Fraction of delocalized eigenstates in the long-range Aubry-André-Harper model
PPrescription for the fraction of delocalized eigenstates in the long-range AAH model
Nilanjan Roy and Auditya Sharma
Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India (Dated: August 25, 2020)We uncover a systematic structure in the single particle phase-diagram of the quasi-periodicAubry-Andr´e-Harper(AAH) model with power-law hoppings ( ∼ r σ ) when the quasi-periodicity pa-rameter is chosen to be a member of the ‘metallic mean family’ of irrational Diophantine numbers.In addition to the fully delocalized and localized phases we find a co-existence of multifractal (lo-calized) states with the delocalized states for σ < σ > Quasi-periodic systems exhibit non-trivial localiza-tion properties [1–4] despite the simplicity of themodels involved. Unlike a random potential as inthe Anderson model [5], a quasi-periodic potentialas in the Aubry-Andr´e-Harper(AAH) model shows adelocalization-localization transition at a non-zero finitestrength of the potential even in one dimension [6, 7].Even the co-existence of delocalized and localized eigen-states (mobility edge as in the Anderson model in threedimensions) has been reported in some variants of theAAH model [8, 9] in one dimension. The AAH poten-tial has been realized in the experiments of ultracoldatoms studying single particle localization [10–12] and‘many body localization’ [13], which has lead to a freshwave of interest [14–16] in interacting quasi-periodic sys-tems [17–20]. There have also been many finite temper-ature studies of the model [21–23] in recent times.The study of Hamiltonians with power-law hoppingsor interactions ( ∝ r σ ) has seen a resurgence of interestafter such Hamiltonians were realized in experiments ofultra-cold systems [24–37]. When the hopping strengthis sufficiently long-ranged, instead of the exponentiallylocalized eigenstates seen in short-range models, one ob-tains algebraically localized eigenstates. In the power-law random hopping model in one dimension algebraiclocalization is found for σ > σ [40, 41].The effect of power-law hoppings on the quasi-periodicAAH potential has also been studied very recently [42].The study has uncovered the systematic appearance ofmultifractal (localized) eigenstates which co-exist withdelocalized eigenstates for σ < σ >
1) [42]. The ir-rationality of the quasiperiodicity parameter ( α ) is whatrenders the Hamiltonian quasiperiodic. The fraction ofdelocalized eigenstates in the different phases of the sys-tem is related to the precise value of α , which is usuallyset to be the ‘golden mean’ ( √ − /
2. The ‘goldenmean’ is a member of a broader class called the ‘metal-
FIG. 1. Schematic of the phases of a single particle in theLRH model for the quasi-periodicity parameter α , shown indifferent colors. The colored phases are also labelled by thefraction of delocalized eigenstates ( η ) as shown in the figure.Here k = 1 , , α is ‘golden mean’, ‘silver mean’ and‘bronze mean’ respectively. The strength of the quasi-periodicpotential and power-law hopping parameter are denoted as λ and σ respectively. lic mean family’, which is a set of irrational Diophantinenumbers and will be discussed in more detail ahead.In this Letter, we chart out the phase diagram of asingle particle in the presence of the AAH potential andpower-law hoppings when α is set to be a member of the‘metallic mean family’, with special attention given to the‘golden mean’, ‘silver mean’ and ‘bronze mean’. In addi-tion to the delocalized and localized phases, we obtainmixed phases where the multifractal (localized) statescan co-exist with delocalized states for σ < σ > a r X i v : . [ c ond - m a t . d i s - nn ] A ug filling fractions which are related to the metallic means.Such behavior for special filling fractions is also found inthe well known nearest neighbor hopping limit [45]. LRH model:
The model of interest is the one dimen-sional long-range AAH (LRH) model given by the Hamil-tonian: H = − N (cid:88) i
2) respectively. Aslowly converging sequence of rational approximations ofthese Diophantine numbers is given by F u − /F u for twosuccessive members in the sequence for a fixed integer k .Each member α of the ‘metallic mean’ family satisfies thefollowing relation:( α ) z = k ( α ) z +1 + ( α ) z +2 , (3)where k = 1 , , .. for α = α g , α s , α b , .. respectively, and z is a non-negative integer. Putting z = 0 in Eq. 3 alsoyields an important case namely, kα + α = 1. Phase diagram : Now we consider a single particle inthe LRH model with different parameters α g , α s and α b (a) (b)(c) σ λ P P P LEP q>3 (d)FIG. 2. (a-c) Fractal dimension D (in color) as a function of λ and increasing fractional eigenstate index n/N starting fromthe ground state for α g and σ = 0 . , . . N = 1000 and δ = 0 .
