Flat bands by latent symmetry
FFlat bands by latent symmetry
C. V. Morfonios, M. R¨ontgen, M. Pyzh, and P. Schmelcher
1, 2 Zentrum f¨ur Optische Quantentechnologien, Fachbereich Physik, Universit¨at Hamburg, 22761 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging, Universit¨at Hamburg, 22761 Hamburg, Germany
Flat energy bands of model lattice Hamiltonians provide a key ingredient in designing dispersionless waveexcitations and have become a versatile platform to study various aspects of interacting many-body systems.Their essential merit lies in hosting compactly localized eigenstates which originate from destructive interfer-ence induced by the lattice geometry, in turn often based on symmetry principles. We here show that flat bandscan be generated from a hidden symmetry of the lattice unit cell, revealed as a a permutation symmetry upon re-duction of the cell over two sites governed by an effective dimer Hamiltonian. This so-called latent symmetry isintimately connected to a symmetry between possible walks of a particle along the cell sites, starting and endingon each of the effective dimer sites. The summed amplitudes of any eigenstate with odd parity on the effectivedimer sites vanish on special site subsets called walk multiplets. We exploit this to construct flat bands by usinga latently symmetric unit cell coupled into a lattice via walk multiplet interconnections. We demonstrate that theresulting flat bands are tunable by different parametrizations of the lattice Hamiltonian matrix elements whichpreserve the latent symmetry. The developed framework may offer fruitful perspectives to analyze and designflat band structures.
I. INTRODUCTION
Wave excitations in a lattice system are governed by theform of its energy band structure and the corresponding eigen-states. Since the dawn of quantum mechanics, substantial ef-forts have been made to understand the response properties ofcrystals in terms of their energy bands. With the technologi-cal advances of the past decades, however, also artificial lat-tice systems have been realized with ever increasing accuracy.This has enabled an unprecedented engineering of bands withtargeted properties. A most intriguing case is that of “flat”bands with vanishing curvature, which have become a sub-ject of intense research for designed lattice setups [1]. Thoserange among various spatial scales and different technologi-cal platforms, such as photonic waveguide or resonator arrays[2–4], optical lattices for trapped atoms [5, 6], superconduct-ing wire networks [7], nanostructured electronic lattices [8],optomechanical setups [9], or electric circuit networks [10].The remarkable features induced by flat bands essentiallyoriginate from the vanishing group velocity—or, equivalently,diverging effective mass—of the eigenstates residing in them.This allows for dispersionless wave excitations over the wholecrystal-momentum range of the flat band [3], which may beexploited for their robust storage and transfer [11]. In turn,transport properties of flat band states can be manipulatedby weak perturbations which set a dominant energy scale forthem [1]. In particular, flat bands have been used, e. g., tomodel certain types of superfluidity [12–18] or topologicalphases of matter [19–21]. Flat bands have also been exploredvery recently to generate many-body localization [22] and“caging” [23, 24] in the presence of interactions, or to con-trol superradiance via synthetic gauge fields [25].Flat bands of discrete lattice Hamiltonians rely on the oc-currence of eigenstates which are strictly localized on a sub-set of sites, with vanishing amplitude in the remainder of thelattice [26]. Such “compact localized states” (CLSs) can beclassified according to the number of unit cells they occupy[27]. Notably, they do not violate the translational invarianceof the lattice since they can, due to their macroscopic degener- acy at the flat band energy, be linearly combined into extendedBloch states. A CLS originates from the destructive interfer-ence of its amplitudes on the neighboring lattice sites coupledto the site subset the CLS occupies. This mechanism may re-sult directly from the geometric symmetry of the lattice unitcell under a site permutation operation [28, 29]. It may alsobe caused by a bipartite (or chiral) symmetry of a lattice com-posed of sublattices [30], or induced “accidentally” by tuningthe Bloch Hamiltonian matrix elements into the CLS condi-tion. Various schemes for generating flat bands from CLSshave been proposed, based e. g. on local permutation sym-metries [29, 31], “origami” rules [32], local basis transforma-tions [28], solving inverse eigenvalue problems [27, 33], and,as shown very recently, using the properties of Gram matri-ces [34] or combining lattice deformations with site additions[35]. Despite the great value of such approaches, the questionremains whether flat bands may be systematically invoked bysymmetry principles beyond the existing paradigms.In the present work, we propose a scheme to create flatbands which is based on a type of hidden symmetry in theunit cell Hamiltonian of a lattice. This so-called latent sym-metry , introduced recently in graph theory [36], is revealed asa permutation symmetry once reducing the unit cell Hamilto-nian over a particular subset of sites to an effective subsystemHamiltonian. Very recently, latent symmetries were proposedas a novel possibility to explain seemingly accidental spectraldegeneracies of generic Hamiltonian matrices [37]. Reduc-tion over a pair of latently exchange-symmetric sites—as wewill focus on here—results in an effective two-site symmetricdimer, and the symmetry-induced parity of this dimer’s eigen-states is inherited in the original unit cell; that is, any of itseigenstates is locally even or odd on the latently symmetricsites. Latent symmetry of two sites can be intuitively inter-preted as a collective symmetry of so-called walks [38] (i. e.,sequential hoppings) along the coupled sites of the unit cell,starting and ending at each of those two sites. Equivalently,the latent symmetry is simply expressed in terms of powersof the Hamiltonian. We here combine latent symmetry withthe occurrence of special subsets of sites called walk multi- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Figure 1.
Left:
An unweighted graph, representing a Hamilto-nian H with unit hopping (thin edge lines) and zero onsite elements,with two cospectral vertices S = { u, v } = { , } forming a la-tently symmetric site pair. Right: H is reduced over S to the ef-fective Hamiltonian ˜ H S describing a symmetric two-site dimer withonsite elements f (visualized as loop edges in the graph) and hop-ping g depending functionally on the eigenvalue E ; in this exam-ple f ( E ) = (8 − E − E + 2 E + 2 E ) /d ( E ) and g ( E ) =( − E + 2 E ) /d ( E ) , where d ( E ) = 7 E − E − E + E . plets . On each such site subset, the amplitudes of any non-degenerate eigenvector with odd parity on the latently sym-metric sites sum to zero. As we show, periodic lattices gen-erated by interconnection of walk multiplets between latentlysymmetric unit cells host flat bands with corresponding CLSswhich occupy single unit cells. Importantly, the underlyinglatent symmetry persists upon the simultaneous variation ofcertain parameters in the lattice Hamiltonian, making the gen-erated flat bands systematically tunable. With our results ap-plicable to arbitrary dimensions, we demonstrate the principlefor one- and two-dimensional lattices with simple prototypecells possessing latent symmetries.After introducing the concepts of latent symmetry and walkequivalence in Sec.II, we show how to combine them to gen-erate flat band lattices in Sec. III, illustrating the principlewith prototype examples. We discuss possible extensions inSec.IV, while Sec.V concludes this work. II. LATENT SYMMETRY, COSPECTRALITY, AND WALKMULTIPLETS
