere N sp s is chosen as the integer nearest to L / (2 π ) so that the length of the equator of the sphere is nearly the same with L . Here N = 12 , N sp s = 10 for bosons and N = 8 , N sp s = 13 for fermions . One can see that in (b) the red dot is slightly lowerthan the black circle at L = 13 , meaning that a non-monotonicdispersion of the edge appears. showing up in related approaches such as the squeezingrules related to Jack polynomials and the patterns of zeros ap-proach. A. Bosons
We now give a more detailed account of the Moore-ReadES in the case of bosons.We first consider the 11 sector and systematically explainthe ES in this sector, which is shown in Fig. 5. The lowest ESlevel is found in the N A = 8 sector at ∆ K A = 0 , correspond-ing to the thin-torus configuration | | . At L = 0 , this is the only entanglement level. We call theedge environment | A B , where the subscript, B, in-dicates the environment is for bosons. Here the subsystem onthe left (right) side of this edge environment is A when weconsider the right (left) edge of A . By definition ∆ N A B A = 0 .(In the following, when we discuss ∆ N X A of an edge environ-ment X , we suppose the subsystem A is on the left side of X . If A is on the right side, one only needs to put a minus |11110202| |0202 FIG. 5. (color online) The ES of bosonic Moore-Read states in the 11 sector at L = 5 . (upper panels) and the 02+20 sector at L = 6 (lowerpanels) for N = N s = 16 . The origin of ∆ K A is chosen to match the Tao-Thouless state. The blue squares represent numerically obtaineddata. The assigned edge modes are labeled by dots with different colors corresponding to different edge environments we describe in the text.The combination of two identical edge environments is marked by the crosses with the color of that edge environment. The combination of twodifferent edge environments is marked by the squares filled with two colors, the left (right) one of which corresponds to the edge environmenton the left (right) edge of the subsystem A . Here we do not differentiate the edge environments D (cid:48) B to I (cid:48) B from the edge environments D B to I B because the former ones are just the mirror symmetries of the latter ones and have a momentum shift ± N s / . sign before the value that we give.) All other levels in the A B A B tower ( A B A B denote the edge environments on the leftand right edge of subsystem A ) are generated from this levelby the momentum-conserving hopping processes, which con-serve N A . For example, a hopping process at the right edgeof subsystem A , | → | , gives the lowestlevel at ∆ K A = 1 .Some processes do not conserve N A ; e.g., | → | or | → | . We call the newkind of edge environment in this example B B . It is clear that ∆ N B B A = ± . Two B B edges create the B B B B tower in the N A = 8 sector, whose dominant thin-torus configurations are: | | | | . Similarly, we can find another new edge environment.When applying a hopping process to the edge environment B B : | → | or | → | ,we obtain the edge environment C B with ∆ N C B A = ± . Two C B edges create the C B C B tower in the N A = 8 sector, whosethin-torus configurations are: | | | | . The different edges can combine with each other to formtowers in other N A sectors. For example, in the N A = 7 sector we predict and observe A B B B , B B A B , B B C B and C B B B towers. In the N A = 6 sector, we can observe another B B B B tower and A B C B and C B A B towers.The A B , B B and C B edges are sufficient to accurately re-produce the ES of the sector up to ξ = 30 for L = 5 . ,as shown in the upper panels of Fig. 5. For larger L moretowers appear and can be explained along the same lines.Now we turn to the (asymmetric cut in the) 02+20 sec-tor, for which the ES possesses more complicated structuresthan that in the 11 sector, as shown in Fig. 5. For simplic-ity, we start our analysis from only one term in the superpo-sition of the thin-torus configuration, for example, from theterm | | . The entire ES in the 02+20 sec-tor is recovered by superposing two mirror images of the ESstemming from the single term.In the thin-torus limit, the only entanglement level at ∆ K A = 0 corresponds to the configuration | | . We refer to this edge environment | as D B . Allother levels in the D B D B tower are generated from this levelby the momentum-conserving hopping processes, which con-serve N A .Through applying leading hopping