Duality and the Modular Group in the Quantum Hall Effect Brian P. Dolan Department of Mathematical Physics, National University of Ireland, Maynooth, Ireland and Dublin Institute for Advanced Studies, 10 Burlington Rd., Dublin, Ireland 9 e-mail: [email protected] 9 (Revised version: 25th December 1998) 9 1 We explore the consequences of introducing a complex conductivity into the quantum Hall effect. n This leads naturally to an action of the modular group on the upper-half complex conductivity a plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding J states,commuteswiththerenormalisationgroupflow,wederivemanypropertiesofboththeinteger 8 and fractional quantum Hall effects including: universality; the selection rule |p1q2−p2q1|=1 for 2 transitions between quantumHall states characterised by filling factors ν1 =p1/q1 and ν2 =p2/q2; critical values of the conductivity tensor; and Farey sequences of transitions. Extra assumptions ] abouttheformoftherenormalisation groupflowleadtothesemi-circlerulefortransitionsbetween l l Hall plateaux. a h - PACS nos: 73.40.Hm, 05.30.Fk, 02.20.-a s e m The purpose of this letter is to explore the conse- is the Landau level addition transformation of [4] while quencesoftheproposal,madein[1]andexaminedfurther X : ν → ν is the flux attachment transformation. t. 2ν+1 a in [2] [3], that the hierarchicalstructure of the zero tem- The particle-hole transformation ν → 1−ν, can be re- m perature integer and fractional quantum Hall effects can alised as the outer auto-morphism σ → 1−σ¯ acting on - be interpreted in terms of the properties of a subgroup the upper-half plane, where σ¯ = σxy −iσxx (it will be d of the modular group, Sl(2,Z) := Γ(1) — specifically assumedthroughoutthatthe electronspins aresplit, for n the subgroup which consists of elements of Γ(1) whose the spin degenerate case one must re-scale σ →2σ). o bottomleft entry iseven, sometimesdenotdΓ (2)inthe The upper-half σ-plane can be completely covered by c 0 [ mathematical literature. This group acts on the upper- copies of a single ‘tile’, or fundamental region (see e.g. half complex plane, parameterised by the complex con- [6]), related to each other by elements of Γ (2). The 2 0 v ductivity,σ =σxy+iσxx,inunitsof eh2,andisgenerated fundamental region has cusps at 0 and 1, linked by a 1 by two operations, T : σ → σ +1 and X : σ → σ . semi-circle of unit diameter, and consists of a vertical 2σ+1 7 a b strip of unit width constructed above this semi-circle. 1 If γ = (cid:18)2c d(cid:19) ∈ Γ0(2), with a,b,c, and d ∈ Z and By assumption all allowed quantum Hall transitions are 5 images of the transition ν = 0 → ν = 1 under some 0 ad−2cb = 1, then γ(σ) = aσ+b . Thus T = 1 1 γ ∈Γ (2), and hence also linked by a semi-circle. 8 2cσ+d (cid:18)0 1(cid:19) 0 9 Each such semi-circle has a special point, in addition 1 0 / and X = . Some consequences of this assump- to the end points, which is a fixed point of Γ (2) in the t (cid:18)2 1(cid:19) 0 a following sense. The point σ = 1+i is left fixed by m tionforthe phasediagramintheσ-planewereexamined ∗ 2 1 −1 in [2] and in the second of these references the author γ = . Similarly the points obtained from σ d- notes that there is a connection with the work of Kivel- ∗ (cid:18)2 −1(cid:19) ∗ bytheotherelementsofΓ (2),σ :=γ(σ ),areleftfixed n son,LeeandZhang[4],butremarksthatthecomparison 0 γ∗ ∗ o between [2] and [4] is not immediate. One of