Consider the eigenvalue problem H | ϕ (cid:105) = E | ϕ (cid:105) for a realsymmetric N × N Hamiltonian matrix H represented in theorthonormal basis of single orbitals | n (cid:105) on N coupled sites, n ∈ H ≡ { , . . . , N } . To introduce the notion of latent sym-metry, let us partition the system into a selected subset S ⊂ H with N S = | S | sites and its complement S = H \ S . Thereduced N S × N S Hamiltonian ˜ H S effectively describing sub-system S under the influence of the rest of the system S is thengiven by [39–41] ˜ H S ( E ) = H S + Γ [ E − H S ] − Γ (cid:62) ≡ H S + Σ S ( E ) , (1)where the diagonal blocks H X = H XX of H are the Hamil-tonians of the isolated subsystems X = S , S and Γ = H SS isthe coupling from S to S . This essentially amounts to Fesh-bach’s projection operators [42] applied to the present discretemodel, while the term Σ S ( E ) can be recognized as the “self-energy” [41, 43] of S induced by its coupling to S . The re-duced eigenvalue problem now has a smaller dimension, but
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21 8 Figure 2. The same graph as in Fig. 1, but now parametricallyweighted in the two different ways which preserve the cospectral-ity of S = { u, v } = { , } . Different edges and loops (hoppingand onsite elements in H ) are visualized by different line numbersand vertex sizes, respectively. Subsets M µ ( µ = 1 , , . . . , ) of siteswith same shading (also with same vertex size) are “walk multiplets”with respect to S (with u, v being “walk equivalent” relative to any M µ ), fulfilling Eq. (7). Here, the doublets are { , } , { , } , { , } ,and the singlets { } , { } . is nonlinear due to the E -dependence of ˜ H S . The spectrum σ ( ˜ H S ) of ˜ H S , given by det( E − ˜ H S ( E )) = 0 , coincides withthat of H after removing E -values which happen to be eigen-values of H S and for which ˜ H S is not defined; symbolically, σ ( ˜ H S ) = σ ( H ) − σ ( H S ) (note that σ is a multiset in the pres-ence of repeated eigenvalues). Most importantly, any eigen-vector | ˜ ϕ (cid:105) of ˜ H S equals the restriction of that of H , with thesame eigenenergy, to the subsystem S [44]: (cid:104) s | ˜ ϕ (cid:105) = (cid:104) s | ϕ (cid:105) for s ∈ S .A latent symmetry is a permutation symmetry Π S of thereduced Hamiltonian ˜ H S such that any extended permutation Π S ⊕ Π S (including the identity Π S = I S ) is not a symme-try of the original Hamiltonian H . Throughout this work, S will consist of two sites u and v , and by ‘latent symmetry’we will always mean symmetry under transposition (i. e., ex-change) of u and v . Then ˜ H S effectively behaves like a two-site dimer with E -dependent onsite potentials and coupling;see Fig. 1. If u and v are latently symmetric in H , this ef-fective dimer is symmetric under exchange of u and v . Ifnon-degenerate, its eigenstates | ˜ ϕ (cid:105) accordingly have definiteparity, (cid:104) u | ˜ ϕ (cid:105) = ± (cid:104) v | ˜ ϕ (cid:105) , and the same holds for the cor-responding eigenstates, with the same eigenenergies, of H : (cid:104) u | ϕ (cid:105) = ± (cid:104) v | ϕ (cid:105) . Such a parity of amplitudes on u and v inthe original, extended system H , is usually traced back to aninvolutory site permutation symmetry. Remarkably, the globalparity in ˜ H S is here inherited to the eigenstates | ϕ (cid:105) as a localparity in H , where there is no permutation symmetry produc-ing it .Latent symmetries were introduced very recently [36, 45]in the context of isospectral graph reductions [46]. There, H is the (weighted) adjacency matrix of a connected graph withvertex set H and edges with weights H mn = (cid:104) m | H | n (cid:105) be-tween vertices. For brevity, we will refer to the graph itselfsimply as H . The isospectral reduction of the graph over asubset S of its vertices is exactly the graph with adjacencymatrix given in Eq. (1). An example graph is shown in Fig. 1,containing the two latently symmetric vertices S = { u, v } = { , } .A crucial fact, promoting the treatment of latent symme-try with the tools of graph theory, is the following [38]:Two latently symmetric vertices u, v of a graph are cospec-tral , meaning that the spectra of the “vertex-deleted” graphs H u = H − u and H v = H − v (where vertices u and v ,as well as edges incident to them, have been deleted, respec-tively) coincide, σ ( H u ) = σ ( H v ) . Alternatively, and of moreuse for our purposes here, cospectral vertices are defined bythe property that their corresponding diagonal entries in anynon-negative power r of H coincide [47], [ H r ] uu = [ H r ] vv ∀ r ∈ N . (2)In general, for an unweighted graph ( H mn ∈ { , } ) theelement [ H r ] mn gives the total number of all possible walks of length r from vertex m to n [48, 49], that is, sequences α = ( a = m, b )( a , b ) · · · ( a r , b r = n ) (3)of r possibly repeated edges ( a i , b i ) with a i +1 = b i . Forexample, the walk with steps → → → → → along the sites of Fig. 1 is denoted as the sequence of edges (1 , , , , , . Note that also loops , with b i = a i , may be included in a walk, representing onsite potentials H a i a i . Eq.(2) concerns the special case of closed walks ( n = m ) starting and ending at each cospectral vertex u or v . Forinstance, Eq.(2) can easily be verified in the unweighted graphof Fig. 1 for the first few powers r , by counting all closedwalks of length r starting at u = 1 or v = 2 . The closed walksof length r = 3 , for example, are (in simplified step notation) → → → and → → → , plus the same inopposite directions, in accordance with [ H ] = [ H ] = 2 .For a weighted graph, a weight w ( α ) is assigned to eachwalk, equal to the product of edge weights along it, w ( α ) = (cid:81) ri =1 w ( a i , b i ) = (cid:81) ri =1 H a i b i . The above interpretation ofmatrix powers in terms of walks is then generalized to a sumover walk weights [50], [ H r ] mn = (cid:88) α ∈A ( r ) mn w ( α ) , r ∈ N (4)where A ( r ) mn is the set of all walks of length r from m to n .Fortunately, there is no need to evaluate Eq.(2) beyond k = N − since, by the Cayley-Hamilton theorem, any higherpowers H k (cid:62) N can be expressed as lower order polynomialsin H . As a consequence, if Eq.(2) holds for r = 0 , .., N − ,it automatically holds for all r . This enables the use of the N × N walk matrix [51, 52] W M of a subset M ⊆ H to encodewalks ending in M , constructed by the action of H r on theindicator vector | e M (cid:105) of M (with (cid:104) m | e M (cid:105) = 1 for m ∈ M and otherwise): W M = [ | e M (cid:105) , H | e M (cid:105) , . . . , H N − | e M (cid:105) ] , (5)also known as the Krylov matrix of H generated by | e M (cid:105) [53, 54]. The r -th column of W M is given by [ W M ] ∗ r = H r − | e M (cid:105) ( ∗ denoting all indices) and its element [ W M ] sr = (cid:88) m ∈ M [ H r − ] sm (6) yields the sum over weighted walks—in the sense of Eq.(4)—of length r − from vertex s to all vertices in M .We call two vertices u, v walk equivalent [55] relative to M if their summed walks to M are equal for any walk length r ,that is, if the corresponding rows of W M are equal, [ W M ] u ∗ = [ W M ] v ∗ . (7)Conversely, we say that M then constitutes a walk multiplet with respect to { u, v } ; specifically, a walk M -let of size M = | M | (singlet for M = 1 , doublet for M = 2 , etc.) [56].Examples of walk singlets ( M = 1 ) and doublets ( M = 2 )are shown in Fig. 2. There, the weights of the graph in Fig. 1have also been parametrized in two different ways such thatthe cospectral pair { , } , and each shown walk multiplet rel-ative to it, are preserved [57]. More specifically, the cospec-trality of { , } and walk multiplets relative to it remain intactfor any arbitrary value—a parameter of H —of edge weights(including loops) which are equal. For instance, in the rightparametrization the equal weights H = H = H = H can be varied together arbitrarily while retaining the cospec-trality and walk multiplets of { , } .As we will see further below, this cospectrality- andmultiplet-preserving parametrization will allow for a flexibletuning of flat bands. In the following, we will first show howthe combination of walk equivalence with cospectrality forvertex pairs may be used to generate CLSs and correspond-ing flat band lattices. III. FLAT BANDS INDUCED BY WALK EQUIVALENTCOSPECTRAL SITES
