Topological Modes Protected by Chiral and Two-Fold Rotational Symmetry in a Spring-Mass Model with a Lieb Lattice Structure
Hiromasa Wakao, Tsuneya Yoshida, Tomonari Mizoguchi, Yasuhiro Hatsugai
JJournal of the Physical Society of Japan
LETTERS
Topological Modes Protected by Chiral and Two-Fold Rotational Symmetry in aSpring-Mass Model with a Lieb Lattice Structure
Hiromasa Wakao , Tsuneya Yoshida , Tomonari Mizoguchi , and Yasuhiro Hatsugai Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
We propose how to realize the topological modes, which correspond to topological zero modes for a quantum system,protected by chiral and rotation symmetry for a mechanical system. Specifically, we show the emergence of topologicalmodes protected by chiral and two-fold rotational symmetry by a spring-mass system with a Lieb lattice structure anddents on the floor. Moreover, comparing the results of a tight-binding model, we have found the additional topologicalmodes for our spring-mass model due to the extra degrees of freedoms. Our approach to realize the topological modescan be applied to other cases with rotation symmetry, e.g., a system of a honeycomb lattice with three-fold rotationalsymmetry.
Introduction:
In this decade, topological phases havebeen extensively analyzed as new quantum states which hostboundary modes protected by topological properties in thebulk.
Remarkably, it turned out that these topologicalphenomena can be observed even beyond the quantum sys-tems (e.g., photonic crystals, mechanical sys-tems etc.). These topological phenomena be-yond the quantum systems originate from the fact that theseclassical systems are described by an eigenvalue equation.One of the advantages of these classical systems is high con-trollability of their parameters. So far thanks to their highcontrollability, a variety of topological phases has been pro-posed for spring-mass systems (SMMs) which are periodicarrangements of the mass points and springs.
For instance, Chern insulators and higher-order topolog-ical phases have been realized for SMMs.Along with the above development, the notion of the topo-logical protection has also been extended to gapless quan-tum systems. A representative example is a Weyl semi-metalshowing the gapless modes in the bulk. These gapless modesare protected by a finite value of the Chern number which isthe topological invariant for two-dimensional systems in classA (no symmetry).
Similar topological gapless modes canalso be found for other local symmetry (e.g. time-reversalsymmetry, particle-hole symmetry, and chiral symmetry).
Furthermore, the analysis of topological gapless states is fur-ther developed by taking into account the spatial symmetry.In particular, the coexistence of chiral symmetry and spatialsymmetry gives topologically stable zero energy modes pro-tected by these symmetry.
To observe the robustness of these topological modes, chi-ral symmetry with a high accuracy is needed. However, unfor-tunately, chiral symmetry for quantum systems such as metalsand insulators can be broken due to the long range hoppingsor the spin-orbit coupling, etc.Under this background, in this paper, we theoretically pro-pose how to realize the above topological zero modes by tak-ing advantage of the high controllability of the classical sys-tems. Specifically, we realize the topological zero modes forthe SMM with a Lieb lattice structure. We note that the sim-ple preparation of the SMM does not preserve chiral sym- metry; by tuning the on-site potential arising from the dentson the floor, we have a SMM preserving chiral symmetry. Inaddition, by comparing the results of a tight-binding model(TBM), we also find that the extra degrees of freedoms of theSMM increase the number of topological zero modes, whichis a unique phenomenon of SMMs. Our approach for the re-alization can be applied to a general two-dimensional tight-binding model (TBM).
The Topological Zero Modes Protected by Chiral andSpatial Symmetry:
Firstly, we briefly review the general ar-guments of the topological zero modes protected by chiral andspatial symmetry for quantum systems.