the aims of by γγ∗γ−1. It can be shown that the imaginary part, c ℑ(σ )≤ 1 or ℑ(σ )=∞, ∀γ. The points σ can be in- : this paper is to explore the relation between these two γ∗ 2 γ∗ γ∗ v approaches. terpretedascriticalpointsforthetransitionγ(0)↔γ(1) i if we further assume that the action of Γ (2) commutes X Following [1]— [3], it will be assumed that the phase 0 withtherenormalisationgroup(RG)flow. Forifσ were diagramforthequantumHalleffectcanbe generatedby γ∗ r a the action of Γ (2) on the upper-half σ plane. This im- notaRGfixedpoint,wecouldmovetoaninfinitesimally 0 closepointφ(σ )6=σ withaRGtransformation,φ. De- mediately implies the ‘law of corresponding states’ of [4] γ∗ γ∗ mandingγ◦φ(σ )=φ◦γ(σ )=φ(σ )thenimplies that and [5]. At Hall plateaux we have σxx = 0 and σxy = s γ∗ γ∗ γ∗ φ(σ ) is also left invariant by γ. But the fixed points where s is a ratio of two mutually prime integers, with γ∗ of Γ (2) are isolated, so there is no other fixed point in- odd denominator (note that s is being used here to la- 0 finitesimally close to σ . Hence φ(σ )=σ and σ must bel the quantum phases and is denoted by sxy in [4]). γ∗ γ∗ γ∗ γ∗ be a RG fixed point. The end points of the arches, at Plateauxcanberelatedtoeachotherbyrepeatedaction σ = ν with ν = p/q rational, are also fixed points of of T and X. At the center of the plateaux, the filling Γ (2). Forq oddthesearestableHallstates. Note,how- factor, ν, is equal to the ratio s=p/q and T :ν →ν+1 0 1 ever that a fixed point of the RG need not necessarily to this rule would be a transition from ν = n → ν = m be a fixed point of Γ (2) — but there is no experimental (n,m∈Z), which could occur by going first from σ =n 0 evidence of such extraneous fixed points of the RG. to σ =n+i∞ and then in to σ =m from σ =m+i∞. Thus the fixed points of Γ (2) must be fixed points of In a real experiment the maximum value of |σ| would 0 the RG,i.e. criticalpoints. Thisleadstothe topologyof presumably be finite, due to impurities. theflowdiagramof[2],reproducedhereinfigures1and2 One can determine sequences of allowedtransitions as where solid lines represent phase boundaries and dashed follows. Suppose ν = p /q , with q odd, is an allowed 0 0 0 0 lines represent quantum Hall transitions. This implies state, with p and q relatively prime. Consider the se- 0 0 the flow diagram proposed in [7], with its experimental quence ν = kn+p0 := pn for n,k,l ∈ Z, where l is even n ln+q0 qn support [8] and is also compatible with the phase dia- (so that q is odd). Then p q − p q = ±1 ⇔ n n+1 n n n+1 gramderivedin [4]. That σ∗ = 1+2i is a critical point for kq0−lp0 =±1. Thus a transition νn+1 →νn is allowed the lowestLandaulevel was arguedin [9]. Phasebound- provided|kq −lp |=1. Inthiswaywecan,forexample, 0 0 aries and transitions are represented by semi-circles in generate the three sequences the figures, but this is not forced on us by the Γ (2) 0 1 2 3 4 5 6 symmetry. They could be distorted from this geometry, → → → → → →... 3 5 7 9 11 13 provided that all phase boundaries are copies of a dis- torted‘fundamental’phaseboundary(runningfrom 1 to 2 7 6 5 4 3 2 1 +i∞) under the action of Γ (2). Similarly the dashed ...