Localization, phases and transitions in the three-dimensional extended Lieb lattices
LLocalization, phases and transitions in the three-dimensional extended Lieb lattices
Jie Liu, ∗ Xiaoyu Mao, † Jianxin Zhong, ‡ and Rudolf A. R¨omer
1, 2, 3 School of Physics and Optoelectronics, Xiangtan University, Xiangtan 411105, China Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom CY Advanced Studied and LPTM (UMR8089 of CNRS),CY Cergy-Paris Universit´e, F-95302 Cergy-Pontoise, France (Dated: April 20, 2020)We study the localization properties and the Anderson transition in the 3D Lieb lattice L (1) andits extensions L ( n ) in the presence of disorder. We compute the positions of the flat bands, thedisorder-broadened density of states and the energy-disorder phase diagrams for up to 4 differentsuch Lieb lattices. Via finite-size scaling, we obtain the critical properties such as critical disordersand energies as well as the universal localization lengths exponent ν . We find that the criticaldisorder W c decreases from ∼ . ∼ . L (1), ∼ . L (2) and ∼ . L (3). Nevertheless, the value of the critical exponent ν for all Lieb lattices studied hereand across disorder and energy transitions agrees within error bars with the generally accepteduniversal value ν = 1 . . , . I. INTRODUCTION
Flat energy bands have recently received renewed at-tention due to much experimental progress in the lastdecade [1]. The hallmark of such flat bands is an absenceof dispersion in the whole of k -space [2–5], implying aneffectively zero kinetic energy. This leads to a whole hostof effects in transport and optical response such as, e.g.localization of eigenstates without disorder [6] and en-hanced optical absorption and radiation. Further stud-ies explorations of flat-band physics have now been donein Wigner crystals [5], high-temperature superconductors[3, 7], photonic wave guide arrays [1, 8–11], Bose-Einsteincondensates [12, 13], ultra-cold atoms in optical lattices[14] and electronic systems [15].Systems that exhibit flat-band physics correspond usu-ally to specially ”engineered” lattice structures such asquasi-1D lattices [6, 16, 17], diamond-type lattices [18],and so-called Lieb lattices,[7, 19–23]. Indeed, the Lieblattice, a two-dimensional (2D) extension of a simple cu-bic lattice, was the first where the flat band structurewas recognized and used to enhance magnetic effects inmodel studies [2, 24, 25]. Most other flat-band systemscited above are also of the Lieb type and exists as ei-ther 2D, quasi-1D or 1D lattices [26]. Less attention hasbeen given to 3D flat-band systems [18] or extended Lieblattices [23, 27]. Furthermore, until recently hardly anywork has investigated the influence of disorder on flat-band systems [16, 28]. Recently, instead of concentratingon the properties of flat-band states, we investigated howthe localization properties in the neighboring dispersivebands are changed by the disorder for 2D flat-band sys-tems [27].In the present work, we extend these studies to theclass of 3D extended Lieb lattices. As is well known ∗ [email protected] † [email protected] ‡ [email protected] [29] the Anderson transition in a simple cubic latticewith uniform potential disorder (cid:15) x ∈ [ − W/ , W/