02. (d) Phase diagram:in addition to extended ( E ) and localized ( L ) phases with η = 1 , P , P , P , ... exist with η = α g , α g , α g , ... . The vertical line separates out the DM edgefor σ < σ > which are members of the ‘metallic mean family’. Inorder to determine the phases we calculate the fractaldimension of the eigenstates for θ p = 0. We employthe box counting procedure to determine the fractal di-mension [52–55]. Dividing the system of N sites into N l = N/l boxes of l sites each, the ‘fractal dimension’ isdefined as: D f = lim δ → f − (cid:80) N l m =1 ( I m ) f ln δ , (4)where I m = (cid:80) i ∈ m | ψ n ( i ) | computed inside the m th boxfor the n th eigenstate | ψ n (cid:105) and δ = 1 /N l . In the per-fectly delocalized (localized) phase D f is unity (zero),whereas for a multifractal state D f shows a non-trivialdependence on f and 0 < D f < D as a function of λ for all thesingle particle eigenstates when the quasiperiodicity pa-rameter is fixed at α g for σ = 0 . , . . σ = 0 .
5, thefraction of delocalized eigenstates decreases and fractalstates (0 < D <
1) appear in blocks as λ increases.We have also checked that in fact D f depends on f forthese blocks of states, which is a signature of multifrac-tality (see Ref. 47). Hence there exists a delocalized-to-multifractal (DM) edge in the eigenstate spectrum. TheDM edge goes down in steps as the fraction of delocal-ized eigenstates decreases with λ . However, the position (a) (b)(c) σ λ P P P P q>3 LE (d)FIG. 3. (a-c) Fractal dimension D (in color) as a function of λ and increasing fractional eigenstate index n/N starting fromthe ground state for α s and σ = 0 . , . . N = 1000 and δ = 0 .
02. (d) Phase diagram:in addition to extended ( E ) and localized ( L ) phases with η = 1 , P , P , P , ... exist with η = α s + α s , α s , α s + α s , ... . The vertical line separates outthe DM edge for σ < σ > of the DM edge remains unchanged within each step asthe fraction of delocalized eigenstates (denoted as η here-after) stays constant in that region. It is found that inthe decreasing step-like regions defined by constant DMedges, η = α g , α g , α g , ... . We denote the step-like regionsas P q ( q = 1 , , , ... ) phases with η = α g , α g , α g , ... re-spectively.Fig. 2(b,c) for σ = 1 . . D ≈
0) withincreasing λ . This implies that there exists a delocalized-to-localized (DL) edge, also well known as the mobilityedge. Similar to DM edges these fixed DL-edge con-taining phases are also denoted as P q ( q = 1 , , , ... )for η = α g , α g , α g , ... respectively. D of all the eigen-states for α s and increasing λ is shown in Fig. 3(a-c)for σ = 0 . , . , . α s one obtains P , P , P , ... phases with η = α s + α s , α s , α s + α s , ... andDM edges (for σ = 0 .
5) and DL edges (for σ = 1 . , . α b and σ = 0 . , . , . P , P , P , ... phases with η =2 α b + α b , α b + α b , α b , ... and DM edges (for σ = 0 .
