Decimation Studies of Bloch Electrons in a Magnetic Field: Higher Order Limit Cycles Underlying the Phase Diagram Jukka A. Ketoja1, Indubala I. Satija2, and Juan Carlos Chaves Department of Physics and Institute of Computational Sciences and Informatics, George Mason University, Fairfax, VA 22030 (February 1, 2008) square lattice where the coupling to next nearest neigh- Adecimationmethodisappliedtothetightbindingmodel bor(NNN) sitesexceededacertainthresholdvalue com- 5 describingthetwodimensionalelectrongaswithnextnearest 9 pared to the nearest neighbor (NN) coupling. [10] Using neighborinteractioninthepresenceofaninversegoldenmean 9 various analytical and numerical tools to study the scal- magnetic flux. The critical phase with fractal spectrum and 1 ingpropertiesofthefractaleigenspectrum,theuniversal- wave function exists in a finite window in two-dimensional n ityclassesoftheCphasewereinvestigated. TheE-Cand parameter space. There are special points on the boundaries a as well as inside thecritical phasewhere therenormalization C-L transition lines were conclusively shown to define a J flowexhibitshigher orderlimit cycles. Ournumerical results new universality class, different from the E-L transition 1 suggest that most of the critical phase is characterized by a line (which belonged to the Harper class). However, the 1 strange attractor of the renormalization equations. analysis in the interior of the C phase was rather incon- 1 clusive. 75.30.Kz, 64.60.Ak, 64.60.Fr v In this paper, we apply decimation methods to the 2 TBM describing Bloch electrons in a field with a NNN 4 interaction. Unlike the previous study [10] which inves- 0 I. INTRODUCTION tigated the scaling properties of the fractal eigenvalues, 1 0 westudythe scalingpropertiesofthefractaleigenstates. 5 The two-dimensionalelectrongaswithirrationalmag- Our study, in agreement with the previous work [10], 9 netic flux is a well-known paradigm in the study of sys- shows that the E-C transition line defines a new univer- / t tems with two competing periodicities. The magnetic sality class while the E-L transition line belongs to the a m fieldresultsinreducingtheproblemtoaone-dimensional universalityclassoftheHarperequation. Inaddition,we tight binding model (TBM) [3] known as the Harper showthatthereexistspecialpointsintheinterioraswell - d equation.[4]TheHarperequationexhibitsbothextended as on the boundary of the C phase which correspond to n (E) and localized (L) states. At the onset of transi- many new universalityclasses. They are associatedwith o tion corresponding to a periodic potential with square higher order limit cycles of the RG equations. Our de- c : symmetry,the statesarecritical(C)with fractalspectra tailed analysis suggests that the rest of the interior of v and wave functions. The scaling properties of the devil the C phase is described by a unique universal strange Xi staircasespectraandthe wavefunctions havebeenstud- attractor of the RG flow. r iedextensivelyusingvariousrenormalizationgroup(RG) InSectionII,wedescribethemodelandbrieflyreview a methods. [5], [6], [7] previously known features of the phase diagram of the Recently, it was pointed out that the Harper equation TBM describing Bloch electrons with NNN interaction also describes the isotropic XY quantum spin model in and make a comparision with the quantum spin prob- a modulating magnetic field of periodicity incommensu- lem. The decimation scheme is reviewed in Section III. rate with the periodicity of the lattice. [8] It was shown Section IV concentrates on higher order symmetric limit that the presence of anisotropy in spin space fattened cycles, and in Section V we describe the phenomenon of the critical point of the Harper equation resulting in a shifted symmetry and doubling of the period of a limit phase diagram where E, L, and C phases all existed in