Let us now consider a graph H with cospectral vertices u, v which are walk equivalent relative to a multiplet M , like inFig. 2 (with M chosen as one of the multiplets M µ ). Due tocospectrality, any non-degenerate eigenvector | ϕ ν (cid:105) has (or, ifdegenerate, can be chosen to have) local parity on { u, v } [58], (cid:104) u | ϕ ± ν (cid:105) = ± (cid:104) v | ϕ ± ν (cid:105) (8)with + ( − ) denoting even (odd) parity. This local parity onthe cospectral pair { u, v } is equivalent to a symmetry Q of H with Q = I which exchanges u and v , that is, Q | e u (cid:105) = | e v (cid:105) , while acting as a general orthogonal transformation onthe complement H \ { u, v } [37, 47], as described in detail inAppendix A.Now, by inserting the spectral decomposition H = (cid:80) ν E ν | ϕ ν (cid:105) (cid:104) ϕ ν | into Eq.(7) and using Eq.(8), one can show[55] that the amplitude sum of any odd { u, v } -parity eigen-state over any walk multiplet relative to { u, v } vanishes, thatis, (cid:88) m ∈ M (cid:104) m | ϕ − ν (cid:105) = (cid:104) e M | ϕ − ν (cid:105) = 0 , (9)where, in the case of degenerate | ϕ − ν (cid:105) , it has been chosen tobe the only { u, v } -odd eigenstate to its eigenvalue E ν , given Figure 3. Schematically depicted graph modifications addressed inSec. III A and Appendix B: (M1) connection of an arbitrary graph G exclusively to a walk singlet c of H , (M2) connection of a walkmultiplet M of H to a single vertex c (cid:48) of an arbitrary graph G (cid:48) , (M3)interconnection of two overlapping walk multiplets X and Y of H . by the projection of the vector | u (cid:105) − | v (cid:105) onto that degeneratesubspace [55]. In particular, (cid:104) m | ϕ − ν (cid:105) vanishes on any walk singlet M = { m } . We note that walk singlets are fixed (thatis, each mapped onto itself) under the action of Q , as shownin Appendix A.The generation of flat bands from a latently symmetric H will ultimately consist in converting it into a Bloch Hamilto-nian by interconnecting any of its walk multiplets within thesame graph H itself via edges with corresponding complexweights. To develop and demonstrate the principle step-by-step in the following subsections, we will first provide thenecessary graph modification rules in Sec. III A; apply themto construct a periodic 1D lattice, or directly its Bloch Hamil-tonian, hosting CLSs in Sec.III B; demonstrate how the corre-sponding flat bands can be parametrically tuned in Sec. III C;and combine the above in a 2D example in Sec.III D. A. Graph modifications preserving walk multiplets
As shown in Ref. [55], certain modifications can be per-formed on a graph H such that the cospectrality of vertex pair { u, v } together with the walk multiplets relative to it remainintact. For clarity, we here focus on their simplest form (seeSec.IV below for related generalizations):The cospectrality of { u, v } , as well as any walk multiplet M of H , are preserved in the new graph H (cid:48) obtained by per-forming the following modifications:(M1) Connection of an arbitrary graph exclusively to anywalk singlet c of H via edges of arbitrary weights,whereby all vertices of the added graph become walksinglets in H (cid:48) ;(M2) Connection of all vertices of any walk multiplet of H to a single vertex c (cid:48) of an arbitrary graph via edges ofuniform weight, whereby all vertices of the added graphbecome walk singlets in H (cid:48) ; (M3) Interconnection of any two walk multiplets of H viaedges of uniform weight between all vertices of onemultiplet and each vertex of the other (added to any al-ready existing edge weights),where any walk multiplet is implied relative to { u, v } . In Ap-pendix B we provide brief proofs of the above properties intheir general form. A generic schematic of the modificationsis given in Fig.3. Note that if the two multiplets in (M3) over-lap (that is, have common vertices), then the vertices in theoverlap are interconnected by double (additional) edges, likethe double loop in Fig.3; see Appendix B.In the following, we will employ the above modifications(M1)–(M3) for the construction of flat band lattices, illus-trated in concrete examples. B. Flat bands via walk multiplet interconnections
In the following principle for constructing flat band lattices,an original latently symmetric Hamiltonian H featuring walkmultiplets will be used as a unit cell of a lattice with Hamil-tonian H (cid:93) . The unit cells are interconnected using the modi-fications described above in Sec. III A. In this way, the latentsymmetry in any copy of H is inherited by the whole latticein the sense that it remains present after the interconnection.In Fig. 4 (top), we illustrate the modifications (M1)–(M3)as applied to our example Hamiltonian H of Fig. 1 to createa periodic lattice H (cid:93) with H as a unit cell. First, for a givenreference cell H (the cell with labeled sites in Fig. 4; simply“cell” will mean “unit cell” from here on), we connect thewalk singlet { } to sites , in the cell above and the doublet { , } to site in the cell below. Thus, the cospectrality ofthe pair { , } and relative multiplets are preserved by simul-taneous application of (M1) (connecting a singlet to the graphabove) and (M2) (connecting a multiplet to the graph below).Note that, after this interconnection, all sites in the remainderof the lattice (outside the reference cell) are singlet relative to { , } in the reference cell. Second, in the resulting graph, weapply (M3) by interconnecting the doublet { , } with the site of the cell above (a singlet relative to { , } in the referencecell) and the singlet { } to sites , of the cell below (bothsinglets relative to { , } in the reference cell).Note that the same interconnections as for the reference cellto adjacent cells can be performed simultaneously for all pe-riodically arranged copies of H , without affecting the cospec-trality of { u, v } = { , } in the reference cell. Thus, sincethe reference cell is chosen arbitrarily, each unit cell in H (cid:93) in-herits the cospectral pair { u, v } (in local labeling for that cell)and its relative walk multiplets from the isolated graph H .The inter-cell connection scheme previously outlined canbe more compactly expressed directly at the level of the BlochHamiltonian H k of the lattice. H k is generally obtained byFourier transformation of the lattice Hamiltonian elements[59] as [ H k ] mn = (cid:88) (cid:96) e i k · (cid:96) (cid:104) m | H (cid:93) | n (cid:96) (cid:105) , (10) Figure 4.