5) andDL edges (for σ = 1 . , . P q phase, the same blocks of multifractal states becomelocalized as one crosses σ = 1 whereas the correspond-ing η remains the same. We chart out the single-particlephase diagram for the parameter α g in Fig 2(d), which is (a) (b)(c) σ λ E P P P P q>3 L (d)FIG. 4. (a-c) Fractal dimension D (in color) as a function of λ and increasing fractional eigenstate index n/N starting fromthe ground state for α b and σ = 0 . , . . N = 1000 and δ = 0 .
02. (d) Phase diagram:in addition to extended ( E ) and localized ( L ) phases with η = 1 , P , P , P , ... exist with η = 2 α b + α b , α b + α b , α b , ... . The vertical line separates outthe DM edge for σ < σ > also obtained in Ref. 42 for a Fibonacci N . Fig 2(d) con-tains weakly multifractal eigenstates, similar to the AAHmodel (see Ref. 47), even in the delocalized regimes dueto the choice of a non-Fibonacci N . It is to be noted thatas σ increases the extent of the mixed phases shrinks asthe LRH model approaches the AAH limit. The phase di-agrams for α s and α b are shown in Fig. 3(d) and Fig. 4(d)respectively. The P q phases (corresponding to α s and α b )in these cases as well, like with α g , contain DM edges for σ < σ >
1. The changes in P q phasesat σ = 1 are denoted by the vertical lines in all the phasediagrams. Fraction of delocalized states : After a careful observa-tion of the phase diagrams, one may propose a sequencewhich dictates the values of η in P q phases correspondingto different quasi-periodicity parameters α , which belongto the ‘metallic mean family’ described in Eq. 2. For any σ > λ = 0), η = kα + α = 1 where k = 1 , , α g , α s , α b respectively and z = 0in Eq. 3. As the quasi-periodic disorder is turned on( λ (cid:54) = 0), η starts decreasing in a sequence according toEq. 3 for the ‘metallic mean family’, which is depictedin Fig. 5 . Eq. 3 implies that one can always express( α ) z as a sum of two bits k ( α ) z +1 and ( α ) z +2 . In theLRH model the bigger bit loses weight at every step be-coming ( k − α z +1 , ( k − α z +1 , ... until it reaches α z +1 ,where it disintegrates again according to the rule defined FIG. 5. Depicts how the fraction of delocalized eigenstates( η ) decreases in a manner that uses the rule defined in Eq. 3.One can express the fraction of the delocalized states as asum of two bits kα z +1 and α z +2 , out of which the biggerbit loses weight at every step until it reaches α z +1 , where itdisintegrates according to the rule defined in Eq. 3 and thenthe bigger bit loses weight at each step. For a specific valueof α , at every step of the sequence one obtains a P q phase. in Eq. 3 and the new bigger bit starts losing weight ateach step. This is a continuous process as depicted bythe sequence in Fig. 5 . For a specific choice of α , oneobtains a P q phase at each step of the sequence. Thetop of the sequence corresponds to the fully delocalized( η = 1) phase. One obtains P , P , .. phases as one goesdown following the sequence. The P q phases possess DM(DL) edges if σ < σ > η in differentphases. Choosing k = 2 in the sequence depicted in Fig. 5leads to the phases labelled by η in Fig. 1 . These phasesare as follows → red: η = 2 α + α = 1 (delocalized);green: η = α + α ( P ); orange: η = 2 α + α ( P ); pur-ple: η = α + α , α + α , ... ( P , P ... respectively) col-lectively, which appear as one proceeds further accordingto the sequence. For large values of σ and λ the localizedphase appears when η = 0, shown in blue. Entanglement entropy : Here we consider noninteract-ing spinless fermions in the LRH model to calculate theentanglement entropy of the fermionic ground states indifferent phases obtained in the previous section. Theentanglement entropy in the ground state of such freefermionic systems is given by [56–58] S A = − L (cid:88) m =1 [ ζ m log ζ m + (1 − ζ m ) log(1 − ζ m )] , (5)where ζ m ’s are the eigenvalues of the correlation ma-trix C A , where C Aij = (cid:68) c † i c j (cid:69) with i, j ∈ subsystem A
20 100 500 L S A λ = 0.1λ = 0.5λ = 1.0λ = 2.0 (a)
20 100 500 L S A λ = 0.1λ = 1.3λ = 2.0λ = 3.0 (b) λ S A σ = 0.5σ = 1.5σ = 3.0 (c) L S A σ = 0.5σ = 1.5σ = 3.0 (d)FIG. 6. (a-b) The subsystem size L dependence of entangle-ment entropy S A with increasing values of λ for fermions athalf-filling and for σ = 0 . .
