Square-root higher-order topological insulator on a decorated honeycomb lattice
SSquare-root higher-order topological insulator on a decorated honeycomb lattice
Tomonari Mizoguchi, Yoshihito Kuno, and Yasuhiro Hatsugai
Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan ∗ Square-root topological insulators are recently-proposed intriguing topological insulators, wheretopologically nontrivial nature of Bloch wave functions is inherited from the square of the Hamilto-nian. In this paper, we propose that higher-order topological insulators can also have their square-root descendants, which we term square-root higher-order topological insulators. There, emergenceof in-gap corner states is inherited from the squared Hamiltonian which hosts higher-order topology.As an example of such systems, we investigate the tight-binding model on a decorated honeycomblattice, whose squared Hamiltonian includes a breathing kagome-lattice model, a well-known exam-ple of higher-order topological insulators. We show that the in-gap corner states appear at finiteenergies, which coincides with the non-trivial bulk polarization. We further show that the existenceof in-gap corner states results in characteristic single-particle dynamics, namely, setting the initialstate to be localized at the corner, the particle stays at the corner even after long time. Suchcharacteristic dynamics may experimentally be detectable in photonic crystals.
I. INTRODUCTION
Since the discovery of the quantum Hall effect [1] andfinding of its topological origin [2–4], topological phasefo matter has attracted considerable interests in the fieldof condensed-matter physics. Topological insulators andsuperconductors (TIs and TSCs) [5, 6] are representativesof such systems of non-interacting fermions, where topo-logically nontrivial nature of the Bloch wave functions orBogoliubov quasiparticles, characterized by topologicalindices [7–9], gives rise to robust gapless states at bound-aries of samples [10, 11]. Such a relation between bulktopological indices and boundary states is called bulk-boundary correspondence [12].It has also been revealed that incorporating crys-talline symmetries [13–18] and/or non-Hermiticity [19–27] makes the topological phases more abundant. Re-markably, the notion of bulk-boundary correspondenceis enriched accordingly [28–38]. Higher-order topologicalinsulators (HOTIs) [28–33, 39–41] are one of the exam-ples exhibiting novel type of bulk-boundary correspon-dence, where topologically-protected boundary modesappear not at d − d − n ( n ≥
2) dimensional boundaries, with d being the spa-tial dimension of the bulk. Nowadays, various theoreticalmodels [30, 32, 39, 42] and possible realizations in solid-states systems [43–48] as well as artificial materials [49–56] have been proposed, and effects of interactions [57–60]and disorders [61] have also been investigated.Besides these developments, Arkinstall et al. recentlyproposed a pathway to realize novel type of TIs, by utiliz-ing the square-root operation [62]. Namely, for a properchoice of positive-semidefinite tight-binding Hamiltoni-ans, referred to as the parent Hamiltonians, their square-root Hamiltonian can be generated by inserting addi-tional sublattice degrees of freedom. Then the Hamil-tonian thus obtained has a spectral symmetry around ∗ [email protected] E = 0, and the topologically protected boundary modeson the parent Hamiltonian is inherited to its square root.Such TIs are referred to as square-root TIs [63], andthey are realized in, e.g., a diamond-chain photonic crys-tal [64].Inspired by these findings, in the present paper, wepropose that the idea of taking square root can be ap-plied to the HOTI. Namely, we can construct the model ofHOTI by taking the square-root of the well-known modelof the HOTI; we term such systems square-root HOTIs.To be concrete, we study a decorated honeycomb-latticemodel. Here a decorated honeycomb-lattice stands for ahoneycomb lattice with one additional site at each edge;such a lattice structure is relevant to several solid-state (I)(II) (i)(ii) (iii) = (a)(b) FIG. 1. (a) The model considered in this paper. Grayarrows represent the lattice vectors: a = (cid:16) , √ (cid:17) and a = (cid:16) − , √ (cid:17) . (b) A schematic figure of the relation be-tween the original Hamiltonian and its square. The squaredHamiltonian is the direct sum of the honeycomb-lattice modelwith sublattice-dependent on-site potentials and the breath-ing kagome-lattice model. a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r systems, such as graphene superstructures [65], metal-organic frameworks [66], and 1T-TaS [67, 68], and thushas long been studied mainly as an example of flat-band models. As for the topological aspect, the possi-bility of the HOTI in a similar model was pointed out inRef. 