One-dimensional 2^n-root topological insulators and superconductors
OOne-dimensional n -root topological insulators and superconductors A. M. Marques, ∗ L. Madail,
1, 2 and R. G. Dias Department of Physics & i3N, University of Aveiro, 3810-193 Aveiro, Portugal International Iberian Nanotechnology Laboratory, 4715-310 Braga, Portugal (Dated: February 26, 2021)Square-root topology is a recently emerged sub-field describing a class of insulators and super-conductors whose topological nature is only revealed upon squaring their Hamiltonians, i.e. , thefinite energy edge states of the starting square-root model inherit their topological features fromthe zero-energy edge states of a known topological insulator/superconductor present in the squaredmodel. Focusing on one-dimensional models, we show how this concept can be generalized to 2 n -roottopological insulators and superconductors, with n any positive integer, whose rules of constructionare systematized here. Borrowing from graph theory, we introduce the concept of arborescenceof 2 n -root topological insulators/superconductors which connects the Hamiltonian of the startingmodel for any n , through a series of squaring operations followed by constant energy shifts, to theHamiltonian of the known topological insulator/superconductor, identified as the source of its topo-logical features. Our work paves the way for an extension of 2 n -root topology to higher-dimensionalsystems. I. INTRODUCTION
Topological insulators (TIs) are one of the most in-tensively studied topics in condensed matter in recentyears. Paradigmatic examples of one-dimensional (1D)TIs, such as the Su-Shrieffer-Heeger (SSH) model , ex-hibit midgap zero-energy edge states under open bound-ary conditions (OBC), which can be related to a non-trivial and quantized topological index characterizing thebulk bands below that energy gap, in what is commonlyreferred to as the bulk-boundary correspondence .Inspired by Dirac’s derivation of his eponymous equa-tion from taking the square-root of the Klein-Gordonequation, Arkinstall et al. proposed a scheme of relat-ing the topological properties of a given 1D model withfinite energy edge states to those of its squared model, aconventional TI with zero-energy edge states. Accord-ingly, these models came to be known as square-rootTIs ( √ TIs). Later on it was realized that the con-cept of √ TIs could be extended to bipartite models,since their squared versions appear in a block diagonalform, with one of them corresponding to the TI fromwhich the topological features are inherited in the start-ing √ TI. Square-root topology was quickly extendedto other 1D models including topological superconduc-tors (TSs) and non-Hermitian systems , to higher-orderTIs [ d -dimensional lattices hosting topological edgestates in ( d − j )-dimensions, with j ≥ ], to topolog-ical semimetals and Chern insulators .The general recipe for the construction of √ TIs andsquare-root TSs ( √ TSs) from their topological squaredcounterparts was developed by Ezawa . It relies on therealization that, upon treating the tight-binding chain asa connected graph, one can construct the square-root ver-sions of a given TI/TS by subdividing its tight-bindinggraph and re-normalizing the resulting hopping param-eters. The main idea of this method is that subdivisionof the tight-binding graph guarantees that it will become bipartite, even if it not so before. In turn, the bipartiteproperty guarantees that the squared Hamiltonian can bewritten in a block diagonal form. Focusing on 1D mod-els, we show here how a further elaboration of the methodin [7] allows one to contruct TIs/TSs of root degree 2 n ( n √ TIs/ n √ TSs), with n any positive integer. Further-more, since the distance between the original TI/TS andthe constructed n √ TIs/ n √ TSs, measured by the numbersuccessive squaring operations that have to be applied tothe Hamiltonian of the latter in order to get to the for-mer, grows with n , we codified the relation between thetwo by introducing the “arborescence of n √ TIs/ n √ TSs”,a term taken from graph theory, that enables one to keeptrack of the original topological features which are inher-ited by the edge states present in the starting model.In a recent work , we already identified a specific sub-class of 1D linear bipartite models, labeled Sine-Cosinemodels, which, after each squaring operation, retrieve asmaller self-similar version of themselves as one of thediagonal blocks, in what we described as a Matryoshkasequence. The results analyzed there can be viewed as anotable subset appearing within the general frameworkof 1D n √ TIs that we draw here.The rest of the paper is organized as follows. In sec-tion II, we review the properties of a 1D √ TI, namelythe diamond chain with π -flux per plaquette, which wewill take as our toy model from which 2 n -root topologyis derived. In section III, we show how to construct thequartic-root TI from the √ TI introduced before, high-lighting the relation between the edge states of the √ TIand the topological state of the original TI. In section IV,we show how the method followed in the previous sectionto find the √ TI can be replicated an arbitrary numberof times to find the n √ TIs with a higher n > n √ TIs is introduced here, as an in-tuitive way of relating any n √ TI with the original TI.In section V, we derive the n √ TSs from the original TS,taken to correspond to the Kitaev chain mapped intoits single-particle tight-binding analog model. Finally, in a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b section VI we present our conclusions. II. DIAMOND CHAIN WITH π -FLUX PERPLAQUETTE Let us begin with a recapitulation of the propertiesof the diamond chain model with a π -flux per plaque-tte, depicted in Fig. 1(a), a known example of a 1D √ TI . Under periodic boundary conditions (PBC), thesquare-root bulk Hamiltonian of this model, written inthe {| A ( k ) (cid:105) , | B ( k ) (cid:105) , | C ( k ) (cid:105)} basis, is given by H √ TI = √ t (cid:32) h †√ TI h √ TI (cid:33) , (1) h †√ TI = (cid:0) − e − ik e − ik (cid:1) , (2)where the lattice constant is set to a ≡ k -dependency is hidden in the termsand will remain so hereafter for convenience, except whendeemed necessary. The block anti-diagonal form of (1) in-dicates that the model is bipartite and, therefore, enjoyschiral-symmetry defined as CH √ TI C − = − H √ TI , with C = diag( − , , , shownin Fig. 1(c), as a consequence of an Aharonov-Bohmcaging effect induced by the π -flux in each plaquette .Even though it would be more correct to speak of a square-root energy spectrum, we will treat all 2 n -rootenergy spectra in this paper as simply energy spectra.Apart from simplifying the notation, this abuse of lan-guage is further justified by the fact that we set anadimensional energy unit for the hopping magnitudes( t ≡