5. (c) S A as function of λ for σ = 0 . , . . L = N/
2. For all the plots in figures (a-c) N = 1024. (d)Entanglement entropy S A as a function of subsystem size L for increasing σ and fixed λ = 2 .
2. For all the plots N = 512for α g and special filling ν = α g . of L sites. For free fermions in d dimensions, typically S A ∝ L d − ln L in metallic phases [59], while it goes as S A ∝ L d − in adherence to the ‘area-law’ in the localizedphases in the presence of disorder.To produce smoother plots, we employ an average of S A over the 100 realizations of θ p uniformly choosingfrom [0 , π ] in all the plots here. We stick to filling frac-tion ν = 0 . α g = ( √ − /
2. The S A vs L plots are shown in Fig. 6(a)and Fig. 6(b) at half-filling with increasing values of λ for σ = 0 . . σ = 0 . λ = 0 . . P phases), theFermi level is delocalized and hence S A ∝ ln L . In thesame figure, when λ = 1 . . P and P phases)the Fermi level is multifractal, S A ∝ ln L but the mag-nitude of S A is drastically low. In Fig. 6(b) for σ = 1 . λ = 0 . . P phases), theFermi level is delocalized and S A ∝ ln L . However, when λ = 2 . . P and P phases) the Fermi level is lo-calized, the magnitude of S A is much lower, and it abidesby the area-law. Transitions of Fermi level at half-fillingare shown in Fig. 6(c) for σ = 0 . , . . σ = 0 .
5, the Fermi level undergoes a DMtransition at λ = 0 .
75. For σ = 1 . . λ = 1 . λ = 1 . S A vs L in the half-filled free fermionic ground statebarely changes in the phase diagram for α s and α b . How-ever, similar to the AAH model [45, 47], the LRH modeltoo shows ‘area-law’ behavior for special fillings ν evenin the delocalized regime. An example of this is shown inFig. 6(d) for λ = 2 . σ = 0 . , . , . ν = α g . In all these plots S A abides by the ‘area-law’.However, the magnitude of S A is significantly smaller for σ = 3 .
0. We point out that while the single particle re-sults depend on whether the system size is a Fibonaccinumber, the many-particle measures do not show such adependence on the system size (see Ref. 47 for details).
Conclusions : We uncover an intricate pattern of thelocalization structure of the AAH potential in the pres-ence of long-range hoppings when the quasi-periodicityparameter is a member of the ‘metallic mean family’.In addition to the fully delocalized and localized phaseswe obtain a co-existence of multifractal (localized) eigen-states with delocalized eigenstates for σ < σ >
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Supplementary Material for “Prescription for the fraction of delocalized eigenstates inthe long-range AAH model”
Nilanjan Roy and Auditya Sharma
Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India
The supplementary material is divided into two sections. First we discuss the results involving the inverse partic-ipation ratio (
IP R ), fractal dimension and entanglement entropy of the AAH model with nearest-neighbor hopping( σ → ∞ limit of the LRH model) and quasi-periodic potential. Next we calculate the IP R and multifractality of theeigenstates in the LRH model which acts as a complementary study to the main article.
