Quantum-spin-Hall topological insulator in a spring-mass system

It is proposed that a lattice, with constituent masses and spring constants, may be considered as a model system for topological matter. For instance, a relative variation of the inter- and intra-unit cell spring constants can be used to create, tune, and invert band structure. Such an aspect is obtained while preserving time reversal symmetry, and consequently emulates the quantum spin Hall effect. The modal displacement fields of the mass-spring lattice were superposed so to yield pseudospin fields, with positive or negative group velocity. Considering that harmonic oscillators are the basis of classical and quantum excitations over a range of physical systems, the spring-mass system yields further insight into the constituents and possible utility of topological material.

It would of much advantage and yield insight, to consider a harmonic oscillator point of view, quite common in physics, for invoking topological phases. In this respect, a discrete spring-mass based mechanical system, may constitute a model system for topological structure as related to phononic materials. For instance, QHE based topological insulators in spring-mass lattices may be created by adding circulating gyroscopes [11,28], Coriolis force [30] or varying spring tension [31]. QVHE has been realized in such systems by alternating the mass at A and B sites of the unit cell of a mechanical graphene-like lattice [29]: figure 1(a). QSHE-like phenomena has also been explored in spring-mass lattices, through coupled pendula [32], and a mechanical granular graphene system [33]. However, many of these systems are difficult to implement in practical applications.
In this paper, we propose a two-dimensional spring-mass system, exemplifying a QSHE topological insulator, in the acoustic domain. Various trivial and non-trivial band structures may be originated by varying the masses (m) and the relative spring constants (k) in the associated lattice. In addition to exhibiting the topological features that have now become familiar to practitioners in the field, we indicate a novel spin degree of freedom. The related pseudospins are observed, in frequency domain analysis as the polarization of modal displacement field of masses in one unit cell: figure 1(a). TRS protected edge modes, incorporating the propagation of such pseudospins, are shown to exist. This structure may be representative of different phases of matter, as the the spring constant can be view as coupling strength between unit cells in various systems. It can be applied as one of the possible practical designs of photonic/phononic topological insulators.
A basis for creating a topological material, based on a spring-mass system, to mimic the QSH effect, is to create intrinsic TRS. We consider a hexagonal lattice of masses and springs arranged in C 6 symmetry. The E and E′ representations are each two-fold degenerate with the individuals being complex conjugates [34]. Consequently, a four-fold degeneracy is required to satisfy TRS and may be enabled through manifesting a double Dirac cone in the band structure. We achieve a four-fold degeneracy, in the band structure of a springmass constituted lattice by the zone-folding method [8].

The spring-mass model and computational methods
We consider a hexagonal lattice with equal masses m connected by linear springs k, as shown in figure 1(a). The unit cell of this hexagonal lattice consists of 2 masses m 1 =m 2 =m, with lattice constants a 1  and a 2  }u is a vector constituted from the two degrees of freedom for each mass-the x and y direction displacements for m 1 and m 2 : u u u u u , , , x y x y = { } and F is the force. We consider a Bloch wave solution of the type u Ue qa la , , x y x y } is the modal displacement, and γ 1 and γ 2 are wave vectors. A dispersion relation is obtained by solving the eigenvalue problem D U U , , The band structure of the hexagonal lattice in figure 1(c) exhibits a single Dirac cone at the K (K′) point. The frequencies are non-dimensionalized as .
k m W = w Subsequently, we fold the first Brillouin zone (BZ) of the hexagonal lattice, twice, to form a new BZ with 1/3 of its original area, as shown in figure 1(b). Consequently, the K (K′) point is mapped to the Γ point at the center of the BZ, creating a double Dirac cone. The smaller BZ corresponds to an expanded unit cell in real space of 3 times of the original unit cell area, with 3×2=6 masses, and lattice constant b 1  ), as indicated in figure 1(a). The band structure based on the expanded unit cell is plotted in figure 1(d), and indicates a double Dirac cone at Γ.
To induce a phase transition, in the topological sense, we break the spatial symmetry of the hexagonal lattice, through changing the spring constants of the connecting masses in the lattice, i.e. distinguishing the intra unit cell spring constant k 1 from the inter unit-cell spring constant: k 2 . Such distinction still preserves the C 6 symmetry of the unit cell. It was found that when k k, 1 2 ¹ the band degeneracy at the Γ point is lifted and yields a band gap, as indicated in figures 2(b) and (c). With k 2 and m constant, we continuously change the value of k 1 from k 1 >k 2 to k 1 <k 2 , through which the band gap at Γ point first closes and then reopens. When k 1 =k 2 , there is no band gap (figures 2(a)-(c)). We study the modes related to this transition for (i) k 1 >k 2 and (ii) k 1 <k 2 .