2] ateach site x is characterized by a critical disorder W c =16 . t [30], with t denoting the nearest neighbor hop-ping strength. The full energy-disorder phase diagram ischaracterized by a simple-connected region of extendedstates ranging from ± t at W = 0 and ending at W c =16 . . , . E = 0 [31]. The critical ex-ponent of the transition has been determined with evergreater precision as close to, e.g., ν = 1 . . , . . II. MODELS AND METHODA. Transfer-matrix method for the 3D Lieb latticesand its extensions L ( n ) We denote the Lieb lattices as L d ( n ) if there are n equally-spaced atoms between two original nearest neigh-bors in a d -dimensional lattice. Here, we shall concen-trate on L (1), L (2) and L (3) as shown in Fig. 1. Toexplore the effects of disorder, we use the standard An-derson Hamiltonian H = (cid:88) x (cid:15) x | x (cid:105)(cid:104) x | − (cid:88) x (cid:54) = y t xy | x (cid:105)(cid:104) y | . (1)The orthonormal Wannier states | x (cid:105) describes electronslocated at sites x = ( x, y, z ) of Lieb lattice with hardboundary condition (we have similar results for periodicboundary conditions as well). The hopping integrals t xy = t only for x , y being nearest neighbors as indi-cated by the lines in Fig. 1, otherwise t xy = 0. a r X i v : . [ c ond - m a t . d i s - nn ] A p r (a) (b) (c)FIG. 1. (a) The Lieb lattice L (1) and its extensions (b) L (2) and (c) L (3). The blue spheres denote the originalnearest-neighbor sites in the underlying cubic lattice whilethe red spheres show the added sites. The solid lines indicatethe cubic structure. The coordinate system is to help identifythe TMM setup used in our study as are the labels A, B, C,and D. For L (1), in order to calculate the localization length λ of the wave function by the transfer-matrix method(TMM), we consider a quasi-one-dimensional bar, withcross area M and length L >> M . A unit length corre-sponds to original site-to-site distances as indicated by Asites in Fig. 1. Along the transfer axis in the z -direction,there are two different slices in L (1). The first slice con-tains the original A sites, and the added B and C sites toform an A-B-C slice, the second (D-)slice only containsthe added D sites as shown in Fig. 1. The TMM equationimplementing H Ψ = E Ψ at energy E for the Hamilto-nian (1) can be written as two parts. First, transferringfrom slice A-B-C to slice D slice, we have Ψ Dz +1 Ψ Az = T A → D Ψ Az Ψ Dz − = E M − (cid:15) z,x − ,y − E t x − − (cid:15) z,x +1 ,y − E t x + − (cid:15) z,x,y − − E t y − − (cid:15) z,x,y +1 − E t y + − M M M Ψ Az Ψ Dz − , (2)where E = (cid:15) z,x,y − Et − t(cid:15) z,x − ,y − E − t(cid:15) z,x +1 ,y − E − t(cid:15) z,x,y − − E − t(cid:15) z,x,y +1 − E , (3)and M , M denote M × M zero and identity ma-trices, respectively. Similarly, t x + , t x − , t y + and t y − are M × M connectivity matrices in the positive/negative x / y directions. With this choice of TMM set-up, we ef-fectively renormalize the added B, C (red) sites shownin Fig. 1(a). Taking M = 3 as an example, we can ex-plicitely write the 3 × matrices t x − = t (4)and t x + = t † x − . Similarly, t y − = t (5) and t y + = t † y − . In Eqs. (4) and (5), the matrix entries(1) can be chosen 0 for hard-wall boundaries and 1 for pe-riodic boundaries. In this way, the effects of sites B andC have been renormalized into effective onsite energies E and hopping terms t x ± , t y ± keeping the transfer matrix T A → D in the standard 2 M × M form. We empha-size that Ψ A,Dz denotes a vector of length M for wavefunction amplitudes in the z th slice [33], either A or D,with x, y = 1 , . . . , M , labelling the position of the orig-inal cubic sites in this slice. In this notation the term (cid:15) z,x,y − Et M ≡ diag (cid:16) (cid:15) z, , − Et , (cid:15) z, , − Et , . . . , (cid:15) z,M,M − Et (cid:17) and similarly for E M and the hopping terms with t x ± , t y ± in Eq. (2). From the D slice to the A-B-C slice, wecan write a more standard TMM form as Ψ Az +1 Ψ Dz = T D → A Ψ Dz Ψ Az − = (cid:15) z,x,y − Et M − M M M Ψ Dz Ψ Az − . (6)in similar notation.The TMM method proceeds by multiplying succes-sively T A → D by T D → A along the bar in z -direction, us-ing M possible starting vector Ψ Az (1) = (1 , , . . . , Az (2) = (0 , , . . . , Az ( M ) = (0 , , . . . ,
