c 2011 Internatonal Press Adv. Theor. Math. Phys. 15 (2011) 449 469 Cyclotron brad group approach to Laughln correlatons Janusz Jacak 1, Ireneusz Jóźwak 2, Lucjan Jacak 1 and Konrad Weczorek 1 1 Insttute of Physcs, Wroc law Unversty of Technology, Wyb. Wyspańskego 27, 50-370 Wroc law, Poland janusz.jacak@pwr.wroc.pl; lucjan.jacak@pwr.wroc.pl; konrad.weczorek@pwr.wroc.pl 2 Insttute of Informatcs, Wroc law Unversty of Technology, Wyb. Wyspańskego 27, 50-370 Wroc law, Poland reneusz.jozwak@pwr.wroc.pl Abstract Homotopy brad group descrpton ncludng cyclotron moton of charged nteractng two-dmensonal (2D) partcles at strong magnetc feld presence s developed n order to explan, n algebrac topology terms, Laughln correlatons n fractonal quantum Hall systems. There are ntroduced specal cyclotron brad subgroups of a full brad group wth 1D untary representatons sutable to satsfy Laughln correlaton requrements. In ths way an mplementaton of composte fermons (fermons wth auxlary flux quanta attached n order to reproduce Laughln correlatons) s formulated wthn unform for all 2D partcles brad group approach. The fcttous fluxes vortces attached to the composte fermons n a tradtonal formulaton are replaced wth addtonal cyclotron trajectory loops unavodably occurrng when ordnary cyclotron radus s too short n comparson to partcle separaton and does not allow for partcle nterchanges along sngle-loop cyclotron brads. e-prnt archve: http://lanl.arxv.org/abs/0910.4250
450 J. JACAK ET AL. Addtonal loops enhance the effectve cyclotron radus and restore partcle nterchanges. A new type of 2D partcles composte anyons s also defned va untary representatons of cyclotron brad subgroups. It s demonstrated that composte fermons and composte anyons are rghtful 2D partcles, not auxlary compostons wth fcttous fluxes and are assocated wth cyclotron brad subgroups nstead of the full brad group, whch may open also a new opportunty for non-abelan composte anyons for topologcal quantum nformaton processng applcatons, due to rcher representatons of subgroup than of a group. 1 Introducton Specfc topologcal propertes of two-dmensonal (2D) N-partcle systems have been recognzed wthn algebrac topology [1] usng homotopy group methods [2, 3]. They turned out to be of partcular sgnfcance n understandng of quantum behavor of 2D electron systems wdely expermentally nvestgated n Hall confguraton snce the dscovery of fractonal quantum Hall effect (FQHE) [4]. A herarchy of Landau level (LL) fractonal fllngs was observed [4 6] and explaned by new topologcal concepts closely related wth planar geometry [7 9]. The exceptonal topology of 2D systems s connected wth nontrval homotopy groups [2, 11 14] descrbng planar trajectores for many-partcle systems. Classes of topologcally nonequvalent closed loops n the confguraton space of a system of N dentcal partcles buld up π 1 homotopy group [1], called n ths case a brad group (full and pure for ndstngushable and dstngushable partcles, respectvely) [2, 11 13]. Brad groups for 2D case are nfnte, whle for hgher dmensons are fnte and equal (full brad groups) to the permutaton group S N (for 1D case ths formalsm s rrelevant) [3]. Untary representatons, n partcular one-dmensonal (1DURs) of the full brad group serve for dentfcaton of quantum partcles correspondng to the same classcal ones [9]. For S N there exst only two dstnct 1DURs: σ e π or σ e 0,(σ the nterchange of th and ( + 1)th partcles) correspondng to fermons and bosons, respectvely. In 2D there s, however, an abundance of dstnct 1DURs of the full brad groups: e θ, θ ( π, π], assocated wth anyons (Abelan) [5,7 9,15]. Anyons reveal a fractonal statstcs the nterchange of two anyons results n a θ phase shft of the wave functon [9, 10]. Crucal for understandng of FQHE was the formulaton by Laughln [7,8] of the wave functon for a ground state of 2D-charged partcle system at strong magnetc feld presence. The Laughln functon [8] corresponds to 1 p (p-odd nteger) fractonal fllng of the lowest LL and s a generalzaton of the Slater determnant. The Slater functon for completely flled lowest LL, for magnetc feld n cylndrcal gauge, has the form (up to an exponental factor) of the Vandermonde determnant, <j (z z j ), z = x + y stands here