Top:
Construction of a lattice Hamiltonian H (cid:93) using theHamiltonian H of Fig. 1 as an isolated unit cell, via modifications(M1)–(M3) of Sec.III A with respect to its cospectral site pair { , } :For each of the copies of H (left) , setting the labeled graph as ref-erence, first we connect the walk singlet { } to sites , of the cellabove using (M1) and the walk doublet { , } to site of the cell be-low using (M2) (middle) , and then connect the walk multiplets { , } and { } to walk singlets of the resulting graph using (M3) (right) ,with inter-cell connections h indicated by dotted lines. Bottom:
TheBloch Hamiltonian H k corresponding to the lattice Hamiltonian H (cid:93) can be constructed from H via (M3) by interconnecting the singlet { } ( { } ) to the doublet { , } ( { , } ), though with complex Her-mitian couplings he ± ikL indicated by purple (green) double-arroweddotted lines, with k = | k | and lattice constant L ; see Sec.III B. where | n (cid:96) (cid:105) is the orbital | n (cid:105) in the cell at position (cid:96) , with | n (cid:105) ≡ | n (cid:96) = (cid:105) for the reference unit cell at (cid:96) = . The eigen-values E ν ( k ) of H k constitute the band structure of the lat-tice. Interconnections between different cells in the latticegraph H (cid:93) (e. g. with some coupling (cid:104) m | H (cid:93) | n (cid:96) (cid:105) = h ∈ R between sites m, n (cid:96) (cid:54) = ) are equivalent to the correspondinginterconnections in the single cell graph H , though addition-ally weighted with conjugate Bloch phases (i. e. coupling [ H k ] mn = [ H k ] ∗ nm = h e i k · (cid:96) between sites m, n ). This isshown in Fig. 4 (bottom) for the example lattice. The result-ing Bloch graph H k is directed, with complex conjugate edgeweights in opposite directions between any vertex pair beinginterconnected. In fact, H k can be seen as resulting from thecospectrality-preserving modification (M3) (interconnectionof two walk multiplets) on H , though with additional uni-form prefactors e ± i k · (cid:96) in either direction of the connection CLS1CLS2 --- -- -
Figure 5. Band structure (left) of the lattice H (cid:93) in Fig. 4, with unitintra-cell couplings and inter-cell couplings h = 2 . It features twoflat bands at E = ±√ (red lines) corresponding to the two latent-symmetry induced CLSs “CLS1” and “CLS2” depicted (right) . TheCLS amplitudes are real with indicated relative sign + (red) and − (blue), with magnitudes proportional to circle areas. The isolated unitcell Hamiltonian H and site labeling are highlighted (green) in themiddle. An example of destructive interference of one of CLS1’s am-plitudes on the connected site of an adjacent cell is indicated by (or-ange) arrows, with ± ϕ denoting the amplitude on the walk doublet { , } . Energies are in units of the uniform intra-cell site couplings,and lengths in units of the lattice constant L = 1 (quasimomentum k in units of /L ). (see Fig.4). In Appendix C we explicate that site pair cospec-trality and corresponding latent symmetry are preserved underwalk multiplet interconnections (M3) with complex Hermi-tian coupling weights.Let us now explain how the multiplet interconnection de-scribed above can induce CLSs and corresponding flat bandsfor the resulting lattice. Specifically, any { u, v } -odd eigen-state | ϕ − ν (cid:105) of the initially isolated unit cell H constitutes aCLS in the lattice H (cid:93) constructed via multiplet interconnec-tion between unit cells. Indeed, consider the infinite-lengthcolumn vector | ϕ − ν ; (cid:96) (cid:105) defined to have the components of | ϕ − ν (cid:105) on the cell at (cid:96) , padded with zeros on all other cells (cid:96) (cid:48) (cid:54) = (cid:96) ,that is: (cid:104) n (cid:96) (cid:48) | ϕ − ν ; (cid:96) (cid:105) = (cid:104) n | ϕ − ν (cid:105) δ (cid:96)(cid:96) (cid:48) . In other words, | ϕ − ν ; (cid:96) (cid:105) is aCLS occupying the cell at (cid:96) .Notice now that | ϕ − ν ; (cid:96) (cid:105) is an eigenstate of the lattice Hamil-tonian H (cid:93) to the eigenenergy E ν (the eigenenergy of | ϕ − ν (cid:105) inthe isolated cell H ). To see this, let us write H (cid:93) in the form H (cid:93) = (cid:77) (cid:96) H + (cid:88) (cid:96) (cid:54) = (cid:96) (cid:48) , X , Y (cid:16) h (cid:96)(cid:96) (cid:48) XY | e X ; (cid:96) (cid:105) (cid:104) e Y ; (cid:96) (cid:48) | + h.c. (cid:17) , (11)where H is repeated on the block-diagonal and | e X ; (cid:96) (cid:105) (cid:104) e Y ; (cid:96) (cid:48) | contains the off-diagonal block coupling multiplet Y in cell (cid:96) (cid:48) to multiplet X in cell (cid:96) with uniform coupling strength h (cid:96)(cid:96) (cid:48) XY ,and zeros otherwise ( | e X ; (cid:96) (cid:105) being the infinite column withcomponents | e X (cid:105) on cell (cid:96) and zeros otherwise)—see e. g. col-ored couplings in Fig. 4 (top right). Now, acting with H (cid:93) on | ϕ − ν ; (cid:96) (cid:105) directly yields H (cid:93) | ϕ − ν ; (cid:96) (cid:105) = E ν | ϕ − ν ; (cid:96) (cid:105) , (12)since H | ϕ − ν (cid:105) = E ν | ϕ − ν (cid:105) for block (cid:96) (corresponding to theonly cell occupied by | ϕ − ν ; (cid:96) (cid:105) ), while (cid:104) e X ; (cid:96) | ϕ − ν ; (cid:96) (cid:105) = (cid:104) e X | ϕ − ν (cid:105) =0 by Eq.(9).As an example, the lattice constructed in Fig.4 features twodifferent CLS types, which are illustrated in Fig.5 (right panel)in two different unit cells of the lattice. The orange arrowsindicate an example of how the CLS amplitudes cancel out(interfere destructively) on a neighboring cell site upon actionof H (cid:93) due to the multiplet condition, Eq.(9).Analogously to the above, | ϕ − ν (cid:105) is an eigenvector of theBloch Hamiltonian H k constructed from H via multiplet in-terconnections. More specifically, H k can be written as H k = H + (cid:88) (cid:96) , X , Y (cid:0) h (cid:96) XY e i k · (cid:96) | e X (cid:105) (cid:104) e Y | + h.c. (cid:1) , (13)summing over all interconnected multiplet pairs X , Y with Y in cell (cid:96) and X in the reference cell (cid:96) ≡ connected withuniform coupling weight h (cid:96) XY . Acting with H k on | ϕ − ν (cid:105) im-mediately yields H k | ϕ − ν (cid:105) = E ν | ϕ − ν (cid:105) , (14)again due to Eq.(9). This holds for any k , so | ϕ − ν (cid:105) correspondsto a flat band at the k -independent eigenenergy E ν in the bandstructure of the lattice.In Fig. 5, the band structure of the lattice constructed inFig. 4 is shown. As we see, there are two flat bands at E = ±√ , corresponding to the two CLSs “CLS1” and“CLS2” depicted on the right, with odd parity on the cospec-tral sites { , } .The above construction of CLSs and flat bands from la-tent symmetry and walk multiplets can be seen as a gener-alization of the construction from local permutation symme-tries Π which are involutory ( Π = I ) and leave certainsites of the unit cell fixed. If n is such a fixed site, i. e. Π | n (cid:105) = | n (cid:105) , then any eigenstate | ϕ (cid:105) with odd parity under Π has (cid:104) n | ϕ (cid:105) = (cid:104) n | Π ϕ (cid:105) = − (cid:104) n | ϕ (cid:105) = 0 , that is, has anode (vanishing amplitude) on the fixed site. Interconnectingunit cells into a lattice by coupling such Π -fixed sites fromcell to cell, any Π -odd eigenstate of the isolated unit cell Figure 6. Band structure (left) of a lattice constructed from H inFig. 1 as a unit cell by interconnecting its walk doublets { , } and { , } as indicated by dotted edges in H k and H (cid:93) (right) , with unitintra-cell couplings and inter-cell couplings h = 2 . The two flatbands at E = ±√ (red lines) correspond to the same CLSs as inFig.5. yields a CLS and thus a corresponding flat band for the lat-tice. This scenario constitutes a special case of the construc-tion described in the present work (based on walk equivalentcospectral sites), where (a) the cospectral sites are related bya common permutation symmetry exchanging those sites and(b) the unit cells are interconnected via walk singlets relativeto the pair of exchanged sites [60]. We note that such a lo-cal exchange symmetry is, in turn, a special case of generallocal permutation symmetries inducing CLSs, as addressed inRef. [29] in terms of so-called equitable partitions of graphs.Relating that approach to latent symmetries involving site sub-sets S of more than two sites is an interesting direction of fur-ther research.We stress that in the present case (Fig. 5), the zeros of theCLSs within the unit cell (on sites , ) are not induced byany permutation symmetry of the cell fixing those nodal sites,but rather by latent symmetry and walk equivalence (of thecospectral sites relative to walk singlets), as described above.For simplicity, we have applied walk singlet-to-doubletinter-cell connections in the above example (Figs. 4 and 5).It is clear from the above, however, that the procedure to gen-erate flat bands applies naturally for any walk multiplet inter-connection as inter-cell coupling; see Eqs.(13) and (14).As an example, in Fig.6 we start with the same graph H (asin Fig. 4) but now interconnect the walk doublets { , } and { , } in H k , i. e. each site , to both , with complex Her-mitian couplings (including Bloch phases), and correspondingreal inter-cell couplings in H (cid:93) , as explained above. This lat-tice maintains the same CLSs and flat bands as before (Fig.5),though generally with modified dispersive bands.In general, any Hermitian walk multiplet interconnection(M3) with complex Bloch phases, applied to a unit cell H , ismapped to an inter-cell connection in the lattice Hamiltonian H (cid:93) preserving the latent symmetry in each cell. This allowsfor great flexibility in generating flat bands with a given la-tently symmetric prototype cell.To summarize, the proposed flat band construction princi-ple consists in(i) starting with a Hamiltonian H in the form of a graphhaving two latently exchange-symmetric, cospectralvertices { u, v } ,(ii) identifying walk multiplets of H relative to { u, v } , and(iii) using H as the unit cell of a lattice constructed by peri-odically interconnecting any walk multiplet of each cellto any walk multiplet of other cells (which can be neigh-boring cells but also more remote ones).The resulting lattice H (cid:93) then features a flat band for eacheigenstate | ϕ − ν (cid:105) of H with odd parity on { u, v } , which be-comes a macroscopically degenerate CLS in H (cid:93) occupyingone unit cell. C. Parametric invariance of latent symmetry flat bands