It is well-knownthat the Schr¨odinger equation is reduced to the eigenvalueproblem. We set a matrix H for the eigenvalue equation. Addi-tionally, we consider a chiral operator Υ and a spatial operator R . Chiral symmetry and spatial symmetry of H are written as { H , Υ } = , (1a)[ H , R ] = . (1b)From (1a), chiral symmetry makes the pair of positive andnegative eigenvalues of H .Let us assume that chirality of lattices is not changed by theoperators R , [ Υ , R ] = . (2)From (1b) and (2), one can see that the matrices H , Υ and R are block-diagonalized by the eigenvectors of R , H = H R H R . . . , (3a) Υ = Υ R Υ R . . . , (3b) R = R R . . . , (3c) a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l . Phys. Soc. Jpn. LETTERS R i = r i ˆ1 n i for i = , , · · · , (3d)where R i , r i ( r i (cid:44) r j for i (cid:44) j ), and n i denote an eigenspace of R , an eigenvalue, and the number of degeneracy of r i , respec-tively. Here, the indices of H and Υ denote the eigenspace of R . Additionally, ˆ1 n is the n × n identity matrix. We note thatchiral symmetry and spatial symmetry are preserved at eacheigenspaces, { H R i , Υ R i } = H R i , R i ] = ν R i = Tr Υ R i , (4)for each eigenspace. This index denotes the di ff erencebetween the number of u + R i and u − R i where Hu ± R i = Υ R i u ± R i = ± u ± R i . Therefore, this index guarantees the | ν R i | topological zero modes protected by chiral symmetry in theeigenspace R i at least. For the total system, the topologicalindex is written as ν = (cid:88) R i | ν R i | . (5)This index guarantees that at least there exist ν the topolog-ical zero modes protected by the operators R and Υ in thetotal system. In the rest of this paper, by the topological zeromodes, we denote topological zero modes protected by chiralsymmetry and rotation symmetry. TBM with a Lieb Lattice Structure:
Based on the abovegeneric argument, we analyze the nearest-neighbor TBM witha Lieb lattice structure under the periodic boundary condition[see Fig. 1 (a)]. Our analysis for the TBM elucidates that thethree-fold degeneracy at M point is protected by chiral sym-metry and two-fold rotational symmetry. For a later use, weset unit vectors as (cid:126) a = (1 , T and (cid:126) a = (0 , T . High symme-try points Γ , M , and X in the first Brillouin zone are located at( k x , k y ) = (0 , π, π ), and ( π, (a)(b) (c) -3-2-10123 (cid:1) M X (cid:1) E n e r gy ✏
In contrast to theTBM, a SMM preserves chiral symmetry without any approx-imation since the coupling appears between the bases con-nected by springs. Therefore, we realize the topological zeromodes for a SMM with a Lieb lattice structure [see Fig. 2(a)] to overcome the di ffi culty in realizing the preservation ofchiral symmetry for the TBM.Let us see the motion of the mass points in the SMM with
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M X (cid:1) F r e qu e n c y ω Γ Γ (a) (b) (a) (c)(b)
Fig. 2. (Color online) (a) The SMM of the Lieb lattice structure. X denotesthe center of the two-fold rotation. (b) The mass points oscillating along thesurface of the dent due to the gravity. (c) The band structure along the lineshown in the panel Fig. 1 (b). a Lieb lattice structure under the periodic boundary condi-tion. For the simplicity, we set the mass of mass points asunit, and unit vectors as (cid:126) a = (1 , T and (cid:126) a = (0 , T . High-symmetry points in the first Brillouin zone are the same as thepoints for the TBM. For a later use, we define a spring con-stant as K , and the displacements of the mass points from theequillibrium point as φ µ(cid:126) k , p . Here, p = , , µ = x , y denotes