1) to form acomplete set. We regularly renorthogonalize these M Ψ states, usually after every 10th multiplication. TheLyapunov exponents γ i , i = 1 , , . . . , M , and their ac-cumulated changes are calculated until a preset preci-sion is reached for the smallest γ min [29, 34–36]. Thelocalization length λ ( M, E, W ) = γ min >
0, the di-mensionless reduced localization length is Λ M ( E, W ) = λ ( M, E, W ) /M . These considerations set out the TMMfor L (1). For the extended Lieb lattices, we follow a sim-ilar strategy, leading to an even more involved renormal-ization scheme which we refrain to review in the interestof brevity. B. Finite-size scaling
The metal-insulator transition (MIT) in the Andersonmodel of localization is expected to be a second-orderphase transition, characterized by a divergence in a cor-relation length ξ ( W ) ∝ | W − W c | − ν at fixed energy E ,and ξ ( E ) ∝ | E − E c | − ν at fixed disorder W [37], where E c is the critical energy and ν , W c as before.We determine the reduced correlation length ξ/M inthe thermodynamic limit assuming the single parame-ter scaling i.e. Λ M ( M, E, W ) = f ( ξ/M ) [30]. For a sys-tem with an MIT this scaling function consists of twobranches corresponding to localized and extended phases.Using finite-size scaling (FSS) [38], we can obtain esti-mates of the critical exponent. Here, we use a method[32, 37] that models two kinds of corrections to scaling:(i) the presence of irrelevant scaling variables and (ii)non-linearity of the scaling variables. Hence one writesΛ = F ( χ r M /ν , χ i M y ) . , where χ r the relevant scalingvariable and χ i the irrelevant scaling variable. We nextTaylor-expand Λ and F up to order n i and n r such thatΛ= n i (cid:88) n =0 χ ni M ny F n ( χ r M /υ ) , F n = n r (cid:88) k =0 a nk χ kr M k/ν . (7)Furthermore, we also expand χ i and χ r by ω = ( W c − W ) /W c (or ( E c − E ) /E c ) to consider the importance ofthe nonlinearities, χ r ( ω ) = m r (cid:88) m =1 b m ω m , χ i ( ω ) = m i (cid:88) m =0 c m ω m . (8)In order to fix the absolute scales of Λ in (7) we set b = c = 1. We then perform the FSS procedure forvarious values of n i , n r , m i , m r , in order to obtain thebest stable and robust fit by minimizing the χ statistic.We quote goodness of fit p values to allow the reader tojudge the quality of our results. III. RESULTSA. Dispersion and disorder-broadened density ofstates for L ( n ) For a clean L (1) system, the dispersion relation canbe derived from (1) as E , = 0 , E , = ± (cid:113) k x + cos k y + cos k z ) , (9)where the k x , k y , k z are the reciprocal vectors correspond-ing to the x , y and z axes, respectively. Fig. 2(a)shows the energy structure of L (1), where we can seetwo dispersive bands which meet linearly at the R point( k x , k y , k z ) = ( π, π, π ) at E = 0. This coincides in energywith the doubly-degenerate flat band. Analogously, wecalculate the energy structures for L ( n ), n = 2 , , L ( n ) lattice has n doubly degenerateflat bands separating n + 1 dispersive bands. Further-more, the two dispersive bands at high and low energiesare separated by energy gaps for these models. We alsonote that for L (3) two dispersive bands again meet lin-early, as for L (1), but in this instance at the Γ point( k x , k y , k z ) = (0 , ,
0) at E = 0. No such linear behaviourcan be found for L ( n ) with n even.We now include the disorder, i.e. W >
0, and we cal-culate the disorder-dependent density of states (DOS)by direct diagonalization for small system sizes M =5 , , and 4 for L ( n ), n = 1 , , ,
4, respectively.The DOS is generated from W = 0 to W = 5 . .
