Flatband generator in two dimensions
FFlatband generator in two dimensions
Wulayimu Maimaiti,
1, 2, 3
Alexei Andreanov,
1, 2 and Sergej Flach
1, 2 Center for Theoretical Physics of Complex Systems, Institute for Basic Science(IBS), Daejeon 34126, Korea Basic Science Program(IBS School), Korea University of Science and Technology(UST), Daejeon 34113, Korea Department of Physics and Astronomy, Center for Materials Theory,Rutgers University, Piscataway, NJ 08854, USA (Dated: January 12, 2021)Dispersionless bands – flatbands – provide an excellent testbed for novel physical phases due tothe fine-tuned character of flatband tight-binding Hamiltonians. The accompanying macroscopicdegeneracy makes any perturbation relevant, no matter how small. For short-range hoppings flat-bands support compact localized states, which allowed to develop systematic flatband generators in d = 1 dimension in Phys. Rev. B d = 2 dimensions. The shape of a compact localized stateturns into an important additional flatband classifier. This allows us to obtain analytical solutionsfor classes of d = 2 flatband networks and to re-classify and re-obtain known ones, such as thecheckerboard, kagome, Lieb and Tasaki lattices. Our generator can be straightforwardly generalizedto three lattice dimensions as well. I. INTRODUCTION
Physical systems with macroscopic degeneracies haveattracted a lot of attention during the last decades. Suchdegeneracies are highly sensitive to even slightest pertur-bations which makes them perfect testbeds for identify-ing and studying various exotic or unconventional cor-related phases of matter. Flatbands are dispersionlessenergy bands of translationally invariant tight-bindingnetworks.
The absence of dispersion implies a macro-scopic degeneracy of the flatband eigenstates. A flatband(FB) results from destructive interference of the hoppingswhich requires their fine-tuning. All the known FB exam-ples with short-range hopping support compact localizedstates (CLS) as eigenstates with strictly compact sup-port. FB networks were extensively studied theoreticallyin d = 1, d = 2 and d = 3 lattice dimen-sions. Models featuring FBs have been experimentallyrealized in a variety of settings, including optical waveguide networks, exciton-polariton condensates, and ultra-cold atomic condensates. Due to their fine-tuned character, flatband systemsare fragile to perturbations that can easily destroy themacroscopic degeneracy. As a consequence, exotic phasesof matter with unusual properties emerge under theeffect of various perturbations: disorder, externalfields, nonlinearities.
Interactions in FB lead toa plethora of interesting phenomena: delocalisation andconserved quantities, disorder free many-body lo-calisation, groundstate ferromagnetism, pair formation for hard core bosons, superfluidity, and superconductivity. However the very defining feature of the flatbands –their fine-tuned degeneracy – makes it difficult to iden-tify them in the relevant Hamiltonian parameter space.A number of methods has been proposed to constructFB lattices: line graph approach, extended cell con-struction method, origami rules, repetitions of mini-arrays, local symmetry partitioning, chiral symme- try, , fine-tuning relying on specific CLS and networksymmetries, using specific properties of FB, etc .All these methods apply to either specific geometries ofthe underlying networks or to networks with particularsymmetries.All the above discussed FB networks support CLS. Itfollows that the properties of the CLS together with anumber of generic network properties form a set of clas-sifiers which will fix a particular FB network class. Thatapproach leads to systematic FB generators based onthese classifiers. The CLS classifiers are its size U (of oc-cupied unit cells) and shape (in dimensions d ≥ d , theBravais lattice type, the hopping range, and the numberof bands ν (i.e. the number of sites per unit cell). Thesimplest generator case U = 1 with arbitrary remainingnetwork classifiers was obtained in Ref. 64. The moresophisticated case U = 2 with d = 1, nearest neighbourunit cell hopping and two bands ν = 2 was solved inclosed form in Ref. 65, with its extension to larger bandnumber ν and CLS size U published in Ref. 66.In this work, we extend the d = 1 FB generator to two dimensions d = 2 and indicate the road to gen-erators in dimension d = 3. We introduce a systematicclassification of d = 2 FB networks using their CLS clas-sifiers - size U and shape. We demonstrate how to findanalytic solutions for some FB classes. We re-generateand classify some of the already known d = 2 FB latticessuch as the checkerboard, kagome, Lieb, and Tasaki lat-tices, along with a multitude of completely new d = 2FB lattices.The paper is organized as follows. In Section II, weintroduce the main definitions and conventions that weuse throught the text. These conventions are direct gen-eralization to d = 2 of the conventions used for the d = 1FB generator. Section II B introduces CLS of d = 2FB Hamiltonians, and the classification of FB Hamilto-nians by their CLS. It also presents an inverse eigenvalueproblem for finding Hamiltonians from a given CLS class. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n An algorithm to solve these inverse eigenvalue problemsis turned into an efficient FB generator in Sec. III B. Inthe following Sec. IV, we apply the FB generator to someclasses of CLS, illustrate how the inverse eigenvalue prob-lem is resolved and show the results. We conclude bysummarising our results and discussing open problems.