67, and here we present a viewpoint from its par-ent Hamiltonian. Namely, the parent Hamiltonian of thepresent system is the direct sum of the honeycomb-latticemodel with different on-site potentials and the breathingkagome-lattice model [69]. The breathing kagome-latticemodel hosts the HOTI in which boundary modes are lo-calized at the corner of the sample, and this higher-ordertopology is succeeded to the decorated honeycomb modelas well. We demonstrate this by numerically calculatingthe eignevalues and eigenvectors to show the existence ofthe corner modes, and by relating them to the topologicalinvariant for both original and squared Hamiltonians.Besides the solid-state systems listed above, our modelof the decorate honeycomb lattice is also experimentallyfeasible in photonic waveguide crystals, as recently re-ported in Ref. 70. Therefore, to detect the topologicallyprotected corner modes in the present system, we calcu-late the single-particle dynamics. Such a single particleinitial state can be easily prepared by injecting an ex-citation beam in photonic crystals, and its propagationpattern, corresponding to the particle dynamics of thequantum mechanical systems, is measurable. We findthat, when the corner states exist, the particle localizedat the corner sites stays at the same corner and does notspread into the bulk.The remainder of this paper is organized as follows.In Sec. II, we explain the decorated hoencycomb-latticemodel we use in this paper, and point out that its squaredHamiltonian corresponds to the direct sum of the hon-eycomb lattice model and the breathing kagome-latticemodel. We also present the exact dispersion relations.In Sec. III, we elucidate the higher-order topology of thismodel, by demonstrating the existence of the in-gap cor-ner states in the finite system under the open boundaryconditions. We also calculate the topological invariant,i.e., the polarization, and show how the HOTI in thesquared Hamiltonian is inherited to the original model.In Sec. IV, we study the single-particle dynamics, andshow its localized nature originating from the existenceof the corner states. Finally, in Sec. V, we present asummary of this paper. II. MODEL
We consider the following tight-binding Hamiltonianon a decorated honeycomb lattice that has five sites perunit cell [Fig. 1(a)]: H = (cid:88) k C † k H k C k , (1) where C k = (cid:0) C k , • , (I) , C k , • , (II) , C k , ◦ , (i) , C k , ◦ , (ii) , C k , ◦ , (iii) (cid:1) T and H k = (cid:18) O , Φ † k Φ k O , (cid:19) . (2)Here O l,m represents the l × m zero matrix, the Φ k is the3 × k = t t t t e − i k · a t t e − i k · a . (3)For the definitions of the lattice vectors a and a , seeFig. 1(a). Note that the model includes only nearest-neighbor (NN) hoppings with two different parameters t and t .The Hamiltonian is chiral-symmetric, i.e., H k satisfies γ H k γ = −H k where γ = (cid:18) I , O , O , − I , (cid:19) . (4)Here I n,n stands for the n × n identity matrix. Thisindicates the existence of the parent Hamiltonian whosesquare root corresponds to H k [62]. The Hamiltonianalso has C symmetry around the sublattice (I), H ( C k ) = U k H k U † k , (5)where U k = e − i k · a . (6)The key feature of this Hamiltonian can be clarified bytaking a square of the Hamiltonian matrix:[ H k ] = (cid:32) h (H) k O , O , h (K) k (cid:33) , (7)where h (H) k = Φ † k Φ k , (8a)is the 2 × h (K) k = Φ k Φ † k , (8b)is the 3 × h (H) k equals to the Hamil-tonian of the honeycomb-lattice model with different on-site potentials on two sublattices [3 t for (I) and 3 t for(II)], while h (K) k equals to the Hamiltonian of the breath-ing kagome-lattice model, as schematically depicted inFig. 1(b). From the real-space viewpoint, this is under-stood as follows. The particle on a white (black) site canmove to the neighboring black (white) sites by operating - - - E ne r g y - - - E ne r g y (a) (b) K KM Γ K KM Γ FIG. 2. The band structures of the Hamiltonian H k for (a) t = 1, t = 1, and (b) t = 1, t = 1 .