05 with 300 samples for L ( n ), n = 1 , ,
3, whilewe have 100 samples for L (4). We also apply a Gaus-sian broadening of the energy levels to obtain a smootherDOS. The results are shown in Fig. 2. For weak disorderswe can clearly identify the large peaks in the DOS withthe flat bands for all L ( n ) models. From W ∼ L ( n ), n = 1 , , ,
4, vanish quicklywith increasing W . B. Phase Diagrams
Fig. 3 shows the energy-disorder phase diagram for L (1). The phase diagram was determined from thescaling behaviour of the Λ( E, W ) for small system sizes M = 6, M = 8 and M = 10 with TMM error ≤ .
1% [37].Data for
W < E = 0 are dif-ferent. In particular, the critical disorder is reduced byabout 50% compared to the Anderson model. This isin agreement with the discussion in section I. Close tothe band edges for small W ≤ L (1) L (2) L (3) L (4)(a) (b) (c) (d)(e) (f) (g) (h)FIG. 2. (a)–(d) Dispersion relations for clean systems and (e)–(h) dependence of the normalized DOS on W for L (1) to L (4).In all cases, the flat bands are doubly degenerate. Different colors in the dispersion relations denote different bands while thecolors in the DOS indicate different DOS values as also emphasized by the contour lines. -2√3-2√3 W=0 (analytical)high precision FSS resultsextended localized W E −2 −1 0 1 2 FIG. 3. Phase diagram for L (1). The three solid and col-ored lines represent the approximate location of the phaseboundary estimated from small M , i.e. the dark yellow lineis constructed by widths M = 6 and M = 8, the blue line by M = 6 and M = 10 and the green line by M = 8 and M = 10.The solid squares ( (cid:3) ) denote high-precision estimates fromFSS for large M . The shaded area in the center containsextended states while states outside the phase boundary arelocalized. The dashed lines on both sides are guides-to-the-eye for the expected continuation of the phase boundary for W <
1. The red short vertical line at E = 0 represents theposition of the two-degenerate flat bands. The diamonds ( (cid:7) )denote the band edges for W = 0, i.e. E min = − √ E max = 2 √
3. The dotted lines are the theoretical band edges ± ( | E min | + − W/
2) and the forbidden areas below those bandedges have been filled by lines.
However, the shoulders that develop at E ∼ ± .
75 and W = 6 are a novel feature. The DOS at such strongdisorder does not exhibit clearly any similar signatures.For L (2) and L (3), we show the phase diagrams inFig. 4, determined with TMM errors of ≤ .
2% and with the same system sizes as for L (1). As before, small dis-order results have to be excluded. Our numerical resultssupport, as for L (1), a mirror symmetry at E = 0 andthe results as shown in Fig. 4 have been explicitly sym-metrized. For both L (2) and L (3), the phase bound-aries of the central dispersive band support a reentrantbehaviour, although this is less so for L (3).The obvious difference between the phase diagrams of L (1), L (2) and L (3) is that the extended region for L (1) lattice is simply connected, while for L (2) and L (3) it is disjoint. This difference can be attributedto the presence of the energy gaps in L (2) and L (3)as in Fig. 2. Let us denote, as in the cubic Andersonmodel, a critical disorder W c as the disorder value at thetransition point from extended to localized behaviour atenergy E = 0. Then we see that the critical disorders are W c ∼ .