CYCLOTRON BRAID GROUP 451 for th 2D partcle coordnate expressed as a complex number. Replacement n ths Slater functon of the Vandermonde determnant wth the Jastrow polynomal, <j (z z j ) p,(poddnteger), results n the Laughln wave functon [7,8] for fllng 1 p. The Laughln functon s stll antsymmetrc but dffers from the Slater functon n the phase shft acqured due to nterchange of a 2D partcle par. For the Vandermonde functon t s π, whle for the Jastrow functon pπ. The dfference n phases s mportant n planar geometry (n hgher dmensons the phase shft has no meanng), but 2π perodcty of the phase factor, e pπ = e π, also n 2D seemngly does not allow for dstngushng of the statstcs mposed by Laughln correlatons from ordnary fermon antsymmetry. Therefore more subtle topologcal atttude the brad group methods, should be here appled n order to grasp the novelty ntroduced by the Laughln functon. The phenomenologcal approach to Laughln correlatons was ntroduced n terms of composte fermons (CFs), regarded as ordnary 2D fermons wth assocated to each partcle even number of magnetc flux quanta [16,17]. The even number, q, of magnetc flux quanta attached to ndvdual partcles does not change antsymmetry of the total system wave functon, but due to Aharonov Bohm effect results n addtonal qπ phase shft durng partcle par nterchange [9]. In ths manner the magnetc feld local fluxes, called as vortces, attached to CFs model the Laughln correlatons [7, 8]. The CF atttude suffers, however, from an artfcal character of the constructon,.e., not explaned source of the magnetc feld fluxes changng fermons nto CFs. Nevertheless, CFs regarded as only weakly (resdually) nteractng, surprsngly well descrbe Hall systems [16, 17] especally wthn the lowest LL (for hgher LLs the nter-level mxng effects perturb a CF pcture). The vortces of CFs, orented oppostely to the external feld are assumed to be able to screen the external magnetc feld, and n the effectve weaker resultant feld, one can deal wth an nteger quantum Hall effect whch yelds p 2p±1 the fractonal herarchy [16, 17]. In other words, the oscllatons n Hall conductvty (FQHE) can be assocated wth Szubnkov de Haas oscllatons n an effectve reduced magnetc feld. The nterestng observaton supportng ths model s the so-called Hall metal state [18] at fllng fracton 1 2, when the total external magnetc feld should be canceled by the averaged nternal feld of CF fluxes. It stll arses, however, an mportant queston of what s the physcal source of these magnetc flux quanta,.e., vortces, attached to charged partcles whch alter orgnal fermons nto CFs and how to understand localzaton of magnetc feld fluxes on ndvdual partcles. In the present paper, we demonstrate the brad structure of CFs, as partcles wth statstcal propertes requred by Laughln correlatons, va assocaton them wth cyclotron brad subgroups nstead of the full brad groups. Introduced below cyclotron brad subgroups reflect the classcal
452 J. JACAK ET AL. brad pcture for 2D N-partcle charged system at the presence of magnetc feld. The quantzaton, va 1DURs of these cyclotron subgroups, allow for natural explanaton of Laughln correlatons, wthout nvokng artfcal vortces. In partcular, ths approach elucdates the CF constructon and the true character of auxlary Jan s vortces [16], whch turned out a useful model of basc trajectory loops unavodably occurrng on cyclotron brads at fractonal LL fllngs 1 p. The mult-loop brads from cyclotron brad subgroups allow for partcle nterchange n the brad pcture, when the sngleloop cyclotron dameter s shorter than the partcle separaton, whch precludes ther exchanges along sngle-loop cyclotron trajectores. In order to enhance cyclotron radus and to restore partcle nterchanges n brad pcture, each partcle must traverse, n classcal brad meanng, a closed p-loop cyclotron trajectory, or n quantum language, each partcle takes away p quanta of the external magnetc feld flux; p 1 of them play the equvalent role as p 1 flux quanta attached to each CF n a tradtonal model, reducng the external feld. Topologcal mplementaton of CFs n brad group terms was not prevously formulated due to perodcty of 1DURs. Assocaton of composte partcles (ncludng CFs) wth a separate cyclotron brad subgroups allows, however, for dstngushng them n terms of untary representatons, despte 2π perodcty of the untary factor. The paper s organzed as follows. In the next paragraph the man lnes of the brad group approach to quantum systems are summarzed. In the followng one, the orgnal dea of cyclotron brad subgroups s developed and appled to descrpton of CFs, and more generally to composte-anyons. The mult-loop structure of cyclotron brads, essental for CF descrpton, s explaned. The role of the Coulomb nteracton s descrbed n a separate paragraph. The possble applcaton of ntroduced composte anyons to topologcal quantum nformaton processng s ndcated. 2 Brad group method for descrpton of N-partcle systems 2.1 Defntons of a full and a pure brad groups Brad group s a frst homotopy group [1], π 1, for confguraton space of N-partcle system. π 1 (A) s a group of topologcally nonequvalent classes of closed trajectores n the space A. In the case of N-partcle system, A s an approprate classcal confguraton space. The brad groups dsplay only a possble classcal moton of N-partcle system and a quantzaton s performed va untary representatons of classcal brad trajectores, as t s descrbed below.