processes, we can gen-erate all edge environments we have observed in the ES (at L = 6 ): D B : 0202 | , ∆ N D B A = 0; E B : 0201 | , ∆ N E B A = − F B : 0200 | , ∆ N F B A = − G B : 0104 | , ∆ N G B A = 1; H B : 0006 | , ∆ N H B A = 2; I B : 0100 | , ∆ N I B A = − . These edges combine with each other to form towers ineach N A sector. For example, in the N A = 8 sector we predictand observe D B D B , E B E B , F B F B , G B G B and H B H B towers, Iithe N A = 7 sector we find D B E B , E B F B , G B D B , H B G B and F B I B towers, and in the N A = 6 sector we find D B F B , G B E B , H B D B and E B I B towers.Similarly, we can start the analysis from the other term | | in the thin-torus configuration. We alsopredict and observe the six edge environments D (cid:48) B to I (cid:48) B ,which are just the mirror symmetries of the edge environments D B to I B for the term | | . For example, theedge environment D (cid:48) B is identified as | , which is themirror symmetry of D B . Moreover, the combination of edgeenvironments D (cid:48) B to I (cid:48) B can form towers in each N A sector.For example, in the N A = 7 sector we can observe the E (cid:48) B D (cid:48) B tower as the mirror symmetry of the D B E B tower.We are also able to understand the counting rule of eachedge environment from a simple exclusion rule in their thin-torus configuration. Here we take the edge environment A B inthe 11 sector and D B in the 02+20 sector as examples. Whenanalyzing the counting rule, we imagine the subsystem on theleft (right) side of the edge environment as a quantum Hallsystem with an open right (left) edge, and then we move parti-cles to the orbitals with higher (lower) momentum to increase(decrease) the momentum of the system. Meanwhile, the gen-eralized exclusion rule of the Moore-Read state, namelyno more than two bosons (fermions) on two (four) consecu-tive orbits, should not be violated. Through the analysis, wecan find that the counting rule of the edge environment A B inthe 11 sector is consistent with that of free bosons plus peri-odic Majorana fermions, while the counting rule of the edgeenvironment D B in the 02+20 sector is consistent with that offree bosons plus antiperiodic Majorana fermions with an even F (see Appendix C). For some edge environments, only oneside of it satisfies the generalized exclusion rule, for examplethe edge environment E B = 0201 | in the 02+20 sector.In this case, we only need to analyze the subsystem on its leftside. Through analysis, we can predict the counting rules ofsome edge environments and compare them with the countingrules that we observe in our numerical data. In the 11 sector, we have A B : 1 , , , , , · · · ; [1 , , , B B : 1 , , , , , · · · ; [1 , , , C B : 1 , , , , , · · · ; [1 , , and in the 02+20 sector, we have D B : 1 , , , , , · · · ; [1 , , , , E B : 1 , , , , , · · · ; [1 , , , , F B : 1 , , , , , · · · ; [1 , , , G B : 1 , , , , , · · · ; [1 , , H B : 1 , , , , , · · · ; [1 , I B : 1 , , , , , · · · ; [1] , where we, for each edge environment, first give the expectedcounting rule and then list the observed result from our nu-merical data, as shown in Fig. 5, in the brackets. For ex-ample, A B : 1 , , , , , · · · ; [1 , , , means that the ex-pected number of edge modes for edge environment A B is , , , , , · · · at ∆ k = 0 , , , , , · · · , while the observednumber of edge modes is , , , at ∆ k = 0 , , , . We seethat the numerically observed count never exceeds the theo-retical expectations (that are derived for an infinite system).There are two reasons for this. First, our data are only numer-ically accurate up to some finite ξ , and thus we count only thelevels that are free of numerical noise. Second, the counting istruncated by the finite system size similar to the situation onthe sphere and should be expected in any geometry. B. Fermions