cycle. Evidence for a strange set is provided in Section a finite measure parameter interval. The existence of a VI. Finally, Section VII contains a summary of our con- fat C phase provided a new scenario for the breakdown clusions. of analyticity in incommensurate systems. Furthermore, basedonnumericalresultsobtainedusinganewdecima- II. TIGHT BINDING MODEL FOR BLOCH tion method [7], [9] it was argued that the fat C phase ELECTRONS IN A FIELD was described by four distinct universality classes char- acterized by limit cycles of the RG flow. The TBM describing Bloch electrons on a square lat- ThefatCphasewasalsoreportedrecentlyintheTBM tice in amagnetic field withboth NN hoppingst andt describing Bloch electrons moving on a tight binding a b 1 and NNN hoppings tab =ta¯b is [10] Although all quantum states are fractal , it is more use- ful to study states at the band edges. This is because 1 ta+2tabcos[2π(σ(i+ )+φ)] ψi+1 the self-similar behavior is usually observed only for the { 2 } minimum and maximum energy states and also for the 1 + ta+2tabcos[2π(σ(i )+φ)] ψi−1 band center if E = 0 is an eigenenergy. In order to get { − 2 } the overall picture, it is sufficient to consider only one +2tbcos[2π(σi+φ)]ψi =Eψi (1) of these states and we will focuss on the quantum state corresponding to E . min Here, σ is the magnetic flux which we choose to be the InRGanalysis,inadditiontofixingthequantumstate inverse golden mean σ = (√5 1)/2. This TBM was − , one has to also fix the phase factor φ in Eq. (1). It has studied in detail in ref. [10]. Fig. 1 shows the phase dia- beenpointedoutinthepreviousstudies[5],[7],thewave gramofthe modelin the space ofthe parametersλ= tb ta function ψi obtained by iterating the TBM diverges un- and α = 2ttaab. The Harper equation corresponds to the less the phase factor φ is tuned to some critical value. limit α = 0 where the NNN coupling term is zero. The In the Harper model, the critical value of the phase fac- phase diagram was obtained [10] using analytical meth- tor is 1 for the negative band edge. For this value, the 2 odstoobtainthescalingbehaviorofthetotalbandwidth main peak is centrally located and the wave function is (TBW) and the Lyapunov exponents and carryingout a symmetric about i = 0. In the study of the quantum numericalmultifractalanalysis. ThelinesAC(E-Ctran- spin model [7], the phase factor had to be varied contin- sition) and CE (C-L transition) were found to be bicrit- uouslyinthefatCphasesothatthe mainpeakcouldbe ical, i.e. the TBW scaled with the system size with the centrallylocatedandtheresultingwavefunctionbecame exponentδ =2. Thiswasincontrastwiththecriticalline bounded. Determinationofthis criticalphase factorwas BC separating the E and L phases where the exponent essentialinordertofindthe RGlimitcyclesandtocom- was known to be unity. Within the region bounded by pute the universal scaling ratios. In general the phase thelinesACandCEandtheα-axis,wheretheNNNcou- factor φ for obtaining symmetric wavefunction need not pling dominated, the multifractal analysis did not lead beidenticaltothephasefactorresultinginboundedwave toconclusiveresultsonthe universality. This wasdue to functions. [7] lackinconvergenceofthe f(α)curvewiththesizeofthe We consider an infinite lattice which extends in both system. Furthermore, their numerical calculation of the positive and negative directions from the i = 0 site. In TBWwascomplicatedbyoscillatorytermssuperimposed the decimation scheme, all sites except those labelled by on the power law. However, this regime was conjectured positive as well as negative Fibonacci numbers are deci- to be critical. mated. TheresultingTBMconnectingthewavefunction The existenceofthe fatC phaseimplies thatthe pres- ψ at two neighboring Fibonacci sites can be written as ence of the NNN coupling in the Bloch electron problem introduces new