2. Γ = (0 , (cid:0) π , (cid:1) , and M= (cid:16) π, π √ (cid:17) are high-symmetry points inthe Brillouin zone. the Hamiltonian. Then, operating the Hamiltonian twicecorresponds to two NN moves of the particles [71, 72],meaning that a particle on a white (black) site can ei-ther goes to the neighboring white (black) sites or comesback to the original site; the former networks the breath-ing kagome (honeycomb) lattice formed by white (black)sites, while the latter corresponds to the on-site poten-tials. We call the squared Hamiltonian, [ H k ] , the parentHamiltonian of this model.As is well-known, the HOTI is realized in the breathingkagome-lattice model [36, 39, 40], and this higher-ordertopology of the parent Hamiltonian is indeed succeededto its descendant H k , as we will show later.By utilizing the square of the Hamiltonian, the disper-sion relations of five bands can be obtained, because thedispersion relations can be obtained for both the hon-eybomb and the breathing-kagome models [73]. To beconcrete, the dispersion relations for h (H) k are ε (H) k = E ± k = 32 (cid:20) t + t ± (cid:113) ( t − t ) + 4 t t | ∆( k ) | (cid:21) , (9)and those for h (K) k are ε (K) k = 0 , E ± k , (10)where ∆( k ) = (cid:0) e i k · a + e i k · a (cid:1) /
3. We note that thedispersion relations for ε (H) k and for h (K) k are the sameexcept for the existence of the zero-energy flat band for h (K) k . It follows from Eqs. (9) and (10) that the disper-sion relations of the decorated honeycomb-lattice modelsare ε k = 0 , ± (cid:113) E ± k . (11)The band structures for several parameters are depictedin Fig. 2. There is a flat band at zero energy regardlessof the choice of parameters, which originates from thechiral symmetry with sublattice imbalance from the con-ventional viewpoint [74–77]. Alternatively, we can regard that this flat band is inherited from the breathing kagomebands of the squared Hamiltonian. When | t | = | t | , thesquared Hamiltonian equals to that for the honeycomb-lattice model with a uniform on-site potential plus the isotropic kagome-lattice model, thus the Dirac cones ap-pear at K and K (cid:48) points [Fig. 2(a)], whereas they aregapped out when | t | (cid:54) = | t | [Fig. 2(b)]. This gap openingmakes it possible to seek the HOTI in this model.We further point out the eigenvectors of H k can beconstructed from those of h (H) k and h (K) k . To be specific,let us focus on the first and the fifth bands, which are rel-evant to the higher-order topology discussed in the nextsection. Let u (H) ( k ) be the eigenvector of h (H) k with theeigenenergy E + k and u (K) ( k ) be that of h (K) k . In fact, u (K) ( k ) can be written as [71, 73] u (K) ( k ) = Φ k u (H) ( k ) . (12)Note that u (H) ( k ) and u (K) ( k ) are not necessarily nor-malized. Then, it follows that the fifth eigenvector of H k ,which we write u ( k ), can be written as u ( k ) = 1 N k (cid:32) (cid:113) E + k u (H) ( k ) u (K) ( k ) (cid:33) , (13)with N k being the normalization constant. One can eas-ily check that u ( k ) is indeed the eigenvector of H k withthe eigenenergy (cid:113) E + k , as H k u ( k ) = (cid:18) O , Φ † k Φ k O , (cid:19) u ( k )= 1 N k (cid:32) Φ † k u (K) ( k ) (cid:113) E + k Φ k u (H) ( k ) (cid:33) = 1 N k (cid:32) Φ † k Φ k u (H) ( k ) (cid:113) E + k u (K) ( k ) (cid:33) = 1 N k (cid:32) E + k u (H) ( k ) (cid:113) E + k u (K) ( k ) (cid:33) = (cid:113) E + k u ( k ) . (14)Further, due to the chiral symmetry, the first eigenvector u ( k ) can be obtained as u ( k ) = γ u ( k ) = 1 N k (cid:32) (cid:113) E + k u (H) ( k ) − u (K) ( k ) (cid:33) . (15) III. CORNER STATES AND THEIRTOPOLOGICAL ORIGIN