1) everywhere.Under open boundary conditions (OBC), there is achiral pair of topological edge states with finite energies E ± edge = ±√ N of unit cells is considered. This feature is re-lated to the fact that, even though the model is inversion( I ) symmetric, its I -axis is shifted in relation to the cen-ter of the unit cell [see Fig. 1(a)]. As a consequence,the I -symmetry operator becomes k -dependent , I : P ( k ) H √ TI ( k ) P − ( k ) = H √ TI ( − k ) , (3)with P ( k ) = diag( e ik , , Z topological index for 1D models , can be expressedin this case as γ n = arg( P n P nπ ) − (cid:90) π dk | u n,A ( k ) | mod 2 π, (4)where P n , P nπ = ± n at the I -invariant momenta k = 0 , π , respectively, and u n,A ( k ) is the A-component of the corresponding eigen-state | u n ( k ) (cid:105) . The second term in (4) appears as a con-sequence of the mismatch between the I -axis and thecenter of the unit cell . Since finite energy eigenstatesin bipartite sublattices have half their weight in each sub-lattice, one can immediately see that | u ± ,A ( k ) | = 1 / k , where n = +( − ) is the index of the top (bottom)band. The first term in (4) is quantized to either 0 or π such that, modulo 2 π , one necessarily has γ ± = π . Thestates of the zero-energy flat band n = 0, on the otherhand, have no weight on the A sublattice , so the usual π -quantization holds for γ .Upon squaring the square-root Hamiltonian in (1) onearrives at H √ TI = (cid:18) H res H par (cid:19) , (5) H res = h †√ TI h √ TI = 2 c , (6) H par = h √ TI h †√ TI = c σ + H TI , (7) H TI = d · σ , (8)where σ is the 2 × σ = ( σ x , σ y , σ z )is the vector of Pauli matrices, d = c (0 , sin k, − cos k )and c = 2 t is a constant energy shift. We label H res ( H par ) as the residual (parent) Hamiltonian and H TI theHamiltonian of the original TI. The real-space represen-tation of H √ TI with a general energy downshift of c ,such that H par → H TI and H res → c , is depicted inFig. 1(b), where it can be seen that H TI models a Creutzladder with π -flux in each triangle in the BC sub-lattice, known to host a non-decaying zero-energy topo-logical edge state at each end for all | t | ≥ H res represents a chain of decoupled A sites. No-tice that the I -axis of the Creutz ladder can be placedat the center of the unit cell, such that the Zak phasesof both its bands recover the usual π -quantization [seeFig. 1(e)]. Through a suitable basis rotation, H TI canbe transformed into the Hamiltonian of a fully dimerizedSSH model in the topological phase .The edge spectrum of the open squared chain, down-shifted by c , for N = 10 complete unit cells is shown inFig. 1(f). At first sight, one might be tempted to iden-tify both midgap states with the topological states of theopen Creutz ladder, our original TI. There is, however,a subtle point that will prove to be of crucial impor-tance throughout the rest of this paper: under OBC andfor N integer, the end sites of the starting square-rootchain [see | A, (cid:105) , | B, N (cid:105) and | C, N (cid:105) in Fig. 1(a)], have alower coordination number than the rest of their equiv-alent bulk sites. This will create an on-site energy offsetupon squaring the Hamiltonian, akin to the offset in theself-energies for states with weight at the edge sites in aA,1 A,2 A,3B,1C,1 B,2C,2aA,1 A,2 A,3B,1 B,2C,1 C,2 A,N-1 A,NB,N-1 B,NC,N-1 C,NA,N-1 A,NB,N-1 B,NC,N-1 C,N√tI-axisI-axis t-t (a)(b) E k ππππ/2π/2 E (c)(e) (x2) -√t (d) i (f ) BC chainA chain I > T > Edge,- > Edge,+ > L T > R res
Figure 1. (a) Diamond chain with π -flux per plaquette, corresponding to the √ TI; (b) squared model of (a), yielding theCreutz ladder on top (the original TI) with a perturbed right end (extra orange hopping) and an independent residual chainof decoupled A sites at the bottom. Shaded regions represent the unit cell [only for the Creutz ladder in (b)], the blue crossits center and the vertical dashed green line the Inversion-axis. (c) Energy spectrum of the model at the left, with t ≡
1, asa function of the momentum under PBC and (d) the state index for the same chain with OBC and N = 10 unit cells. (e)-(f)Same as is (c)-(d), respectively, but for the model in (b). The Zak phases for each band are indicated in the bulk spectra[cumulative for the degenerate bands in (e), with π for the top band of the Creutz ladder and 0 for the band of decoupled Asites]. second-order perturbation theory , (cid:104) A, n | H √ TI | A, n (cid:105) = 2 c , n = , , , . . . , N, (9) (cid:104) A, | H √ TI | A, (cid:105) = c , (10) (cid:104) ν, n | H √ TI | ν, n (cid:105) = c , n = , , , . . . , N − , (11) (cid:104) ν, N | H √ TI | ν, N (cid:105) = c , (12)with ν = B, C , and an extra hopping at the right edgeof the Creutz ladder depicted in orange in Fig. 1(b), (cid:104)
B, N | H √ TI | C, N (cid:105) = t = c . (13)As a result, the left and right topological states of theCreutz ladder, written as | T L (cid:105) = √ ( | B, (cid:105) + | C, (cid:105) ) and | T R (cid:105) = √ ( | B, N (cid:105) − |
C, N (cid:105) ), respectively, will have dif-ferent energies, on account of the correction terms at theright edge, H √ TI | T L (cid:105) = c | T L (cid:105) , (14) H √ TI | T R (cid:105) = 0 , (15)There is an impurity state at the left edge of the Achain, labeled | I res (cid:105) ≡ | A, (cid:105) , which is degenerate with | T L (cid:105) by comparing (10) and (14). These are the two midgap states of Fig. 1(f), while | T R (cid:105) becomes degener-ate with the bulk states of the lower band of the Creutzladder, becoming a non-topological edge state due to theeffect of the extra hopping term at the right end. Re-turning now to the open square-root model, the finiteenergy states shown in Fig. 1(d) can be written, in the {| A, (cid:105) , | B, (cid:105) , | C, (cid:105)} basis, as | Edge , ±(cid:105) = 12 ±√ = 1 √ (cid:18) ± | I res (cid:105)| T L (cid:105) (cid:19) , (16)that is, in relation to the squared model, each edge stateof the starting square-root model can be said to be “half”topological and “half” impurity. As we will see below, for n √ TIs the ratio of topological-to-impurity component foreach edge state of the 2 n -root model, relative to the orig-inal TI, will change in favor of the impurity componentas n increases. III. QUARTIC-ROOT TOPOLOGICALINSULATOR A √ TI of our original TI [the Creutz ladder along theB and C sites of Fig. 1(b)] would have to be a givenmodel that, upon squaring, would contain the √ TI model[the diamond chain of Fig. 1(a)] as one of its diagonalblocks, apart from an overall constant energy shift. Inother words, one has to ensure that the √ TI chain: (i)keeps the same flux pattern as the √ TI model; (ii) isbipartite, and therefore has chiral symmetry and can bewritten in a block anti-diagonal form; (iii) has a sub-lattice composed of the sites that will become the √ TIchain upon squaring; (iv) the sites in this sublattice havethe same onsite energy upon squaring (constant energyshift). The first three conditions are met by followingthe method outlined by Ezawa which, in essence, treatsthe problem as one of graph theory: one constructs thesplit graph through subdivision of the original graph(chain) by adding nodes (sites) at the middle of each link(hopping term), as illustrated by the added grey sites inFig. 2, which guarantees that the higher-degree root TIis bipartite, with one of its sublattices corresponding tothe sites of the starting model (the blue nodes in the √ TIof Fig. 2 form a sublattice given by the sites of the √ TI).In the process, the hopping terms are doubled in numberand transformed as √ te iφ → √ te iφ/ , i.e. , one takes thesquare-root of the magnitude and divides the phase bytwo, in order to keep the same π -flux per plaquette. a √t-√ta √t √t i √TI√TI Figure 2. Construction of the √ TI from the √ TI. The shadedregions indicate the respective unit cell. The gray sites appearfrom subdivision of the √ TI (a site is included in the middleof each link). The resulting hopping terms have, in relationto the corresponding ones at the √ TI, the square root of the(constant) magnitude and halved phase factors. Green extrasites are included in the √ TI to keep the same coordinationnumber for all sites of the blue sublattice, which implies thatsites | , N + 1 (cid:105) and | , N + 1 (cid:105) have to be included at the rightend under OBC. If we view the squaring of the Hamiltonian as a two-step quantum walk, where the on-site potential is given by the weighted sum of the paths that a particle canmake from a given site to its connected neighbors in thefirst step and then hop back in the second step, it be-comes clear that the last condition listed above is not metconsidering the blue and grey sites alone in Fig. 2 sincethe coordination number of the spinal sites | , l (cid:105) is four,where | j, l (cid:105) , with j = 1 , , . . . ,