II. MAIN DEFINITIONSA. Models
We consider a d = 2 translational invariant tight-binding lattice with ν sites per unit cell. We use thesame notation as in our previous works, Ref. 65 and66: we label wavefunctions by the unit cell index n , sothat the full wavefunction reads Ψ = ( (cid:126)ψ , . . . , (cid:126)ψ n , . . . ),where the ν -component vector (cid:126)ψ n is the wave function ofthe n th unit cell. For simplicity we restrict to nonzerohopping between nearest neighbour unit cells only, andnote that the generalization to longer range hoppings iscumbersome but straightforward. Nearest neighbor unitcells are defined along (combinations of) primitive latticetranslation vectors. A set of matrices H χ describes thehopping between different pairs of unit cells. The index χ encodes the direction of the hopping - x, y , etc . Theresult is illustrated in Fig. 1 for a square lattice and atriangular lattice, the only possible cases for the n.n. di-rections in d = 2. A unit cell on the square lattice has4 neighbours and 2 different directions of n.n. hopping- both along the primitive lattice translations vectors (cid:126)a and (cid:126)a . A site on a triangular lattice has 6 neighboursand 3 possible directions of n.n. hopping: (cid:126)a i =1 , , onlytwo of which corresponds to the primitive lattice trans-lation vectors. With these conventions the Hamiltonianeigenvalue problem reads H (cid:126)ψ n + (cid:88) χ (cid:16) H † χ (cid:126)ψ n (cid:48) χ + H χ (cid:126)ψ n χ (cid:17) = E (cid:126)ψ n , n ∈ Z . (1)Here H describes the intracell hopping and H χ is thenearest neighbor hopping matrix for the χ th direction,with n χ and n (cid:48) χ being the respective indices of the twonearest neighboring unit cells along the χ th direction.Because of the translation invariance the Floquet-Blochtheorem applies and the eigenstates of Eq. (1) can be ex-pressed as (cid:126)ψ n = (cid:126)u ( k ) e − i k · (cid:126)R n , where the Bloch polariza-tion vector (cid:126)u ( k ) has ν components u µ ( k ) , µ = 1 , . . . , ν .Finally k = ( k x , k y ) is the wave vector, and (cid:126)R n is theposition of the n th unit cell. Then the eigenvalue prob-lem (1) in momentum space reads H k (cid:126)u ( k ) = E ( k ) (cid:126)u ( k ) . (2)The eigenvalues E ( k ) provide the band structure of theHamiltonian H k . (a) (b) ⃗ a ⃗ a ⃗ a ⃗ a ⃗ a FIG. 1. Example of nearest neighbor unit cells for 2D lattices,where (cid:126)a , (cid:126)a are primitive translation vectors and (cid:126)a = (cid:126)a + (cid:126)a . In our conventions: (a) a square lattice has only 2 n.n.hopping directions: (cid:126)a , (cid:126)a , (b) a triangular/hexagonal latticehas 3 n.n. hopping directions: (cid:126)a i =1 , , . B. Classification of compact localised states
In our previous work we have shown that the size U of a CLS is the only CLS-related flatband classifier in di-mension d = 1 dimension. For d = 2 dimensions, size andshape of the CLS plaquette turn into relevant flatbandclassifiers. The size of a CLS is given by two integers U and U which define its plaquette size along the twoprimitive lattice translation vectors (cid:126)a , (cid:126)a . The shape ofa CLS plaquette is encoded by a U × U matrix T withinteger entries 0 or 1. The zero elements of the matrix T prescribe the locations of unit cells with zero wave-function amplitudes in the CLS plaquette. The numberof all possible nontrivial matrices T is finite and can besorted and counted using an integer s ≥
0. Thereforewe arrive at the extended CLS flatband classifier vector U = ( U , U , s ).For U = 1 or U = 1 there is only one possible shapeand we can shorten the classifier U to (1 , U ) or ( U , U = U = 2, and we choose the integer s to count the numberof zeros in the above matrix T : s = 0 : T = (cid:18) (cid:19) , (3) s = 1 : T = (cid:18) (cid:19) , (4) s = 2 : T = (cid:18) (cid:19) . (5)The case U = 1 discussed in the introduction corre-sponds to U = (1 , d = 2 is U = (2 ,
1) and will be discussed below. Thenext and less trivial set of cases in d = 2 is U = (2 , , U = (2 , ,
1) and U = (2 , , U = (2 , , U = (2 , , U = (2 , , U = (2 , , (a) (b)(c) (d) ⃗ a ⃗ a ⃗ a ⃗ a ⃗ a ⃗ a ⃗ a ⃗ a ⃗ a ⃗ a ⃗ a FIG. 2. The U -classification of 4 specific and known 2D lat-tices with a flatband. Open circles denote the lattice sites,black lines nonzero hopping elements of equal value. Shadedareas indicate the U × U CLS plaquette, and the darkershaded regions show the occupied unit cells (black circlesshow the nonzero wavefunction amplitudes of the flatbandCLS). Red boxes denote the unit cell, and (cid:126)a , (cid:126)a are prim-itive translation vectors, and (cid:126)a = (cid:126)a ± (cid:126)a . Lattices andrespective (extended) classifier vectors of the respective CLS:(a) Lieb: U = (2 , , U = (2 , , U = (2 , , U = (2 , , We will show that these known flatband networks aremembers of vast families of flatband networks, each witha number of continuously tunable control parameters.For U , U ≤ (cid:126)ψ n in Eq. (1) arenonzero for a maximum of four unit cells. We label theseCLS components as (cid:126)ψ i =1 ,..., as shown in Fig. 3(e). Wewill use the vector (cid:126)ψ i and the bra-ket | ψ i (cid:105) notations in-terchangeably throughout the text. III. THE FLATBAND GENERATORA. The eigenvalue problem
Just as in the d = 1 case we construct FB Hamil-tonians from their CLS, considering the latter as inputparameters and reformulating the problem of finding aFB Hamiltonian into an inverse eigenvalue problem forthe hopping matrices H χ . To achieve this we rewritethe eigenvalue problem (1) for the nonzero amplitudes (cid:126)ψ i and supplement it with destructive interference con-ditions which ensure the strict compactness of the eigen-state. Overall the way we solve this system in d = 2 is very similar in spirit to but is in general more complexand involved than the d = 1 case. U = 1 The U = 1 case assumes a CLS which occupies onlyone unit cell with wave function (cid:126)ψ and leads to a simpleset of equations: H (cid:126)ψ = E FB (cid:126)ψ ,H i (cid:126)ψ = H † i (cid:126)ψ = 0 , i = 1 , , . (6) U = 2 Two hopping matrices – The simplest case of two hop-ping matrices H , H can be always related to a squarelattice geometry. One example is the Lieb lattice shownin Fig. 2(a). The possible CLS shapes and the hoppingsfor this case are shown in Fig. 3. The eigenvalue problemand destructive interference conditions read: H (cid:126)ψ + H (cid:126)ψ δ U , = ( E FB − H ) (cid:126)ψ ,H † (cid:126)ψ + H (cid:126)ψ δ U , δ s, = ( E FB − H ) (cid:126)ψ , (cid:16) H (cid:126)ψ δ s, + H † (cid:126)ψ (cid:17) δ U , = ( E FB − H ) (cid:126)ψ δ U , , (cid:16) H † (cid:126)ψ δ s, + H † (cid:126)ψ δ s, (cid:17) δ U , = ( E FB − H ) (cid:126)ψ δ U , δ s, ,H (cid:126)ψ = H † (cid:126)ψ = 0 ,H (cid:126)ψ = H (cid:126)ψ = 0 ,H (cid:126)ψ δ U , = H † (cid:126)ψ δ U , = 0 ,H (cid:126)ψ δ U , δ s, + H † (cid:126)ψ = 0 ,H † (cid:126)ψ δ U , δ s, = H † (cid:126)ψ δ U , δ s, = 0 . (7)For specific values of U , U and s the above systemgives the eigenvalue problem and destructive interferenceconditions for a CLS with the extended classifier vector U = ( U , U , s ). Three hopping matrices – Several known flatband lat-tice models have three hopping matrices which connectdifferent unit cells, for example checkerboard, kagome,and dice lattices (see Fig. 2(b-d)). The three respectivehoppings matrices H , H , H as shown in Fig. 4. It isinstructive to note that the freedom in choosing differentunit cells has consequences in our classification scheme.As an example, the dice lattice with its unit cell choice inFig. 2(d) falls into the category U = (2 , ,
0) with threenontrivial hopping matrices. The unit cell choice usedin Fig. 1 in Ref. 38 leads to a much larger CLS plaque-tte with classifier U = (3 ,