The thin-torus and edge analysis of the fermion ES (Fig. 6)is entirely analogous to the boson case, and thus we only pro-vide a condensed exposition of the analysis here. Note how-ever, that the edge assignment would have been much trickierin the fermion case if we would have started out by consid-ering the large L regime where the edge dispersion is non-monotonic .In the 0101 sector, we can observe three edge environments: A F : 01010101 | , ∆ N A F A = 0; B F : 01010100 | , ∆ N B F A = − C F : 00011111 | , ∆ N C F A = 1 . Their counting rules are A F : 1 , , , , , · · · ; [1 , , , , B F : 1 , , , , , · · · ; [1 , , , C F : 1 , , , , , · · · ; [1 , , which are the same as those for the 111 bosonic state. Thecombinations of edge environments can form towers in each N A sector: A F A F , B F B F and C F C F towers in the N A = 8 sector, A F B F and C F A F towers in the N A = 7 sector, and C F B F tower in the N A = 6 sector. |0101010101100110| |0110011010011001| |1001100111001100| |11001100 FIG. 6. (color online) The ES of fermionic Moore-Read states in the 0101 sector at L = 7 (upper panel), the 0110+1001 sector at L = 8 (middle panel) and the 1100+0011 sector at L = 8 (lower) for N = 16 , N s = 32 . The origin of ∆ K A is chosen to match the thin-torusground state. The blue squares represent numerically obtained data. The assigned edge modes are labeled by dots with different colorscorresponding to different edge environments as described in the text. The combination of two identical edge environments is marked by thecrosses with the color of that edge environment. The combination of two different edge environments is marked by the squares filled withtwo colors, the left (right) one of which corresponds to the edge environment on the left (right) edge of the subsystem A . Here we do notdifferentiate the edge environments F (cid:48) F to I (cid:48) F from F F to I F because the former ones are just the mirror symmetries of the latter ones and havea momentum shift ± N s / . In the 0110+1001 sector, first we start our analysis from theterm | | . We find twoedge environments: D F : 01100110 | , ∆ N D F A = 0; E F : 01001111 | , ∆ N E F A = 1;01100100 | , ∆ N E F A = − , whose counting rules are expected as D F : 1 , , , , , · · · ; [1 , , , , , E F : 1 , , , , , · · · ; [1 , , . If we start from the other term, | | , we also findthe following two edge environments: D (cid:48) F : 10011001 | , ∆ N D (cid:48) F A = 0; E (cid:48) F : 10011000 | , ∆ N E (cid:48) F A = − | , ∆ N E (cid:48) F A = 1 , with counting rules being D (cid:48) F : 1 , , , , , · · · ; [1 , , , , E (cid:48) F : 1 , , , , , · · · ; [1 , , , . The combination of these edge environments can generate D F D F , E F E F , D (cid:48) F D (cid:48) F and E (cid:48) F E (cid:48) F towers in the N A = 8 sector; D F E F , E F D F , D (cid:48) F E (cid:48) F and E (cid:48) F D (cid:48) F towers in the N A = 7 sector;and E F E F and E (cid:48) F E (cid:48) F towers in the N A = 6 sector.In the 0011+1100 sector, if we start from the term | | , we find the follow-ing four edge environments: F F : 00110011 | , ∆ N F F A = 0; G F : 00110010 | , ∆ N G F A = − H F : 00110000 | , ∆ N H F A = − I F : 00011111 | , ∆ N I F A = 1 , with counting rules being F F : 1 , , , , , · · · ; [1 , , , , G F : 1 , , , , , · · · ; [1 , , , , H F : 1 , , , , , · · · ; [1]; I F : 1 , , , , , · · · ; [1] . The combination of edges generates towers in each N A sec-tor: F F F F and G F G F towers in the N A = 8 sector, F F G F , G F H F and I F F F towers in the N A = 7 sector, and F F H F and I F G F towers in the N A = 6 sector. If we start from the other term, | | , we find edge envi-ronments F (cid:48) F to I (cid:48) F which are just the mirror symmetries of F F to I F with the same counting rules. For example, the edgeenvironment F (cid:48) F is identified as | , whichis the mirror symmetry of F F . Moreover, the combinationof edge environments F (cid:48) F to I (cid:48) F forms towers in different N A sectors. For example, in the N A = 7 sector we predict andobserve the G (cid:48) F F (cid:48) F tower as the mirror symmetry of the F F G F tower. IV. ENTANGLEMENT ENTROPY
The entanglement entropy of the Moore-Read state wasstudied earlier on the sphere and disk geometry , and thetopological part, γ , has been reported to be consistent, al-beit not in perfect agreement, with the theoretical predictions.However, the limitations of these geometrical setups make itvery hard to verify whether the scaling regime (2) is reachedor if the approximate agreement with theory is accidental. Inaddition, there are large finite size effects on the disk due tothe (large) physical edge of the system. Here we revisit this is-sue using the torus setup that allows for superior control of the entanglement scaling properties as demonstrated for AbelianFQH states in Ref. 21. This method of partitioning impliestwo disjoint edges between the blocks, each of length L , sothe entanglement entropy should satisfy the