directions in the two-dimensionalspace. Additionally, we define the ratio of a natural length ofa spring to the length of the spring in equillibrium as η whichdenotes the tension added to the springs and contributes theinteraction between longitudinal and transverse waves. Wenote that η is the independent parameter in our mechanicalmodel since our system is in equilibrium for any K . In thispaper, we focus on the case of the tension for 0 ≤ η ≤ (cid:126)φ = (cid:18) φ x (cid:126) k , , φ y (cid:126) k , , φ x (cid:126) k , , φ y (cid:126) k , , φ x (cid:126) k , , φ y (cid:126) k , (cid:19) T , theNewtonian equation of motion describing the vibrations ofmass points under the periodic boundary condition is writtenas d dt (cid:126)φ = − D ( (cid:126) k ) (cid:126)φ, (9)where D ( (cid:126) k ) is the positive semi-definite matrix called amomentum-space dynamical matrix whose dimension is sixsince there are three sublattices and two spatial coordinates.This matrix can be divided into two terms, D ( (cid:126) k ) = D s ( (cid:126) k ) + D h . D s ( (cid:126) k ) and D h are positive semi-definite and positive definitematrices, respectively. D s ( (cid:126) k ) denotes the coupling describedby spring constants and tensions of the springs. However, D s ( (cid:126) k ) has the non-uniform diagonal elements. Therefore, thissystem does not preserve chiral symmetry because D s ( (cid:126) k ) doesnot anticommute with a chiral operator.Thus, in order to maintain chiral symmetry, we introducethe on-site potential D h arising from dents on the floor. In this paper, we assume that the shape of the dents is aparaboloid. From the constraint force by the dents, the po-tential energy due to the gravity is coverted to the vibrationalenergy. Therefore, mass points oscillate along the surfaces ofthe dents because of the gravity, which gives rise to the on-site potential [see Fig. 2 (b)]. This on-site potential only af-fects the diagonal elements of D ( (cid:126) k ), and makes diagonal el-ements of D ( (cid:126) k ) constant. Moreover, this on-site potential has high controllability since the potential depends on the shapeof the dents.Specifically, the explicit form of D s ( (cid:126) k ) and D h in this sys-tem is written as D s ( (cid:126) k ) = D s D s D s (cid:16) D s (cid:17) † D s ˆ0 (cid:16) D s (cid:17) † ˆ0 D s , (10) D h ( (cid:126) k ) = diag (2 K η, K η, K , K (1 + η ) , K (1 + η ) , K ) , (11)with D s = diag (4 K − K η, K − K η ) , (12a) D s = diag (2 K , K (1 − η )) , (12b) D s = diag (2 K (1 − η ) , K ) , (12c) D s = diag (cid:16) − K (1 + e − ik x ) , − K (1 + e − ik x )(1 − η ) (cid:17) , (12d) D s = diag (cid:16) − K (1 + e − ik y )(1 − η ) , − K (1 + e − ik y ) (cid:17) . (12e)Here, ˆ0 n denotes the n × n zero matrix. For a later use, wechoose D h so that the diagonal elements of D ( (cid:126) k ) become 4 K .We note that there does not exist the coupling between thedisplacements along x and y direction since the springs arealigned horizontality along the x and y direction.Assuming that the mass points oscillate with a frequency ω ,one can write φ µ(cid:126) k , p = e − i ω t ξ µ(cid:126) k , p . Substituting this into Eq. (9),we obtain − ω (cid:126)ξ + D ( (cid:126) k ) (cid:126)ξ = , (13)where (cid:126)ξ = (cid:18) ξ x (cid:126) k , , ξ y (cid:126) k , , ξ x (cid:126) k , , ξ y (cid:126) k , , ξ x (cid:126) k , , ξ y (cid:126) k , (cid:19) T is the eigenvector of D ( (cid:126) k ). As a result, the problem is reduced to the eigenvalueequation analogous to the Schr¨odenger equation. By solvingEq. (13), we obtain the dispertion relation. Figure 2 (c) showsthe band structure for K = η = .