3) (and three empty unit cellsin the CLS), but also to a reduced number of only twonontrivial hopping matrices.For three hopping matrices the eigenvalue problemand the corresponding destructive interference conditions (a) (d) (b)(c) (e) =0 =0=0=0
FIG. 3. Classification of compact localised states for caseswith two hopping matrices. Each square represents a unitcell. Directions of the hopping and respective destructive in-terference conditions are indicated by arrows. Where twohopping terms (arrows) meet, both will contribute to the de-structive interference cancellation. (a) U = (1 ,
1) single unitcell ( U = 1) CLS. (b) U = (2 ,
1) case. (c) U = (2 , ,
2) case.(d) U = (2 , ,
1) case. (e) U = (2 , ,
0) case. (a) (b)(c) (d) (e) =0=0=0 =0=0=0
FIG. 4. Classification of compact localised states for caseswith three hopping matrices. Each square represents a unitcell. Directions of the hopping and respective destructive in-terference conditions are indicated by arrows. Where two ormore hopping terms (arrows) meet, all will contribute to thedestructive interference cancellation. (a) U = (1 ,
1) singleunit cell ( U = 1) CLS. (b) U = (2 ,
1) case. (c) U = (2 , , U = (2 , ,
1) case. (e) U = (2 , ,
0) case. read: H (cid:126)ψ + H (cid:126)ψ δ U , = ( E FB − H ) (cid:126)ψ ,H † (cid:126)ψ + H (cid:126)ψ δ U , δ s, + H † (cid:126)ψ δ U , = ( E FB − H ) (cid:126)ψ , (cid:16) H (cid:126)ψ δ s, + H † (cid:126)ψ + H (cid:126)ψ (cid:17) δ U , = ( E FB − H ) (cid:126)ψ δ U , , (cid:16) H † (cid:126)ψ + H † (cid:126)ψ (cid:17) δ U , δ s, = ( E FB − H ) (cid:126)ψ δ U , δ s, ,H (cid:126)ψ = H (cid:126)ψ = 0 ,H † (cid:126)ψ δ U , δ s, = H (cid:126)ψ δ U , δ s, = 0 ,H † (cid:126)ψ = H (cid:126)ψ δ U , = 0 ,H (cid:126)ψ + H † (cid:126)ψ = 0 ,H (cid:126)ψ δ U , + H (cid:126)ψ δ U , = 0 ,H † (cid:126)ψ δ U , + H (cid:126)ψ δ U , δ s, = 0 ,H † (cid:126)ψ + H (cid:126)ψ δ U , δ s, = 0 . (8) For specific values of U , U , s we obtain the eigen-value problem and destructive interference conditions fora CLS with the extended classifier vector U = ( U , U , s ).In order to apply the generator defined below, we definethe transverse projectors Q i , Q ij , Q ijk on { (cid:126)ψ i } , { (cid:126)ψ i , (cid:126)ψ j } and { (cid:126)ψ i , (cid:126)ψ j , (cid:126)ψ k } respectively. B. The generator
The sets of equations (7,8) are the starting point of ourflatband generator. Our goal is to generate all possiblematrices H χ which allow for the existence of a flatband,given a particular choice of the CLS shape, E FB and H .We arrive at the following protocol:1. Choose the number of bands ν .2. Choose a hopping range.3. Choose a plaquette shape of the CLS.4. Choose an arbitrary Hermitian H .5. Choose a flatband energy E FB .6. Exclude H , H and H from the equations (7,8) toget nonlinear constraints on the CLS components (cid:126)ψ i , and solve these constraints to find all CLS com-ponents (cid:126)ψ i .7. With the chosen H , E FB and the CLS (cid:126)ψ i obtainedat the previous step, solve equations (7,8) for H χ .The above protocol admits variations which can simplifythe task. In some cases the nonlinear constraints allowto skip item 5 and keep the flatband energy E FB a freeparameter to be fixed when executing item 6, as we willshow below. Step 6 requires solving nonlinear equationswhich may have either no CLS solutions, or a manifoldof CLS solutions with freely tunable parameters, or evenseveral such manifolds. Using a CLS solution from step6, and executing step 7, will in general yield a solutionmanifold of hopping matrices with freely tunable param-eters as well, which correspond to the freedom of fixingonly the flatband states and not constraining the rest ofthe spectrum, bands, and eigenvectors. IV. SOLUTIONS
We consider CLS sizes restricted to U , U ≤
2, unlessstated otherwise. A. U = 1 Without loss of generality we assume that H is diag-onal with first diagonal entry E FB . From the first linein Eq. (6) we conclude (cid:126)ψ = (1 , , , ..., H i ) ,µ =( H i ) µ, = 0, i.e. the hopping matrices have zeros on theirfirst row and column, and freely choosable entries else-where. The entries parametrize the remaining dispersivedegrees of freedom. These entries as well as an over-all multiplicative scaling factor and energy gauge are thefree parameters of the corresponding manifold of U = 1flatband Hamiltonians. These solutions correspond tothe detangled basis of U = 1 flatband networks. Addi-tional manifold parameters are obtained from entanglingthe CLS with the dispersive network through commutinglocal unitary operations. For two bands ν = 2 and H = 0 it follows that theonly solutions are either U = 1 flatbands, or decoupled1D networks with H = 0 and U = 2 (see details of thederivation in Appendix A). B. U = (2 , A schematic of the CLS and the destructive inter-ference conditions for this case are shown in Fig. 3(b)and Fig. 4(b) for the two and three hopping matricescases respectively. The eigenvalue problem (7), respec-tively (8), involves only one hopping matrix H , whilethe matrices H , enter additional destructive interfer-ence conditions only. Therefore the eigenvalue problemreduces to the 1D case solved in Ref. 66. It follows (seeAppendix B 1 and Ref. 66 for details) H = ( E FB − H ) | ψ (cid:105) (cid:104) ψ | ( E FB − H ) (cid:104) ψ | E FB − H | ψ (cid:105) . (9)The CLS components (cid:126)ψ , (cid:126)ψ are subject to nonlinear con-straints (cid:104) ψ | E FB − H | ψ (cid:105) = 0 , (cid:104) ψ | E FB − H | ψ (cid:105) = (cid:104) ψ | E FB − H | ψ (cid:105) , (10)that are resolved similarly to the 1D case. For twohopping matrices the additional destructive interference a n b n c n xy (a) (b) FIG. 5. Example of a U = (2 ,
1) CLS FB network and ν = 3.(a) Tight binding lattice with two hopping matrices. Linesindicate nonzero hoppings, filled circles show the position ofa CLS. a n , b n , c n indicate the sites in one unit cell. See Ap-pendix B 1 for details. (b) Band structure corresponding to(a). yields H = Q M Q , (11)where M is an arbitrary ν × ν matrix. Figure 5 providesan example network for the two hopping matrices case.For three hopping matrices the derivation of H and H is given in Appendix C 1. C. U = (2 , , The case U = (2 , ,
2) is shown in Figs. 3(c) and 4(c),where the two CLS-occupied unit cells just touch oneanother. Equations (7-8) reduce to simple eigenproblemsfor the amplitudes in each individual unit cell: H | ψ i =1 , (cid:105) = E FB | ψ i =1 , (cid:105) . (12)We choose some H and thereby fix E FB and (cid:126)ψ , . If E FB is nondegenerate, then (cid:126)ψ ∝ (cid:126)ψ and the problemreduces to a U = 1 CLS. If E FB is degenerate we needat least ν = 3 bands. The amplitudes (cid:126)ψ , can then bepicked as distinct linear combinations of the eigenvectorscorresponding to E FB . The intra-cell hopping matrices H , H , H are reconstructed from the destructive inter-ference conditions, Eqs. (7-8): H | ψ (cid:105) = H | ψ (cid:105) = 0 (cid:104) ψ | H = (cid:104) ψ | H = 0 H † | ψ (cid:105) + H | ψ (cid:105) = 0 H † | ψ (cid:105) + H | ψ (cid:105) = 0 H | ψ (cid:105) = H † | ψ (cid:105) = H | ψ (cid:105) = H † | ψ (cid:105) = 0The last line implies that H = Q M Q where M is anarbitrary ν × ν matrix. The first two lines of the aboveequation constitute a coupled inverse problem of finding H , H from their known action on (cid:126)ψ , (cid:126)ψ . This problemcan be decoupled into inverse problems for H and H by defining: H | ψ (cid:105) = Q | z (cid:105) H | ψ (cid:105) = Q | w (cid:105) . (13)The inverse problems for H , H have been solved inRef. 66. D. U = (2 , , A schematic of the CLS and destructive interferenceconditions for this case are shown in Fig. 3(d) andFig. 4(d) for the two and three hopping matrices respec-tively.