530 for the cubic lattice [31], ∼ . L (1), ∼ . L (2) and ∼ . L (3). Hence as expected,in the Lieb lattices the last extended states vanish alreadyat much weaker disorders and the trend becomes strongerwith increasing n in each successive L ( n ). C. High-precision determination of criticalproperties for the Lieb models
1. Model L (1) In order to determine the critical properties at thephase boundaries for the Lieb models, we have to go tolarger system size for a reliable FSS. In all cases, theresults are collected up to M = 20 and with TMM con-vergence errors ≤ . W at theband centre at constant E = 0 and outside the band cen-tre at E = 1. Furthermore, we also study two transitions (a) W=0 (analytical)high precision FSS resultsextendedextended extendedlocalized W E −3 −2 −1 0 1 2 3 (b) √6-√6 √2-√2 2√2-2√2 W=0 (analytical)high precision FSS results extendedextended extendedlocalized W E FIG. 4. Phase diagrams for (a) L (2) and (b) L (3) lattices. The symbols, lines and colors are as in Fig. 3, i.e. representingsmall M estimates with M = 6 , (cid:3) ) denote high-precision FSS results from Λ M with an TMMerror ≤ .
1% for width M ≤
16 and ≤ .
2% for width M = 18. The diamonds ( (cid:7) ) denote the maximal band edges from W = 0at ± L (2) and ± √ L (3). as function of E corresponding to the point marking thereentrant behaviour at constant W = 3 and the kink inthe phase boundary at constant W = 6. In Fig. 5, weshow the Λ M ( E, W ) data, the resulting scaling curvesand the variation of the scaling parameter ξ for typicalexamples of FSS results.In Table I we present fits for all 4 cases shown in Fig. 5with higher expansion coefficients n r and m r that showthat our results are stable with respect to an increase inan expansion parameter. We have also checked that theyare stable with respect to slight changes in the choiceof parameter intervals δW and δE for fixed energy andfixed disorder transitions, respectively. The reader willhave noticed, however, that the accuracy of the data isnot good enough to reliably fit irrelevant scaling con-tributions and hence the results in Table I are all for n i = m i = 0 although we have indeed performed our FSSallowing for these additional parameters. Furthermore,one can see in Fig. 5 that the accuracy of the TMM databecomes worse for the fixed disorder transitions at W = 3and especially W = 6. The reason for this behaviour isin principle well understood since at the points, the DOShas an appreciable variation which leads to extra correc-tions not well captured in the FSS [39]. Usually, largersystem sizes M can reduce these variations but this isnot possible here due to computational limitations.
2. Models L (2) and L (3) We follow a similar strategy as in the previous sectionin order to finite-size scale the localization lengths for L (2) and L (3). The TMM convergence errors werechosen as ≤ .
1% up to M = 16 and, due to the increasedcomplexity of these models, as ≤ .
2% for the largestsystem size with M = 18. Fig. 6(a) shows Λ M ( E = 0 , W )and the scaling curve for L (2) at energy E = 0 with n r = 3 , m r = 3. From the panel with the Λ M ( E = 0 , W )data, it is very hard to observe a clear crossing at W c .The situation improves for Λ M ( E, W = 4) in Fig. 6(b)which exhibits a clear crossing of Λ M around E c ∼ . L (3) shown in Fig. 6(c) the crossing for Λ M ( E =0 , W ) is again somewhat less clear. Nevertheless, in allthree cases, the FSS results produce stable and robustfits with estimates for W c , E c and ν as shown in Table I.As for L (1), the FSS fits L (2) and L (3) do not resolvepotential irrelevant scaling corrections. IV. CONCLUSIONS
