Matryoshka approach to Sine-Cosine topological models
AA Matryoshka approach to Sine-Cosine topological models
R. G. Dias and A. M. Marques Department of Physics & i3N, University of Aveiro, 3810-193 Aveiro, Portugal (Dated: February 2, 2021)We address a particular set of SSH( n ) models ( n being the number of sites in the unit cell)that we designate by Sine-Cosine models [SC ( n ) ], with hopping terms defined as a sequence of n sine-cosine pairs of the form { sin( θ j ) , cos( θ j ) } , j = 1 , · · · , n . These models, when squared, generatea block-diagonal matrix representation with one of the blocks corresponding to a chain with uniformlocal potentials. We further focus our study on the subset of SC (2 n − ) chains that, when squared anarbitrary number of times (up to n ), always generate a block which is again a Sine-Cosine model, if anenergy shift is applied and if the energy unit is renormalized. We show that these n -times squarablemodels [SSC ( n ) ] and their band structure are uniquely determined by the sequence of energy unitrenormalizations and by the energy shifts associated to each step of the squaring process. Chiralsymmetry is present in all Sine-Cosine chains and edge states levels at the respective central gapsare protected by it. The extension to higher dimensions is discussed. The characterization of squared-root topological insu-lators ( √ TI) relies on the fact that the square of thematrix representation of √ TI Hamiltonian in the Wan-nier basis is a block diagonal matrix, more precisely, itis the direct sum H = H T I ⊕ H of two blocks H T I and H that have the same finite energy spectrum, afterapplying a constant energy downshift, but different eigen-states ( H T I being the Hamiltonian of a known topologi-cal insulator) . This reflects the fact that √ TI Hamil-tonian H is defined in a bipartite lattice [lattice withsublattices A and B, such that the Hamiltonian can bewritten as a sum of hopping terms (which imply finiteHamiltonian matrix elements) between different sublat-tices, H = H AB + H BA ].As very recent examples of squared-root topologicalinsulators, one may cite the diamond chain in the pres-ence of magnetic flux or our work on the t t t t tight-binding chain (where a modified Zak’s phase, a sublat-tice chiral-like symmetry, modified polarization quanti-zation, etc., were found ) where H T I corresponds to thewell-known Su-Schrieffer-Heeger (SSH) model . In thesecases, the topological invariants and symmetries of theSSH Hamiltonian H T I map into modified topological in-variants of the original Hamiltonian (see Ref. ). The t t t t tight-binding chain is a particular case of theSSH(4) model which is a generalization of the topologi-cal SSH chain .Recently, several methods of generating the square rootHamiltonian of a given Topological insulator Hamiltonianin 1D and 2D have been proposed. Thesemethods do not allow its consecutive application due tothe appearance of non-uniform local potentials and theconsequent loss of the bipartite property. This also re-flects the fact that the square-root lattice and the originalone are not self-similar.In this paper, we consider a particular subset of 1DSSH( N ) models, N being the number of sites in the unitcell, that we designate by Sine-Cosine models, such thatthe consecutive squaring of the Hamiltonian has alwaysa block-diagonal matrix representation with one of the blocks corresponding to a bipartite chain (apart from anenergy shift) which is self-similar to the original chain,that is, it is again a sine-cosine model provided that theenergy unit is renormalized. The sequence of energy unitrenormalizations associated to each step of the squaringprocess determines the energy gaps in the spectrum ofthe original chain. The higher dimension generalizationsof these 1D models will also have the energy gaps at theinversion-invariant points determined by the renormal-ization factors. We show that a square-root Hamiltonianof these higher dimensional models can be also obtainedfron the 1D counterparts introducing a π -flux per pla-quette. Sine-Cosine chains : Assume an SSH( n ) chainwith a unit cell with n sites and with nearest-neighborhopping terms t i , i = 1 , · · · , n , for some positive integer n . The Sine-Cosine model of order n , SC ( n ) , is definedimposing that t j − = sin( θ j ) and t j = cos( θ j ) , with j = 1 , · · · , n (see top diagram in Fig. 1).By squaring this bipartite