universality classes. The same does not ψ(i+Fn+1)=c+n(i)ψ(i+Fn)+d+n(i)ψ(i) (2) happen if the cosine term in the Harper equation is just − − replaced by more generic periodic functions. [11] ψ(i−Fn+1)=cn(i)ψ(i−Fn)+dn(i)ψ(i). (3) Eq. (1)involvingbothdiagonalandoff-diagonaldisor- The index n above refers to the level of decimation. derbearssomeresemblancetotheTBMdescribingquasi- For the Harper model it suffices to define only one set particlefermionexcitationsinaquantumXYspinchain. of the ”decimationfunctions” c (i) and d (i) because of n n [8] Unlike Harper, both the anisotropic XY spin chain the symmetry of the wave function about i = 0. This and Eq. (1) exhibit C phase in a finite parameter inter- implies that the scaling ratios are the same on the pos- val. However,intheTBM(1)thefatCphaseisobserved itive and negative side. This was also asymptotically beyond a critical value of the NNN hopping whereas in true in the quantum spin model everywhere else except the spin model the fat C phase can be seen even with along the C-L transition line [7], where the wave func- infinitesimal spin space anisotropy. tionswereasymmetricabouti=0resultinginvanishing Motivated by the success of our decimation scheme to scaling ratios on one side. [12] The reason why we now confirm the existence of a fat C phase and obtaining the have to introduce separate decimation functions for the universalbehaviorinthequantumspinchain[7],wenow positive (+) andnegative ( ) side is that sometimes the apply the method to Eq. (1). Our main focuss is to asymptotic (n ) c+(0−),d+(0) appear to be shifted obtain the universality classes of the fat C phase. compared to c−→(0)∞,d−(n0). Thnis type of shifted symme- n n try turns out to be helpful in locating higher order limit cycles and is discussed in detail in Section V. III. DECIMATION SCHEME Using the defining propertyofthe Fibonaccinumbers, Fn+1 = Fn+Fn−1, the following recursion relations are Our decimation approach describes the scaling prop- obtained for c and d (we will omit the +, indices if n n ertiesofthe wavefunctions for aspecific valueofenergy. the equations do not depend upon them) [9],−[7]: 2 cn+1(i)=cn(i+Fn)cn−1(i+Fn)−d−n1(i)dn+1(i) (4) IV. HIGHER ORDCEYRCLSEYSMMETRIC LIMIT dn+1(i)= dn(i)[dn(i+Fn)+ − cn(i+Fn)dn−1(i+Fn)]c−n1(i). (5) Fig. 1showstheE,C,andLphasesofthemodelinthe For a fixed i, the above coupled equations for the deci- λ αspace. Theiterationofdecimationequationsshows − mation functions define a RG flow which asymptotically that the BC critical line defining the boundary between (n )convergeonanattractor. Inourearlierstudies, the E and L phases is described by a 3-cycle of the RG the→E,∞C,andLphasesweredistinguishedbythedistinc- flow. On this line (with the exception of the point C) tions in the attractors of the RG flow. In the Harper as the decimation functions flow to the same limit cycle as well as in the quantum spin case, the C phase was char- for the critical Harper equation (point B). However, the acterizedbyanontrivial asymptoticp-cycleattheband pointC is describedby adifferent3-cycledefining a new edges with p equal to 3 or 6. universalityclass. TableIcomparesthe universalscaling Theexistenceofanontrivialp-cycleforthedecimation ratios in these two universality classes. Fig. 2(a-b) show functions often implies that the wave function is neither the wave function in these two cases. In both cases, a extendednorlocalizedandexhibitstheself-similarityde- bounded wave function was obtained for φ = 1/2 and scribed by the wave function as well as decimation functions were symmetric about the center of the lattice: i.e. c+ = c− ψ(i) ψ([σpi+1/2]) (6) and d+ =d−. ≈ The bicritical line AC was found to exhibit two new where [ ] denotes the integer part. The p-cycle of the universalityclasses: AtthepointA,thedecimationfunc- self-similar bounded wavefunction can be used