IPR, FRACTAL DIMENSION AND ENTANGLEMENT ENTROPY IN THE AAH MODEL
The AAH model has a self-dual point at λ = 2, where the Hamiltonian in position space maps to itself inmomentum space. As a consequence all the single-particle eigenstates are delocalized for λ < λ > IPR : The inverse participation ratio (
IP R ) is a key quantity for studying delocalization-localization transitions. Itis defined as I n = N (cid:88) i =1 | ψ n ( i ) | , (6)where the n th normalized single particle eigenstate | ψ n (cid:105) = (cid:80) Ni =1 ψ n ( i ) | i (cid:105) is written in terms of the Wannier basis | i (cid:105) , n / N -3 -2 -1 I n (a) n / N -3 -2 I n α g α s α b (b) n / M -4 -2 ∆ n (c) n / M -6 -4 -2 ∆ n (d)FIG. 7. (a) IP R of the single particle eigenstates I n for different values of α and fixed N = 1024. (b) Similar plots for N = 610 ,
408 and 360 for α g , α s and α b respectively. For these plots n/N in the x-axis stands for the fractional index ofeigenstates. (c) Consecutive level-spacings ∆ n = E n +1 − E n for different values of α and fixed N = 1024. (b) ∆ n ’s for N = 610 ,
408 and 360 for α g , α s and α b respectively. n/M in the x-axis stands for the fractional index of level-spacings, wheretotal number of spacings M = N −
1. For all the plots λ = 1 in the AAH model. The legend shown in figure (b) applies alsoto figures (a), (c) and (d). N -3 -2 -1 I n α g3 α g α g (a) N -3 -2 I n (b)FIG. 8. (a) IP R of the special eigenstates with fractional index n/N = α g , α g , α g as a function of system size N , which is anon-Fibonacci number. (b) Similar plots for N , which is a Fibonacci number corresponding to α g . For all the plots λ = 1. Thedashed line represents 1 /N dependence of IP R of the non-special delocalized eigenstates. representing the state of a single particle localized at the site i of the lattice. For a delocalized eigenstate I n ∝ N − whereas for a localized eigenstate I n ∝ N . For a critical state I n shows intermediate behavior. IP R of all the singleparticle eigenstates for λ = 1 (delocalized phase) is shown in Fig. 7(a) for a non-Fibonacci N = 1024 and differentvalues of α . There exist eigenstates with high IP R for fractional index n/N = α g , α g , α g ( ≈ . , . , . IP R eigenstates are also found for the cases of ‘silver mean’ ( α s ) and ‘bronzemean’( α b ) at n/N = α s + α s , α s , α s + α s , α s , ... ( ≈ . , . , . , . , ... ) etc. and n/N = 2 α b + α b , α b + α b , α b , ... ( ≈ . , . , . , ... ) respectively. The single-particle energy spectra of these systems show large gaps at the positionswhere the high- IP R states exist [45] as shown in Fig. 7(c). In this figure the level-spacing ∆ n = E n +1 − E n with E n being the energy of the n th eigenstate. Total number of level-spacings M = N − IP R eigenstates seem to vanish if N is chosen to be a Fibonacci number as shown in Fig. 7(b) for λ = 1 and N = 610 ,
360 and 408 for α g , α s and α b respectively. However, we remark that the large gaps still continueto persist in the energy spectra as also shown in Fig. 7(d). The high- IP R eigenstates show an anomalous system sizedependence. As an example we show the scaling of
IP R of the special eigenstates with N in Fig. 8 for λ = 1 . α g .Here N is restricted respectively to be non-Fibonacci and Fibonacci in Fig. 8(a) and (b). For non-Fibonacci N the scal-ing behavior is severely anomalous and deviates from 1 /N . For Fibonacci N the scaling behavior is less anomalous andclose to 1 /N although not exactly 1 /N which is represented by the dashed line for non-special delocalized eigenstates. Fractal dimension : The fractal dimension D is calculated for each single particle eigenstate for λ = 1 and differentparameters α g , α s and α b in a system of non-Fibonacci number of sites N = 1000 as shown in Fig. 9(a). In the n / N D (a) n / N D α g α s α b (b)FIG. 9. (a) Fractal dimension D of the single particle eigenstates for different values of α and fixed N = 1000. (b) Similarplots for N = 610 ,