→ → → → → → →1 2 0 13 11 9 7 5 3 transition trajectories must all be copies of a distortion of the ‘fundamental’ arch spanning 0 and 1. Note, how- 2 5 8 11 everthatthefixed points are immovable. Ausefulaspect → → → →... (1) 3 7 11 15 of the semi-circular arches used in the figures is that the plus higher sequences obtained by adding an integer to intersection of any solid phase boundary with a dashed each term in these sequences. Such sequences are called transition is a fixed point of Γ (2), as are the end points 0 Fareysequencesandtheirrelevancetothe quantumHall of the arches (which are rational numbers or points at effectwasexaminedin[10]. Notethatagivenexperiment σ = r + i∞ for integral or half-integral r). Any dis- may jump from one sequence to another. Thus tortion from semi-circular geometry must leave the end points and intersections of phase boundaries and transi- 3 2 5 ...→ → → →... tion trajectories pinned at the fixed points of Γ0(2). 5 3 7 As in[4],the phasediagramgeneratedby Γ (2)deter- 0 is observed in [11]. mines which transitions are allowed and which are not. Eachtransitioncontainsacriticalpointgivenbyγ(σ ). Thus, for example, s : 1 → 0 is allowed while s : 1 → 1 ∗ 3 3 7 a b is not. All allowed transitions are generated by acting Thus if γ = , the critical point is at (cid:18)2c d(cid:19) on the arch passing through σ = 0 and σ = 1 by some, γ ∈Γ (2). This allows the derivation a selection rule for 2ac+2bc+ad+2bd+i (p q +p q )+i a tran0sition s1 = p1/q1 → s2 = p2/q2, where q1 and q2 σγ∗ = 2d2+4cd+4c2 = 1 (1q12+2q222) (2) are odd, and the pairs p and q (i = 1,2) are relatively i i when the transition goes from ν = γ(0) = b/d = p /q prime (for brevity we shall not always distinguish below 1 1 1 to ν = γ(1) = a+b = p /q . The parameters of γ can between s, labelling the quantum Hall phase, and ν, the 2 2c+d 2 2 be related to physical parameters as follows. Following fillingfactor,exceptwherenecessaryforcomparisonwith [5], let η be the effective charge of the quasi-holes of a [4]—ontherealaxis,whenσ =0,theyarethesame). xx Hall state, e = η, θ the statistical parameter (i.e. the We shall see that a transition is allowed if and only if ∗ phaseofthe twoquasi-particlewavefunction changesby p q −p q =±1. 1 2 2 1 πθwhenthepositionsofthetwoparticlesareexchanged) From the above assumptions we have (relabeling if necessary) ν = γ(0), ν = γ(1). Thus p1 = b and and s be the Hall state, with magic filling factor ν = s. 1 2 q1 d Thenthecriticalconductivityforatransitionfroms=ν a b pq22 = 2ac++bd where γ = (cid:18)2c d(cid:19) ∈ Γ0(2). Since to s′ = ν − η2/θ is given by equation (26) of [5] (in dimensionless units) ad − 2cb = 1, b and d are mutually prime, by an el- ementary result of number theory, hence (taking plus η2 η2 σ = , σ =s−θ (3) signswithout lossofgenerality)b=p1,d=q1. Similarly xx 1+θ2 xy 1+θ2 (a+b)d−(2c+d)b=1 implies that a+b and 2c+d are Equating these with the critical values in equation (2), mutually prime, hence a+b=p and 2c+d=q . Thus 2 2 there are two possibilities, depending on whether ν = p −p p γ = 2 1 1 and the condition detγ = 1 then γ(1) or γ(0), (cid:18)q2−q1 q1(cid:19) requires p q −p q = 1. The only possible exception a+b p d q 2 1 1 2 i) ν = = 2 , θ = = 1, 2c+d q 2c+d q 2 2 2 η2 = (2c+1 d)2 = q122 (4) σpe=rim1e+n2i√ta3l,lyu.nTdhere γap∈peΓa(r1a)n,cwehoifchΓ0a(r2e)nisotduoebsteorvthede eexx-- tension of Kramers-Wannier duality σ →1/σ to the xx xx whole complex plane. It was argued in [17] that this b p (2c+d) q 1 2 ii) ν = = , θ =− =− , extension leads naturally to Γ(1) acting on the upper- c p d q 2 1 half complex plane, for a coupled clock model. This was applied to the quantum Hall effect