CYCLOTRON BRAID GROUP 453 The confguraton space of N dentcal partcles located on a manfold M (e.g., R n, or compact manfolds) s defned as: Q N (M) =(M N \ Δ)/S N, for ndstngushable dentcal partcles, and as: F N (M) =M N \ Δ, for dstngushable dentcal partcles; M N s the Nth Cartesan product of the manfold M, Δ s the set of dagonal ponts (when coordnates of two or more partcles concde), subtracted n order to preserve conservaton of the partcle number, S N s the permutaton group the quotent structure s ntroduced n order to account for ndstngushablty of quantum partcles. Note, that ndstngushablty of partcles s here artfcally ntroduced n the defnton of confguraton space, whch ndcates that ths property s ndependent of quantum uncertanty prncples. For these confguraton spaces two types of brad groups are defned [2]: a full brad group and a pure brad group. π 1 (Q N (M)) = π 1 (M N \ Δ)/S N ), (2.1) π 1 (F N (M)) = π 1 (M N \ Δ), (2.2) For M = R n, n>2 the brad group have a smple structure. The full brad group, for n>2, equals to a permutaton group S N (note, that ths group s a fnte group, of rank N!). For M = R 2 (and for compact locally 2D manfolds, as a sphere or a torus n 3D) the brad groups are nfnte hghly nontrval groups. It s convenent to llustrate a structure of the brad groups for the plane va a smple presentaton usng geometrcal brads [2, 12] cf. Fgure 1. In ths fgure there are depcted: (a) geometrcal brad correspondng to the generator σ of the full brad group (nterchange of the th and ( + 1)th strngs representng partcle trajectores), (b) geometrcal brad correspondng to the nverse element of the generator, σ 1, (c) geometrcal brad for the square of the generator (σ ) 2 e (e the neutral element of the group). In Fgure 1: Geometrcal brad presentaton for B N : (a) the generator σ ;and (b) ts nverse, σ 1 ; (c) square of the generator σ 2.
454 J. JACAK ET AL. 3D (σ ) 2 = e, whch smplfes the brad structure to ordnary permutaton group S N, whle n 2D (σ ) 2 e and t causes complcated (of nfnte type) structure of planar brads. One can lst formal condtons mposed on generators σ,=1,...,n 1, n order to defne the full brad group for the plane, n an abstract manner [2, 11]. These condtons are wrtten below, σ σ +1 σ = σ +1 σ σ +1, for 1 N 2, (2.3) σ σ j = σ j σ. for 1, j N 1, j 2. (2.4) The ntal orderng of partcles s not conserved for brads from a full brad group, whle for brads from a pure one, the orderng must be conserved. The generators l j of a pure brad group [2] correspond to double exchanges of partclepars, j, however, wthout any perturbaton of the assumed orderng of partcles, and have the followng form n terms of σ generators: l j = σ j 1 σ j 2 σ +1 σ 2 σ 1 +1 σ 1 j 2 σ 1 j 1, 1 j N 1. (2.5) The pure group s a subgroup of the full group snce the generators l j are expressed by means of σ generators. For defnng relatons for generators of the pure group cf. [2, 12]. Note that the connecton between the full brad group and the pure one s gven by the quotent relaton [2], B N /π 1 (F N (R 2 )) = S N (B N stands here for commonly used notaton for the full brad group for the plane) [11]. For the sphere S 2 the addtonal condton for generators, beyond those gven by equatons (2.3) and (2.4), s mposed [2], σ 1 σ 2 σ n 2 σ 2 n 1 σ n 2 σ 2 σ 1 = e, (2.6) whch dsplays the fact that on the sphere a loop of a selected partcle embracng all other partcles s contractble to a pont. For the torus T addtonal relatons [14] correspond to two nonequvalent paths of each partcle on ths not smple-connected manfold. 2.2 Quantzaton n brad group pcture Quantzaton of the system of N dentcal ndstngushable partcles can be performed by applcaton of the Feynman ntegral over trajectores, leadng
CYCLOTRON BRAID GROUP 455 to a propagator (probablty for a transton from a pont a toapontb n the confguraton space): I a b = dλe S[λ a,b]/, (2.7) where S[λ a,b ] s the classcal acton for the trajectory λ a,b n the classcal confguraton space of N-partcle system, dλ s a measure n a trajectory space. To each trajectory lnkng a and b ponts n the N-partcle confguraton space, one can attach, however, addtonal closed loops whch are elements of the full brad group. Thus resultng trajectores fall nto separated topologcally nonequvalent classes, represented by elements of the full brad group. Therefore an addtonal untary factor (the weght of the separated trajectory class) should be added [9, 10] n the formula for ntegraton over trajectores, together wth the addtonal sum over the brad group elements (snce each element of the full brad group can be attached to a loop-less smple trajectory λ a,b ): I a b = l π 1 e α l dλ l e S[λ l(a,b)]/, (2.8) π 1 represents here the full brad group. These factors e α l forma1durof the full brad group. Dstnct representatons correspond to dstnct types of quantum partcles, lnked to the same classcal ones. As was mentoned n the Introducton secton, for S N, whch s the full brad group for 3D manfolds (and for hgher dmensons), there exst only two dstnct 1DURs, { e 0, σ e π (2.9), correspondng to bosons and fermons, respectvely (leadng to a symmetry and antsymmetry propertes of relevant wave functons). For 2D space (the plane), the brad group (consderably rcher than S N ) has an nfnte number of 1DURs [12,13], wrtten for the group generators as σ e θ, θ ( π, π], where each θ enumerates a dfferent type of so-called anyons [5, 7 10, 15]. Note that elements of 1DUR of the full brad group do not depend on the ndex (of the generator σ ) owng to condton (2.3) mposed on generators. Because the 1DUR elements commute, then from equaton (2.3) t follows that e θ = e θ +1, where σ e θ, whch gves ths -ndex ndependence of 1DUR elements.