7. We note that the mo-mentum space dynamical matrix corresponds the Hamiltonianof the TBM since the SMM is reduced to two copies of theTBM for η =
0, i.e., the tension is infinitely strong.Let us discuss the symmetry for this mechanical system.We define a chiral operator Υ S as Υ S = Υ T ⊗ ˆ1 = diag (1 , , − , − , − , − . (14a)Additionally, we define a two-fold rotational operator R S around the point X shown in Fig. 2 (a) in this mechanical sys-tem as R S ( (cid:126) k ) = R T ( (cid:126) k ) ⊗ ( − ˆ1 ) , = diag (cid:16) − , − , − e − ik x , − e − ik x , − e − ik y , − e − ik y (cid:17) . (14b)These operators correspond to the ones for the TBM, re-spectively. However, dimensions of these matrix are doubledsince the operations are applied not only to the sublattice de-grees of freedoms, but also to directions of the displacements.Two-fold rotational operator is written as R S ( (cid:126) k ) D ( (cid:126) k ) R − S ( (cid:126) k ) = D ( − (cid:126) k ).The diagonal elements of the momentum-space dynamical
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LETTERS matrix just shift the band structure, ω ( (cid:126) k ). Therefore, this sys-tem preserves “chiral symmetry” since D ( (cid:126) k ) subtracted the di-agonal elements anticommute with Υ S , { D ( (cid:126) k ) − K ˆ1 , Υ S } =
0. Topological zero modes for a SMM corresponding to onesfor a TBM can be observed independently of the constantcontribution except the shift of these modes. To emphasizethis shift, in the rest of this paper, topological zero modes fora SMM are denoted by topological modes. In addition, thissystem preserves two-fold rotational symmetry at the high-symmetry points Γ , M , and X ; [ D ( (cid:126) k ) , R S ( (cid:126) k )] = R S ( (cid:126) k ) since (cid:104) Υ S , R S ( (cid:126) k ) (cid:105) = Υ S can be block-diagonalized by the eigen-vectors of R S ( (cid:126) k ). Then, the number of the topological modesprotected by two-fold rotational symmetry for this system is ν = Γ M X , (15)at least. The above results can be obtained by noticing Υ S =Υ T ⊗ ˆ1 . We note that the number of topological modes forthe SMM is doubled compared to that for the TBM, which isunique properties of the Lieb lattice structure. These topolog-ical modes are observed as two flat bands at Γ and X points,and two flat bands and two Diarc cones at M point in the bandstructure [see Fig. 2 (c)]. We note that these topological modesare stable for any η since Υ S and R S ( (cid:126) k ) do not depend on η .Therefore, these modes for the SMM are stable even whenthe longitudinal and the transverse waves are coupled, whichdemonstrates the robustness of topological modes against theperturbation preserving chiral symmetry.Additionally, the extra topological modes protected by ro-tational symmetry are unique topological phenomena for themechanical system. These additional topological modes orig-inate from the internal degrees of freedoms (i.e., dispal-cements along the x - and y -direction) which may also shiftthe eigenspace of R due to the angular momentum.We finish this part with a remark on generality of our ap-proach. Our approach to realize the topological modes pro-tected by chiral symmetry and rotation symmetry can be ap-plied to generic two-dimensional SMMs. Indeed, we can re-alize the topological modes protected by chiral symmetry andthree-fold rotational symmetry for the SMM with a honey-comb structure. Summary:
We have proposed how to realize the topo-logical modes protected by chiral and rotation symmetry fortwo-dimensional systems. Specifically, by taking advantageof the high controllability of SMMs, we have realized topo-logical modes protected by chiral symmetry and two-fold ro-tation symmetry. We note that in our approach, introducingthe on-site potential arising from the dents on the floor playsan essential role in preserving the chiral symmetry. In addi-tion, compared to the results of the TBM, the number of topo-logical modes increases for the SMM which is due to the pres-ence of the extra degrees of freedoms. Our approach by theSMM and the dents on the floor can be applied to a generaltwo-dimensional TBM.
Acknowledgment:
This work is partly supported by JSPS KAKENHI Grants Nos. JP17H06138, JP20H04627,and JP20K14371. We thank Senri Suzuki for discussion inan early stage of the work.
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