In this section, we discuss the topological nature of thepresent model, focusing on the higher-order topologicalphase and its relation to the squared Hamiltonian.We first study the finite sample shaped into triangleunder the open boundary conditions, shown in Fig. 3(a).For this system, the energy spectrum as a function of t /t is plotted in Fig. 3(b); without loss of generality,we set t / ≥ t /t <
1, whereas they vanish for t /t > t /t = 1 corre-sponds to the gap-closing point, indicating the topologi-cal phase transition at this point. Figure 3(c) is the en-ergy spectrum for fixed values of t and t with t /t < (d) (e)(a) (c) - - - E ne r g y (b) - - - t / t E / t FIG. 3. (a) The finite system under the open boundary con-ditions, consisting of 45 black and 36 white sites. (b) Theenergy spectrum for as a function of t /t . The lines coloredin red correspond to the in-gap corner states. (c) The energyspectrum for t = 0 . t = 1. The in-gap corner states areencircled by cyan and magenta ellipses. The charge densityof (d) the lower three in-gap states and and (e) the higherthree in-gap states. The radii of yellow dots represent theprobability density. FIG. 4. The polarization of Eq. (18) as a function of t /t .Blue dots and orange triangles are for the first and the fifthbands of the decorated honeycomb-lattice model, respectively.Purple squares are for the upper dispersive band of the break-ing kagome-lattice model h (K) k , and the green diamonds arefor the upper dispersive band the honeycomb-lattice model h (H) k . states, we plot the probability distribution for the in-gapstates: N νj = | φ ( ν ) j | , (16)where j denotes the sites and φ ( ν ) j is the real-space wavefunction defined for the eigenmode γ ν as γ ν = (cid:88) j (cid:16) φ ( ν ) j (cid:17) ∗ C j , (17)with C j being the annihilation operator at the site j .The results are shown in Fig. 3(d) for the negative-energymodes and Fig. 3(e) for the positive-energy modes [79],respectively; here we take averages over the three quasi-degenerate states. We clearly see that the ing-gap statesare indeed the corner states, manifesting that the HOTIis realized in the present model. It is worth noting thatthe corner states have large amplitude on both black sitesand white sites at the corners. In fact, the proper choiceof the corner-site termination is necessary to obtain thein-gap corner states, which is a ubiquitous feature of two-dimensional HOTIs. The present corner termination ischosen such that the square of the Hamiltonian of thefinite system corresponds to the triangular geometry forthe breathing kagome lattice which was used in Ref. 39.Next, we discuss the topological origin of the cornerstates. To this end, we employ the polarization as atopological invariant. For C -symmetric systems, it canbe written in a concise form [33, 52, 80]:2 πp n ≡ arg θ n (K) (mod 2 π ) , (18)where θ n ( k ) = u † n ( k ) · ( U k u n ( k )) . (19) (a) (b) (c) (d) FIG. 5. The charge density N (cid:96) ( t ) of Eq. (23) at t = 500 represented by the radii of green dots. The panels (a) and (b) are for t = 1, t = 0 .