It is important to notice that the generation of CLSs and re-sulting flat bands from latent symmetry and walk multipletsof a graph H is not restricted to a fixed set of edge weightvalues H mn . Indeed, there is a certain freedom in changing H ’s elements parametrically while still inducing flat bandsfrom the same latent symmetry and walk multiplets. Specif-ically, this parametrization means that there exist groups ofthe elements H mn which can be set to a common arbitraryreal value per group, without breaking the given latent sym-metry and selected walk multiplets. For example, the weightparametrizations shown in Fig.2 preserve the cospectrality of { u, v } as well as the multiplets interconnected to form the lat-tice in Fig.4. Thus, when varying the weight parameters (thatis, the common value of each group of elements H mn ), flatbands are still induced for the constructed lattice. Their en-ergy positions, however, generally depend on the weight pa-rameters, which allows for tuning the flat bands relative to therest of the band structure.We demonstrate this parametric invariance of the flat bandsfor our navigating example graph in Fig. 7, where the bandedges for the lattice in Fig.5 are plotted for a continuous vari-ation of selected couplings in the unit cell. Specifically, us-ing the cospectrality- and multiplet-preserving edge weight Figure 7. Band edges (black lines) and band projections (cid:83) k ∈ BZ E ν ( k ) (gray shades) of the bands E ν ( k ) over the Brillouinzone (BZ) for the lattice in Fig. 5 with varying coupling parameter H = H = H = H = p indicated (orange lines) in theschematic on the left; p = 1 corresponds to the band structure inFig. 5. The two flat bands induced by latent symmetry of sites , occur at any p , with energies varying in p (thick red lines). parametrization of Fig.2 (right), a selected subset of couplingsis set to a common varying value p (see Fig. 7 caption). Aswe see, while the dispersive band widths vary with p , the flatbands constructed by latent symmetry for p = 1 remain flatfor any p (see red lines, whose vertical cross sections at any p are single points at the corresponding E ν ). This is in contrastto flat bands that may appear “accidentally” when varying p ,as seen e. g. for the second lowest band which becomes flat ata single point around p ≈ . .Further, in this example the upper (lower) flat band en-ergy increases (decreases) linearly with p across the disper-sive bands and the gaps between them. This demonstratesthe possibility to tune the flat band positions relative to dis-persive bands without invoking any apparent symmetry of theunit cell. D. Flat bands via walk singlet augmentation
Another variation of using the graph modifications inSec. III A for flat band construction is to first modify a la-tently symmetric graph H itself, before interconnecting it intoa lattice. In particular, using (M2) we can augment H by con-necting new vertices to walk multiplets relative to a cospectralpair { u, v } . In the resulting graph H (cid:48) , each such new vertex c (cid:48) will be a walk singlet, which will in turn have vanishing am-plitude in any non-degenerate eigenvector with odd parity on { u, v } ; see Eq. (9). This ‘singlet augmentation’ may be used,e g., to bring a given unit cell into a more preferable shape for -3-2-1 0 001234
758 6 1 234 910 --- -- -
Figure 8. Band structure (left) of a 2D lattice (bottom right) con-structed by repetition of an unweighted -vertex graph H augmentedby two vertices and connected to the graph’s cospectral pair { , } and the walk doublet { , } , respectively (top right) , with dot-ted double-arrowed edges indicating complex couplings he ± ik x ( y ) L in ± x ( y )-direction in the Bloch Hamiltonian H (cid:48) k (see text). Inter-cell edges with unit weight h = 1 (dotted lines) connect the walksinglets { , , , } of the graph in x - and y -direction, preservingits two CLSs (depicted in the lattice; colormap as in Fig. 5) whichcorrespond to two flat bands at E = ±√ − . connection into a lattice.We demonstrate this procedure by constructing a 2D flatband lattice in Fig. 8. The original -vertex graph H (upperright of figure) has four doublets relative to the cospectral pair { u, v } = { , } . The graph has two eigenvectors with odd { u, v } -parity which vanish on those singlets. Note that, likethe graph in Fig. 2, also this one can be parametrized in itsedge weights while keeping its latent symmetry and corre-sponding compact eigenvectors, as we will see below. Forsimplicity, we first keep its unweighted version. We now con-nect two new vertices and to two doublets using modi-fication (M2), which thus yields two more singlets on whichthe previous compact eigenvectors also vanish. Then, we con-nect the new graph H (cid:48) into a 2D lattice—similarly to the pro-cedure in Sec. III B—via its four corner singlet vertices, asshown, described by the corresponding Bloch Hamiltonian Figure 9. Band edges and projections as in Fig. 7 but for the 2Dbands of the lattice in Fig.8 for varying parameter p in two differentcases of the edge weight parametrization as shown at the top; p = 1 corresponds to the band structure in Fig.8. H (cid:48) k . The resulting band structure E ν ( k ) features two flatbands at E = ±√ − , with the corresponding CLSs de-picted in two unit cells of the lattice.We emphasize that the CLSs are induced by the latent sym-metry of the site pair { , } , and not by a permutation symme-try of the lattice cell. Specifically, the cell is indeed reflectionsymmetric about one diagonal (the line passing through sites and ), and the CLSs are odd under this reflection with nodeson this diagonal, as expected (recall discussion on permutationsymmetry Π in Sec. III B). This symmetry does not explain,however, the other two CLS nodes at sites and . Eachof those are instead fixed under the latent symmetry opera-tion Q (see Appendix A) induced by the cospectrality between u, v . In fact, a general weight parametrization preserving thewalk multiplet structure violates the cell’s reflection symme-try, though retains the compactness of the CLSs, that is, theirnodes on the singlet sites, and the corresponding flat bands.The latter is demonstrated in Fig. 9, where a cospectrality-and multiplet-preserving parametrization of the edge weightsby real parameters p i =1 , ,..., is considered (top panel).Parametrization of the onsite elements, or loops, is also pos-sible but not shown for simplicity. The band edge evolutionfor two parametrical variations is plotted. In the first case (leftplot), we set p = p = p (other intracell hoppings to unity)and vary p , whereby the flat bands (red lines) are preservedwith linearly varying energy. In the second case (right plot),we set p = p = p = p (other intracell hoppings againequal unity), whose variation modifies the dispersive bandsbut leaves the flat band energies fixed. We thus see that suchparametrizations of the unit cell Hamiltonian preserving itslatent symmetry, together with the chosen inter-cell couplings(whose variation, not shown here, evidently also preserves thelatent symmetry), can be used to tune the induced flat bandsflexibly in relation to the surrounding band structure.Finally, we note that in this example we interconnected theunit cells via their corner walk singlets for simplicity. Onecould instead, or additionally, interconnect larger multipletsbetween the cells, still preserving the same CLSs and con-comitant flat bands—though generally changing the disper-sive bands. For example, the walk doublet { , } (see Fig. 8)of the cell at each (cid:96) could be connected diagonally in the lat-tice to the doublet { , } of the cell at (cid:96) + L ˆ x + L ˆ y . IV. DISCUSSION
Having demonstrated how latent symmetry, in combinationwith walk multiplets, may be employed to induce flat bands,let us now discuss some aspects and extensions of the pre-sented framework.