11 and l = 1 , , . . . , N ,is the state of a particle occupying site j of unit cell l ,while it is two for the same sublattice sites | , l (cid:105) and | , l (cid:105) ,resulting in (cid:104) , l | H √ TI | , l (cid:105) = 4 √ t, (17) (cid:104) β, l | H √ TI | β, l (cid:105) = 2 √ t, (18)with β = 2 ,
3. As such, the on-site potentials at the rele-vant blue sublattice do not form a constant energy shiftupon squaring. There are two fundamental methods that -4-202 E -π 0 π k -π 0 π k -π 0 π k (x5)(x5) (a) (b) (c) Figure 3. Energy spectrum, with t ≡
1, as a function of themomentum for: (a) the √ TI model given in (20); (b) and(c) the diagonal block of the √ TI model corresponding tothe √ TI model given in (1) and to the √ res , (cid:48) model givenin (24), respectively. The five-fold degenerate bands are indi-cated in the plots. allow one to circumvent this limitation and impose a con-stant energy shift upon squaring the Hamiltonian: (i) byre-normalizing some of the hopping terms, which is moreeconomical at the level of the number of sites per unitcell, at the cost of requiring successive re-normalizionsof the hopping terms after each squaring operation (thereader is referred to Appendix A for further details onthis method, which is expected to become the easiest ofthe two to handle as the root degree of the n √ TI in-creases); (ii) the magnitude of the hopping term is keptconstant (that is, n √ t for all hoppings in n √ TI) andextra sites are added and connected to the sites of therelevant sublattice with lower coordination number, inorder to compensate for the difference on their on-siteenergies upon squaring the Hamiltonian . The effecton the energy spectrum of the extra green sites in theunit cell will simply be to originate the same number ofextra zero-energy flat bands which, according to Lieb’stheorem , is given by the imbalance in the number ofsites in each sublattice (eight for the gray and green sub-lattice and three for the blue sublattice, yielding a total offive zero-energy bands). We will follow this latter methodin the following sections of the main text, which consistsof adding the extra green sites depicted in Fig. 2, suchthat (18) becomes now (cid:104) β, l | H √ TI | β, l (cid:105) = 4 √ t, (19)yielding the same result as (17), as required.Finally, there is an ambiguity in the definition of theunit cell of √ TI, since sites | , (cid:105) and | , (cid:105) can be addedeither to the left, as in Fig. 2, or to the right. The am-biguity is somewhat resolved, however, when OBC areconsidered, in which case one has to add these two sitesat the right of the last unit cell also (see sites 4 and 5 atthe incomplete N + 1 unit cell in Fig. 2), in order to keepthe coordination number at the blue sites of both edgesthe same as for the bulk blue sites. Since the extra sitesperturb the right edge physics, preventing in general theemergence of right edge states, one could argue that adirect bulk-edge correspondence is lost, since the lowernumber of edge states under OBC will not have a corre-spondence with the cummulative Zak phases of the bulkbands below the energy gap where each of them lies. Onthe other hand, even without the extra sites at the rightedge under OBC the usual bulk-edge correspondence isstill broken, since the I -axis does not cross the center ofthe unit cell of the √ TI, meaning that the Zak phases ofthe bulk bands are not π -quantized in general.Under PBC, the bulk Hamiltonian of the √ TI, in theordered {| j ( k ) (cid:105)} basis, where j = 1 , , . . . ,
11 refers tothe j th component within the unit cell, has the form H √ TI = √ t (cid:32) h † √ TI h √ TI (cid:33) , (20) h † √ TI = i − ie ik e ik , (21)whose squared version therefore becomes H √ TI = (cid:18) H √ par , H √ res , (cid:19) , (22) H √ par , = h †√ TI h √ TI = c I + H √ TI , (23) H √ res , = h √ TI h †√ TI = c I + H √ res , (cid:48) , (24)where c = 4 √ t is a constant energy shift, explic-itly included also in H √ res , to keep the energy spec-trum of H √ TI and H √ res , (cid:48) leveled, I m is the m × m identity matrix and H √ TI is given in (1) with {| k ) (cid:105) , | k ) (cid:105) , | k ) (cid:105)} → {| A ( k ) (cid:105) , | B ( k ) (cid:105) , | C ( k ) (cid:105)} . Di-agonalization of H √ TI , H √ TI and H √ res , (cid:48) yields the en-ergy spectra of Fig. 3. It can be seen that the √ TI and √ res , (cid:48) models share the same spectrum, apart from the five-fold degenerate lower band in the latter, correspond-ing to squaring and downshifting by c the degeneratezero-energy bands of the √ TI model, originated by theimbalance in the number of sublattice sites, as explainedabove. As expected, the spectrum of Fig. 3(b) exactlymatches the one on Fig. 1(c), since both of them corre-spond to the same √ TI model.The energy spectrum of the open √ TI chain at thebottom of Fig. 2, for N = 10 complete unit cells plussites 4 and 5 of the N + 1 unit cell, is shown in Fig. 4(a),where two chiral pairs of non-degenerate edge states withenergies E √ TIedge = ± (cid:113) E √ TIedge + c = ± √ t (cid:113) √ ± , (25)where E √ TIedge = ±√ t [see edge states in Fig. 1(d)]. Thespatial profile of these edge states is shown in Fig. 4(d).After squaring this spectrum and taking out the con-stant shift c , we arrive at the spectrum of Fig. 4(b),where the edge states now appear in two doubly degen-erates pairs and the E = 0 states in Fig. 4(a) become the E = − √ t = − c states in Fig. 4(b). Lastly, this squaredand shifted spectrum is in turn squared and downshiftedby c = 2 t in Fig. 4(c). All four edge states are nowdegenerate at zero energy, with only one of them corre-sponding to the topological state | T L (cid:105) of the original TImodel, defined above (14), while the other three are im-purity states, one stemming from the H res (cid:48) = H res − c [the | I res (cid:105) state defined below (15)] and the other two,labeled (cid:12)(cid:12) I , (cid:11) and (cid:12)(cid:12) I , (cid:11) , from H res , (cid:48) = H √ res , (cid:48) − c ,with H √ res , (cid:48) defined in (24), involving only the sites inthe green and gray sublattice depicted at the bottom ofFig. 2.We define the ancestor Hamiltonian as the direct sumof all the terms contained in Fig. 4(c), H anc := H TI ⊕ H res (cid:48) ⊕ H res , (cid:48) , (26)which is the Hamiltonian describing the ancestor chain.The four edge states of the √ TI chain in Fig. 4(a), labeled | T , j (cid:105) , with j = 1 , , , | (cid:104) T L | T , j (cid:105) | = 14 , (27)that is, the edge states of the √ TI can be said to be, inrelation to the ancestor chain, “one quarter” topological[notice the same | T L (cid:105) component appearing in all edgestates of Fig. 4(d)] and “three quarters” impurity, whichshould be compared with the case of the edge states ofthe √ TI chain in (16). - - E i - -
202 0 20 40 60 80 100 i - ic shift c shift (a) (b) (c) (d) T L > | component Figure 4. Energy spectrum, in units of t ≡
1, as a function of state index i , obtained from diagonalization of: (a) H √ TI , theHamiltonian of the open √ TI chain at the bottom of Fig. 2 with N = 10 complete unit cells plus extra sites 4 and 5 at unitcell N + 1; (b) H (cid:48) √ TI = H √ TI − c , with c = 4 √ t ; (c) H (cid:48) √ TI = H (cid:48) √ TI H (cid:48) √ TI − c , with c = 2 t . Bulk (edge) states are coloredin blue (red). (d) Profile of the edge states in the √ TI chain, where the radius of the circle represents the amplitude of thewavefunction at the respective site and the color represents its phase, coded by the color bar at the right.