to define tions were found to asymptotically convergeto a period- the universal scaling ratios 6 limit cycle for φ = 1/2. The regime bounded by the points A and C on the line AC (excluding the points A ζj =nl→im∞|ψ(Fpn+j)/ψ(0)|; j =0,...,p−1. (7) and C) appears to be described by a unique symmetric limit cycle of the period 12 for φ =1/2. The 12-cycle is This equation describes the decay of the wave func- particularlyclearforthemiddle pointM(λ=.5,α=1). tion with respect to the central peak. A well-defined However, for other points on this line, the RG flow may limit ζj exists for an integer p for which asymptotically exhibit long transients before settling on the limit cycle ψ(Fn+p) ψ(Fn). For an even p, it often happens that of the point M. This is illustrated in Fig. 3 for the point ≈ ψ(Fn+p/2) ψ(Fn) so that the above equation de- MM (λ = .25, α = 1). The RG iterations begin close to | | ≈ | | finesactuallyonlyp/2differentscalingratios. Whenever the 6-cycle of the point A and then follow the 3-cycle of any scalingratioζj takesa finite value betweenzero and the point C for a while before approaching the 12-cycle unity, the wave function over the infinite system is nei- of the point M . There is additional complexity involved therlocalizednorextended. Inordertofullycharacterize because inorder to observethe approachwe hadto shift the self-similarity of a wave function, an infinite number the data for MM by 6 decimation levels. We used the of scaling ratios have to be defined. [7] However, using quadruple precision to confirm that the MM limit cycle theonesdefinedaboveonecanalreadyseparatedifferent asymptotically converged to the limit cycle of M. universality classes from each other. Symmetric period-12 cycles were also observed at the Thenumericstodemonstrateap-cyclewasratherchal- points F, G, and H which fall on the line t =t . How- a ab lengingforthecasewhereE =0wasnotaneigenenergy. ever,the limit cycle at each point was different implying This was because at many points in the phase diagram, differentscalingpropertiesandhencedifferentuniversal- the energy was required with 16 digits precision (ma- ity classes. chine double precision) in order to see the asymptotic Itshouldbenotedthatwiththeexceptionofthepoints cyclewithtwoormoredigitsofprecision. Evenfortridi- B, C, and H, all the limit cycles observed for φ = 1/2 agonal matrices, we were able to determine the energy resulted in asymptotically diverging wave functions (see only up to 12digits. With this precision,the decimation Fig. 2). The decimation functions corresponding to di- equations could be iterated only about 16 times. The vergingwavefunctionscannotbeusedtoobtainuniversal conjecture for the existence of a limit cycle provided a scaling ratios for the new universality classes. In section very efficient Newton method where the energy and the V,wedetermininetheboundedwavefunctionsbytuning limit cycle were determined self-consistently. Diagonal- the phase factor. ization routines provided a good starting value of the Wealsostudiedsystematicallyotherpointsonthelines energy which in principal could be improved to an ar- CE and FH and points inside the C phase, but we did bitrary precision. At many points where the transients not see any evidence of limit cycles for φ = 1/2. Since were rather long, the quadruple precision was used to the numerics should be sufficient to show at least cycles confirm the existence of a limit cycle. of length 12 (unless there are very very long transients), 3 wecanconjecturethattherearenocyclesoftheorder12 VI. STRANGE SET OF THE or shorter on these lines with the phase 1/2. In section RENORMALIZATION FLOW VI,wepresentsomeevidencethattherestoftheCphase is described by a strange attractor of the RG flow. Weexploredtheideaofdescribingthe regionbounded by the lines AC andCE by a strange set ofthe RG flow. Fig. 5 