in [18] and [19]. It 1 1 appearsto havebeen notedfirstin[1]that the subgroup η2 = = . (5) d2 q12 Γ0(2) has the special property of preserving the parity of the denominator for rational ν = p/q. The subgroup Inbothcaseswereproducetheresult,thatη =±1/q,[12] Γ(2),consistingofallelementsofΓ(1)whicharecongru- and[13]. Noteinpassingthatthetransitionfrombosonic entto the identity, mod 2, wasalsoconsideredin[2]and to fermionic conductivities givenby equation(14) of ref- has been further investigated in [20]. Note however that erence [4] is implemented by the action of an element there is no element of Γ(2) which leaves σ = 1+i fixed, ∗ 2 of Γ(1) which is not in Γ0(2). Thus σ = γ(σ(b)) where indeed there is no element of Γ(2) which leaves any σ σ(b) = σ(b) + iσ(b) is the complex conductivity of the with ∞>ℑ(σ)>0 fixed. xy xx η2−θs s Itisapleasuretothank JanPawlowskifordiscussions bosonic Chern-Simons action and γ = 1 . η (cid:18) −θ 1(cid:19) about the RG flow of the quantum Hall effect. The above discussion gives the explicit connection be- tween the Chern-Simons analysis of [4] and the group References theory analysis of [2]. Wemakeafinalcommentaboutthe‘semi-circle’lawof reference [14] - [16]. By assumption, each quantum Hall transitioncanbeobtainedfromtheonebetween0and1, passingthroughσ = 1+i,bytheactionofsomeelement ∗ 2 of Γ (2). Since Γ (2) maps semi-circles built on the real 0 0 [1] C. A. Lu¨tken and G. G. Ross, Phys. Rev. B48, 2500 axisintoothersuchsemi-circleswecandeducethe‘semi- (1993) circle law’ of reference [14] - [16] by making one extra [2] C. A.Lu¨tken, Nuc.Phys. B396, 670 (1993); assumption—thatthe‘fundamental’archbetween0and J. Phys.A :Math. Gen. 26, L811-L817 (1993) 1 is a semi-circle. This implies that all other transitions [3] C. P. Burgess and C. A. Lu¨tken, Nuc. Phys. B500, 367 are semi-circles andallows predictions to be made of the (1997) maximumvaluesofσxxandρxxinanyallowedtransition, [4] S. Kivelson, D-H.Leeand S-C.Zhang, Phys. Rev. B46, ν → ν , as well as the values of σ and ρ at which 1 2 xy xy 2223 (1992) theyoccur. Thusthemaximumvalueofσ isatσmax = xx xx [5] D-H. Lee, S. Kivelson and S-C. Zhang, Phys. Rev. Lett. ν1−2ν2, where σxy = ν1+2ν2 (where ν1 > v2). In general, 68, 2386 (1992) this does notcoincide withthe criticalvalueσ =γ(σ ), γ∗ ∗ [6] R. A. Rankin, Modular Forms and Functions, C.U.P. except for the integer transitions (table 1). (1977) The maximum value of ρ is found by constructing the semi-circlethrough 1 anxdx 1 (providedneither van- [7] D. E. Khmel’nitskii, Pis’ma Zh. Eksp. Teor. Fiz 38, 454 ν1 ν2 (1983) (JETP Lett. 38, 552 (1983) ishes). Thusρmax = 1 1 − 1 ,whereρ = 1 1 + 1 . xx 2 ν2 ν1 xy 2 ν2 ν1 [8] H. P. Wei, A.M. Chang, D. C. Tsui, A. M. M. Pruisken Some representative ex(cid:0)amples(cid:1)are shown in tab(cid:0)le 1. (cid:1) and M. Razeghi, Surf.Sci. 170, 238 (1986) Tosummarise,assuming(asin[3])thatthe phase and [9] Y. Huo, R. E. Hetzel and R. N. Bhatt, Phys. Rev. Lett. flow diagram for the upper-half complex conductivity 70, 481 (1993) plane can be generated from an action of Γ (2) which 0 [10] J. ZangandJ.L. Birman,Phys.Rev.B47,16305 (1993) commutes with the RG, one deduces: (i) that all crit- ical points are given by γ σ , where σ = 1+i, with [11] R. Willett, J. P. Eisenstein, H. L. St¨ormer, D. C. Tsui, ∗ ∗ 2 A. C. Gossard and J. H. English, Phys. Rev. Lett. 59, γ ∈ Γ0(2); (ii) the phase di(cid:0)agr(cid:1)am of [4], [2] and [8]; (iii) 1776 (1987) the laws of corresponding states [4], [5]; and (iv) the se- [12] R. B. Laughlin, Phys.Rev.Lett. 50, 1395 (1983) lectionrule |p q −p q |=1,dictating whichtransitions 1 2 2 1 F. D.M. Haldane, Phys.Rev.Lett. 51, 605 (1983) are allowed and which are forbidden. Lastly, the semi- [13] R.G.Clark,J.R.Mallett,S.R.Haynes,J.J.Harrisand circlelawof[14]-[16]iscompatiblewith,butnotimplied C. T. Foxon, Phys.Rev.Lett. 60, 1747 (1988) by, Γ0(2). [14] A. M. Dykhne and I. M. Ruzin, Phys. Rev. B50, 2369 It should be noted that the full modular group does (1994) not provide the correct phase structure, since it would [15] I. Ruzin and S. Feng, Phys.Rev.Lett. 74, 154 (1995) imply further critical points at the images of σ = i and 3 [16] I.M. Ruzin. N.R. Cooper and B.I. Halperin, Phys. Rev. B53, 1558 (1996) [17] J. L. Cardy and E. Rabinovici, Nuc. Phys. B205, 1 (1982) J. L. Cardy, Nuc.Phys.B205, 17 (1982) [18] A.ShapereandF.Wilczek,Nuc.PhysB320,669(1989) [19] C. A. Lu¨tken and G. G. Ross, Phys. Rev. B45, 11837 (1992) [20] Y. Georgelin and J-C. Wallet, Phys. Lett. A224, 303 (1997); Y. Georgelin, T. Masson and J-C. Wallet, J. Phys.A30, 5065 (1997) [21] D. Shahar, D. C. Tsui, M. Shayegan, E. Shimshoni and S.L. Sondhi,(cond-mat/9611011) [22] D. Shahar, D. C. Tsui, M. Shayegan, R. N. Bhatt and J. E. Cunningham, Phys. Rev.Lett. 74, 4511 (1995) D.Shahar,D.C.Tsui,M.Shayegan,J.E.Cunningham, E. Shimshoni and S. L. Sondhi, (cond-mat/9607127) Solid StateComm. 102, 817 (1997) [23] M. Hilke, D. Shahar, S. H. Song, D. C. Tsui, Y. H. Xie and D. Monroe, (cond-mat/9708239) [24] L.W.Wong,H.W.Jiang,N.TrivediandE.Palm,Phys. Rev.B51, 18033 (1995) [25] H. W. Jiang, R. L. Willett, H. L. Stormer, D. C. Tsui, L. N.Pfeiffer and K.W. West, Phys. Rev.Lett. 65, 633 (1990) 4 Table 1. Some examples of allowed transitions. The matrix γ maps the points σ =0 and σ =1 to the transition indicated in the leftmost column. Some representative experimental support (not exhaustive) is also indicated. The last two columns assume the semi-circle law (ρ is the resistivity). Transition γ Critical Critical σ at σMax ρ at ρMax ν →ν Conductivity Resistivity xx xx 1 2 n+1→n 1 n (2n+1)+i (2n+1)+i (a) (2n+1)+i (2n+1)+i(b) (cid:18)0 1(cid:19) 2 2n2+2n+1 2 2n(n+1) 1 →0 1 0 (2n+1)+i (c) (2n+1)+i(d) 1+i (2n+1)+i∞ 2n+1 (cid:18)2n 1(cid:19) 2(2n2+2n+1) 2(2n+1) n → n+1 1 n (4n2+6n+3)+i (4n2+6n+3)+i (4n2+6n+1)+i (4n2+6n+1)+i(e) 2n+1 2n+3 (cid:18)2 2n+1(cid:19) 2(4n2+8n+5) 2n2+2n+1 2(2n+1)(2n+3) 2n(n+1) 3n+2 → 3n+5 3 3n+2 (24n2+58n+41)+i (24n2+58n+41)+i (24n2+58n+29)+i (24n2+58n+29)+i(f) 4n+3 4n+7 (cid:18)4 4n+3(cid:19) 2(16n2+40n+29) 18n2+42n+29 2(4n+3)(4n+7) 2(3n+2)(3n+5) (a) These points all lie on the semi-circle ρ=i−eiθ0≤θ ≤π. For n=1 see [21]. (b) Assumes n6=0. (c) These points all lie on the semi-circle σ = 1(i−eiθ), 0≤θ ≤π. 2 (d) For n=0 see [22] and [23], for n=1 see [22] and [24], for n=2 see [25]. (e) Assumes n6=0. For n=1,...,5 and n=−3,...,−7 see [14]. (f) For n=0 see [14]. σ σ 0 12 1 32 2 52 3 72 4 0 1 1 1 1 3 3 5 1 7 5 7 3 5 7 9 1 10 8 6 4 10 8 12 2 12 8 10 4 6 1010 1 1 1 2 1 2 3 4 3 2 5 4 6 8 9 7 5 7 3 5 7 7 5 3 7 5 7 9 Fig. 1 The phase structure in the upper-half complex Fig. 2 A magnified view of the phase structure in the σ plane. The solid curves represent phase boundaries upper-half complex σ plane. and the dotted curves allowed transitions. Points where dotted and solid lines cross are critical points. 5