456 J. JACAK ET AL. For the sphere S 2 1DURs have the form [12,13], e θ, where θ = kπ/(n 1), k =0, 1, 2,...,2N 3. It s nterestng to notce, that for two partcles on the sphere (.e., for N = 2 one has only k =0, 1) only bosonc or fermonc statstcs are avalable (actually because of equaton (2.6)), and anyons may occur on the sphere for three partcles, at least. In the case of a torus T,for an arbtrary number N of partcles, θ =0orπ are admtted only [13,14] thus on a torus any anyons do not exst, except for fermons and bosons. Ths result was generalzed [13] also for all compact locally 2D manfolds wth excepton for the sphere. The classcal trajectores from the full brad group have no quantum meanng. Quantum partcles do not traverse any brad trajectores snce they do not have trajectores at all. In agreement wth the general rules of quantzaton [19, 20], N-partcle wave functon must transform accordng to 1DUR of an approprate element of the brad group when the partcles traverse classcally a closed loop n the confguraton space of N-partcle system correspondng to ths brad element. As brads from the full brad group descrbe nterchanges of partcles, thus correspondng 1DURs dsplay statstcs phase factors. Note that mportant are also mult-dmensonal untary rreducble representatons (MDURs) of brad groups. Accordng to an dea of Ktaev [15, 21], an arbtrary untary evoluton of mult-qubt system (e.g., of a double qubt gate for QIP) [15, 22] can be approxmated by an MDUR (of an approprate rank) of a full brad group, provded the suffcent densty level of MDURs n the untary matrx space [15]. MDURs can be lnked wth degenerated low-energy exctatons (quaspartcles/quasholes, typcally regarded as anyons) above the ground state for some fractonal LL fllngs. Snce elements of MDUR do not commute, as matrces, these degenerate states of anyons are referred as non-abelan anyons [15]. Unfortunately, the non-abelan anyons recently nvestgated n partcular low excted states for 5/2 and 12/5 LL fllng factors correspond probably to not suffcently dense MDURs (for non-abelan anyons n 5 2 case the MDURs are not dense enough to approxmate needed qubt gates [15], and another consdered now state 12 5 s stll dsputable [23]). Thus searchng for other opportuntes for fractonal statstcs systems wth more dense MDURs assocated wth non-abelan anyons s of hgh sgnfcance. In the next secton, we wll ntroduce a cyclotron brad subgroups of a full brad group. As subgroups have usually rcher representatons than a group, thus one can expect that the cyclotron brad subgroups would be convenent for topology methods for QIP, snce the relevant MDURs of cyclotron subgroups would be more dense n comparson to representatons of a full brad group.
CYCLOTRON BRAID GROUP 457 3 Cyclotron brad groups at magnetc feld presence Let us emphasze that the brad groups descrbed above are constructed n the absence of the magnetc feld. Elements of the full brad group were all trajectores wthout any modfcatons caused by the magnetc feld. Incluson of the magnetc feld consderably confnes, however, the varety of admtted trajectores. All trajectores must be of cyclotron shape at the presence of the magnetc feld and ths property hghly modfes the brad group structure. Instead of a full brad group, cyclotron trajectores form a brad subgroup a cyclotron subgroup, n partcular at 1/p fractonal LL fllng. It leads to an opportunty for an mplementaton of CFs (2D partcles at a strong magnetc feld presence) va cyclotron subgroups of the full brad group. Followng ths dea, at magnetc feld presence the summaton n the Feynman propagator must be confned to the subgroup elements only,.e., to selected, sutably to cyclotron moton, classes of trajectores nstead of arbtrary elements of the full brad group. The 1DURs of the cyclotron brad subgroups wll thus substtute the 1DURs of the full brad group n the path ntegral (2.8). Let us consder 2D-charged partcle system wth planar densty N S (N s the number of partcles, S s the surface of a sample) and at presence of a perpendcular magnetc feld B. Topology of a manfold where the partcles are located s assumed here the same as of the plane R 2 (t would be consdered as an compact subset of R 2, wthout a boundary) [2]. For ths manfold one can defne the full brad group [2,11 13], beng the π 1 homotopy group [1] of the confguraton space for N ndstngushable partcles on R 2. Ths brad group s commonly called as B N (the Artn group) [11] and s generated by nterchanges of neghborng partcles at chosen ther orderng [2, 12], σ, =1,...,N 1, (3.1) wth defnng relatons gven by equatons (2.3) and (2.4). 3.1 Defnton of the cyclotron brad subgroup Let us defne the cyclotron brad subgroup by means of ts generators b (p) of the followng form: b (p) = σ p, p =1, 3, 5, 7, 9,...; =1,...,N 1, (3.2) where each p corresponds to a dfferent type of the cyclotron subgroup and σ are generators of the full brad group.