shows the two-dimensional projection of the at- V. SHIFTED SYMMETRY AND PERIOD tractor obtained by plotting an inverse decimation func- DOUBLING tionfortwosubsequentdecimationlevelsinvariousparts of the phase diagram. Having E up to 12 digits, the min Fig. 4 shows the bounded wave functions obtained by RG equations are estimated to give correct decimation tuning the phase factorφ to a certaincriticalvalue. Un- functions up to about 16 levels. Transients were taken like the symmetric wave functions of section IV, these into account by excluding the first six decimation levels wave functions are not symmetric about i = 0. It turns fromthe data. It is interesting to note that the iteration out that this asymmetry is due to a constant shift be- of the decimation functions in three different parameter tween the wave functions on the positive and negative regimes namely the line FH, the region CHG (excluding sides. Numerical iteration of the TBM (1) shows that in the line CH),andthe regionACGFappearto asymptot- addition to Eq. (7), the wave functions on the positive ically converge on roughly the same set. Similar figures and negative sides are asymptotically related by were obtained also on the line CG, GD and the lines FG and GH. The fact that different parts of the phase dia- ψ(Fn) ψ( Fn+s) (8) gram are described by similar invariant sets makes us to ≈ − exclude the possibility that the observedbehavior is due Instead of the symmetric solution for the decimation to long transients. However, although the possibility of functionscorrespondingtoc+(0)=c−(0),d+(0)=d−(0) n n n n a very long limit cycle cannot be completely ruled out, foralldecimationlevelsn,wefoundtheasymptoticequa- we believe that the observed behavior suggests that the tions interior of the fat C phase ( excluding the special points c+(0) c− (0) (9) whichexhibitlimitcycle)isattractedtoauniqueinvari- n ≈ n+s ant set of the RG equations. We conjecture that the set d+(0) d− (0), (10) n ≈ n+s is a strange attractor. The iterates of the RG flow on the CE line seem to lie i.e. asymptotically there was a shift of s levels between on the inner boundary of the invariant set correspond- thepositiveandnegativedecimationfunctions. Morover, ing to the interior of the C phase (see Fig. 5). In the in above + and could be interchanged with the same shift s, which im−plied that c+(0) c− (0) c+ (0), previous studies [10], the CE line defining the boundary n ≈ n+s ≈ n+2s of C and L phases was found to be bicritical. In anal- i.e. the asymptotic period p=2s. The shift s was found ogy with AC line, we would expect that the decimation to be equal to the period of the symmetric limit cycle functions on the CE line converge to a limit cycle of the discussed in section IV. Therefore, the phenomenon of order24forφ=1/2. Wedidnotseeanyevidenceofthis shifted symmetry resulted in doubling the period of a cycle. However, its existence can not be ruled out spe- limit cycle for the decimation functions. ciallyinviewofthe possibility oflongtransientsandthe The points F and G, which respectively fall on the factthateventhesymmetriclimitcyclecouldbeoforder intersection of the line t = t and the lines t = 0 and a ab b 24. Therefore,the problemofdetermining the universal- the self-dual line BD, exhibit the phenomenon of shifted ity class along the CE line describing the C-L transition symmetry with s=12 (see Table II). This value implies remains open. the asymptotic cycle-length 24. The phenomenon of shifted symmetry is very crucial in locating the limit cycles of period 24. This is because VII. CONCLUSIONS withdoubleprecisionarithematics,theRGequationscan be iterated only about 24 times. Without having the In this paper, we demonstrate that our decimation shifted symmetry we could not have deduced that the scheme is an extremely useful tool to study general