following specificscaling relation S A ≈ αL − γ + O (1 /L ) , where γ is the topological entropy whose theoretical value is ln( √ for bosons and ln( √ for fermions . (See Ref. 51for the scaling relation of entanglement entropy in variousphysical systems.)Figs. 7 and 8 show the entropy S A , its derivative dS A /dL ,and the intercept of its linear approximation, S A − L × dS A /dL , as functions of L in different sectors for differentsystem sizes. Arguably the boson results (Fig. 7) look morepromising. In this case, the entropy in the different sectors dif-fers for small L , as can be expected from the thin torus limitwhere the ± states have an entropy of ln 2 while the sector has zero entropy. However, from L ≈ the entropies, S A , in the three sectors are very similar (left panel), althoughthe more sensitive indicators dS A /dL (middle panel) and inparticular S A − L × dS A /dL (right panel) show some dif-ferences. The density entropy in the bosonic More-Read stateappears to be about α ≈ . . In the case of fermions (Fig.8), we find that the scaling regime is not yet reached, eventhough one may make a crude estimate of the entropy density, α ≈ . .The entropy density of a state is an indicator of how chal-lenging it is to simulate the state on a classical computer,through a one-dimensional algorithm such as DMRG ,which has already been applied to the FQHE problem , orthrough recently-proposed true two-dimensional algorithmslike PEPS or MERA . The larger entropy densities ofMoore-Read states imply that they are more difficult to simu-late than the Laughlin states.Even for our largest system sizes, where we have obtaineddata for a range of L values ( N = 14 for bosons and N = 16 for fermions), we cannot extract a reliable topological entropy(see Figs. 7 and 8). However, we can observe some interest-ing phenomena. First, the entropy densities α = dS A /d (2 L ) of Moore-Read states are significantly larger than that of thefermionic Laughlin state at ν = 1 / and the bosonic Laugh-lin state at ν = 1 / . Second, the entropy properties ofbosonic Moore-Read states in the 11 sector and those in the02+20 sector become similar at large L . Their curves of S A , dS A /dL and S A − L × dS A /dL overlap after L ≈ .Let us also highlight some of the finite size features. Forsmall L the finite size convergence is essentially perfect andthe curves for different system sizes are on top of each otherin a given sector. At larger L , the curves show a strongerdependence on N s . The N s -dependence shows up first for thesmallest system size and at increasing L for progressivelylarger system sizes. This reflects the fact that, for any finite-size system, at very large L the edges of block A get tooclose and cannot be thought of as independent. In particular,once L exceeds some value, we enter the thick-torus limit,and the entanglement entropy goes to some ( N dependent)saturation value. Corresponding to the saturation of S A , the0 FIG. 7. (color online) The bosonic Moore-Read state entropy S A , its derivative dS A /dL and the intercept of its linear approximation S A − L × dS A /dL as functions of L for N = N s = 12 and N = N s = 14 in the 11 sector, 02+20 sector and 02-20 sector. The theoreticalvalue of the topological entropy γ = − ln 4 is indicated by the black line. In a rather large window of L , the entropy properties of theMoore-Read states in the 11 sector and 02+20 sector are quite similar. derivative dS A /dL drops to zero after some L . Therefore,the appropriate scaling regime of the entropy, S A , may be ex-pected to be valid only in a window of L , after the O (1 /L ) term is small enough but before S A saturates. This analy-sis was shown to provide excellent results for abelian FQHstates in Ref. 21. However, as already mentioned, we findthat the finite size corrections to the scaling are too large tofaithfully determine the topological part, γ , of the entropy forthe Moore-Read state. Given the limitations encountered alsoin other geometries, we conclude that an accurate and reliabledetermination γ for the Moore-Read state remains a challengefor the future. V. DISCUSSION