1. Two hopping matrices
The case of two hopping matrices and an arbitrarynumber of bands can be resolved following a similarderivation as for U = (2 , ν = 3 bands. We can consider two cases: (a) theCLS amplitudes are linearly independent, or (b) the CLSamplitudes are dependent. The latter case includes theknown cases of the Lieb and Tasaki lattices.Case (a) has one of the destructive interference condi-tions reading (cid:104) ψ | H + (cid:104) ψ | H = 0. For ν = 3 the num-ber of components of each of the vectors Ψ , , is alsoequal to three. Therefore it is straightforward to showthat the destructive interference condition splits into two: (cid:104) ψ | H = (cid:104) ψ | H = 0 . (14)Then the eigenvalue problem Eq. (7) reads H (cid:126)ψ + H (cid:126)ψ = ( E FB − H ) (cid:126)ψ ,H † (cid:126)ψ = ( E FB − H ) (cid:126)ψ ,H † (cid:126)ψ = ( E FB − H ) (cid:126)ψ ,H (cid:126)ψ = H † (cid:126)ψ = H (cid:126)ψ = H † (cid:126)ψ = 0 ,H (cid:126)ψ = H † (cid:126)ψ = H (cid:126)ψ = H † (cid:126)ψ = 0 . (15)We eliminate H , H from the eigenproblem and obtainthe nonlinear constraints on the CLS amplitudes: (cid:104) ψ | H | ψ (cid:105) = E FB (cid:104) ψ | ψ (cid:105) , (cid:104) ψ | H | ψ (cid:105) = E FB (cid:104) ψ | ψ (cid:105) , (cid:104) ψ | H | ψ (cid:105) = E FB (cid:104) ψ | ψ (cid:105) , (cid:104) ψ | E FB − H | ψ (cid:105) + (cid:104) ψ | E FB − H | ψ (cid:105) = (cid:104) ψ | E FB − H | ψ (cid:105) . (16)Finally we obtain the hopping matrices: H = ( E FB − H ) | ψ (cid:105) (cid:104) ψ | ( E FB − H ) (cid:104) ψ | E FB − H | ψ (cid:105) ,H = ( E FB − H ) | ψ (cid:105) (cid:104) ψ | ( E FB − H ) (cid:104) ψ | E FB − H | ψ (cid:105) . (17)Case (b) assumes (cid:126)ψ i =1 , , to be linearly dependent | ψ (cid:105) = α | ψ (cid:105) + β | ψ (cid:105) . (18) a n
1) CLS FB networks. (a)Tight binding lattice with two hopping matrices and ν = 3.Lines indicate nonzero hoppings, filled circles show the posi-tion of a CLS. a n , b n , c n indicate the sites in one unit cell. SeeAppendix B 2 for details. (b) Band structure correspondingto (a). (c) Same as (a) but for three hopping matrices and ν = 2. a n , b n indicate the sites in one unit cell. See AppendixC 2 for details. (d) Band structure corresponding to (c). This yields the following solution (see details in Ap-pendix B 2): H = Q | a (cid:105) (cid:104) ψ | ( E FB − H ) (cid:104) ψ | Q | a (cid:105) ,H = Q | b (cid:105) (cid:104) ψ | ( E FB − H ) (cid:104) ψ | Q | b (cid:105) , (19)where | a (cid:105) , | b (cid:105) are arbitrary vectors, and E FB , H , (cid:126)ψ , (cid:126)ψ are chosen respecting the constraints (cid:104) ψ | E FB − H | ψ (cid:105) = 0 , (cid:104) ψ | E FB − H | ψ (cid:105) = 0 , (cid:104) ψ | E FB − H | ψ (cid:105) = 0 , ( E FB − H ) ( α | ψ (cid:105) + β | ψ (cid:105) ) = 0 . (20)Equations (19, 20, 18) provide a complete solution to thisspecial U = (2 , ,
1) case with ν = 3 bands. The knownexamples such as Lieb lattice and Tasaki’s lattice can beconstructed from our generator as demonstrated in Ap-pendix B 2. Figure 6(a,b) shows one generated example(see details in Appendix B 2).