Hamiltonian, one obtainsa block-diagonal matrix (one for each sublattice of thebipartite chain) and one of the blocks [shown in the mid-dle diagram of Fig. 1(a)] corresponds to a tight-bindingmodel with uniform local potentials ε j = sin( θ j ) +cos( θ j ) = 1 and hopping terms t j = cos( θ j ) sin( θ j +1 ) .The uniform potentials can be removed applying an en-ergy shift of one. Note that if the hopping terms areglobally multiplied by a hopping factor t , one can stillrecover the Sine-Cosine form for the Hamiltonian settingthis parameter t as the unit energy so that the energyshift is again one (in units of t ).In the simple case of a uniform chain with hoppingparameter t , one has θ j = π/ , for all j , and the hoppingparameter is t = t/ √ , so the energy shift necessaryremove the uniform potentials (which is one in units of t ) becomes t (see Fig. 2). Obviously, the sublatticeHamiltonian corresponds to another uniform chain with t = t cos( π/
4) sin( π/
4) = t / t . Note that if weconsider the respective inverse operation, the square-rootof the t chain, the bottom zero energy level (in red) in a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b (b) H(k)=SC(4) s i n θ s i n θ c o s θ c o s θ s i n θ s i n θ cos θ c o s θ e − ik (a) unit cell H H H s i n θ s i n θ c o s θ c o s θ c o s θ n s i n θ n s i n θ c o s θ n s i n θ c o s θ s i n θ c o s θ s i n θ s i n θ c o s θ c o s θ c o s θ n s i n θ c o s θ n s i n θ s i n θ c o s θ Figure 1. (a) The top diagram illustrates the Sine-Cosinechain with a unit cell of n sites and hopping terms t j − =sin( θ j ) and t j = cos( θ j ) , with j = 1 , · · · ,n . Upon squar-ing, the Hamiltonian for one of the sublattices has the formshown in the bottom diagram, with uniform local potentials ε j = sin( θ j ) +cos( θ j ) = 1 , as illustrated in the simple case ofthree-site chains in the bottom diagrams of (a) (b) Schematicrepresentation of a SC (4) bulk Hamiltonian. The sites cor-respondent to A and B sublattices are coloured in light blueand light red, respectively. Fig. 2 is shifted by √ t in the top spectrum.These arguments apply to an infinite chain as well asto chains with periodic boundary conditions. In the caseof a finite chain with open boundary conditions with anarbitrary number sites, impurity-like potentials may begenerated at the edge sites of the sublattices when squar-ing the Hamiltonian (the local potentials are not uni-form). However, for particular values of the system size,the same reasoning can be applied. This will be discussedafter we discuss the spectra of Sine-Cosine chains withperiodic boundary conditions (PBC) in the next subsec-tion. Application to bulk Hamiltonians : In this sub-section, we show that, when the unit cell of PBC Sine-Cosine chains has n sites, for certain choices of the θ j ,the squaring process can be applied n times and, at eachstep, one of the Hamiltonian blocks is again a Sine-Cosinechain (and this is why we call it a Matryoshka sequence),if an energy shift is applied and if the energy unit isrenormalized. We label these n -times squarable Sine-Cosine chains, SSC ( n ) [they are a subset of the SC (2 n − ) chains]. Furthermore, the sequence of energy shifts andenergy unit renormalizations determine the energy gapsin the respective spectrum.A bulk Hamiltonian H ( k ) is the Hamiltonian of theunit cell closed onto itself with a twisted boundary (re- energy shift √ t energy shift − t H √ H ε in units of t ε in units of t t = t ++ chiralchiral - - - - Figure 2. The squaring process for a uniform chain (that is, θ j = π/ , for all j ) with PBC. The top spectrum is for a chainwith 32 sites and hopping parameter t and the bottom fora chain with 16 sites and hopping parameter t . The coloredlines indicate the folding energies in the following steps of thesquaring sequence. flecting the e ik phases in the hopping terms connectingone unit cell to the next), see Fig. 1(b). If the real spaceHamiltonian is bipartite and the unit cell has more thanone site, then H ( k ) is also bipartite and the squaringprocess will generate a block diagonal matrix.Our Matryoshka sequence of Sine-Cosine chains is con-structed starting from the last Hamiltonian in the squar-ing process which is that of a uniform chain with a singlesite in the unit cell and applying successively a squareroot operation (see Fig. 2). At each step of the squareroot process, we obtain a Hamiltonian with a new