408 and 360 for α g , α s and α b respectively. For all the plots λ = 1 in the AAH model. n/N in the x-axisstands for fractional index. Here δ = 1 /N l = 0 . delocalized phase D ≈ D ≈ n/N ≈ α g , α g , α g etc. for α g . Similar deviations can be seen at n/N ≈ α s + α s , α s , α s + α s , α s etc. for α s , and n/N ≈ α b + α b , α b + α b , α b etc. for α b . For these special eigenstates 0 < D < D vanish and D ≈ N . f D f n / N = α g3 n / N = α g2 n / N = α g FIG. 10. Fractal dimension D f as a function of f for the single particle eigenstates with fractional index n/N = α g , α g , α g .The solid lines represent plots for Fibonacci N = 610 whereas the dashed lines represent plots for non-Fibonacci N = 1000.For all the plots, λ = 1, α = α g and δ = 1 /N l = 0 . In Fig. 10 we show the fractal dimension D f as a function of f for the eigenstates with fractional index n/N = α g , α g , α g for λ = 1 and ‘golden mean’ α g . In this figure the solid lines represent the plots for Fibonacci N = 610 whereas the dashed lines represent the plots for non-Fibonacci N = 1000. We observe that the solid lineschange very little with f and are close to 1. Here D f deviates a little from 1 because these eigenstates are notperferctly delocalized as depicted in Fig. 8(b). On the other hand the dashed lines show a small variation with f andtheir typical value is just a fraction of one. This indicates that for non-Fibonacci N , the special eigenstates with high IP R are weakly multifractal whereas for Fibonacci N , the special eigenstates become (almost) delocalized like allthe non-special eigenstates in the system. This is true even for ‘silver mean’ α s and ‘bronze mean’ α b (not shown here). Entanglement entropy : The ground state entanglement entropy S A of half the system (subsystem L = N/
2) as afunction of filling fraction ν for λ = 1 is shown in Fig. 11(a) for a non-Fibonacci N = 256 and different values of α . Here ν = N p /N where N p and N are the number of particles and number of sites respectively. Similar to high IP R in Fig. 7(a), significantly low S A is found at ν ≈ α g , α g , α g etc. for α g ; ν ≈ α s + α s , α s , α s + α s , α s etc. for α s ; ν ≈ α b + α b , α b + α b , α b etc. for α b . But in contrast to Fig. 7(b) of IP R , the low S A regions seem to persistas shown in Fig. 11(b) even for Fibonacci N = 610 , ,
360 for α g , α s , α b respectively. In the delocalized phase, S A ∝ ln L [45] for all values of ν except for the special values of ν where S A abides by the ‘area law’ with significantlysmaller magnitudes. The signature of criticality in the model is absent for special ν . These properties of the special ν have been shown earlier in Ref. 45 for α g and hold good for α s and α b also. However, the non-special half-filled( ν = 0 .
5) ground state shows S A ∝ ln L both in the delocalized phase and at the critical point (almost ln L ) whereas S A ∝ L in the localized phase [64]. Although the value of S A at the critical point is larger than that in the localizedphase, it is smaller than that in the delocalized phase. ν S A α g α s α b (a) ν S A (b)FIG. 11. (a) Entanglement entropy S A of the ground state as a function of fermionic filling ν for different values of α and fixed N = 256. (b) Similar plots for N = 610 ,
408 and 360 for α g , α s and α b respectively. For all the plots λ = 1 in the AAH modeland size of subsystem A is L = N/ IPR AND FRACTAL DIMENSION IN THE LRH MODEL
Here we calculate
IP R of the eigenstates for the LRH model with finite σ . To get a hint about the phases in themodel, here we choose a fixed λ = 2 . σ → ∞ limit) and differentvalues of σ = 0 . , . , . α g , α s and α b . The IP R of all the single particle eigenstatesfor α g are shown in Fig. 12(a),(b) and (c) for σ = 0 . , . . n / N -3 -2 -1 I n N = 256N = 512N = 1024 (a) n / N -3 -2 -1 I n N = 256N = 512N = 1024 (b) n / N -3 -2 -1 I n N = 256N = 512N = 1024 (c) n / N -3 -2 -1 I n N = 256N = 512N = 1024 (d) n / N -3 -2 -1 I n N = 256N = 512N = 1024 (e) n / N -3 -2 -1 I n N = 256N = 512N = 1024 (f) n / N -3 -2 -1 I n N = 256N = 512N = 1024 (g) n / N -3 -2 -1 I n N = 256N = 512N = 1024 (h) n / N -3 -2 -1 I n N = 256N = 512N = 1024 (k)FIG. 12. (a-c) The inverse participation ratio I n of the single-particle eigenstates for α g = ( √ − / N = 256 , , σ = 0 . , . . α s = ( √ −
1) with increasing N for σ = 0 . , . . α b = ( √ − / N for σ = 0 . , . . λ is kept fixed at λ = 2 . n/N is the fractional eigenstate index. are delocalized ( I n ∝ N − ) as long as the fractional index n/N < α g . The IP R of the remaining eigenstates for n/N > α g shows an intermediate dependence on N i.e. N − < I n < N . It turns out that these eigenstates aremultifractal [42](see Fig. 13(a)). Hence a DM edge exists at n/N = α g for σ = 0 . λ = 2 .