458 J. JACAK ET AL. Fgure 2: (a) The generator σ of the full brad group and the correspondng relatve trajectory of partcle th and ( + 1)th exchange; (b) the generator of the cyclotron brad subgroup, b (p) = σ p (n the fgure p = 3), corresponds p 1 to addtonal 2 loops when the th partcle nterchanges wth the ( + 1)th one (an addtonal loop results n 2π phase shft; 2R 0 nter-partcle separaton). The full brad group element b (p) (the generator of the cyclotron brad subgroup of type p) represents the nterchange of th and ( + 1)th partcles wth p 1 2 loops. It s clear due to the defnton of the sngle nterchanges by the generators σ of the full brad group, cf. Fgure 2. The generators b (p) create the subgroup of the full brad group as they are expressed by generators σ of the full brad group. The b (p) do not, however, satsfy condton (2.3) (cf. Fgure 3(c)), whle condton (2.4) s mantaned for b (p) : b (p) b (p) j = b (p) j b (p), for 1, j N 1, j 2(cf. Fgure 3(d)). Condton (2.3) resulted n ndependence of 1DUR of the brad group generator ndex. Dsappearance of ths condton for the cyclotron brad subgroup leads to possble dependence of the subgroup 1DUR on the ndex, n general. The 1DURs of the full group, σ e α, confned to the subgroup, do not depend, however, on and yeld: b (p) e pα, (3.3) p an odd nteger, α ( π, π]. These 1DURs of the cyclotron brad subgroup, numbered by the pars (p, α), descrbe composte anyons, and, n a partcular case, CFs for α = π. 3.2 Mult-loop cyclotron brad structure For the above constructon of the cyclotron subgroups the N-partcle wave functon acqures an approprate phase shft due to a pecular type of partcle
CYCLOTRON BRAID GROUP 459 Fgure 3: Formal condtons defnng a full brad group for R 2, cf. equatons (2.3) and (2.4); volaton of condton (2.3) for the cyclotron subgroup generators b (3) (c) (condton (2.4) s mantaned for the cyclotron subgroup generators (d)). nterchanges n the brad pcture,.e., we replace the Aharonov Bohm phase of fcttous fluxes by addtonal brad loops (each loop adds 2π to the total phase shft cf. Fgure 2). It s notceable f one takes nto account the rules of quantzaton n the brad group framework [19, 20]. In agreement wth them, N-partcle wave functon must transform accordng to 1DUR of an approprate element of the brad group, when the partcles traverse classcally a closed loop n the confguraton space of N-partcle system correspondng to ths brad element. For the cyclotron brad subgroup generated by b (p), =1,...,N 1 (defned by equaton (3.2)), we obtan for partcle par nterchange the total wave functon phase shfts pπ (for α = π n the representaton gven by equaton (3.3)), as s requred by Laughln correlatons [7, 8], wthout modelng them by fcttous vortces. 3.3 Defnton of an ndvdual partcle cyclotron trajectory Note, that each addtonal loop of a relatve trajectory for partcle par nterchange (such a trajectory s needed for defnton of the subgroup
460 J. JACAK ET AL. Fgure 4: Cyclotron (left) and correspondng relatve (rght) trajectores for nterchanges of th and ( + 1)th 2D-partcles at strong magnetc feld, (a) for ν = 1; (b) for ν = 1 3, respectvely (3D for better vsualzaton); n both cases, ν =1, 1 3, the approprate cyclotron radus R c fts wth the nter-partcle separaton 2R 0 =2R c,2r 0 nter-partcle separaton s fxed by the Coulomb repulson. generators b (p) ) reproduces an addtonal loop n ndvdual cyclotron trajectores for both nterchangng partcles cf. Fgure 4. In ths fgure, the cyclotron moton of partcle par s depcted for the nterchange of th and ( + 1)th partcles separated by double cyclotron radus 2R c, wthout any addtonal loops (a) and wth the addtonal loop (b), respectvely. The cyclotron trajectores are repeated n the relatve trajectory (rght) wth a double radus n comparson to the ndvdual partcle trajectores (left). In quantum language, wth regard to classcal mult-loop cyclotron trajectores, one can conclude only on the number, BS N / hc e, of flux quanta per sngle partcle n the system, whch for the LL fllng 1 p s p,.e., the same as the number of cyclotron loops of each partcle. Thus a smple pctoral rule could be here formulated: an addtonal loop on a brad correspondng to partcle nterchange, ntroduced n accordance wth generators of the cyclotron brad subgroup, results n two addtonal flux quanta percng the ndvdual partcle cyclotron trajectores. It mmedately follows from the defnton of the cyclotron trajectory. One can defne ths trajectory as the ndvdual partcle trajectory correspondng to a double nterchange of the partcle par (cf. Fgure 5). In ths way, the cyclotron trajectores of both nterchangng partcles are closed, smlarly as closed the relatve trajectory for double nterchange s. If the nterchange s smple,.e., wthout any addtonal loops, the correspondng ndvdual partcle cyclotron trajectores are also smple, sngle-loop