IV. n -ROOT TOPOLOGICAL INSULATORS The procedure followed in the last section to find the √ TI model from the √ TI can be readily generalized.More concretely, one can find the n √ TI, for any n ∈ N ,from the n − √ TI in four steps:1. One starts by subdividing the graph made by the n − √ TI chain, that is, a new node (site) is introducedat the middle of each link (hopping term). The resultingchain is therefore bipartite.2. The magnitude of the new hopping parameters isgiven by the square-root of the corresponding one in the n − √ TI chain, while their phases are divided in half tokeep the same flux pattern in the chain, e iφ n − √ t → e iφ/ n √ t. (28)Note that each link in the n − √ TI chain generates twoafter subdivision.3. One has to ensure that the same squared on-sitepotential appears at the sites in the sublattice thatdecouples from the other one upon squaring of theHamiltonian to generate the n − √ TI chain. First, onehas to identify the sites of the relevant sublattice with the highest squared on-site potential (defined as c n − ) andthe others, with a lower squared on-site potential. Tolevel the squared on-site potentials of the sites of thesetwo subsets, one has to compensate for the differenceof those in the latter subset by (i) re-normalizing thehopping terms (the approach followed in Appendix A),and/or (ii) introducing connections to extra sites, theapproach followed in the previous sections. The phasesof the extra hopping terms constitute an additionaldegree of freedom that one can use to guarantee thatthe relevant sublattice remains in itself bipartite uponsquaring the model, as will be illustrated in the nextsection for a √ TS.4. This last step only applies under OBC. Theextra sites of the n √ TI that result from subdivisionof the n − √ TI create an ambiguity in the definitionof the unit cell, since the outermost new sites can beplaced either to the right or to the left in the unit cell.We chose to place these sites at the leftmost regionwithin the unit cell (see sites | , (cid:105) and | , (cid:105) of the √ TI chain at the bottom of Fig. 2). However, underOBC these outermost sites have to be included alsoat the right of the last complete unit cell, in order toavoid a lower squared on-site potential at the rightmostsites of the relevant sublattice (see sites | , N + 1 (cid:105) and | , N + 1 (cid:105) at the bottom of Fig. 2, keeping at fourthe coordination number of the two rightmost blue sites). ++ H √TI H √TI H √res, H √res, H res, H √TI H √TI H √TI H TI H √res,4 H √res,4 H res,4 H √res,2 H res,2 H res c c c c c c c c c c c n-1 c n-1 H anc = + + + edge = + + + + + n-1 edge = n ’’’’’’ ’’ ’ : Figure 5. Outwards directed rooted tree (arborescence) of n √ TIs. The root node on top corresponds to the Hamiltonian ofthe starting n √ TI model. From each level to the next the Hamiltonian is squared and shifted down in energy by c n − j , with j = 1 , , . . . , n the level index. Each link selects one of the diagonal blocks of the resulting Hamiltonian, with the lower rootdegree TIs placed along the hypotenuse (each generating two children nodes from its block diagonal squared form) and thesuccessive residual Hamiltonian placed at the right of those (each generating a single child node). The process is stopped atthe level where H TI is reached along the hypotenuse. The ancestor Hamiltonian H anc is defined as the direct sum over allleaf nodes. edge , indicated on the root and leaf nodes, is the number of edge states appearing from diagonalization of theHamiltonian of the respective node. In H anc only the red encircled edge state is topological, while all others are impurity states. By applying n successive squaring operations to theHamiltonian of the n √ TI chain, and taking out the con-stant energy shift given by the on-site potential energyat the relevant sublattice after each operation (in orderto keep this sublattice bipartite), a cascade of block diag-onal Hamiltonians appear, containing both the residualblocks and the lower degree roots of n √ TI. A way to visu-alize this is by constructing the outwards directed rootedtree (arborescence ) of n √ TIs, depicted in Fig. 5. Theroot node at the top represents the Hamiltonian of thestarting n √ TI chain, hosting edge = 2 n non-degenerateedge states. Each link connects a Hamiltonian to one ofits block diagonal terms upon squaring and shifting downby the constant energy term of the respective level, with n + 1 levels in total. Nodes along the hypotenuse corre-spond to a given root degree of the original TI and havetwo child nodes, since their squared Hamiltonians yield,apart from a constant energy shift, a residual block andthe block with the lower root degree TI [see, e.g., the squared √ TI model in (22)]. Nodes living outside thehypotenuse, however, correspond to the successive resid-ual Hamiltonians that one has to continue squaring andshifting down by the constant of the respective level, suchthat each originates only one child node. The processstops when we reach the TI model along the hypotenuse.We define the ancestor Hamiltonian as the direct sum ofall leaf nodes, H anc := H TI ⊕ H res (cid:48) ⊕ H res , (cid:48) ⊕ · · · ⊕ H res , n − (cid:48) . (29)The number of zero-energy edge states edge originatingfrom each of the terms in (29) is indicated at the bottomof Fig. 5. Only the encircled one stemming from H TI is topological in nature [the | T L (cid:105) state defined in (14)],while all others are impurity states of the residual Hamil-tonians that appear due to the detuned on-site potentialsat their respective left edge sites. As expected, there isa total of 2 n degenerate zero-energy states in H anc , thesame as in the root node H n √ TI , with each of the | T n , j (cid:105) states, with j = 1 , , . . . , n , appearing in a different en-ergy gap of the latter. If we define the projector onto theimpurity subspace of the ancestor chain,ˆ I = n (cid:88) m =0 2 m (cid:88) j =1 | I res,2 m , j (cid:105) (cid:104) I res,2 m , j | , (30)with (cid:12)(cid:12) I res,2 , (cid:11) ≡ | I res (cid:105) defined below (15), then we candetermine the topological and impurity weights of theedge states of our starting n √ TI chain, relative to theedge states of the ancestor chain, as | (cid:104) T L | T n , j (cid:105) | = 12 n , (31) (cid:104) T n , j | ˆ I | T n , j (cid:105) = 2 n − n , (32)for all j = 1 , , . . . , n . We can conclude that, as n in-creases, the weight of the starting edge states on the topo-logical state of the ancestor chain gets diluted, in favor ofan increasing weight over the sum of all impurity states.For the case of the model we have been considering,whose original TI is given by the Creutz ladder depictedin Fig. 1(b), the values of the constant energy shifts, cor-responding to the on-site potential at the sites formingthe diagonal block in the squared Hamiltonian yieldingthe lower root degree TI, can be readily found to be c = 2 t, (33) c m = 4 m √ t, m = 1 , , . . . , n − , (34)which, setting t ≡