2. Three hopping matrices
The configuration of this case is shown in Fig. 4(d).The eigenvalue problem and the destructive interference a n
0) CLS FB network, twohopping matrices, and ν = 3. (a) Tight binding lattice. Linesindicate nonzero hoppings, filled circles show the position ofa CLS. a n , b n , c n indicate the sites in one unit cell. See Ap-pendix B 3 for details. (b) Band structure corresponding to(a). conditions read H | ψ (cid:105) + H | ψ (cid:105) = ( E FB − H ) | ψ (cid:105) ,H † | ψ (cid:105) + H † | ψ (cid:105) = ( E FB − H ) | ψ (cid:105) ,H † | ψ (cid:105) + H | ψ (cid:105) = ( E FB − H ) | ψ (cid:105) ,H | ψ (cid:105) = H | ψ (cid:105) = H | ψ (cid:105) = 0 , (cid:104) ψ | H = (cid:104) ψ | H = (cid:104) ψ | H = 0 ,H | ψ (cid:105) + H | ψ (cid:105) = 0 ,H | ψ (cid:105) + H † | ψ (cid:105) = 0 , (cid:104) ψ | H + (cid:104) ψ | H = 0 . (21)The family of flatband solutions for two bands ν = 2is derived in Appendix C 2, with a particular memberchoice of the family shown in Fig. 6(c,d). Notably thecheckerboard lattice is also part of the family, asoutlined in Appendix C 2. For ν = 3 a more cumbersomederivation yields the family of flatband solutions whichcontains the kagome lattice (not shown here). E. U = (2 , , The CLS and destructive interference conditions forthe case of two hopping matrices and the square shapedCLS are shown in Fig. 3(e). For simplicity we restrictourselves to three bands and use a direct parameteriza-tion of the CLS amplitudes (cid:126)ψ i to solve Eq. (7). The fullanalytical solution is reported in Appendix B 3. Thereare three free parameters in this solution, and Fig. 7shows an example FB Hamiltonian for this case.To conclude we note that the increased complexity ofthe equations (7-8) for fully 2D shapes as compared tothe 1D case is an expected and generic feature. Howeverwhile it does not seem to be possible to work out fullsolutions in general – the nonlinear constraints on theamplitudes and complex destructive interference condi-tions typically being the main obstacle – we believe that such solutions can be found in individual cases. V. CONCLUSIONS
In this work, we extended the systematic 1D flatbandgenerator to two dimensions. Two important addi-tional classifiers have been identified and added to makethe 2D generator complete. First, we need to specify theunderlying Bravais lattice class. Second and most im-portantly we need to specify the shape of the compactlocalized states at otherwise fixed CLS plaquette size.We derived analytical solutions for a number of differentFB classes, and reproduced some of the well-known FBlattices: Lieb, Tasaki, kagome and checkerboard, alongwith a number of new 2D FB lattice examples. Our gen-erator results in the possibility of counting free continu-ously variable parameters after fixing one particular 2DFB class. These existing parameters demonstrate thatFB Hamiltonians, while being fine-tuned models, emergeas members of finite dimensional Hamiltonian manifoldswith an additional rich structure. Our results can bestraightforwardly extended to larger compact localizedstates in 2D, and also to 3D cases, no matter how cum-bersome the derivations could turn. Therefore our FBgenerator provides a direct way to search for flatbandsfor fixed lattice geometries in any lattice dimension. ACKNOWLEDGMENTS
This work was supported by the Institute for BasicScience in Korea (IBS-R024-D1).
Appendix A: ν = 2 and H = 0 We consider two bands and H = 0 and demonstratethat any possible flatbands always either reduce to class U = 1 or decouple into 1D networks.First we consider the U = (2 ,
1) case in Fig. 3(b).The eigenvalue problem and the destructive interferenceconditions (7) read H (cid:126)ψ + H (cid:126)ψ = E FB (cid:126)ψ ,H (cid:126)ψ + H † (cid:126)ψ = E FB (cid:126)ψ ,H (cid:126)ψ = H † (cid:126)ψ = 0 ,H (cid:126)ψ s = H † (cid:126)ψ s = 0 , s = 1 , . (A1)The last line enforces that either (i) H = 0 or (ii) (cid:126)ψ ∝ (cid:126)ψ . Case (i) reduces the system to a set of disconnected1D networks which were completely studied in Ref. 65.Case (ii) yields that (cid:126)ψ , are an eigenvector to H (up toa normalization factor) and form a complete U = 1 CLS,and the considered U = (2 ,
1) case reduces to a linearcombination of two U = 1 CLS states.For the U = (2 , ,
0) case in Fig. 3(e) the destructiveinterference conditions in Eq. (7) read H (cid:126)ψ = H (cid:126)ψ = 0 ,H † (cid:126)ψ = H † (cid:126)ψ = 0 ,H (cid:126)ψ = H (cid:126)ψ = 0 ,H † (cid:126)ψ = H † (cid:126)ψ = 0 . (A2)The first line enforces that either (i) H = 0 or (ii) (cid:126)ψ ∝ (cid:126)ψ . Case (i) reduces the system to disconnected1D networks. The third line results in (iia) H = 0 or(iib) (cid:126)ψ ∝ (cid:126)ψ . Case (iia) reduces the system to discon-nected 1D networks. Case (iib) implies (cid:126)ψ ∝ (cid:126)ψ andreduces the problem to U = 1.The case U = (2 , ,
1) shown in Fig. 3(d) yields thefollowing destructive interference conditions: H (cid:126)ψ = H (cid:126)ψ = 0 ,H † (cid:126)ψ = 0 ,H (cid:126)ψ = H (cid:126)ψ = 0 ,H † (cid:126)ψ = 0 ,H † (cid:126)ψ + H † (cid:126)ψ = 0 . (A3)The first line enforces that either (i) H = 0 or (ii) (cid:126)ψ ∝ (cid:126)ψ . Case (i) reduces the system to disconnected1D networks. The third line results in (iia) H = 0 or(iib) (cid:126)ψ ∝ (cid:126)ψ . Case (iia) reduces the system to discon-nected 1D networks. Case (iib) reduces the problem to U = 1.The case U = (2 , ,
2) shown in Fig. 3(c) is slightlymore involved. The eigenproblem in this case reads: H (cid:126)ψ , = E FB (cid:126)ψ , . There are two possible solutions: (i) ψ ∝ ψ or (ii) H = E FB I and (cid:126)ψ ⊥ (cid:126)ψ . Case (i) reduces the system to U =1. In case (ii) we consider the destructive interferenceconditions H (cid:126)ψ = H (cid:126)ψ = 0 H † (cid:126)ψ = H † (cid:126)ψ = 0 H (cid:126)ψ + H † (cid:126)ψ = 0 H (cid:126)ψ + H † (cid:126)ψ = 0From the first two lines and orthogonality of (cid:126)ψ , we con-clude that H , ∝ | ψ (cid:105) (cid:104) ψ | . However this is incompati-ble with the remaining two destructive interference con-ditions as verified by direct substituion and taking intoaccount the mutual orthogonality of (cid:126)ψ , . Appendix B: FB generation for two hoppingmatrices and ν ≥ U = (2 , From Eq. (7), we get the eigenvalue problem and de-structive interference conditions H | ψ (cid:105) = ( E FB − H ) | ψ (cid:105) , (cid:104) ψ | H = (cid:104) ψ | ( E FB − H ) ,H | ψ (cid:105) = H | ψ (cid:105) = H | ψ (cid:105) = 0 , (cid:104) ψ | H = (cid:104) ψ | H = (cid:104) ψ | H = 0 . (B1)Since H only appears in the destructive interference con-ditions, we can express it as H = Q M Q where M is an arbitrary ν × ν matrix. The remaining problemis identical to the 1D problem discussed in our previouswork so that we only sketch the solution. Using the de-structing interference conditions, we eliminate H fromthe eigenvalue problem and find the following nonlinearconstraints on the CLS (cid:104) ψ | E FB − H | ψ (cid:105) = 0 , (cid:104) ψ | E FB − H | ψ (cid:105) = (cid:104) ψ | E FB − H | ψ (cid:105) . (B2)The destructive interference conditions suggest the fol-lowing ansatz H = Q | u (cid:105) (cid:104) v | Q , where | u (cid:105) , | v (cid:105) are vec-tors to be fixed. Plugging this ansatz into Eq. (B1) wefind the vectors (cid:126)u, (cid:126)v and the final expression for H H = ( E FB − H ) | ψ (cid:105) (cid:104) ψ | ( E FB − H ) (cid:104) ψ | E FB − H | ψ (cid:105) . (B3) a. ν = 3 example We choose H and we parameterise the CLS ampli-tudes and H as follows H = , (cid:126)ψ = abc , (cid:126)ψ = def ,H = Q | u (cid:105) (cid:104) v | Q ,(cid:126)u = ( u , u , u ) , (cid:126)v = ( v , v , v ) . The nonlinear constraints (B2) yield b = −√ f, d = f, c = e = 0. Then it follows that H = − √ − a f − √ fa − a f − a f √ fa − a f √ a f √ ,H = √ Af aAf −√ Af aAf a A √ − aAf −√ Af − aAf √ Af , where A = (cid:0) √ au + 2 f ( u − u ) (cid:1) ( av + 2 f ( v − v ))( a + 4 f ) . The FB energy E FB is then obtained from the first non-linear constraint (B2).The specific lattice structure of the Hamiltonian inFig. 5(a) corresponds to the following choices of free pa-rameter: x = x , y = y , x = 1 , y = 5 , u = u ,v = v , u = 1 , v = − , f = − , a = 1 . The hopping matrices and CLS amplitudes read H = − √ − √ −
32 32 √ − √ −
12 1 √ ,H = − √
25 25 √ − − √ − √ − ,(cid:126)ψ = √ , (cid:126)ψ = − − .