chi-ral symmetry as illustrated in Fig. 2. The first iterationdeviates from the general expressions for the followingones since the uniform chain has a single-site unit celland therefore the respective bulk Hamiltonian cannot bewritten in the sine-cosine form described in the beginningof this section. Therefore we describe this first iterationbefore presenting the general expressions.1. From the SSC (0) to the SSC (1) chain . The Sine-Cosine chain SSC (1) (corresponding to the SSHmodel) has hopping terms { sin θ, cos θ } and whensquared, generates a set of two equal bands with en-ergy relation ε ( k ) = 1 + sin(2 θ ) cos k that correspondsto the spectrum of the uniform tight-binding SSC (0) chain with an energy shift equal to one and a hop-ping parameter t (0) = sin(2 θ ) / √ (that determines thebandwidth). So the SSC (0) chain has as band limits ±√ t (0) and the chiral level ε (0) SSC (0) (folding level un-der the squaring operation) is zero. These values alsodetermine the band structure of the SSC (1) chain: theband limits are ± (cid:112) ± √ t (0) and a new chiral sym-metry is present with chiral level ε (1) SSC (1) = 0 . The chi-ral level of the SSC (0) chain is present at the SSC (1) spectrum at the energies ε (1) SSC (0) = ± . Note that -π π- - - folding energiesband limits (a) left edge link LLL RLRRRRR LLL LLLLL(b) +++++++-++--++-++---+---+-+++-+--+++-++--+---+-+----------++--+-
Figure 3. (a) Band structure of the SSC (3) chain with t (0) =sin(0 . π ) / √ , t (1) = 0 . / √ , and t (2) = 0 . / √ that gen-erate unit cell hopping constants { sin θ , cos θ , sin θ , cos θ , sin θ , cos θ , sin θ , cos θ } ≈ { } . The folding levels (that inter-sect energy curves at ± π/ ) are shown as well as the bandlimits ε ±±±± = ± (cid:114) ± t (2) (cid:113) ± t (1) (cid:112) ± √ t (0 (only thesigns are indicated). (b) Right (R) and left (L) edge levels ofa SSC (3) chain with OBC and N = 2 n p − sites, with n = 3 and integer p > , for all the possible choices of the leftmosthopping term that allow the squaring into SSC ( j ) chains. the notation ε ( n ) means a level in the spectrum of theSSC ( n ) chain.If we introduce a global factor t (1) in the hopping con-stants of the SSC (1) chain so that the hopping pa-rameters become { t (1) sin θ, t (1) cos θ } , then the bandlimits become ± t (1) (cid:112) ± √ t (0) and ε (1) SSC (0) = ± t (1) .Note that the uniform chain band energy shift and itsbandwidth (for any choice of energy unit) determinethe hopping parameters of the SSC (1) chain and thesame will occur if we repeat the square root operation[applying it to the SSC (1) chain, then to SSC (2) andso on].2. From the SSC ( n − chain to the SSC ( n ) chain . Whensquaring the SSC ( n ) Hamiltonian, the condition that it generates a block corresponding to a SSC ( n − chain is written as t ( n − sin θ ( n − j = cos θ ( n )2 j − sin θ ( n )2 j (1) t ( n − cos θ ( n − j = cos θ ( n )2 j sin θ ( n )2 j +1 (2)for j = 1 , · · · , n − with n − + 1 ≡ . This impliesthat the global hopping factor in the SSC ( n − chainis given by t ( n − = (cid:113) (cos θ ( n )2 j − sin θ ( n )2 j ) + (cos θ ( n )2 j sin θ ( n )2 j +1 ) (3)for any value of j . These equations determine (almostuniquely) the set { θ ( n ) j } of the SSC ( n ) if t ( n − and { θ ( n − j } are known.Similarly to what was explained in the case SSC (0) → SSC (1) , any level ε ( n − in the SSC ( n − spectrumbecomes a pair of levels , ±√ t ( n − ε ( n − , in theSSC ( n − spectrum. It is simple to conclude that theband structure of the SSC ( n ) is characterized by thefollowing sequence of energy values that give the topand bottom energies of each band, ε ± ± ± · · · ± ± (cid:124) (cid:123)(cid:122) (cid:125) n = ± (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) ± t ( n − (cid:118)(cid:117)(cid:117)(cid:116) ± t ( n − (cid:115) . . . (cid:114) ± t (1) (cid:113) ± √ t (0) , (4)where all the possible combinations of signs must beconsidered. The folding levels associated with the chi-ral symmetries that appear at each step of the squar-ing process are, in the SSC ( n ) spectrum, given by theordered sequence of the values (all the possible com-binations of signs must be considered) ± , ± (cid:112) ± t ( n − , ± (cid:113) ± t ( n − (cid:112) ± t ( n − , ... , ± (cid:118)(cid:117)(cid:117)(cid:116) ± t ( n − (cid:115) ± t ( n − (cid:114) . . . (cid:112) ± t (1) . To summarize, the set { θ ( n ) j } in the SSC ( n ) Hamiltonianis determined by the sequence of hopping factors t ( j ) , j = 1 , · · · , n − , which are the energy units for each stepof the construction of the SSC ( n ) chain, starting from theuniform chain. Obviously, all the bandwidths and gaps inthe spectrum are also determined by this sequence. Notethat we assumed that all hopping parameters are positiveand this places all angles in the first quadrant. A gaugetransformation can change the sign of the hopping termsmaintaining the spectrum. Even with this condition, theset { θ ( n ) j } in the SSC ( n ) is not unique given the sequenceof hopping factors t ( j ) , j = 1 , · · · , n − , because there arestill the two possible choices for the SSC ( j ) sublattice ateach step of the construction of the SSC ( n ) Hamiltonian(but the two choices generate the same spectrum).Note that the SSC ( n ) chain can be viewed as a n -rootof an uniform tight-binding chain, but a generalized onedue to degrees of freedom associated with global hoppingfactors t ( n ) . Finite systems with open boundaries : In thissection, we show how to generate edge states in any ofthe gaps in the band structure of the SC(n) chain whichwill be protected by the chiral symmetry of a particularstep of the construction of the SSC ( n ) Hamiltonian. Notethat the edge states are replicated in the correspondentgaps in the unfolding process. For example, the gaps inFig. 3 from top to bottom are SSC (1) , SSC (2) , SSC (1) ,SSC (3) , SSC (1) , SSC (2) , SSC (1) gaps and an edge stateassociated with the SSC (1) chain will be present in allSSC (1) gaps.Let us first explain the appearance of edge states in theusual SSH(2) chain [equivalent to the SC (1) ≡ SSC (1) chain]. Edge states appear in an open boundary SSH(2)chain when a weak link is present at the boundaries. Ourdefinition of weak link is a hopping term in the unit cellthat can be adiabatically increased from zero (with allthe other hopping terms in the unit cell finite, constantand larger) without closing the central gap. If one ofthe sublattices of the bipartite chain has one more site,an edge state is always present (it changes from a rightedge state to a left edge state at the topological transi-tion, reflecting the fact that there is always one weak linkat one of the boundaries) and has finite support only inthis sublattice. Furthermore, its energy is exactly zero,a value protected by the chiral symmetry. If both sub-lattices have the same number of sites and we can splitthe chain in two halves, each of them with a weak link atthe boundary, two edge sates will be present with nearlyzero energy which are protected by the chiral symmetryand the band gap.In order to generate edge states in a chosen gap of theSSC ( n ) chain, one chooses the boundaries in such a waythat the squaring process will generate the SSC ( j ) chainthat has that gap as the central one with a weak linkat its boundaries. Also, in order to guarantee that oneof the blocks at each squaring step is that of a bipartitechain (apart from an energy shift), we impose that thenumber of sites of the SSC ( n ) chain is N = 2 n p − ,with integer p > , so that the inner sublattice is theone corresponding to the bipartite OBC SSC ( j ) chainin all steps (the number of sites at each step is of theform N = 2 j p − ). That way, all sites of the OBCSSC ( j ) chain will have the same local potential (equalto one). In the case of the SSC ( n ) chain, there are n − +- H s i n θ s i n θ c o s θ c o s θ x x x x s i n θ y c o s θ y s i n θ y c o s θ y cos θ x sin θ x c o s θ y s i n θ y Figure 4. A square-root Hamiltonian of a 2D SSC ( n − Hamiltonian is possible if π -flux per plaquette is introducedin the 2D SSC ( n ) lattice. This flux implies the existence of4 blocks (four decoupled sublattices) in the squared Hamilto-nian, one of them being the 2D SSC ( n − one (sublatticewith squared sites). possible choices of chain terminations given a system size N = 2 n p − . In Fig. 3(b), we show the edge state levelsin the case of a OBC SC(3) chain with N = 2 n p − sites for all the possible choices of the leftmost hoppingterm ( sin θ , sin θ , sin θ , sin θ ) which agree with theprevious argument.Interestingly, for these system sizes, the squaringmethod can be extended until we reach a single leveland this implies that the spectrum of the OBC SSC ( n ) chain is the combination of the spectrum obtained froma single level with zero energy (applying