asymptoticperiodis24aswecouldnotgotobigenough quasiperiodic TBMs. The Bloch electron with NNN in- decimation levels to see the full cycle on one side only. teraction and the anisotropic quantum XY chain in a This is shown explicitly in Table II. The same shift and transverse field are two known examples where the C period is observed for the critical phase at the point M phase exists in a finite parameter range. In the spin in the middle of the line AC. At the point A the shift is problem,the C phasewascharacterizedby four different 6 and the asymptotic period therefore 12. limit cycles of the RG flow. [7] The present study shows that in the Bloch electron case the situation is lot more complex: in addition to six new universal limit cycles, 4 which correspond to self-similar wave functions, there is [6] D. J. Thouless and Q. Niu, J. Phys. A 16, 1911 (1983); strong numerical evidence of a RG strange set. To best R. B. Stinchcombe and S. C. Bell, J. Phys. A 20, L739 ofourknowledge,thisistheonlyexampleofaquasiperi- (1987); 22, 7171 (1989); L. Suslov, Sov. Phys. JETP odic model where the golden mean incommensurability 56, 612 (1982); 84, 1972 (1983); 57, 1044 (1983); M. Wilkinson,J.Phys.A20,4357(1987);D.Dominguez,C. does notresultinself-similarwavefunctions atthe band Wiecko, and J. V. Jose, Phys. Rev.B 45, 13919 (1992). edges. Even in the regime where no limit cycles exist, [7] J.A.KetojaandI.I.Satija,Phys.Lett.A194,64(1994); the RG scheme provides a clear distinction between the Conformal, Subconformal and Spectral Universality in E, C, and L phases: in the E and L phases, the dec- Incommensurate Spin Chains, submitted to Phys. Rev. imation functions are trivial, and in the C phase they B. assume finite non-trivial values. [7] It should be noted [8] I. I. Satija, Phys. Rev. B 48, 3511 (1993); Phys. Rev. B that in this regime where the C phase is not described 49,3391(1994); I.I.SatijaandJ.C.Chaves,Phys.Rev. by the limit cycles of the RG equations, the previous B 49, 13239 (1994). studies [10] showed lack of convergence in f(α) curve. [9] J. A.Ketoja, Phys.Rev.Lett. 69, 2180 (1992). The general case of TBM (1) where the NNN inter- [10] J. H. Han, D. J. Thouless, H. Hiramoto, and M. actions tab and ta¯b are not equal provides an interesting Kohmoto, Critical and bicritical properties of Harper’s equationwithnextnearestneighborcoupling(preprint); limit of the Bloch electron on a triangular lattice. [13] Y. Hatsugai and M. Kohmoto, Phys. Rev. B 42, 8282 Our studies have shown that [14] the universality class (1990). of this model is related to the subconformal universality [11] H.J.SchellnhuberandH.Urbschat,Phys.Rev.Lett.54, classoftheIsingmodel.[7]Thisestablishesaninteresting 588(1985);R.C.BlackandI.I.Satija,Phys.Lett.A157, relationshipbetween the anisotropic Blochelectron with 246 (1991). NNN interaction and anisotropic quantum spin chains. [12] The C-L transition line in the quantum spin model is In the quantum spin problem, the fattening of the special as it also corresponds to the onset of magnetic C phase is due to the broken O(2) symmetry in spin long range order and to the conformal invariance of the space which translates to the broken U(1) symmetry in system. [8] the quasiparticle fermion Hamiltonian. For the Bloch [13] F.H.ClaroandG.H.Wannier,Phys.Rev.B,6068(1979); electron, the existence of C phase is due to a NNN in- D. J. Thouless, Phys.Rev.B 28, 4272 (1983). [14] J.C.Chaves,I.I.Satija, andJ.A.Ketoja, unpublished. teraction. Although at this point we are unable to pin point the commonality between the symmetric breaking in these two problems, we believe that the origin of the FIG. 1. The phase diagram of the anisotropic electron fat C phase and new universality classes may be tied to gas with t = t . The solid lines BC, AC, and CE are ab ab certain broken symmetries of the models. respectively the E-L, E-C, and