CYCLOTRON BRAID GROUP 461 Fgure 5: Indvdual partcle closed cyclotron trajectores correspondng to double relatve trajectores for nterchanges of th and ( + 1)th 2D-partcles at strong magnetc feld, (a) for ν = 1; (b) for ν = 1 3, respectvely (3D for better vsualzaton); the number of B feld flux quanta per partcle s ndcated n both cases, ν =1, 1 3 ; the resultng cyclotron radus R c fts wth the nter-partcle separaton 2R 0 =2R c n both cases. (crcles on 2D plane). But when the nterchange of partcles s mult-loop, as assocated wth p-type cyclotron subgroup (p >1), the double nterchange relatve trajectory has 2 p 1 2 + 1 closed loops and the ndvdual cyclotron trajectores are also mult-loop, wth p loops. It s llustrated n Fgure 5. It s worth to emphasze the dfference between turns of wndngs (e.g., of a wre) and mult-loop 2D cyclotron trajectores. The latter ones cannot enhance a percng total magnetc feld flux BS (thus the number of flux quanta per partcle concdes wth the number of loops of closed cyclotron ndvdual partcle trajectory), whle n the former case, each turn of wndngs adds a new porton of the flux as a new turn adds a new surface n fact (whch s no case n 2D). 3.4 Relaton of cyclotron brad subgroups wth CFs We wll explan below that the mult-loop shape of the relatve trajectory for nterchanges, as defned by the subgroup generators (3.2) (and correspondng mult-loop form of ndvdual partcle cyclotron trajectores), s an unavodable property n the case when nter-partcle separaton (resulted from the densty N S and fxed by the Coulomb repulson) s greater than the double value of sngle-loop cyclotron radus. In ths case, n partcular at 1 p LL fllng fracton, any exchanges along smple sngle-loop cyclotron trajectores are mpossble, because the correspondng cyclotron radus s too short. In order to restore a possblty of partcle nterchanges (necessary, on the other hand, for brad structure defnton and thus for statstcs determnaton), too short cyclotron radus must be enhanced. The way to enhance the effectve cyclotron radus, whch would agan ft to nter-partcle separaton, s the mult-loop character of cyclotron moton and smultaneously resultng mult-loop brads for partcle nterchanges (represented by generators of the
462 J. JACAK ET AL. cyclotron brad subgroups, equaton (3.2)). The addtonal cyclotron loops take away a part of the external feld flux and thus reduce the effectve feld whch leads to an expected growth of a resultng cyclotron radus. The total flux of the external feld through the surface S s BS. For p type of CFs, f one consders the relatve trajectory of double nterchange of th and ( + 1)th partcles (thus closed and wth 2 p 1 2 +1=p loops), one gets the ndvdual partcle closed cyclotron trajectores wth the same number p of loops (cf. Fgure 5), embracng the total flux p hc e (each loop takes away a sngle flux quantum n accordance wth the above presented nterpretaton). Thus for p type of CFs we deal wth closed p-loop cyclotron trajectores of partcles,.e., p flux quanta per partcle, BS = Np hc e. On the other hand, the degeneracy of the LL equals to N 0 = SBe hc, (neglectng spn) and for fractonal fllng ν, N 0 = N ν. BS N = hc 1 e ν gves 1 ν flux quanta per partcle, whch fts wth the prevous estmaton only for ν = 1 p. In the case of p-loop trajectory each loop has ts sze adjusted to the external magnetc feld flux dmnshed by p 1 quanta per partcle taken away by remanng loops, exactly as n the case of the Jan s model. Indeed, f BS = hc e pn, then hc e = B p S N and S N corresponds to p tmes lowered feld. Followng an analogy wth Jan s model, one could argue that for ν = 1 2 and p = 3, two loops per partcle take away the total B feld flux and the thrd loop has to be of nfnte radus (Hall metal [18]) for zero rest-feld. The addtonal loops take away flux quanta smultaneously dmnshng the feld; ths gves an explanaton of the fcttous Jan fluxes screenng the feld B. Thus the presented cyclotron subgroup mplementaton of CFs can be addressed to Jan s theory wth all advantages of the related conclusons [17], n partcular of the nteger quantum Hall effect n the rest-feld, leadng to herarchy of FQHE [16, 17]. One can thus summarze why 2D-charged partcles must be assocated wth classcal mult-loop brads for felds correspondng to fractonal fllng of LL. For ν = 1 one has R c = R 0 (where R c s the cyclotron radus, πrcb 2 = hc e and 2R 0 s the separaton of partcles, adjusted to the densty and fxed by the short-range part of the Coulomb repulson, πr0 2 = S N ). For ν<1 the radus of cyclotron trajectory wthout addtonal loops R c <R 0, and then R c s too short for partcle nterchange along these trajectores. Addtonal loops can, however, enhance R c and agan allow nterchanges, snce for p-loop cyclotron trajectores hc e = πr2 c B p,andr c grows n comparson to sngle-loop trajectores; for ν = 1 p, agan R c = R 0, though the external feld s p tmes bgger than for ν = 1 (at constant N). The fcttous fluxes of Jan s CFs played actually the smlar role leadng to an ncrease of cyclotron radus