1, simplifies to c = 2 and c m =4. The energy of the 2 n edge states coming from thediagonalization of the root node H n √ TI can be obtainedrecursively through E n √ TI = ± (cid:113) E n − √ TI + c n − , (35) E TI = 0 . (36)As a final example, let us follow the four steps out-lined above to construct the √ TI model. Its unit cell isdepicted in Fig. 6. The blue sites form a sublattice corre-sponding to the unit cell of the √ TI shown at the bottomof Fig. 2, while the gray sites are a consequence of step1, the subdivision of √ TI. The magnitude of the newhopping parameters becomes √ t , while the phases of thedashed hopping terms are now π , that is, half the phaseof the corresponding hoppings in the √ TI, in agreementwith step 2, which keeps the same π -flux per plaquettepattern. The green extra sites are introduced in orderto keep the same coordination number at the sites in theblue sublattice such that, upon squaring H √ TI , the on-site energy is the same for all blue sites, as required bystep 3. As a general rule of thumb, one identifies the siteswithin a unit cell that are positioned at the left of thespinal site as the set of sites that has to be included alsoat the right end under OBC, such that step 4 is fulfilledwhen, under OBC, the open sites in Fig. 6 are includedin the incomplete N + 1 unit cell for it to yield exactly, a √t √t e i π / Figure 6. 43-sites unit cell of the √ TI model. Blue sitesform a sublattice yielding the √ TI model at the bottom ofFig. 2 upon squaring, gray sites come from the subdivisionof √ TI and green sites are extra sites introduced to keep thecoordination number constant for sites in the blue sublattice.The 10 open sites at the right belong to the next unit cell andhave to be included at unit cell N + 1 under OBC. as one of its diagonal block and apart the c energy shift,the √ TI chain at the bottom of Fig. 2.The energy spectrum for the total Hamiltonian at eachlevel of the arborescence of the √ TI, i.e. , with H √ TI asthe root node in Fig. 5, for N = 5 complete unit cellsplus 10 extra sites at unit cell N + 1, is shown in Fig. 7.The progression shows how the eight non-degenerate edgestates coming from diagonalization of H √ TI [in-gap redstates in Fig. 7(a)] end up as the eight-fold degeneratezero-energy state coming from diagonalization of H anc [midgap states in Fig. 7(d)].By applying steps 1 through 4, one can construct the √ TI from the √ TI in Fig. 6. We have been assumingin the main text that step 3 is satisfied by adding extrasites, while keeping the magnitude of the hopping termsthe same everywhere, such that the number of sites perunit cell for the n √ TI follows the recurrence relation n uc = n − uc + 2 n − , n ≥ , (37) = 3 , (38)meaning that one has = uc = 171, = 683, etc.In other words, as n increases it becomes impractical tofollow this method, with the one followed in Appendix A,which relies on re-normalizing the hopping parameters inplace of adding extra sites, progressively gaining tractionas the most tractable of the two. - - c shift c shift c shift i i i iE (a) (b) (c) (d) Figure 7. Energy spectrum, in units of t ≡
1, as a function of state index i , obtained from diagonalization of: (a) H √ TI , theHamiltonian of the open √ TI chain with the unit cell of Fig.6, with N = 5 complete unit cells plus the 10 open extra sites at unitcell N + 1; (b) H (cid:48) √ TI = H √ TI − c , with c = 4 √ t ; (c) H (cid:48) √ TI = H (cid:48) √ TI H (cid:48) √ TI − c , with c = 4 √ t ; (d) H (cid:48) √ TI = H (cid:48) √ TI H (cid:48) √ TI − c ,with c = 2 t . Bulk (edge) states are colored in blue (red). Only a partial spectrum is shown in (b)-(d). It should be noted that the n √ TIs studied so far pro-vide a practical way of generating 1D models with anall-bands-flat spectrum , alternatively to a recent pro-posal of generating these spectra by taking advantage ofa non-Abelian AB caging effect, related to the period-icity with which a π -flux appears in the plaquettes ofthe diamond chain . Another type of n √ TI, whoseenergy spectrum is in general dispersive, is analyzed inAppendix B. This other type of n √ TI is labeled there as n √ SSH, since the original TI is in this case taken to bethe SSH model. V. n -ROOT TOPOLOGICALSUPERCONDUCTORS In this section, we show how the same method for con-structing n √ TIs can be applied for the construction of n √ TSs, using the Kitaev model as the original TS,whose √ TS is already known . We will go one stepfurther and determine the √ TS, from which one canstraightforwardly generalize for n > p -wave supercon-ductor, reads as H TS = (cid:88) k ( c † k c − k ) (cid:18) t cos k − µ
2∆ sin k
2∆ sin k − t cos k + µ (cid:19) (cid:18) c k c †− k (cid:19) , (39)where c k is the particle annihilation operator acting onthe state with momentum k , ∆ is the superconductingpairing term induced in the chain by proximity effect withthe superconductor and µ the chemical potential. Diago-nalization of this Hamiltonian yields two symmetric flatbands for t = ∆, as shown in Fig. 8(d). The presenceof an annihilation operator in the Nambu pseudospinor Ψ k = ( c † k c − k ) prevents a real-space representation ofthe model by direct application of the inverse Fourier transform to (39). However, one can construct the single-particle tight-binding analog of the Kitaev chain throughthe mapping Ψ k = ( c † k c − k ) → Ψ k = ( c † ,k c † ,k ), thatis, one treats each component of Ψ k as if corresponding,e.g., to different atomic species or internal states. Underthis mapping, the real-space representation of the single-particle tight-binding analog model can be readily foundto have the form of the Creutz ladder in Fig. 8(a), with its“electron channel” at the top leg and its “hole channel”at the bottom leg. Notice that the superconducting pair-ing is translated into an inter-leg crossed hopping termcarrying a π phase factor. Thus, this system becomesamenable to the same four steps treatment outlined inthe previous section for the construction of the n √ TI.We will further set µ = 0 below, otherwise the successive n √ TS models would not be bipartite and would there-fore lack chiral symmetry, such that their squared mod-els could not be written in a block diagonal form andour method would not be applicable. For convenience wewill also set t ≡ √ TS of Fig. 8(b),whose bulk Hamiltonian, written in the ordered {| j ( k ) (cid:105)} basis, where j = 1 , , . . . , j th componentwithin the unit cell, is given by H √ TS = (cid:32) h †√ TS h √ TS (cid:33) , (40) h √ TS = √ t (1 + e − ik ) 0 e − i π √ ∆ e − i ( k − π ) √ ∆ e − i ( k − π ) √ ∆ e − i π √ ∆0 − i √ t (1 − e − ik ) . , (41)whose diagonalization, for t = ∆, yields the energy spec-trum in Fig. 8(e), where the two-fold degenerate zero-0 (b) t √ ∆ √e i π /4 t-t i ∆ (a) t √ i (c) t √ ∆ √e i π /8 ∆ √ i e i π /4 t √ - - E -π π k -π π k - - E -π π k - - E (x2) (x6)(x2)(x2) (d) (e) (f ) Figure 8. Single-particle tight-binding analog representation of the: (a) Kitaev TS model; (b) √ TS model; (c) √ TS model.Shaded regions represent the unit cell. Gray sites come from the subdivision of the chain to the left, while the green sites in √ TS were included to keep the on-site potential constant in the blue sublattice sites upon squaring H √ TS . (d)-(f) Energyspectrum as a function of the momentum for the respective models on top under PBC and with ∆ = t ≡ energy band comes as a consequence of Lieb’s theorem ,equaling in number the imbalance within the unit cell be-tween sites in the gray sublattice (four) and in the bluesublattice (two).Let us suppose we want to construct the √ TS fromthe √ TS in Fig. 8(b), whose sites become now the rel-evant blue sublattice of √ TS. We start by applyingsteps 1 and 2, subdivision of √ TS and subsequent re-normalization of the magnitudes and phases of the hop-ping parameters (remember that the superconductingpairings were converted into hopping terms), leading tothe model depicted in Fig. 8(c) without the green sites.Upon squaring the Hamiltonian, the