successivelyenergy shifts, energy unit renormalizations and squareroots to each level, that is, each level ε ( j − generates ±√ t ( j − ε ( j − levels) with a spectrum that has lev-els at the folding energies (due to the extra site in theother sublattice relatively to the SSC ( j ) sublattice).The presence of edge states in the central band ateach step of the construction can be confirmed addingthe Zak’s phase of the positive bands leading as usual to π in the non-trivial topological phase. Extension to 2D : The 2D version (for higher dimen-sions the reasoning is similar) of the SSC ( n ) chain can beconstructed in the same way as the 2D SSH model is con-structed from the SSH chain, that is, the hopping termsin the x direction are those of a x -SSC ( n ) chain and thesame for hopping terms in the y direction [the y -SSC ( n ) hopping terms can be different from the x -SSC ( n ) ones].This 2D model will have a band structure that can becharacterized by the band limits at the inversion invari-ant momenta and these limits will be the sum of twoterms of the form of Eq. 4, ε x ±±±···±± + ε y ±±±···±± .One may be tempted to try to construct the 2D-SSC ( n ) model following the method given for the SSC ( n ) chain sothat, when squaring the Hamiltonian, 2D-SSC ( j ) blocksare generated. Despite the fact that the lattice is bi-partite, one faces one difficulty: the dimension of the2D-SSC ( n − model is one fourth of that of the 2D-SSC ( n ) model. When squaring the Hamiltonian, the bi-partite property guarantees the appearance of two diago-nal blocks, each one corresponding to different sublattice(in Fig. 4, the two sublattices have different colors). Anextra factor is required in order for one of these blocksto become a diagonal sum of two smaller blocks, reflect-ing the division of the sublattice in two other sublattices(with pink circular sites and pink squared sites in Fig. 4,respectively). So we are only able to find a single square-root of a 2D SSC ( n − model, and that is the 2D SSC ( n ) model with π -flux per plaquette, with flux introduced bymultiplying the x -hopping terms by ( − at every otherrung. This π flux generates destructive interference inthe hopping terms from pink circular sites to/from pinksquared sites in Fig. 4 and therefore one may interpret itas an additional “bipartite” property. Conclusion : Square-root topological insulators haveattracted attention due to the presence of finite energytopological edge states in non-central gaps of the chiralspectrum that cannot be characterized using the usualtopological invariants. In this paper, we extend the con-cept of SRTI by introducing a particular 1D Hamilto-nian [that we label n -times squarable Sine-Cosine model,SSC ( n ) ] of the family of the SSH( n ) chains that canbe squared multiple times generating at each step a self-similar Hamiltonian (with a smaller unit cell) in what wecall a Matryoshka sequence, each of them with its own chiral symmetry. Edge states at any gap of the originalchain are protected by one of these chiral symmetries.This sequence of chiral symmetries is lost in general ifthe Hamiltonian is perturbed away from the Sine-Cosineform, but as long as the band gaps remain open, the edgestates should survive.These models are determined by the sequence of en-ergy unit renormalizations in the squaring process andtheir spectrum has a very simple form in terms of theseparameters. This fine tuning of their band structureas well as the control over the presence or absence ofedge states in any of the spectrum gaps makes thesemodels very appealing in the context of artificial latticessuch as photonic , optical lattices , topoelectri-cal circuits or acoustical lattices , where the effec-tive hopping terms can be adjusted in order to reproducethe necessary set of angles { θ j } . Acknowledgments : This work was developedwithin the scope of the Portuguese Institute for Nanos-tructures, Nanomodelling and Nanofabrication (i3N)projects UIDB/50025/2020 and UIDP/50025/2020.RGD and AMM acknowledge funding from FCT - Por-tuguese Foundation for Science and Technology throughthe project PTDC/FIS-MAC/29291/2017. AMM ac-knowledges financial support from the FCT through thework contract CDL-CTTRI-147-ARH/2018. J. Arkinstall, M. H. Teimourpour, L. Feng, R. El-Ganainy,and H. Schomerus. Topological tight-binding models fromnontrivial square roots.
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