C-L transition lines. With theexception of the point C, theBC line is described bythe Harper universality class. The bicritical line AC is described ACKNOWLEDGMENTS by three different universal limit cycles corresponding to the pointsA,C,andtheregimeinbetweenAandC.Inaddition, The research of IIS is supported by a grant from Na- the points F, G, and H are also described by limit cycles of the RG flow. The period of the limit cycle ( see section IV tionalScienceFoundationDMR093296. JAKisgrateful and V ) is indicated in a bracket close to the point. The for the hospitality during his visit to the George Mason twoentriesinsidethebracketdescribethesymmetricandthe University. IISwouldliketoacknowledgethe hospitality shifted-symmetry periods. For example, (12) near the point of National Institue of Standard and Technology where Hshowsthatitexhibitsonlythesymmetriclimitcycle(with part of this work is done. bounded wave function) with period p = 12. The (12,24) near the point G shows that it exhibits both the symmetric aswellastheshifted-symmetrylimitcyclesofperiods12and 24, respectively. FIG.2. (a-g) shows the wave function at the points B, C, [1] Present address: Department of Physics, ˚Abo Akademi, A,M,F,G,andHcorrespondingtothephasefactorφ=1/2 Porthansgatan 3, FIN-20500 ˚Abo, Finland. such that the reflection symmetry about i = 0 is preserved. [2] e-mail: [email protected]. ThiscausesthewavefunctiontodivergeforthepointsA,M, [3] For a review, see J. B. Sokoloff, Phys. Rep. 126, 189 F, and G while the wave function at the point B, C, and H (1985). is bounded. In these plots, the maximum value of the wave [4] P.G.Harper,Proc.Phys.Soc.LondonA68,874(1955). function is scaled to unity. [5] S.OstlundandR.Pandit,Phys.Rev.B29,1394(1984); S.Ostlund,R.Pandit,D.Rand,H.Schellnhuber,andE. D.Siggia, Phys.Rev.Lett. 50, 1873 (1983). 5 FIG. 3. (a) The inverse decimation function 1/cn(0) vs. TABLE II. The decimation functions for the point G n (φ = 1/2) along the bicritical line AC: The data for the (φ = 1/3) at site i = 0 showing the shifted symmetry with point MM (λ = .25,α = 1; shown by crosses), shifted by 6 s=12 and thus indirectly implying the limit cycle of length decimation levels,eventuallyfollows the12-cycleofthepoint 24. Weseethatc+n(0)≈c−n±12(0)andalsod+n(0)≈d−n±12(0). M (λ = .5,α = 1; solid line with small crosses showing the locations of theperiodic orbit). n c+(0) c−(0) d+(0) d−(0) n n n n FIG. 4. (a-d) show the absolute value of the wave func- 6 -0.817 -2.477 -3.499E-02 0.109 tion at the points A (φ = 1/4), M (φ = .3202185) , F 7 2.852 -6.304 -1.257 -3.952 (φ = .2777...), and G (φ = 1/3) . The phase factor φ is 8 -4.149 5.509 1.439 -0.501 chosen sothatthemain peakiscentrally located resultingin 9 1.990 1.176 0.191 -4.571E-02 a bounded wavefunction. 10 -4.463 -1.166 -1.685 0.302 11 2.733 -4.122 -1.168 -1.808 12 -3.255 42.235 0.498 -9.109 FIG.5. Atwo-dimensional projections oftheinversedec- 13 1.517 1.559 0.140 -1.328E-02 imation functions inside the fat C phase. The data has been 14 4.982 -6.744 3.607 0.250 obtained by sampling on the line FH (a), in the region CHG 15 -3.738 -0.831 -0.427 -1.720E-02 (b), and (c) of the whole C phase which includes (a) and (b) 16 -1.961 -1.957 0.467 8.547E-02 andalsothesquareACGFandthepointsaboveit. Thedark 17 -12.536 9.281 -6.087 0.944 dots correspond to the data obtained along to the line CE 18 -2.507 -0.809 0.109 -3.454E-2 (excludingthe pointsC and H). 19 -6.266 2.874 -3.942 -1.26 20 5.532 -4.132 -0.502 1.436 21 1.174 1.996 -4.567E-02 0.191 TABLEI. Theuniversalscaling ratios ζj (j =1,3) at the 22 -1.168 -4.457 0.303 -1.684 point B (Harper) and at the point C (bicritical). 23 -4.119 2.738 -1.807 -1.168 24 42.288 -3.256 -9.106 0.498 j ζj(B) ζj(C) 0 (0, 2, 8, 34, 144, 610,...) 0.2107 0.1712 1 (1, 3, 13, 55, 233,... ) 0.2107 0.2353 2 (1, 5, 21, 89, 377,... ) 0.2107 0.2387 6