2. The U = (2 , , case and ν = 3 The eigenvalue problem and destructive interferenceconditions in Eq. (7) become H (cid:126)ψ + H (cid:126)ψ = ( E FB − H ) (cid:126)ψ ,H † (cid:126)ψ = ( E FB − H ) (cid:126)ψ ,H † (cid:126)ψ = ( E FB − H ) (cid:126)ψ ,H (cid:126)ψ = H † (cid:126)ψ = H (cid:126)ψ = 0 ,H (cid:126)ψ = H (cid:126)ψ = H † (cid:126)ψ = 0 ,H † (cid:126)ψ + H † (cid:126)ψ = 0 . (B4)Using the destructive interference conditions, we elimi-nate H , H from the eigenproblem and obtain the non-linear constraints on the CLS amplitudes: (cid:104) ψ | H | ψ (cid:105) = E FB (cid:104) ψ | ψ (cid:105) , (cid:104) ψ | H | ψ (cid:105) = E FB (cid:104) ψ | ψ (cid:105) , (cid:104) ψ | H | ψ (cid:105) = E FB (cid:104) ψ | ψ (cid:105) , (cid:104) ψ | E FB − H | ψ (cid:105) + (cid:104) ψ | E FB − H | ψ (cid:105) = (cid:104) ψ | E FB − H | ψ (cid:105) . (B5) a. Linearly independent CLS components In this case, as explained in the main text, the de-structive interference condition involving both H and H decouples into two separate conditions. Then theeigenvalue problem Eq. (B4) reads H (cid:126)ψ + H (cid:126)ψ = ( E FB − H ) (cid:126)ψ ,H † (cid:126)ψ = ( E FB − H ) (cid:126)ψ ,H † (cid:126)ψ = ( E FB − H ) (cid:126)ψ ,H (cid:126)ψ = H † (cid:126)ψ = H (cid:126)ψ = H † (cid:126)ψ = 0 ,H (cid:126)ψ = H † (cid:126)ψ = H (cid:126)ψ = H † (cid:126)ψ = 0 . (B6)The nonlinear constraints on the amplitudes of the CLSare given by the same Eq. (B5). Assuming that the non-linear constraints are resolved, we provide below the so-lution to Eq. (B6). We use the following single projectorchoices H = Q | x (cid:105) (cid:104) y | Q and H = Q | v (cid:105) (cid:104) w | Q ,where the transverse projectors are enforced by the de-sctructive interference conditions. Plugging in these ex-pression into the eigenproblem (B6) and using the non-linear constraints we find the hopping matrices: H = ( E FB − H ) | ψ (cid:105) (cid:104) ψ | ( E FB − H ) (cid:104) ψ | E FB − H | ψ (cid:105) ,H = ( E FB − H ) | ψ (cid:105) (cid:104) ψ | ( E FB − H ) (cid:104) ψ | E FB − H | ψ (cid:105) . (B7) b. Linearly dependent CLS components The linear dependence of the CLS amplitudes (cid:126)ψ , (cid:126)ψ , (cid:126)ψ reads | ψ (cid:105) = α | ψ (cid:105) + β | ψ (cid:105) . (B8)This CLS is not necessarily reducible to the U = 1class as long as the amplitudes are not all mutually pro-portional: For example, the Lieb lattice falls under the U = (2 , ,
1) case and has (cid:126)ψ = (cid:126)ψ + (cid:126)ψ . The constraintson the CLS (B5) in this case become (cid:104) ψ | E FB − H | ψ (cid:105) = 0 , (cid:104) ψ | E FB − H | ψ (cid:105) = 0 , (cid:104) ψ | E FB − H | ψ (cid:105) = 0 , ( E FB − H )( α | ψ (cid:105) + β | ψ (cid:105) ) = 0 . (B9)For the given E FB , H and CLS amplitudes, satisfyingthe above constraints, the eigenvalue problem and de-structive interference conditions (B4) become β (cid:104) ψ | H = (cid:104) ψ | ( E FB − H ) ,α (cid:104) ψ | H = (cid:104) ψ | ( E FB − H ) ,H | ψ (cid:105) = H | ψ (cid:105) = 0 ,H | ψ (cid:105) = H | ψ (cid:105) = 0 , (cid:104) ψ | H = 0 , (cid:104) ψ | H = 0 . (B10)These are two decoupled inverse eigenvalue problems for H and H that we resolve in the same way as before. H = Q | u (cid:105) (cid:104) ψ | ( E FB − H ) (cid:104) ψ | Q | u (cid:105) ,H = Q | v (cid:105) (cid:104) ψ | ( E FB − H ) (cid:104) ψ | Q | v (cid:105) , (B11)where E FB , H , (cid:126)u = ( u , u , u ) , (cid:126)v = ( v , v , v ) are freeparameters; (cid:126)ψ , (cid:126)ψ are coinstrained by Eq. (B9) while (cid:126)u should not to be proportional to (cid:126)ψ , and (cid:126)v should not beproportional to (cid:126)ψ . c. Three band examples for the linearly dependent case We choose H and parameterise the CLS amplitudesas follows H = (cid:15) , ψ = abc , ψ = efg . (B12) a. The Tasaki and Lieb lattice families Here wedemonstrate how the Lieb lattice with a flatband at E FB = 0 enforced by the chiral symmetry, and the re-lated Tasaki lattice are derived from our solution. Thenonlinear constraints (B9) give ψ = ac √ (cid:15) ( ± i ) c , ψ = eg √ (cid:15) ( ± i ) g To reproduce the Lieb lattice Hamiltonian the followingunitary transformation is used H (cid:48) = U H U † , ψ (cid:48) = U ψ , ψ (cid:48) = U ψ where U = cos( θ ) cos( ϕ ) cos( θ ) sin( ϕ ) sin( φ ) − sin( θ ) cos( φ ) cos( θ ) sin( ϕ ) cos( φ ) + sin( θ ) sin( φ )sin( θ ) cos( ϕ ) sin( θ ) sin( ϕ ) sin( φ ) + cos( θ ) cos( φ ) sin( θ ) sin( ϕ ) cos( φ ) − cos( θ ) sin( φ ) − sin( ϕ ) cos( ϕ ) sin( φ ) cos( ϕ ) cos( φ ) The Lieb lattice is recovered for the following choices ofparameters: b = i a √ (cid:15) √ , f = − iaα √ (cid:15)β √ , g = − aαβ √ ,c = a √ , e = a θ = − π , φ = 3 π ,ϕ = − π , (cid:15) = − , c = a √ , β = α. Then the CLS amplitudes are given by ψ (cid:48) = −√ a , ψ (cid:48) = √ a , ψ (cid:48) = α −√ a √ a . With the choice θ = − π , φ = π , (cid:15) = − H (cid:48) = ϕ ) cos( ϕ )sin( ϕ ) 0 0cos( ϕ ) 0 0 . The hopping matrices H (cid:48) , H (cid:48) are given by Eq. (B11): H (cid:48) = − u sin( ϕ ) u α − sin( ϕ ) α , H (cid:48) = v cos( ϕ ) v α cos( ϕ ) α The above hopping matrices correspond to the family ofTasaki lattices. For u = v = 0, we retrieve the hoppingmatrices of the Lieb lattice family. b. Obtaining the example in Fig. 6(a,b) We choose E FB = (cid:15) , (cid:15) (cid:54) = 0 , a = − b (cid:112) − f ( (cid:15) − f √ (cid:15) , e = − (cid:112) f − f (cid:15) √ (cid:15) , β = − αbf . Then Eq. (B11) yields the following solution ψ = − bDf √ (cid:15) bc , ψ = − D √ (cid:15) fg , ψ = αψ + βψ ,H = bBD(cid:15)Af − bB ( (cid:15) − √ (cid:15)A bCD √ (cid:15)Af bC (1 − (cid:15) ) A b D √ (cid:15)αcf − αbfg b − b (cid:15)αcf − αbg ,H = − DfG(cid:15)F f G ( (cid:15) − √ (cid:15)F − DfH √ (cid:15)F − f H (1 − (cid:15) ) F − Df √ (cid:15)αcf − αbg f ( (cid:15) − α ( cf − bg ) , D = (cid:112) − f ( (cid:15) − ,A = α ( cf − bg ) (cid:0) bf u + cDu √ (cid:15) − cf u (cid:15) (cid:1) ,B = b (cid:0) Du + f u √ (cid:15) (cid:1) + bcDu + c f u √ (cid:15),C = b (cid:0) Du √ (cid:15) + f u (1 − (cid:15) ) (cid:1) − bcf u (cid:15) + c f u (cid:15),F = α ( cf − bg ) (cid:0) Dgv √ (cid:15) + f v − f gv (cid:15) (cid:1) ,G = Df v + g (cid:0) Dv + gv √ (cid:15) (cid:1) + f v √ (cid:15),H = Df v √ (cid:15) + f ( v − v (cid:15) ) − f gv (cid:15) + g v (cid:15). We use the following parameter values u = 0 , u = 0 , u = 2 , α = 1 , β = 1 , v = 0 , v = 0 ,v = 1 , a = 0 , b = − , g = 0 , (cid:15) = 12 , c = 2 , f = − , finding the CLS and hopping matrices ψ = − − , ψ = − − , ψ = − − ,H = , H = − − − , H = −
14 14 The lattice structure and band structure correspondingto the above hopping matrices is shown in Fig. 6(a,b).
3. The U = (2 , , case with three bands Putting the values U = 2 , s = 0 to Eq. (7), we findthe following eigenvalue problem H ψ , + H ψ , = ( E FB − H ) ψ , H † ψ , + H ψ , = ( E FB − H ) ψ , ,H ψ , + H † ψ , = ( E FB − H ) ψ , ,H † ψ , + H † ψ , = ( E FB − H ) ψ , , (B13)and destructive interference conditions H ψ , = H ψ , = H ψ , = H ψ , = 0 ,H † ψ , = H † ψ , = H † ψ , = H † ψ , = 0 . (B14)We assume that H , H have two zero modes and pa-rameterize the hopping matrices H , H in the followingway: H = | x (cid:105) (cid:104) y | = ad ae afbd be bfcd ce cf , (B15) H = | u (cid:105) (cid:104) v | = gr gs gthr hs htlr ls lt , (B16) where | x (cid:105) = abc , | y (cid:105) = def , | u (cid:105) = ghl , | v (cid:105) = rst . The zero more of H , H are given by (the top two linescorrespond to H , the bottom two – to H ; in every rawthe first two elements are the right eigenvectors, whilethe last two are the left ones) − f d , − ed , − c a , − ba , − t r , − sr , − l g , − hg . Next we enforce the constraints on H , H and theCLS amplitudes by the destructive interference condi-tions (B14). Since ψ , is the right zero eigenmode ofboth H and H , it has to be perpendicular to both (cid:126)y and (cid:126)v , or equivalently it is parallel to the cross prod-uct of (cid:126)y and (cid:126)v and also parallel to one of the right zeroeigenvectors of H , H : (cid:126)ψ , = α ( y × v ) (cid:107) − f d (cid:107) − t r . where we have introduced the proportionality factor α ,that we set to 1 for convenience. Treating the other CLSamplitudes in the same way (and setting the proportion-ality factors to 1 as well) we arrive at the following set ofconstraints on the elements of the CLS amplitudes: t = f, b = s, e = h, c = l,d = r = a = g. (B17)Then the expressions for all ψ reduce to the followingequations: ψ = ( − bf + ef ) α ab − ae ) α } = α (cid:48) − f a ,ψ = ( − bc + bf ) β ( ac − af ) β = β (cid:48) − ba ,ψ = ( ce − ef ) γ ( − ac + af ) γ = γ (cid:48) − ea ,ψ = ( bc − ce ) η − ab + ae ) η = η (cid:48) − c a . (B18)where α (cid:48) , β (cid:48) , γ (cid:48) , η (cid:48) are given by α (cid:48) = ( b − e ) α, β (cid:48) = ( c − f ) β,γ (cid:48) = − cγ + f γ, η (cid:48) = − bη + eη. (B19)2We can set one of the pre-factors to 1: we choose η = 1.Then Eq. (B18) becomes ψ = − ( b − e ) f α a ( b − e ) α , ψ = − b ( c − f ) βa ( c − f ) β ,ψ = − e ( − cγ + f γ ) a ( − cγ + f γ )0 , ψ = − c ( − b + e )0 a ( − b + e ) . We choose H as H = (cid:15) . Putting Eqs. (B17) and (B19) into Eq. (B16), we get thehopping matrices H = a ae afab be bfac ce cf , H = a ab afae be efac bc cf , (B20)The eigenvalue problem (B13) becomes ( b − e ) (cid:0) − a ( c − f )( β + γ ) + f αE FB (cid:1) − a ( b − e )( c − f )( bβ + eγ ) a ( b − e )( − c ( c − f )( β + γ ) + α ( (cid:15) − E FB )) = 0 , ( c − f ) (cid:0) a ( b − e )(1 + α ) + bβE FB (cid:1) a ( c − f )(( b − e ) e (1 + α ) + β − βE FB ) a ( b − e )( c − f )( c + f α ) = 0 , ( c − f ) (cid:0) a ( b − e )(1 + α ) − eγE FB (cid:1) a ( c − f )( b ( b − e )(1 + α ) + γ ( − E FB )) a ( b − e )( c − f )( c + f α ) = 0 , ( b − e ) (cid:0) − a ( c − f )( β + γ ) − cE FB (cid:1) − a ( b − e )( c − f )( bβ + eγ ) a ( b − e )( − ( c − f ) f ( β + γ ) − (cid:15) + E FB ) = 0 . Assuming that a (cid:54) = 0 , c (cid:54) = f, b (cid:54) = e we solve the aboveequations (according to Eqs. (B18-B19) a = 0 makes the ψ i =1 ,..., proportional, i.e. we find U = 1 CLS; while c (cid:54) = f, b (cid:54) = e enforces ψ i =1 ,..., = 0). Setting a = 1 forconvenience the solution is b = sgn( E FB ) √ E FB − | α | (cid:113) − (cid:112) − α (1 − E FB ) E (4(1 + α ) − αE ) + ( E FB − E FB ( αE − α ) ) ,c = (cid:112) α ( E FB − (cid:15) √ E FB , f = − √ E FB − (cid:15) √ αE FB ,e = − | α | − E FB √ (cid:114) − (cid:113) − α ( E FB − E (4(1 + α ) − αE ) + ( E FB − E FB ( αE − α ) ) ,β = − α ( E FB − E + (cid:112) − α (1 − E FB ) E (4(1 + α ) − αE )2(1 + α )( E FB − E ,γ = − α ( E FB − E + sqrt − α ( E FB − E (cid:0) α ) − αE (cid:1) α )( E FB − E . Pluging these solutions into Eq. (B20) we get the follwing hopping matrices H = − B √ − √ E FB − (cid:15) √ α √ E FB √ E FB − BE FB E FB − − √ E FB − √ E FB − (cid:15) √ αBE / √ α √ E FB − (cid:15) √ E FB − √ αB √ E FB − (cid:15) √ √ E FB (cid:15)E FB − , H = √ E FB − BE FB − √ E FB − (cid:15) √ α √ E FB − B √ E FB − B √ E FB − (cid:15) √ √ α √ E FB √ α √ E FB − (cid:15) √ E FB √ √ α ( E FB − √ E FB − (cid:15)BE / (cid:15)E FB − where A = (cid:113) − α ( E FB − E (4( α + 1) − αE ) ,B = (cid:115) ( E FB − E FB ( αE − α + 1) ) − A ( α + 1) E This solution has three free parameters (cid:15), E FB , α . Wechoose the following values (cid:15) = − , E FB = − , α = 1 togenerate the example shown in Fig. 7: A = 0, B = (cid:112) / H = − ,H = − √ − √ √ − − √ √ − √ − ,H = √ − √ − √ − √ √ √ − Appendix C: FB generation for three hoppingmatrices1. U = (2 , This case is shown in Fig. 4(b). The eigenvalue prob-lem and destructive interference conditions for H Eq. (8)are identical to the case of two hopping matrices and aresolved in Appendix B 1. The only difference are the de-structive interference conditions for H , : H | ψ (cid:105) = H | ψ (cid:105) = 0 (cid:104) ψ | H = (cid:104) ψ | H = 0 ,H | ψ (cid:105) + H † | ψ (cid:105) = 0 , (cid:104) ψ | H + (cid:104) ψ | H † = 0 . (C1)We define: H = Q M Q , H = Q M Q , | x (cid:105) = Q | ψ (cid:105) , | y (cid:105) = Q | ψ (cid:105) . Then the last two equations (C1) become: Q M † Q | x (cid:105) = − Q M Q | y (cid:105) = − Q | a (cid:105) , (cid:104) y | Q M † Q = − (cid:104) x | Q M Q = − Q (cid:104) b | , where we have introduced two arbitrary vectors (cid:126)a and (cid:126)b .For two bands ν = 2 the above equations imply that the last two conditions in (C1) decouple, and therefore theproblem reduces to U = 1 as in the case of two hoppingmatrices. For the number of bands ν > H and H reduces to two independent inverseeigenvalues problems: one for M ( H ) Q M Q | y (cid:105) = Q | a (cid:105) , (cid:104) x | Q M Q = Q (cid:104) b | , and a similar problem for the matrix M ( H ). This is alinear problem: we search for a particular solution as Q M Q = | u (cid:105) (cid:104) y | + | x (cid:105) (cid:104) v | , where we choose the overlined vectors so that: (cid:104) x | ∝ (cid:104) x | , (cid:104) x | x (cid:105) = 1 and (cid:104) y | ∝ (cid:104) y | , (cid:104) y | y (cid:105) = 1. We also assume that (cid:126)u ⊥ (cid:126)x and (cid:126)v ⊥ (cid:126)y . We find upon substitution of the ansatzinto the inverse problem: Q | u (cid:105) = Q | a (cid:105) , (cid:104) v | Q = (cid:104) b | Q . These (cid:126)u and (cid:126)v have the assumed previously orthogonalityproperties. It then follows that the full solution of ( ?? )is given by: H = Q | a (cid:105) (cid:104) y | Q + Q | x (cid:105) (cid:104) b | Q + Q K Q . The inverse problem for H is resolved the same way withminimal modifications. U = (2 , , and ν = 2 We choose the following H and parameterise the CLSamplitudes as follows H = (cid:18) (cid:19) , (cid:126)ψ = (cid:18) pr (cid:19) , (cid:126)ψ = (cid:18) st (cid:19) , (cid:126)ψ = (cid:18) uv (cid:19) . We parameterize the hopping matrices and solve theeigenvalue problem and destructive interference condi-tions in Eq. (21): H = (cid:18) a acb b c (cid:19) , H = (cid:18) d e dfe f (cid:19) , H = (cid:18) g glh h l (cid:19) . a = r ( bp − s ) s , c = − bpr ,d = i (cid:112) b ( p + s ) ( r − s ) − bpr s + r s s , f = 0 ,g = − i ( bp − s ) (cid:112) b ( p + s ) ( r − s ) − bpr s + r s bs ,l = u = 0 ,h = − i (cid:112) b ( p + s ) ( r − s ) − bpr s + r s rs ,v = − i (cid:112) b ( p + s ) ( r − s ) − bpr s + r s bs ,e = − ip (cid:112) b ( p + s ) ( r − s ) − bpr s + r s rs ,t = r (cid:18) b − ps (cid:19) , E FB = b (cid:0) p + s (cid:1) rs . 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