We have investigated the entanglement spectrum (ES) andthe von Neumann entropy of bosonic and fermionic Moore-Read states on the torus. The ES on the torus is much moreintricate and the analysis thereof poses a number of challengescompared to the sphere geometry where there is a single edgeand a unique ground state. One such challenge is that in agiven particle number sector, several towers appear due to pos-sible compensating charge transfer across the two boundaries.The study of the entanglement in this geometry is neverthe-less well motivated as it provides new insights, for instanceby connecting to the vicinity of the microscopically well un-derstood thin-torus limit, and also because it may provideguidance for future studies of entanglement in other many-body systems where no natural analog to the quantum Hallsphere exists. In particular, the recently suggested fractionalChern insulators are most naturally studied using periodicboundary conditions, i.e. on a torus.In this work we have suggested a procedure in order to re-solve the problem of the non-trivial ground-state degeneracyon the torus: we used exact diagonalization and choose to cal-culate the entanglement in the pure (simultaneous) eigenstatesof H , T and T q . This is different from the mixed-state recipeof Refs. 33 and 42, for which we expect a superimposed en-tanglement spectrum and a shifted prediction of the topologi- cal entropy γ .For the ES, we found a tower structure similar to, but sig-nificantly richer than, what was found earlier in the ES of theLaughlin state. We used two complementary ideas in orderto disentangle the ES by extending the results of Ref. 28 tonon-Abelian states. The first approach is based on a combi-nation of two chiral CFT edges. Each of these is individu-ally similar to the edge spectrum previously extracted fromES studies on the sphere. This interpretation is powerful as itreproduces the entire ES through the assignment of a few lev-els. It also reflects the intricate structure of the correlations inthe Moore-Read state: Even for one cut of our system, edgescorresponding to different topological sectors with differentcounting rules combine to form towers. Our second approachuses the adiabatic connection to the thin-torus limit: A per-turbation analysis away from the thin-torus states yields thelocations of the towers, and the counting rule of each edge en-vironment follows from a generalized exclusion principle inthe occupation number basis.A further difficulty encountered when disentangeling thetorus ES is the non-monotonic dispersion that appears forfermions at large L , as the lack of a natural vacuum levelat the bottom/center of each tower severely increases the diffi-culty of the assignment of the edge modes. In the present case,this difficulty can be circumvented by following the smoothdependence of the edge levels to the small L regime, wherethe dispersion is always monotonic.For the von Neumann entropy, we found that the area-lawentropy density, α = dS A /dL ≈ . − . (per magneticlength), is larger than in the Laughlin states for both bosonsand fermions. However, the comparable smallness of α is nev-ertheless encouraging regarding the possibilities of simulatingthe Moore-Read state using entanglement-based algorithms.Our results also show that an accurate and reliable determina-tion γ for the Moore-Read state on the torus remains a chal-lenge for the future. It is likely that alternative methods, suchas DMRG in a cylinder setup, will be needed to reach thisgoal.The generalization of the analysis given here for the Moore-Read state to more generic non-abelian FQH states should be1 FIG. 8. (color online) The fermionic Moore-Read state entropy S A , its derivative dS A /dL and the intercept of its linear approximation S A − L × dS A /dL as functions of L for N s = 28 and N s = 32 in the 0101 sector, 0110+1001 sector and 0110-1001 sector. Thetheoretical value of the topological entropy γ = − ln 8 is indicated by the black line. For the cut in the ± sector is equivalentto a translation of the cut in the ± sector. Therefore we make an average over their entropies and only show the averaged results(referred to as the ± sector in above). straight forward, but nevertheless interesting. The generaliza-tion to fractional Chern insulators is more challenging, but islikely to be rewarding. ACKNOWLEDGMENTS
Z. L. gratefully acknowledges the financial support fromthe MPG—CAS Joint Doctoral Promotion Programme (DPP)and the Max Planck Institute of Quantum Optics. E. J. B. issupported by the Alexander von Humboldt Foundation. E. J.B. and A. M. L. thank Juha Suorsa and Masud Haque for re-lated collaborations. H. F. is supported by the “973” program(Grant No. 2010CB922904).