on-site potential insites 1 and 2 in the blue sublattice can be easily seen toyield c = 2 √ ∆ + 2 √ t , while for sites 3 and 6 (4 and 5) ityields only 2 √ t (2 √ ∆) because they are missing, in rela-tion to sites 1 and 2, two √ ∆ ( √ t ) hopping connections. Since we want the same c on-site potential at all sitesin the blue sublattice in order to satisfy step 3, we intro-duce the extra green sites in Fig. 8(c) with judiciouslychosen hopping terms to sites 3,4,5 and 6 such that theyexactly compensate the difference in their respective on-site potential, upon squaring, in relation to c . Notice the π included in the √ ∆ hoppings connected to the greensites, introducing a π -flux around these small loops. Thisis required in order to prevent finite couplings betweensites 3 and 4 (5 and 6) upon squaring the Hamiltonian(for instance, there are two two-step hopping processesbetween sites 3 and 4, with √ t ∆ magnitude but symmet-ric ± π phase factors, thus canceling out), which wouldbreak the bipartite condition and, therefore, not yield the √ TS in Fig. 8(b) as one of its diagonal blocks. The bulkHamiltonian of the √ TS, written in the ordered {| j ( k ) (cid:105)} basis, where j = 1 , , . . . ,
18 refers to the j th componentwithin the unit cell, is given by1 H √ TS = (cid:32) h † √ TS h √ TI (cid:33) , (42) h √ TS = e − ik √ t √ t e − i ( k − π ) √ ∆ 0 0 0 e − i π √ ∆ 00 e − i ( k − π ) √ ∆ 0 e − i π √ ∆ 0 00 e − i ( k − π ) √ t e − i π √ t − i √ ∆ √ t √ t i √ ∆0 0 i √ ∆ √ t √ t − i √ ∆ √ t √ t e − i π √ ∆ 0 0 e i π √ ∆ 0 00 e − i π √ ∆ 0 0 e i π √ ∆ 00 e − i π √ t e i π √ t , (43)whose diagonalization, for t = ∆, yields the energy spec-trum in Fig. 8(f), where the six-fold degenerate zero-energy band comes, similarly to the √ TS, as a conse-quence of Lieb’s theorem (gray and green sites belong tothe same sublattice).As with the n √ TI studied above, there remains anambiguity in the definition of the unit cell of the n √ TS.For instance, in the √ TS of Fig. 8(c), sites 3 through18 have been placed at the left of sites 1 and 2 [thoseultimately forming the original TS in Fig. 8(a)] withinthe unit cell, but they could just as well been placed atthe right. If choosing one of the options dissipates theambiguity, one must remember that step 4 requires that,under OBC, sites other than 1 and 2 have to appear atboth ends. Since we chose to grow the unit cell to theleft, extra sites 3 through 18 have to be included at theincomplete unit cell N + 1 at the right edge under OBC.The energy spectrum of the open √ TS with N = 50complete unit cell plus the 16 extra sites at unit cell N +1is shown in Fig. 9(a). Four doubly degenerate (red) bandsof edge states are present at different energy gaps forfinite ∆.Diagonalization of the squared Hamiltonian of theopen √ TS chain, followed by a c = 2 √ t + 2 √ ∆ en-ergy shift, yields the energy spectrum of Fig. 9(b), wherethe two red bands of edge states become four-fold degen-erate. Due to the c shift, the six-fold degenerate bandat E = 0 of Fig. 9(a) becomes the lower E = − c bandin Fig. 9(b), which looses its flatness since c depends on∆.Upon squaring once more the Hamiltonian correspond-ing to the spectrum of Fig. 9(b) and subtracting a con-stant energy shift given by c = 2 t + 2∆, one obtains theancestor Hamiltonian H anc of the √ TS. Its diagonaliza-tion yields the energy spectrum of Fig. 9(c). The six-folddegenerate lowest energy band has E = − c and thefour-fold degenerate highest energy one has E = c − c .The bands of edge states all collapse onto the eight-folddegenerate Majorana zero-energy red band. For a finite ∆, the √ TS chain is in the topologicalphase and harbors edge = 8 edge states, double thenumber of the √ TI chain [compare with Fig. 4(a)]. Ingeneral, one has edge = 2 n +1 edge states in the n √ TSchain. The arborescence of n √ TSs is the same as inFig. 5, only doubling the number of edge states presentin the root node and, consequently, also in all the leafnodes. Whereas the original TI we considered previouslyonly contributed with one topological edge state to all n √ TI, localized at the left edge, since the impurity hop-ping and on-site potential terms appearing at the rightend send the right edge state to the bulk [see the Creutzladder in Fig. 1(b) and energy spectrum of Fig. 1(f)], ouroriginal TS, the Kitaev chain depicted in Fig. 8(a), con-tributes with one topological edge state at each end toall n √ TS. Note that these two oppositely located edgestates would correspond, in the Majorana basis of γ op-erators, to the two free γ operators living at each end ofthe chain that combine to form a Majorana fermion. Infact, it is because we want to keep this correspondencethat, in contrast with the right edge perturbed Creutzladder in Fig. 1(b), the original ladder in Fig. 8(a) is un-perturbed, requiring that gray sites from the incompleteunit cell N + 1 to be already present at the right edge atthe square-root level shown in Fig. 8(b). This is why wefind double the number of edge states here, in relation tothe same level of the TI studied in the previous sections.It is clear that a √ TS can be constructed from the √ TS in Fig. 8(c) by application of steps 1 to 4 and,by further repetition of the process, any n √ TS can beconstructed. The energy of the edge states can be de-termined recursively, as in (35), by changing only thedegeneracy of the initial value in (36), E n √ TS = ± (cid:113) E n − √ TS + c n − , (44) E TS = 0 ( × , (45)where the constant energy shifts are now given by c m = 2 m √ t + 2 m √ ∆ , (46)2 - - - - - - - - ∆E ∆ ∆(a) (b) (c) c shift c shift Figure 9. Energy spectrum, in units of t ≡
1, as a function of ∆ obtained from diagonalization of: (a) H √ TS , the Hamiltonianof the open √ TS chain with the unit cell of Fig. 8(c), with N = 50 complete unit cells plus extra sites 3 to 18 at unit cell N + 1; (b) H (cid:48) √ TS = H √ TS − c , with c = 2 √ t + 2 √ ∆; (c) H (cid:48) √ TS = H (cid:48) √ TS H (cid:48) √ TS − c , with c = 2 t + 2∆. Bulk (edge) states arecolored in blue (red). Each red edge band is two-fold degenerate in (a), four-fold degenerate in (b) and eight-fold degeneratein (c). with m = 0 , , , . . . , n −
1. Analogously to the case of a n √ TI, all edge = 2 n +1 edge states present in a n √ TShave a diluting weight in the two topological edge statesof the TS as n increases and, conversely, an increasingweight in the impurity edge states that stem from theresidual Hamiltonian terms in H anc .Remarkably, for ∆ = t the model of Fig. 8(a) becomesthe Creutz ladder in Fig. 1(b) without the right edgehopping and on-site potential perturbations. This showsthat modifications to the same original TI/TS can gen-erate different types of n √ TIs/ n √ TIs, given that beforethe diamond chain with π -flux per plaquette was identi-fied as the square-root version of the right edge perturbedCreutz ladder combined with a chain of decoupled sites,whereas here we constructed the √ TS and the √ TS fromthe unperturbed Creutz ladder only. Since it would behard to guess a priori that the very specific combinationof Fig. 1(b) would be the one to generate a √ TI, some-times it is best in practice to start with a presumptivecandidate for square-root topology, whose admission cri-teria are (i) being bipartite, (ii) having finite energy edgestates and (iii) having non-quantized topological indices,and work from there in both directions, finding its origi-nal TI/TS along one and the higher root degree versionsalong the other.