CYCLOTRON BRAID GROUP 463 n the reduced resultant feld. One can conclude thus that for ν = 1 the cyclotron trajectores are sngle-loop and brads are generated by b (p=1) = σ, whle for ν = 1 p, p>1, the cyclotron trajectores must be mult-loop, smultaneously resultng n brads generated by b (p) = σ p. Note fnally that for a fxed magnetc feld orentaton the one-sde cyclotron rotaton s admtted, thus the cyclotron subgroup should be confned to ts semgroup structure only. It does not cause, however, any perturbatons of relevant 1DURs of cyclotron subgroups, whch are crucal for dentfcaton of composte partcles. The addtonal loops assocated wth the approprate subgroup generators lead to the phase shfts for partcle nterchanges, just as for Jan s CFs and permt correspondng Laughln-type functon requrements to be satsfed. These loops replace the fcttous screenng fluxes. Note once more that mult-loop trajectores (smlarly as sngle-loop ones) have only meanng n classcal brad terms. Quantum partcles do not traverse any trajectores, also any mult-loop cyclotron trajectores. The correspondng wave functons transform, however, n an agreement wth 1DURs of the brad group or of the subgroup [19, 20], resultng n approprate statstcs behavor. Let us emphasze that though CFs actually are not compostons of partcles and vortces, we have not modfed the orgnal name composte fermons. Moreover, we use the smlar name composte anyons for partcles assocated wth fractonal 1DURs (.e., wth fractonal pα n equaton (3.3)) of the cyclotron subgroup nstead of the full brad group. The phase shft θ can be calculated as the Berry s phase along closed trajectory n confguraton space for model mult-partcle wave functon correspondng to lowenergy exctatons above the ground state at fractonal fllng of LL. These exctatons quaspartcles/quasholes were tradtonally assocated wth anyons n the case of a fractonal Berry s phase. It s, however, clear that t s mpossble to dstngush between fractonal θ and pα both these phase shfts can be the same fracton. As consdered quaspartcles/quasholes are exctatons at the magnetc feld presence, thus these states should be rather assocated wth cyclotron brads 1DURs, and therefore are composte anyons and not ordnary anyons, as prevously regarded. Ths change, anyons for composte anyons, would result n convenent for QIP more dense relevant MDURs correspondng to brad subgroup nstead of the full brad group. 3.5 A role of the short-range part of the Coulomb nteracton The crucal character of the short-range part of the Coulomb nteracton for Laughln correlatons s vsble from the fact that the Laughln functon s an
464 J. JACAK ET AL. accurate ground state wave functon at 1 p LL fllng, f to confne the Coulomb nteracton represented by the so-called Haldane s pseudopotental [6, 24], V = >j m V mpm, j (Pm j s the projector on the states of th and jth partcles wth relatve angular momentum m), to the components V m,wth m =1,...,p 2 only. These V m terms, the Coulomb nteracton energy of an partcle par wth relatve angular momentum m, contrbute the shortrange part of the nteracton of electrons, and the remanng terms longrange nteracton tal, correspondng to greater partcle separaton,.e., wth m = p,..., do not nfluence strongly the Laughln functon [6, 24, 25]. The Laughln correlatons are assocated wth the ncompressble states whch correspond to dscrete spectrum of Coulomb nteracton projected on LL states,.e., nteracton expressed n terms of Haldane s pseudopotental wth components assgned by relatve angular momentum of partcle pars. Ths property, essental for FQHE, was even named by Laughln as a quantzaton of partcle separaton [6, 8]. Quantzaton of the Coulomb nteracton after projecton on relatve angular momentum of partcle pars n LL Hlbert subspace results n ncompressble FQHE states numbered by ntegers (egenvalues of relatve angular momentum of partcle pars), the same whch occur n the Laughln functons (the exponent n the Jastrow polynomal). It s mportant to note that accordng to an atttude to FQHE usng Haldane s pseudopotental (confned to the short-range part of the Coulomb repulson), the Laughln correlatons revealed n the mult-partcle wave functon are unambguous possblty for accurate ground-state at fractonal LL fllng, not only a varatonal result of the ground state modelng [6,24]. It supports an dea that Laughln correlatons are a fundamental topology-orgnated property of nteractng charged 2D partcles. One can thus expect that ths Landau quantzaton behavor of nteractng 2D-charged system must also manfest tself wthn brad group quantzaton approach to the same system, va the ntroduced cyclotron subgroup structure. Snce the Laughln correlatons can be expressed wthn CF approach, thus the Coulomb repulson (the short-range part of Haldane s pseudopotental) s of a fundamental sgnfcance also for the CF constructon. It should be, however, emphaszed that the Coulomb nteracton wth the dscrete spectrum,.e., wth separaton by energy gaps the dstnct relatve angular momenta of partcle pars for suffcently hgh magnetc feld (notceable va projecton of the nteracton on fractonal flled LL as n the defnton of Haldane s pseudopotental) does not play a role of standard dressng of partcles wth nteracton, typcal for quaspartcles n solds, just because the nteracton has not a contnuous spectrum n ths projecton. An effectve descrpton of a local gauge feld attached to partcles s suppled by the Chern Smons (Ch S) feld theory (chral feld,.e., breakng