There are two ways to understand the Lieb lattices asoriginating from the normal simple cubic lattices: (i) asshown in Fig. 1, one can view the L ( n ) lattices as a cu-bic lattice with additionally added sites between the ver-tices of the cube, effectively allowing for additional back-scattering and interference along the original site-to-siteconnections and hence potentially leading to more local-ization. On the other hand, one might argue that (ii) the L ( n ) lattices can be constructed by deleting sites froma cubic lattice, for example a central site in Fig. 1(a) andthe 6 face-centered sites. In this view, the decrease of pos-sible transport channels should give rise to stronger ef-fective localization. Both constructions lead to the samepredictions and agree with what we find here, namely,the localization properties in all L ( n ) lattices show anincreased localization with respect to the cubic Ander-son lattice and become stronger when n increases. Thisis, for example, clear from looking at the behaviour of W c ( n ) in Table I. It is instructive to study the behaviouras n → ∞ . From Fig. 2, we see that the overall bandwidth decreases as n increases. At the same time, thenumber of flat bands increases and the extremal energyof these bands extends as well towards | E | = 2. Hence (a) Λ M W L og ( Λ M ) −0.3−0.200.1 Log ( ξ/M ) ξ W (b) Λ M W L og ( Λ M ) −0.3−0.200.10.2 Log ( ξ/M ) ξ W (c) Λ M E L og ( Λ M ) −0.5−0.4−0.2−0.1 Log ( ξ/M ) ξ E (d) Λ M E L og ( Λ M ) −0.12−0.1−0.06−0.04 Log ( ξ/M ) ξ E FIG. 5. (a) FSS of the localization lengths for L (1) with E = 0, (b) E = 1, (c) W = 3, and (d) W = 6. System sizes M are14 ( (cid:3) ), 16 ( (cid:13) ), 18 ( ♦ ), 20 ( (cid:52) ). The left half in each panel denotes a plot of Λ M versus disorder W or energy E , the solid linesare fits to the data acquired by Eqs. (7)–(8) with (a+b) n r = 3, m r = 1, (c) n r = 2, m r = 1 and (d) n r = 1, m r = 1. Theright half in each panel shows the scaling function F (solid line) and the scaled data points with the same n r and m r as in thecorresponding left half while each inset gives the scaling parameter ξ as a function of disorder strength W , in (a) and (b), orenergy E in (c) and (d). The parameters of the fits are shown in detail in Table I. for very large n , where L ( n ) is simply a 2 renormalizedlattice, but n renormalized sites apart, with proliferatingflat bands. Our results for the critical exponent then sug-gest that as n increases and the dispersive bands becomesmaller, the critical properties in each band still retainthe universality of the 3D Anderson transition — at leastup to n = 3 that we have been able to compute (cp. Fig.7. This is in good agreement with previous results inloosely coupled planes of Anderson models in which theuniversal 3D behaviour was also retained [40]. However,for loosely coupled planes, the MIT was retained evenfor small interplane coupling — a truly 2D localizationbehavior only emerged when the interplace coupling waszero. The point of view of this work is different, i.e. thechange from 3D dispersive bands with an MIT to a solely1D system without MIT is not a continuous change, butrather an eventual replacement and shrinking of disper-sive bands by a proliferation of flat bands as n grows. ACKNOWLEDGMENTS
We wish to acknowledge the National Natural Sci-ence Foundation of China (Grant No. 11874316), theProgram for Changjiang Scholars and Innovative Re-search Team in University (Grant No. IRT13093), andthe Furong Scholar Program of Hunan Provincial Gov-ernment (R.A.R.) for financial support. This work alsoreceived funding by the CY Initiative of Excellence (grant”Investissements d’Avenir” ANR-16-IDEX-0008) and de-veloped during R.A.R.’s stay at the CY Advanced Stud-ies, whose support is gratefully acknowledged. We thankWarwick’s Scientific Computing Research TechnologyPlatform for computing time and support. UK researchdata statement: Data accompanying this publication areavailable from the corresponding authors. L (1)∆ M E δW n r m r W c CI( W c ) ν CI( ν ) p [8 . , . [1 . , .
65] 0 . .
598 [8 . , . .
55 [1 . , .
63] 0 . .
595 [8 . , . .
57 [1 . , .
66] 0 . . . M E δW n r m r W c CI( W c ) ν CI( ν ) p [8 . , . [1 . , .
65] 0 . .
439 [8 . , . .
57 [1 . , .
62] 0 . .
438 [8 . , . .
57 [1 . , .
62] 0 . . . M W δE n r m r E c CI( E c ) ν CI( ν ) p [3 . , . [1 . , .
82] 0 . .