2. As shown inFig. 12(b) and Fig. 12(c) the eigenstates are delocalized for n/N < α g whereas the eigenstates are localized ( I n ∝ N )for n/N > α g for the same λ and σ = 1 . . λ = 2 . σ = 1 . , .
0. Also we notice that the fraction of the delocalized eigenstates can change with σ for afixed λ . However, the occasional fluctuations of IP R as discussed for the AAH model, especially in the delocalizedregimes are also visible for the LRH model, since values of N are chosen to be non-Fibonacci numbers in all the plotsof Fig. 12.We also show the results obtained from the LRH model for the silver and bronze means. Plots obtained using α s and λ = 2 . σ = 0 . , . . n/N ≈ α s + α s for σ = 0 . n/N ≈ α s for σ = 1 . .
0. Fig. 12(g-k) areobtained using fixed α b , λ = 2 . σ = 0 . , . . n/N ≈ α b for σ = 0 . n/N ≈ α b for σ = 1 . .
0. We see that inevery plot of Fig. 12 the fraction of delocalized eigenstates can always be expressed as a function of the parameter α . However, the IP R fluctuations in the delocalized regime continue to persist in these cases also, although theyvanish if N is a Fibonacci number. It is noticeable that the IP R fluctuations increase in the delocalized regime with σ . Fractal dimension : As evidence for multifractality we plot (cid:104) D f (cid:105) as a function of f for the P phase (with α g fractionof delocalized states) for σ = 0 . σ = 1 . α g . Here (cid:104) D f (cid:105) denotes D f averaged over α g fraction of delocalized and (1 − α g ) fraction of non-delocalized eigenstates.We chose a Fibonacci system size N = 987 to avoid fluctuations due to high- IP R eigenstates in the delocalized phase.In Fig. 13(a) and Fig. 13(b) (cid:104) D f (cid:105) averaged over α g fraction of eigenstates shows a similar small variation with f with (cid:104) D f (cid:105) being close to 1, which implies these states are delocalized. (cid:104) D f (cid:105) averaged over (1 − α g ) fraction of eigenstatesis a fraction and shows a non-trivial dependence on f for σ = 0 . (cid:104) D f (cid:105) is close to 0 and shows almost nodependence on f for σ = 1 .
5. This indicates that these states are multifractal for σ = 0 . σ = 1 . P q phases corresponding to α s and α b . f < D f > (a) f < D f > (b)FIG. 13. (a) Averaged (cid:104) D f (cid:105) as a function of f for λ = 1 . σ = 0 . P phase with a DMedge. (b) Similar plots for λ = 2 . σ = 1 . P phase with a DL edge. (cid:104) D f (cid:105) is calculated byaveraging over α g fraction of delocalized and (1 − α g ) fraction of multifractal/localized eigenstates. For all the plots systemsize N = 987 and δ = 1 /N l = 0 ..