A. Number and spatial extension of CLSs
In each of the above examples, Figs.5 and 8, there were twoCLSs per unit cell associated with a cospectral site pair { u, v } in H . As mentioned above, the number of such CLSs dependson the structure of the graph used as a cell. Specifically, thenumber of eigenstates of H with odd { u, v } -parity is given bythe dimension of the Krylov subspace generated by the vector |−(cid:105) ≡ | u (cid:105) − | v (cid:105) [58], that is, the rank of the correspondingKrylov matrix [ |−(cid:105) , H |−(cid:105) , H |−(cid:105) , . . . , H N − |−(cid:105) ] . Also,there may be more than one cospectral pair in the graph H ,each of which may induce different CLSs in a correspond-ing multiplet-interconnected lattice. Of course, such latentlysymmetric cospectral pairs may further coexist with cospec-tral pairs corresponding to permutation symmetries swappingtwo vertices u, v . Clearly, for such pairs all other sites in H are walk singlets, with corresponding CLSs confined to { u, v } in each lattice cell. In the examples shown here, we have cho-sen cell graphs having only latent symmetries for clarity.Note, further, that in the above flat band constructionscheme (see Sec.III B) we have explicitly considered the orig-inal graph H (or some augmented one, see Sec. III D) as theunit cell of the generated lattice H (cid:93) . The induced CLSs thenoccupy U = 1 unit cell each, using the number of occupiedunit cells U as a flat band classifier [27] (recently generalizedaccordingly for lattice dimensions d > [61]). One could inprinciple, however, start with a supercell H (cid:48) of a target lat-tice H (cid:93) , consisting of U > interconnected copies of H , andlook for new cospectral pairs { u (cid:48) , v (cid:48) } which are not cospectralin H . Then, CLSs induced by { u (cid:48) , v (cid:48) } -odd eigenstates of H (cid:48) will generally occupy U > primitive unit cells within thesupercell. The key challenge here would be to design inter- H connections which coincide with walk multiplet intercon-nections between supercells H (cid:48) . We leave this endeavour forfuture work. B. Generalizations of walk multiplets
The concept of walk multiplets can be generalized [55] byreplacing the indicator vector of M in Eq. (5) with a nonuni-form version | e γ M (cid:105) , with a tuple γ of generally different ampli-tudes γ m = (cid:104) m | e γ M (cid:105) (15)for m ∈ M and otherwise. γ m = 1 , up to a global factor,corresponds to the uniform walk multiplets considered so far.If a new vertex c (cid:48) is connected to M via those weights γ m ,then the associated cospectrality is preserved and c becomesa walk singlet if Eq. (7) is fulfilled—now with the walk ma-trix generated by | e γ M (cid:105) . In other words, the modification (M2)of Sec. III A is generalized to such nonuniform walk multi-plets, as is, similarly, the multiplet-interconnection (M3); seeAppendix B. A particular case is that of overlapping uniformmultiplets M µ ( µ = 1 , , . . . ), whose union yields a nonuni-form multiplet with indicator vector | e γ M (cid:105) = (cid:80) µ | e M µ (cid:105) , where | e M µ (cid:105) is the usual indicator vector of multiplet M µ . An exam-ple is schematically shown as overlapping multiplets X and Y in Fig.3.Another variation is to consider walk anti-equivalence byreplacing Eq. (7) with [ W M ] u ∗ = − [ W M ] v ∗ . In this case, M is a walk anti-multiplet relative to { u, v } and the role of par-ity is swapped: Now the eigenvectors | ϕ + ν (cid:105) with even parityon { u, v } become CLSs, with vanishing amplitudes on anti-singlets [55]. These generalizations of the concept of walkmultiplets offer an even larger flexibility in generating flatband lattices from graphs with latent symmetries. C. Occurrence and construction of latently symmetric graphs
In all of the above, we have assumed that the original graph H is latently exchange-symmetric, that is, features some pairof vertices u and v which are cospectral but not exchange-symmetric in H . We also assumed the given graph to fea-ture some walk multiplets relative to { u, v } . The aim was toshow how these properties, when given, can be used instead ofcommon symmetries—that is, permutation symmetries com-muting with H —to induce CLSs and corresponding flat bandsfor periodic lattice structures.The systematic construction of latently symmetric graphsis far from trivial. To date, and to the best of our knowl-edge, there is indeed no general procedure for constructingundirected, latently symmetric graphs; it is rather a subjectof ongoing research. One approach is based on “unpacking”the isospectrally reduced form of a graph [38], by applyingpartial fraction decomposition to its functional dependence on0the eigenvalue E , and then accordingly constructing a gen-erally directed graph with complex weights. Another recent,semi-empirical approach [62], starts from a graph with triv-ially cospectral vertices—that is, induced by some permuta-tion symmetry—which is then modified by adding verticesand edges such that the permutation symmetry is broken whilethe cospectrality is not.In fact, the defining property of vertex cospectrality, Eq.(2)evaluated up to r = N − , makes it straightforward toresort to numerical iteration for verifying it. In this spirit,Ref. [62] reports on the occurrence of latently symmetricgraphs out of all possible unweighted graphs of given smallsize. Specifically, we have created a database of all un-weighted graphs (adjacency matrices) of size up to N = 11 which have at least one cospectral vertex and no permuta-tion symmetry. For N (cid:54) , there is no such graph. For N = 8 , , , , there are , ,
78 489 , suchgraphs, respectively. Although this is, in each case, a smallportion ( ≈ . , . , . , . (cid:104) ) of all possible graphs, theanalysis shows that there is a substantial number of latentlysymmetric unweighted graphs even for such small sizes. Thismeans that latent symmetry would in principle not be hard todesign in a targeted setup, consulting e. g. the above database.For larger graphs ( N (cid:29) ), there is numerical evidencethat the occurrence of latent symmetries is correlated to that ofcommon permutation symmetries, in the sense that their per-centage has been found to follow the same trend when vary-ing a structural parameter for a class of randomly generatedgraphs, as stated in Ref. [45]. The exact reason for this behav-ior is an open question.In the same manner as cospectral vertices, we identify walkmultiplets relative to a cospectral pair { u, v } of a given graphby scanning through all vertex subsets of all possible sizes forthose that fulfill Eq.(7). For the graphs available in the abovedatabase, we have observed that, typically, the graphs havemultiple walk multiplets (relative to the featured cospectralpair(s)) for each multiplet size—although there are e. g. caseswhere walk singlets are absent—with the number of multi-plets typically increasing with their size. Also, there is alwaysat least one walk doublet, namely the cospectral pair itself.The walk multiplet structure is further enriched by consider-ing their generalized version (nonuniform and anti-multiplets,see Sec. IV B above), as described in detail in Ref. [55]. Therelation of general walk multiplets to the structure of eigen-vectors of graphs with cospectral vertices is an interestingtopic to be pursued. V. CONCLUSIONS
We have shown how flat bands can be induced by latentsymmetry between a pair of sites in the unit cells of discretelattices. This symmetry is revealed as an exchange permu-tation symmetry of the effective Hamiltonian upon reductionof the cell over the site pair subsystem, and imposes odd oreven local parity of the original Hamiltonian eigenstates onthose two sites. Using recent concepts and tools from graphtheory, where latent symmetry takes the form of cospectrality between two vertices, we propose a framework for generatingflat bands from the structural properties of graphs lacking per-mutation symmetries. The key ingredient is the occurrence of walk equivalence of cospectral vertices relative to vertex sub-sets called walk multiplets . This signifies a collective symme-try between possible walks along the edges of a graph fromits cospectral vertices to a given walk multiplet, expressed interms of corresponding walk matrices . Crucially, the ampli-tude sum on walk multiplets vanishes for any non-degenerateeigenvector with odd parity on cospectral vertices.When connecting the graph as a unit cell into a lattice viaits walk multiplets, those eigenvectors constitute compact lo-calized states (CLSs) forming flat bands within an otherwisedispersive band structure. We illustrate the scheme for 1Dand 2D lattices using simple graphs with cospectral sites. Ageneralization to more complex cell geometries, possibly withmultiple latent symmetries, and to higher dimensional latticesis straightforward. As we demonstrate, the latent symmetrypersists over flexible parametrizations of the lattice Hamilto-nian elements, making the induced flat bands systematicallytunable. This should allow for a feasible generation of flatbands from latent symmetries in various realization platformssuch as, e. g., photonic waveguide arrays or electric circuitnetworks, with tailored inter-site connections. We thus offera fundamental insight into a class of CLSs originating fromhidden Hamiltonian symmetries, which may also provide avaluable tool in designing flat band setups.