748 [3 . , . .
76 [1 . , .
84] 0 . .
748 [3 . , . .
75 [1 . , .
82] 0 . . . M W δE n r m r E c CI( E c ) ν CI( ν ) p [3 . , . [1 . , .
01] 0 . .
076 [3 . , . .
54 [1 . , .
99] 0 . .
077 [3 . , . .
54 [1 . , .
00] 0 . . . L (2)∆ M E δW n r m r W c CI( W c ) ν CI( ν ) p [5 . , . [1 . , .
92] 0 . .
965 [5 . , . .
70 [1 . , .
89] 0 . .
963 [5 . , . .
75 [1 . , .
92] 0 . . . M W δE n r m r E c CI( W c ) ν CI( ν ) p [1 . , . [1 . , .
68] 0 . .
705 [1 . , . .
56 [1 . , .
70] 0 . .
703 [1 . , . .
53 [1 . , .
66] 0 . . . L (3)∆ M E δW n r m r W c CI( W c ) ν CI( ν ) p [4 . , . [1 . , .
78] 0 . .
791 [4 . , . .
63 [1 . , .
78] 0 . .
791 [4 . , . .
63 [1 . , .
78] 0 . . . L ( n ), n = 1 , , M , fixed E (or W ), range of W (or E ), expansion orders n r , m r are listed as well as resulting critical disorders W c (or energies E c ), their 95%confidence intervals (CI), the critical exponent ν , its CI, and the goodness of fit probability p . The averages contain the meanof the three preceding W c (or E c ) and ν values, with standard error of the mean in parentheses. The bold W c , E c and ν valueshighlight the fits used as examples in Figs. 5 and 6. Appendix A: Dispersions
For completeness, we here include the dispersion rela-tions shown in Fig. 2. For L (2), we have E , = 1 , E , = − , E = ρ + + ρ − , (A1a) E = ωρ + + ω ρ − , E = ωρ − + ω ρ + , (A1b) where ω = − i √ , ρ ± = (cid:115) − q ( k )2 ± (cid:114)(cid:16) q ( k )2 (cid:17) − (cid:0) (cid:1) and q ( k ) = 2 (cos k x + cos k y + cos k z ). For L (3), wefind E , = √ , E , = −√ , E , = 0 , (A2a) E , , , = ± (cid:113) ± (cid:112)
10 + q ( k ) . (A2b)Last, for L (4), the four doubly degenerate flat bands aregiven as E , , , , , , , = 12 (cid:16) ± ± √ (cid:17) , (A3a)and the remaining five dispersive bands are the solutionsof the 5th order equation E − E + 13 E − q ( k ) = 0 . (A3b) [1] D. Leykam and S. Flach, APL Photonics , 070901(2018).[2] H. Tasaki, Progress of Theoretical Physics , 489(1998).[3] S. Miyahara, S. Kusuta, and N. Furukawa, Physica C:Superconductivity , 1145 (2007).[4] D. L. Bergman, C. Wu, and L. Balents, Physical ReviewB - Condensed Matter and Materials Physics , (2008).[5] C. Wu, D. Bergman, L. Balents, and S. Das Sarma, Phys-ical Review Letters , 070401 (2007).[6] D. Leykam, J. D. Bodyfelt, A. S. Desyatnikov, and S.Flach, The European Physical Journal B , 1 (2017).[7] A. Julku et al. , Physical Review Letters , 045303(2016).[8] R. A. Vicencio et al. , Physical Review Letters ,245503 (2015).[9] S. Mukherjee et al. , Physical Review Letters , 245504(2015).[10] D. Guzm´an-Silva et al. , New Journal of Physics ,(2014).[11] F. Diebel et al. , Physical Review Letters , (2016).[12] F. Baboux et al. , Physical Review Letters , (2016).[13] S. Taie et al. , Science Advances , (2015).[14] R. Shen, L. B. Shao, B. Wang, and D. Y. Xing, PhysicalReview B , 041410 (2010).