VI. CONCLUSIONS
We outlined here the procedure to go beyond square-root topology , towards topology with a root degree ofany positive integer power of two (2 n -root topology, with n ∈ N ). This was demonstrated for one-dimensionaltopological insulators and supercondutors, where theoriginal models from which higher root degree versionswere constructed were taken to be the Creutz ladder, forthe former, which generated a family of extended dia-mond chains with a π -flux per plaquette, and the Kitaevchain mapped onto a single-particle tight-binding model, for the latter. The connection between a 2 n -root topo-logical insulator/superconductor and its original topolog-ical insulator/superconductor was established through agraph shaped as an outwards directed rooted tree, wherethe Hamiltonian of the former is identified as the rootnode and the Hamiltonian of the latter as one of the leafnodes.Concerning the experimental realization of the 2 n -roottopological insulators studied here, several platforms ap-pear as suitable candidates for their implementation,most notably photonic , solid-state and opticallattices . In what concerns the 2 n -root topological su-perconductor model introduced here, on the other hand,its direct implementation in a superconducting systemseems out-of-reach with current technology, as it wouldrequire some very fine local tuning of the phases of the su-perconducting and hopping terms. Nevertheless, and aswe explained in section V, one can simulate this model bytranslating it into its single-particle tight-binding analogmodel, which can be realized in the venues listed aboveor, as recently proposed, in topoelectrical circuits .Our work invites future studies searching for deeperconnections between a Hamiltonian and any of its pos-itive integer powers. We expect 2 n -root topology to begeneralizable to simply n -root topology. In particular,odd-root topology would come as a natural extension ofour results, since a block anti-diagonal Hamiltonian willremain so when raised to any odd power (one shouldonly be careful in taking out the constant energy shiftsafter each squaring operation). In turn, raising this odd-power Hamiltonian to an even power will render it blockdiagonal. For instance, one can treat 6-root topologyas product of cubic-root and square-root topology. Itmay be possible to construct, on this basis, a networkof topological insulators such that, as with mathematicaltheorems, the set of those we elevate to an “axiomatic”status, like that implicitly granted to the Su-Schrieffer-Heeger model, would become a matter of convention.The results concerning the extension of 2 n -root topol-3ogy to encompass two-dimensional Chern, weak andhigher-order topological insulators, as well as topologicalsemimetals, are being finalized and will be the subject ofa forthcoming article. ACKNOWLEDGMENTS
This work was developed within the scope of the Por-tuguese Institute for Nanostructures, Nanomodelling andNanofabrication (i3N) projects UIDB/50025/2020 andUIDP/50025/2020 and funded by FCT - PortugueseFoundation for Science and Technology through theproject PTDC/FIS-MAC/29291/2017. AMM acknowl-edges financial support from the FCT through the workcontract CDL-CTTRI-147-ARH/2018. LM acknowl-edges financial support from the FCT through the grantSFRH/BD/150640/2020. AMM would like to thank RuiAm´erico Costa for useful clarifications on graph theory.
Appendix A: Alternative construction of the n -rootCreutz ladder Figure 10. Unit cells of the n = 2 , , n -root models usingthe re-normalization method starting from the diamond chainonwards to (a) √ TI with 7-sites, (b) √ TI with 15-sites and(c) √ TI with 31-sites. Black sites appear from subdivisionof the n − √ TI (a site is included in the middle of each link).Gray sites belong to the next unit cell and have to be includedat unit cell N + 1 under OBC. In this section we introduce a more economic construc-tion of the 2 n -root Creutz ladder by re-normalization ofthe hopping parameters. This approach follows the samesteps as those described in Section IV except that weadopt step 3(i) to ensure evened on-site energies uponsquaring the Hamiltonian.We start by introducing nodes in the middle of eachlink in the n − √ TI where the magnitude of the new two-fold hopping terms is transformed as n − √ te iφ/ n − → n √ te iφ/ n − . We then identify the highest squared on-sitepotential c n − which will determine the re-normalizationof the hopping terms connected to sites with lowersquared on-site potential (often corresponds to the sub-set of lower coordination number). At this point, there is also an ambiguity in the definition of the unit cell underOBC (the outermost extra sites can be placed either tothe right or to the left within the unit cell). In orderto preserve a uniform squared on-site potential along thechain these outermost sites have to be included in bothboundaries.This method was followed for the construction of the2 n -root Creutz ladder starting from its √ TI of Fig. 1(a)and up to n = 4, as depicted in Fig. 11. White andblack sites respectively set apart the old sublattice of theoriginal model for n − √ TI and the subset of extra sitesadded to the latter to model the n √ TI chain.Predictably, the only hopping parameters that do notrequire re-normalization are those connected to the spinalsites with highest coordination number with squared on-site potential of c n − = 4 n − √ t , provided that n > √ n √ TI. The num-ber of sites per unit cell for the n √ TI follows the recur-rence relation n uc = n − uc + 2 n , n ≥ , (A1) = 3 , (A2)where we begin with the 3-site diamond chain unit cell.The closed form solution of Eq. A1 by iteration is explic-itly written in terms of the initial condition as n (i) = + n (cid:88) i =2 i = − n +1 , (A3)with (cid:80) ni =2 i = − n +1 being the general formula forthis geometric progression of n terms. The use of recur-rence relations to describe the overall running time of aproblem of size- n in terms of the running time on smallerinputs often pairs with asymptotic notation as a meansto evaluate the relative efficiency of different algorithms.We may probe the efficiency for both methods describedin this report, namely (i) re-normalization approach and(ii) site increment method of Section IV, from the per-spective of the n √ TI-Hamiltonian recursive dimensions.For Eq. A3 the growth rate in order notation is O (2 n ).For the alternative method, the solution for the recur-rence relation yields n (ii) = + n (cid:88) i =2 i − = 13 (1 + 2 n +3 ) , (A4)with a growth rate of O (2 n ). At this point we can saythat O ( n (ii) ) > O ( n (i) ) and the ratio for the site incre-ment as n (cid:29) n (i) n (ii) = 3( − n +1 )1 + 2 n +3 ≈ n (cid:29) n . (A5)This relation asymptotically approaches zero in the limit n → + ∞ . Thus, for this criteria, the re-normalization4 Figure 11. Energy spectrum, in units of t ≡