ACKNOWLEDGMENTS
We thank Jens Kwasniok for helpful discussions regarding Q -matrices and walk multiplets. Funding by the DeutscheForschungsgemeinschaft under grant DFG Schm 885/29-1 isgratefully acknowledged. M. P. is thankful to the ‘Studien-stiftung des deutschen Volkes’ for financial support in theframework of a scholarship. Appendix A: Cospectrality from walk matrices and orthogonalsymmetry
We here give a brief account on the orthogonal symme-try matrix Q describing vertex cospectrality. The purpose isto provide an insightful connection between the latent sym-metry of a graph, upon reduction over two cospectral ver-tices, and the underlying symmetry operation exchangingthose vertices in the original graph. The description is adaptedfrom Ref. [51] to a graph with N vertices H and symmetricweighted adjacency matrix H .First, consider two arbitrary subsets U , V ⊆ H with walkmatrices W X = [ | e X (cid:105) , H | e X (cid:105) , . . . , H N − | e X (cid:105) ] (A1)for the indicator vectors | e X (cid:105) , X = U , V . If W V is invertible(that is, has full rank N ), then the matrix Q UV = W U W − V (A2)1commutes with H , thus representing a general symmetrytransformation. To see this, recall that H N = N − (cid:88) r =0 c r H r (A3)by the Cayley-Hamilton theorem which states that H fulfillsits own characteristic equation χ ( x ) = (cid:80) Nr =0 a r x r = 0 ,where c r = − a r /a N . Therefore, we have that HW X = W X C, X = U , V (A4)where C = . . . c . . . c . . . c ... ... . . . ... ... . . . c N − (A5)is the companion matrix [53] for H . Thus, HQ UV = W U CW − V = W U ( W − V HW V ) W − V = Q UV H. (A6)Further, if both W U and W V are invertible and fulfill W (cid:62) U W U = W (cid:62) V W V , (A7)then Q UV is orthogonal: Q (cid:62) UV = [ W − V ] (cid:62) W (cid:62) U == [ W − V ] (cid:62) W (cid:62) V W V W − U = W V W − U = Q − UV . (A8)With invertible W X ( X = U or V ), H has simple eigenvalues[51] E ν (no degeneracies) and then, because Q UV commuteswith H , it is a polynomial in H . Thus, if H is symmetric, sois Q UV , and since it is also orthogonal, we obtain that Q UV = Q (cid:62) UV = Q − UV . (A9)Now, if U = { u } and V = { v } constitute two cospectralvertices, we have [47] W (cid:62) u W u = W (cid:62) v W v . (A10)Then, if both W u and W v are invertible, Eq. (A9) holds for Q { u } , { v } ≡ Q of the main text, that is, Q = Q (cid:62) Q = I . Inparticular, since QW v = W u and QH = HQ , we have that Q | v (cid:105) = | u (cid:105) and Q | u (cid:105) = | v (cid:105) . Thus, being also orthogonal, Q is block-diagonal with one block being the antidiagonal ma-trix J S = (cid:20) (cid:21) (A11)swapping u and v in S = { u, v } .Further, for a walk multiplet M the condition [ W M ] u ∗ =[ W M ] v ∗ , Eq. (7), yields (cid:104) u | H r | e M (cid:105) = (cid:104) v | H r | e M (cid:105) = (cid:104) v | H r Q (cid:62) Q | e M (cid:105) = (cid:104) u | H r Q | e M (cid:105) for k = 0 , . . . , N − , where we used H = H (cid:62) , QH = HQ , and Q | v (cid:105) = | u (cid:105) .Since W u has full rank, the N columns H r | u (cid:105) span an N -dimensional column space, meaning that Q | e M (cid:105) = | e M (cid:105) (bothvectors have equal projections in all N dimensions). Thus,if H has a vertex subset F consisting of walk singlets, then [ W s ] u ∗ = [ W s ] v ∗ ∀ s ∈ F , so another block of Q is the | F |×| F | unit matrix I F leaving the singlet vertices fixed (causing theodd- { u, v } -parity eigenvectors to vanish on them).The remaining orthogonal block Q O operates on the re-maining vertices within O = H \ ( S ∪ F ) , transforming thecorresponding rows of W v into those of W u : [ W u ] O , ∗ = Q O [ W v ] O , ∗ . With vertices labeled accordingly, Q thus hasthe form Q = J S ⊕ I F ⊕ Q O . (A12)As an example, for the graph of Fig. 1 the | O | × | O | block isgiven by Q O = (cid:104) A BB A (cid:105) , with A = (cid:104) (cid:105) and B = (cid:104) − − (cid:105) ,where O = { , , , } .Notice here that, since Q commutes with H , its eigenvec-tor matrix block-diagonalizes H accordingly under similaritytransformation. Such a transformation can be seen as reminis-cent of the “Fano detangling” procedure of Ref. [28], thoughhere for a cospectral site pair { u, v } (instead of a single site)and determined from the walk structure of H .If the spectrum of H is degenerate or has any eigenvectorwith vanishing amplitudes on { u, v } , then W u and W v do nothave full rank [47, 52] and are thus not invertible. Hence, al-though a Q matrix still exists, which is unique under the con-vention of treating eigenvectors vanishing on { u, v } as { u, v } -even [47], it cannot be obtained directly from Eq.(A2) [63].Alternatively, the following expression can be used for a Q -matrix (obeying, Q = Q (cid:62) Q = I and Q | u (cid:105) = | v (cid:105) ) [37]: Q = P + − P − = I − P − , (A13)where P ± = (cid:80) ν | ϕ ± ν (cid:105) (cid:104) ϕ ± ν | is the projector onto eigenvec-tors with ± -parity on { u, v } ( P + also including eigenvectorsvanishing on { u, v } ), chosen in case of degeneracy such thatthere is at most one eigenvector of each parity non-vanishingon { u, v } for any given eigenvalue. This expression is not di-rectly derived from the structure of the graph (specifically, itswalk matrices W u , W v ) but rather invokes the spectral prop-erties of H —that is, one first needs to find its eigenvectors. Appendix B: General cospectrality-preserving graph extensionsand intraconnections
We here show that the cospectrality of a pair { u, v } andthe walk multiplets relative to it are preserved by the mod-ifications (M1), (M2), and (M3) listed in Sec. III A. Like inRef. [55], the modifications are now stated in a more generalform for nonuniform walk multiplets with weighted indicatorvector | e γ M (cid:105) (see Sec.IV).For an original weighted adjacency matrix of a graph H ,the modified one H (cid:48) will have the form of a sum H (cid:48) = A + B (B1)2with A = H ⊕ G (B2)generally being a block-diagonal matrix (including the case ofabsent or × block G ) and B = | b (cid:105) (cid:104) b | + | b (cid:105) (cid:104) b | (B3)being a symmetric sum of rank-one coupling matrices. Set-ting the | b , (cid:105) to be site subset indicator vectors below, B willexpress the interconnection of those subsets in the modifiedgraph H (cid:48) .The powers of H (cid:48) , appearing in the corresponding modifiedwalk matrices W (cid:48) M , are given by [ A + B ] r = r (cid:88) p =0 (cid:88) π ( A,B ) { A r − p B p } , (B4)where (cid:80) π ( A,B ) { A r − p B p } denotes the sum of all distinct per-mutations of A ’s and B ’s in matrix products with r − p A ’s and p B ’s; for instance, AAB + ABA + BAA for r = 3 , p = 1 . H (cid:48) r is thus generally a weighted sum of products of the ma-trices H r − p ⊕ G r − p , [ | b (cid:105) (cid:104) b | ] n , [ | b (cid:105) (cid:104) b | ] n , [ | b (cid:105) (cid:104) b | ] n , [ | b (cid:105) (cid:104) b | ] n with p ∈ { , , . . . , r } and n i ∈ { , , . . . , p } .In the following, we briefly prove preservation of cospec-trality and walk multiplets under modifications (M1), (M2),(M3), which are depicted schematically in Fig.3.