[15] M. R. Slot et al. , Nature Physics , 672 (2017).[16] P. Shukla, Physical Review B , 184202 (2018).[17] A. Ramachandran, A. Andreanov, and S. Flach, PhysicalReview B , 1 (2017).[18] M. Goda, S. Nishino, and H. Matsuda, Physical ReviewLetters , (2006).[19] W.-X. Qiu et al. , Physical Review B , 241409 (2016).[20] R. Chen, D.-H. Xu, and B. Zhou, Physical Review B ,205304 (2017). [21] M. Ni, B. Ostahie, and A. Aldea, Physical Review B -Condensed Matter and Materials Physics , 1 (2013).[22] M. Sun, I. G. Savenko, S. Flach, and Y. G. Rubo, Phys-ical Review B , 161204 (2018).[23] A. Bhattacharya and B. Pal, Physical Review B ,235145 (2019).[24] E. H. Lieb, Physical Review Letters , 1201 (1989).[25] A. Mielke and H. Tasaki, Communications in Mathemat-ical Physics , 341 (1993).[26] C.-C. Lee, A. Fleurence, Y. Yamada-Takamura, and T.Ozaki, Physical Review B , 045150 (2019).[27] X. Mao, J. Liu, J. Zhong, and R. A. R¨omer, 1 (2020).[28] P. Shukla, Physical Review B , 1 (2018).[29] B. Kramer and A. MacKinnon, Reports on Progress inPhysics , 1469 (1993).[30] A. MacKinnon and B. Kramer, Physical Review Letters , 1546 (1981).[31] A. Rodriguez, L. J. Vasquez, K. Slevin, and R. A. R¨omer,Physical Review B , 134209 (2011).[32] K. Slevin and T. Ohtsuki, Physical Review Letters ,382 (1999).[33] Strictly speaking, the labels A and D are not neededto label Ψ A,Dz , but we retain them here for the readersconvenience.[34] V. Oseledets, Trans. Moscow Math. Soc. , 179 (1968).[35] K. Ishii, Progress of Theoretical Physics Supplement ,77 (1973).[36] C. W. Beenakker, Random-matrix theory of quantumtransport, 1997.[37] A. Eilmes, A. M. Fischer, and R. A. R¨omer, PhysicalReview B , 245117 (2008).[38] A. MacKinnon and B. Kramer, Zeitschrift f¨ur Physik BCondensed Matter , 1 (1983).[39] P. Cain, R. A. Roemer, and M. Schreiber, Ann. Phys. ,507 (1999).[40] F. Milde, R. R¨omer, M. Schreiber, and V. Uski, TheEuropean Physical Journal B , 685 (2000). (a) Λ M W L og ( Λ M ) Log ( ξ/M ) ξ W (b) Λ M E L og ( Λ M ) Log ( ξ/M ) ξ E (c) Λ M W L og ( Λ M ) Log ( ξ/M ) ξ W FIG. 6. FSS of the localization lengths for (a) L (1) at E = 0and (b) W = 4 as well as for (c) L (3) at E = 0. System sizes M are 10 ( (cid:47) ), 12 ( (cid:46) ), 14 ( (cid:3) ), 16 ( (cid:13) ), 18 ( ♦ ). The arrangementin each panel is as in Fig. 5, i.e. scaling curves (solid lines)and scaled Λ M data (symbols) in the left half of each panel,scaling curve F (lines) with scaled data (symbols) in the righthalf and ξ in the inset. The chosen expansion coefficients are(a) n r = 2, m r = 2, (b) n r = 2, m r = 1 and (c) n r = 2, m r = 1 as highlighted in Table I. FIG. 7. Variation of the averaged critical exponent ν corre-sponding to L (1) (red), L (2) (blue) and L (3) (green) forthe seven averages from Table I. The green horizontal dashlines indicate ν = 1 . . , . ν = 1 ..