3, and (c) and (d) are for t = 0 . t = 1. Red circles represent the initial position of the particle. For (a) and(b), the green dots are smaller than black and white symbols representing the sites. The C symmetry enforces the quantization of p n as p n = l · l with l = 0 , ,
2. To characterized the corner states,we focus on the first and the fifth bands. In Fig. 4, weplot the polarization. Clearly, both p and p take for t /t <
1, where the corner states emerge, and 0 for t /t >
1, where the corner states do not emerge. Thejump of p n occurs at t = t , where the band gap closesand the topological phase transition occurs.How the polarization is related to the squared Hamil-tonian? To see this, we calculate the polarization for u (H) ( k ) and u (K) ( k ). Note that the C rotation matrix U k is block-diagonalized into the kagome sector and thehoneycomb sector [see Eq. (6)], thus in the calculationof p K / H we employ each sector of U k as a C rotationmatrix. The results are plotted in Fig. 4. We see that p changes from ( t /t <
1) to 0 ( t /t >
1) for h (K) k ,whereas it is 0 for h (H) k . This indicates that the higher-order topology of the present model is inherited from thebreathing kagome-lattice model. However, as we havepointed out, the corner modes have amplitudes on bothhoneycomb (white) and kagome (black) sites, meaningthat the actual corner states are not described by thekagome lattice alone. One can also find that the fol-lowing relation of the topological invariants between theoriginal model and the squared model holds:2 πp ≡ πp ≡ π ( p K + p H ) (mod 2 π ) , (20)which follows from Eqs. (13) and (15) [81]. Note that,unlike the case of the square-root TI in the diamondchain [64], the fractionalization of the topological invari-ant does not occur in the present model, because the keysymmetries are not broken by the square-root operation. IV. DYNAMICAL PROPERTIES
In this section, we address the dynamics of the single-particle state associated with the localized corner states.Although the following formulation is written in thelanguage of the second quantization, one can employthe same scheme to describe the dynamics of electro-magnetic waves in photonic crystals [62, 64, 82–85]. To be concrete, time dependence of single-particle wavefunctions in tight-binding models can be translated intothe z dependence of electro-magnetic waves in photoniccrystals, where z stands for the spatial direction in whichthe wave propagates. Thus, the characteristic dynamicsproperties we show below may experimentally be realiz-able by using the photonic crystal setup.Suppose that the single particle is localized at the site j in the initial state: | Ψ(0) (cid:105) = C † j | (cid:105) , (21)where | (cid:105) represents the vacuum state. Then, the wavefunction at time t can be obtained by the unitary timeevolution: | Ψ( t ) (cid:105) = e − iHt | Ψ(0) (cid:105) = (cid:88) ν (cid:16) φ ( ν ) j (cid:17) ∗ e − iε ν t γ † ν | (cid:105) , (22)where ε ν denotes the eigenenergy of ν -th mode, and weset (cid:126) = 1. Thus the density at the site (cid:96) is obtained as N (cid:96) ( t ) = (cid:104) Ψ( t ) | C † (cid:96) C (cid:96) | Ψ( t ) (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) ν e − iε ν t (cid:16) φ ( ν ) j (cid:17) ∗ φ ( ν ) (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (23)We emphasize that this quantity can be easily observedin the photonic crystal. In Fig. 5, we plot N (cid:96) ( t ) for large t (compared with the band width of the system). Here theinitial state is chosen such that the particle is localized ateither the white site or the black site on the left-bottomcorner. In the case without the corner states [Figs. 5(a)and 5(b)], the particle spreads over the entire system,thus the amplitude at individual sites becomes small. Incontrast, in the case with the corner states [Figs. 5(c)and 5(d)], the particle stays at the left-bottom cornereven after long time. V. SUMMARY
To summarize, we have proposed the HOTI analogueof the square-root TIs, which we term the square-rootHOTI. We study the decorated honeycomb-lattice modelas an example, which can be regarded as a square rootof the direct sum of the honeycomb-lattice model with asublattice-imbalanced on-site potential and the breathingkagome-lattice model, the latter of which is a representa-tive model for the HOTI. We indeed find that the HOTIin the breathing kagome-lattice model is succeeded to thismodel, as manifested by the existence of the corner statesat finite energies and the nontrivial bulk polarization.Seeking experimental realization of the square-rootHOTI and the corner states in decorated honeycombsystems will be an interesting future problem. Aswe have emphasize, implementation of the decoratedhoneycomb-lattice structure in photonic crystals hasbeen reported [70], which will serve as a promising plat-form. To realize the HOTI, the imbalance of two hop- pings, t (cid:54) = t , is essential, and such a situation can be re-alized by placing the sites on the edges of hexagons closeror farther to the sublattice (I) than to the sublattice (II).Under such a setup, we expect that characteristic local-ized dynamics associated with the corner states can beobserved. In addition, the nontrivial bulk polarization inthe square-root HOTI may be experimentally accessibleby observing the beam propagation if one can prepare awell-tailored beam profile as an initial state [85]. ACKNOWLEDGMENTS
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