CYCLOTRON BRAID GROUP 465 tme reverson and party). Ths approach revved [26, 27] n the area of FQHE successfully descrbng partcles wth fluxes, n partcular anyons and Jan s CFs [17]. It stll, however, does not explan, what the spontaneously arsng fluxes are. It was demonstrated [6, 25] that the short-range part of the Coulomb nteracton stablzes CFs aganst acton of Ch S feld (ts anthermtean term), whch mxes states wth dstnct angular momenta wthn LL [25] n dsagreement wth the Jan s CF model n Ch S feld approach [17, 25]. The Coulomb nteracton removes the degeneracy of these states and results n energy gaps whch stablze CF pcture, especally effectvely wthn the lowest LL. For hgher LLs CFs are not so useful due to possble mxng between LLs nduced by nteracton [28]. The short-range part of the Coulomb nteracton stablzes CFs also n our descrpton, smlarly as t removes nstablty caused by Ch S feld for angular momentum orbts n LL [25]. Indeed, f the short-range part of the Coulomb repulson was reduced, the separaton of partcles would not be rgdly kept (adjusted to a densty only n average) and then another cyclotron trajectores, addtonal to those for fxed partcle separaton (multloop at ν = 1 p ), would be admtted, whch volates the subgroup constructon. Thus the short-range part of the Coulomb nteracton turns out to be crucal for CF formaton n any descrpton. Confnng of the full brad group to the subgroup wth mult-loop structure of cyclotron moton s justfed only for partcle separaton adjusted to the double cyclotron radus. It s a role of the short-range part of the Coulomb repulson whch does not allow closer nter-partcle separaton than that whch follows form the densty. In ths manner the short-range part of the Coulomb nteracton s nvolved n the constructon of the cyclotron brad structure. The long-range tal of the Coulomb nteracton s left as a resdual nteracton of partcles, whch agrees wth the Jan s model of weakly nteractng CFs [16, 17]. 4 Conclusons We have developed the brad group descrpton for the case of N-charged 2D partcle system at strong magnetc feld presence, va defnton of the cyclotron brad subgroups. Ths formalsm allowed for nterpretaton of the Laughln correlatons of 2D-charged systems wthn the brad group approach to N partcle quantum systems. In ths manner we formulated a new mplementaton of CFs employng brad group methods. Brad descrpton of CFs was not prevously establshed because of 2π perod of 1DURs.
466 J. JACAK ET AL. In the present paper we have avoded ths problem va reducton of 1DURs to specally chosen brad subgroups selected n accordance wth a 2D cyclotron moton. These cyclotron brad subgroups, generated by the new generators = σ p (p =1, 3, 5,... enumerates a sort of composte anyons, σ are generators of the full brad group), are separated brad objects whch allow for dstngushng n statstcs of CFs (wth p>1) from ordnary fermons. It supports an dea that CFs are rghtful 2D quantum partcles whch cannot be mxed wth ordnary fermons, or wth other sorts of CFs (although all correspond to antsymmetrc functons). Dstngushng of CFs from fermons s mportant n partcular for numercal dagonalzaton of nteracton of CFs (not all antsymmetrc functons can be admtted n dagonalzaton procedure, but only those whch have the same phase shft due to partcle nterchanges, unless the mxng of varous sorts of CFs took place [ths s prohbted, smlarly to the mxng of fermons and bosons n 3D]). b (p) CFs turn out thus to be real 2D partcles and not quaspartcles,.e., they are not fermons dressed wth nteracton only, but are arranged as separate partcles n topologcal terms. Identfcaton of the specal brad group object, the subgroup of the full brad group, assocated wth CFs, resolves also the problem of fcttous magnetc flux quanta, vortces, attached to these partcles wthn the standard Jan s model. The Aharanov Bohm phase shfts caused by hypothetcal fluxes are replaced wth the phase shfts due to addtonal p 1 2 loops durng nterchanges of partcles (descrbed n classcal brad terms). These loops are an unavodable property of nterchanges of unformly dstrbuted (due to the Coulomb repulson) 2D partcles n a strong external magnetc feld when ordnary cylotron radus s too short for partcle nterchanges (each partcle traverses, n a classcal brad pcture, a closed p-loop cyclotron trajectory or n quantum language, t takes away p quanta of the B feld flux; p 1 of them play the equvalent role as p 1 screenng flux quanta attached to each CF n Jan s model). The 1DURs, b (p) e pα, α ( π, π], of the cyclotron brad subgroups generated by b (p) (p odd nteger) supply, more generally, an mplementaton of composte anyons, ncludng CFs of rank p, forα = π. In partcular, CFs (for α = π) gan the phase shft pπ (due to the addtonal loops) the same as requred by Laughln-type correlatons. The composte partcles wthn the presented mplementaton are thus not connected wth the full brad group but wth ther cyclotron subgroups. It makes CFs descrbed rghtfully wth other types of 2D quantum partcles wthn the unform brad group approach, despte the 2π perod lmtaton for 1DURs. An mportant role of the short-range part of the Coulomb nteracton s ndcated. Ths nteracton fxes the nter-partcle separaton, (only n
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