Appendix A: Hamiltonian generating Moore-Read states
The bosonic and fermionic Moore-Read states on the torusare the unique zero-energy ground states of translational in-variant three-body interaction Hamiltonians H = (cid:88) { k } δ (cid:48) k + k + k ,k + k + k V { k } a † k a † k a † k a k a k a k , (A1)where { k } = k , k , k , k , k , k , a k ( a † k ) annihilates (cre-ates) a boson or a fermion in the state ψ k in Eq. (2), V { k } = + ∞ (cid:88) { s } , { t } = −∞ δ (cid:48) s ,k − k δ (cid:48) s ,k − k P ( { s } , { t } ) × exp (cid:110) − π L ( s + s + s s ) − π L ( t + t + t t ) (cid:111) × exp (cid:110) i πN s t (2 k − k + 2 s + s ) (cid:111) × exp (cid:110) i πN s t (2 k − k + s + 2 s ) (cid:111) , and δ (cid:48) is the periodic Kronecker delta function with period N s . P ( { s } , { t } ) is a certain polynomial of s , s , t , t , exactform of which depends on the targeted filling fraction.We use exact diagonalization to obtain the ground statesof (A1) after choosing a proper form of P . Up to a con-stant factor, when P = 1 , (A1) can generate the bosonicMoore-Read states at filling factor ν = 1 , while when P = − (4 π ) ( s /L + t /L )[( s − s ) /L + ( t − t ) /L ] ,(A1) can generate the fermionic Moore-Read states at fillingfactor ν = 1 / . Appendix B: Edge excitation of Moore-Read state
Compared with the Laughlin state, the edge excitations ofthe Moore-Read state are richer. It has one branch of freebosons and one branch of Majorana fermions obeying eitherantiperiodic (B1) or periodic boundary conditions (B2).For free bosons plus antiperiodic Majorana fermions, theexcitation spectrum is described by the Hamiltonian H APedge = (cid:88) m> [ E b ( m ) b † m b m + E f ( m − / c † m − / c m − / ] , (B1)where b and b † ( c and c † ) are standard boson (fermion) cre-ation and annihilation operators, E b ( m ) [ E f ( m ) ] is the dis-persion relation of bosons (fermions) and the total momen-tum operator is defined as K = (cid:80) m> [ mb † m b m + ( m − / c † m − / c m − / ] . The counting rule of the edge excita-tions, namely the number of energy levels at each K , de-pends on the parity of the number of fermions ( − F , F = (cid:80) m> c † m − / c m − / . For even F , the counting rule is , , , , , · · · at ∆ k = 0 , , , , , · · · ; while for odd F ,the counting rule is , , , , , · · · at ∆ k = 0 , , , , , · · · .Here ∆ k is defined as K − K where K is the lowest mo-mentum ( K = 0 for even F and K = 1 / for odd F ).For free bosons plus periodic Majorana fermions, the edge2 TABLE I. In this table, we analyze the counting rule of the edge environment A B in the 11 sector, which is , , , , , · · · at ∆ k =0 , , , , , · · · . ∆ k = 0 ∆ k = 1 ∆ k = 2 ∆ k = 3 ∆ k = 41111111111 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | TABLE II. In this table, we analyze the counting rule of the edge environment D B in the 02+20 sector, which is , , , , , · · · at ∆ k =0 , , , , , · · · . ∆ k = 0 ∆ k = 1 ∆ k = 2 ∆ k = 3 ∆ k = 40202020202 | | | | | | | | | | | | | | | | | | | | excitation Hamiltonian is H Pedge = (cid:88) m> [ E b ( m ) b † m b m + E f ( m − c † m − c m − ] , (B2)for which the total momentum is K = (cid:80) m> [ mb † m b m +( m − c † m − c m − ] . Through a similar analysis with thatfor the antiperiodic case, one can get that the counting ruleis , , , , , · · · at ∆ k = 0 , , , , , · · · for both even andodd F = (cid:80) m> c † m − c m − .The counting of each edge environment observed in our ES should be consistent with one of the four sectors here beforethe finite size effect truncates the series after some ∆ k de-pending on the system size. Appendix C: The counting rules of edge environments
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