1. Singlet extension (M1)
For a singlet c ( (cid:54) = u, v ) of H connected symmetrically—i. e., so that H (cid:48) is symmetric—to an arbitrary graph G withvertices G , we have | b (cid:105) = | c (cid:105) , which is the indicator vectorof c in H (cid:48) , and | b (cid:105) = | e γ G (cid:105) , which is the arbitrarily weightedindicator vector of G , in Eq.(B3).From Eqs. (B1) and (B4), elements [ H (cid:48) r ] uu = (cid:104) u | H (cid:48) r | u (cid:105) thus only have contributions involving | u (cid:105) in factors [ H q ] uu and [ H q ] uc , [ H q ] cu for different powers q . For instance,with (cid:104) c | e γ G (cid:105) = 0 , we have A B A = A ( (cid:104) e γ G | e γ G (cid:105) | c (cid:105) (cid:104) c | + | e γ G (cid:105) (cid:104) e γ G | ) A , whose uu -element becomes [ A B A ] uu =[ H ] uc (cid:104) e γ G | e γ G (cid:105) H cu .Since { u, v } are cospectral in H and c is a walk singlet,those factors [ H q ] uu , [ H q ] uc , [ H q ] cu remain equal under thereplacement u → v , as do, trivially, factors not containing theindex u . This yields [ H (cid:48) r ] uu = [ H (cid:48) r ] vv , so { u, v } remaincospectral in H (cid:48) .Similarly, walk matrix elements [ W (cid:48) M ] ur for any walk mul-tiplet M of H only have contributions involving | u (cid:105) in factors [ W M ] uq and [ H q ] uc . Thus, since [ W M ] uq = [ W M ] vq ( M walkmultiplet in H ) and [ H q ] uc = [ H q ] vc ( c walk singlet in H ),we have [ W (cid:48) M ] u ∗ = [ W (cid:48) M ] v ∗ , that is, M is a walk multiplet alsoin H (cid:48) .
2. Multiplet extension (M2)
For a walk multiplet M of H connected symmetrically to asingle vertex c (cid:48) of an arbitrary graph G , we have | b (cid:105) = | c (cid:48) (cid:105) , which is the indicator vector of c (cid:48) in H (cid:48) , and | b (cid:105) = | e M (cid:105) inEq. (B3). With similar arguments as in Appendix B 1 above,again we get [ H (cid:48) r ] uu = [ H (cid:48) r ] vv and [ W (cid:48) X ] u ∗ = [ W (cid:48) X ] v ∗ forany walk multiplet X of H .
3. Multiplet interconnection (M3)
If two disjoint walk multiplets X and Y of H are symmet-rically and fully interconnected—that is, each vertex of oneis connected to all of the other, with weights added to anyalready existing connection—we have A = H in Eq. (B2),with G now being absent, and | b (cid:105) = | e X (cid:105) , | b (cid:105) = | e Y (cid:105) inEq.(B3). With similar arguments as in Appendix B 1, cospec-trality of the pair { u, v } and any walk multiplet M relative toit are preserved in H (cid:48) . Using the same form of the intercon-nection matrix B , this also holds if X and Y overlap, that is,have common vertices. Appendix C: Symmetry versus Hermiticity
In this appendix we briefly comment on the relationbetween vertex cospectrality and latent symmetry whenconsidering a complex Hermitian—as opposed to a realsymmetric—Hamiltonian H . Note that, for complex Hermi-tian H , the cospectrality condition for a pair { u, v } in termsof walk matrices, Eq.(A10), is replaced with W † u W u = W † v W v , (C1)with ( ) † = ( ) (cid:62)∗ denoting Hermitian conjugation.It was recently shown [38] that cospectrality of a vertexpair { u, v } of a graph H is equivalent to latent symmetry be-tween u and v —that is, the × reduction ˜ H { u,v } of H isbisymmetric—if H is symmetric, that is, its graph is undi-rected. Therefore, to relate vertex cospectrality to latent sym-metry, we have assumed a symmetric unit cell Hamiltonianmatrix H , which was also chosen real to generally possess areal eigenvalue spectrum.Nevertheless, if H is modified into a Bloch Hamiltonian H k exclusively by interconnecting walk multiplets with self-adjoint complex weights (a special case of a directed graph;see H k in Fig. 4 with each dotted line indicating complexconjugate weights he ± i k · (cid:96) in either direction), then vertexpair cospectrality does imply corresponding latent symme-try, and vice versa. Indeed, the multiplet interconnectionin Appendix B above remains valid in the same form (with (cid:104) x | = | x (cid:105) † and “symmetric” replaced by “self-adjoint”) forwalk multiplets with indicator vector | e γ M (cid:105) weighted by a com-plex tuple γ (see Eq. (15)). For instance, in H k in Fig. 4 thesinglet { } is connected with complex weight γ = he ikL =[ H k ] m = [ H k ] ∗ m ( m = 1 , ) to the doublet { , } and thesinglet { } is connected with complex weight γ = he − ikL =[ H k ] m = [ H k ] ∗ m ( m = 3 , ) to the doublet { , } , for somereal h .Now, since such walk multiplet interconnections preserve { u, v } -cospectrality and relative multiplets (as shown in Ap-pendix B), in particular { u, v } itself remains a walk doublet3in H k : [ H r k ] uu + [ H r k ] uv = [ H r k ] vv + [ H r k ] vu (C2)for all powers r ∈ N . As a consequence, [ H r k ] uv = [ H r k ] vu ∈ R ∀ r ∈ N . (C3)Thus, the restriction of each power H r k to the cospectral pairis bisymmetric, that is, commutes with the × exchangematrix J S = { u,v } , Eq.(A11).As we showed very recently in Ref. [37], a necessary andsufficient condition for a latent symmetry transformation T upon reduction to a vertex subset S is that all powers of H restricted to S have the same symmetry: T ˜ H S = ˜ H S T ⇐⇒ T [ H r ] S = [ H r ] S T ∀ r ∈ N . (C4) In the present case S = { u, v } is a cospectral pair and T = J S , with [ H r k ] S J S = J S [ H r k ] S implying ˜ H k ; S J S = J S ˜ H k ; S ,meaning that H k has a latent J S -symmetry in its reductionover { u, v } .To summarize: For a general directed graph H , { u, v } -cospectrality is necessary but in general not sufficient forcorresponding latent symmetry [38]; but for a complex self-adjoint H (cid:48) (in our case the Bloch Hamiltonian H k ) con-structed from an undirected H via Hermitian interconnectionof walk multiplets relative to { u, v } , it is both necessary andsufficient. [1] D. Leykam, A. Andreanov, and S. Flach, Artificial flat bandsystems: From lattice models to experiments , Adv. Phys. X ,1473052 (2018).[2] D. Leykam and S. Flach, Perspective: Photonic flatbands , APLPhotonics , 070901 (2018).[3] S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman,P. ¨Ohberg, E. Andersson, and R. R. Thomson, Observation ofa Localized Flat-Band State in a Photonic Lieb Lattice , Phys.Rev. Lett. , 245504 (2015).[4] R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real,C. Mej´ıa-Cort´es, S. Weimann, A. Szameit, and M. I. Molina,
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