1, as a function of state index i , obtained from diagonalization of (a) H √ TI , theHamiltonian of the open √ TI chain obtained by the re-normalization method with N = 5 complete unit cells plus 14 openextra sites at unit cell N + 1, (b) H (cid:48) √ TI = H √ TI − c , with c = 4 √ t , (c) H (cid:48) √ TI = H (cid:48) √ TI H (cid:48) √ TI − c , with c = 4 √ t , (d) H (cid:48) √ TI = H (cid:48) √ TI H (cid:48) √ TI − c , with c = 4 √ t , and (e) H (cid:48) √ TI = H (cid:48) √ TI H (cid:48) √ TI − c , with c = 2 t . Bulk (edge) states are coloredin blue (red). The y-axis scaling in plots (b)-(e) was adjusted in agreement with the energy spectrum of √ TI (a) [(b)-(d) weremultiplied by a factor of 1 / √ / method is clearly more tractable in the realization ofhigher n √ TI.The results of Fig. 11 were simulated for the √ TIchain with N = 5 unit cells with (i) = 31 sites by ap-plying n = 4 successive squaring operations to its Hamil-tonian, taking out the constant energy shift given by theon-site potential energy at the relevant sublattice aftereach operation (white sites in Fig. 10). In agreementwith the above, the outermost new (14) sites were placedin the leftmost region within the unit cell and includedalso at the right-side of the last complete unit cell [seeFig. 10(c)].As expected, there is a total of 2 = 16 degen-erate zero-energy states [in the midgap of Fig. 10(e)]arising from the diagonalization of H anc and the topo-logical weight of the edge states of our starting √ TIchain [in-gap red states in Fig. 10(a)], relative to theedge states of the ancestor chain is given by the factor | (cid:104) T L | T , j | T L | T , j (cid:105) | = 1 /
16, with | T L (cid:105) defined above(14). Both the values of the constant energy shifts deter-mined by the highest squared on-site potential and theenergies of the 2 edge states can be readily found tobe equivalent to those described in the main text, thusvalidating this alternative method. Appendix B: The n -root SSH model We will validate the proposed method for the construc-tion of energy dispersive 2 n -root topological insulators using the well-known SSH model. We start with the al-ready formulated √ SSH by Ezawa , corresponding to anSSH model whose solution both for PBC and OBC canbe exactly found , and recursively obtain n √ SSHfor n = { , } (see Fig. 12).Under PBC, the bulk Hamiltonian for the √ SSHmodel, written in the ordered {| j ( k ) } basis, where j =1 , , ..., th Fourier transformed component withinthe unit cell [see Fig. 12(a), where we define site indexa-tion, within the unit cell and from left to right, with thesequence { , , , } ], is given by H √ SSH = (cid:32) h †√ SSH h √ SSH (cid:33) , (B1) h †√ SSH = (cid:18) √ t √ t e − ik √ t √ t (cid:19) , (B2)Considering OBC and assuming an integer number N of unit cells, there is a topological non-trivial phase inthe t < t regime with 2 non-degenerate edge statesof energy E ± edge = ±√ t + t [red curves in Fig. 13(c)].The resulting Hamiltonian from the squaring operation of H √ SSH describes an SSH chain with alternating { t , t } parameters and on-site local potentials (cid:15) par = t + t , plusan impurity chain with uniform t = √ t t and staggeredon-site potentials t , t , t , t , . . . . For the constructionof the higher root-degree SSH models from the steps de-scribed in the main text, we identify the sublattice withlower squared on-site potentials has the subset of sitesof the previous n − √ SSH model (blue in Fig. 12). Ineach site, two extra connections were included (green in5 (a)(b) (c) n n TI TI TI Figure 12. Unit cells of the n = 1 , , n -root models usingthe re-normalization method starting from (a) √ SSH with 4-sites, (b) √ SSH with 16-sites and (c) √ SSH with 64-sites.Blue sites form the sublattice from the previous n − √ SSHchain, gray sites come from the subdivision of the latter, greensites are extra sites introduced to keep the coordination num-ber constant for sites in the blue sublattice and white sites atthe right belong to unit cell N + 1 under OBC.
Fig. 12) with alternating hopping parameters. This en-sures a constant energy shift for each step n defined as c n − = 2 ( n − √ t + n − √ t ) , n > , (B3) c = t + t , (B4)Recall that there are other options for leveling thesquared on-site potentials and the previous alternativemaintains consistent the magnitudes of the original hop-ping terms. The constant shifts c n − are included in the squaring operation, written in the generic form as H n √ SSH = (cid:32) H √ par , n − H √ res , n − (cid:33) , (B5) H √ par , n − = h † n √ SSH h n √ SSH = c n − I par + H n − √ SSH , (B6) H √ res , n − = h n √ SSH h † n √ SSH = c n − I res + H √ res , n − (cid:48) , (B7)where I par(res) is the m × m identity matrix with m =dim( H √ par(res) , n − ).The energy spectrum for the total Hamiltonian at eachsquaring operation for the root node √ SSH with N = 3complete unit cells plus 5 extra sites at unit cell N + 1, isshown in Fig. 13. The progression shows how the eightnon-degenerate edge states coming from diagonalizationof H √ SSH [in-gap red curves in Fig. 13(a)] end up as theeight-fold degenerate zero-energy state (in the topologi-cal regime t < t ) coming from diagonalization of H anc [midgap red curve in Fig. 13(d)]. Of these, only one corre-sponds to the topological state of the original perturbedSSH model.The spatial profile of four selected edge states of H √ TI are shown at the bottom of Fig. 13. After the first squar-ing operation, these edge states appear in four doubly de-generates pairs in the topological regime of t < t
2, whilethe zerp-energy states in Fig. 13(a) become the E = − c states, not seen in Fig. 13(b) because they appear belowthe lowest band, outside the range of the y-axis. Theenergies of the edge states can be obtained recursivelythrough E n √ TIedge = ± (cid:114) E n − √ TIedge + c n − = (B8) E TIedge = 0 . (B9)These 8 edge states of √ SSH can be written as linearcombinations of the degenerate edge states in Fig. 13(d),all with an equal weight of 1 / ∗ [email protected] J. K. Asb´oth, L. Oroszl´any, and A. P´alyi,
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