Results and discussions
3.1. Modal displacement fields in hexagonal spring-mass lattices: the case for pseudospins The modal displacement and its x and y components, of the masses in the unit cell, at the Γ point of the k 1 >k 2 lattice are shown in figures 3(a)-(d). The labeling of the modes in figures 3(a)-(d) follows the nomenclature for the lower to higher band degeneracy corresponding to figure 2(b). The modal displacements for a given mass in p 1 (/d 1 ) are orthogonal to p d , 2 2 ( ) / respectively. The constituent x and y direction displacements are plotted successively below. Since each mass has two degrees of freedom-the displacements in the x-and the ydirections, in considering the parities of modal displacements in figure 3, we consider the x-and the y-direction modal displacement fields separately. We find that the x/y direction displacements fields at Γ are of odd and even spatial parities-of the p x (/ p y ) and d x y 2 2 -(/d xy ) variety, as inferred both from the sense of the symmetry of the displacements and stated relationships in the C 6 character table [34]. For instance, the p x (/ p y ) character is antisymmetric with respect to the center, even symmetric to the x-(/y-) axis, and odd symmetric to the y-(/x-) axis, while the d x y 2 2 -(/d xy ) parity is symmetric with repsect to the center, and even(/odd) symmetric to both the x and y axes.
Hybridizing the p d 1 1 / and p d 2 2 / modes in a symmetric and antisymmetric manner yields pseudospins [8]

Figures 3(e)-(h)
illustrates the related phase distribution of p + , p − , d + and d − in the range of p to π (see appendix B). Clearly seen from the phase relationship that harmonic wave propagation in p + /d + and p -/dhave opposite polarizations. Taking the time harmonic component e t iw into consideration, due to the orthogonality of displacements in p 1 /d 1 and p 2 /d , 2 each mass corresponding to the hybridized mode p + /d + rotates in the one direction, while each mass in p -/drotates in the opposite direction. The incorporation of the relative motions of the six masses in the unit cell leads to rotation of the whole displacement field. Such rotation may be considered as one manifestation of a pseudo-spin. One can follow the motion in d + during one time period T: figure 4, indicating such clockwise orientability of the displacement field.
We find that for the case of k 1 <k , 2 the modal displacement fields have exactly the same odd and even spatial parities, but d 1 and d 2 are now associated with the higher two degenerate bands, while p 1 and p 2 corresponds to the lower two bands (figure 2(c)). This demonstrates that band inversion happens at the Γ point during the process of closing and reopening the band gap, and a change in topology of the band structure. Such a change has  been previously quantified through the spin Chern number [35]. The Hamiltonian on the basis states of p d p d , , , ] can be obtained (see appendix C) to be of the following form: x y indicates the relative energy of p and d bands, which is positive in the lattice of k 1 > k , 2 and negative in the lattice of k 1 <k , 2 respectively. The spin Chern number can be calculated from Since B is negative, C s depends on the sign of M, which leads to C 0 s = when M 0, > and C 1 s =  when M 0.
< This means that for the lattice with k 1 >k , 2 C 0, s = and the band gap is topologically trivial ( figure 2(b)). When we decrease k 1 to k 1 <k , 2 the band gap becomes topologically non-trivial (figure 2(c)) and C 1. s =  Therefore, from the topological band theory [1] it would be expected that there would exist pseudospin-dependent edge modes at the boundary between topologically trivial and topologically non-trivial lattices.

Propagating edge modes
The pseudospin-dependent edge modes are vividly illustrated through simulations on a ribbon-shaped lattice that is periodic in one direction and of the width of one unit cell in the other direction:  ) and (c), we clearly see two additional states appear within the bulk band gap connecting the lower bands to the higher bands, as illustrated by red and green lines in figure 5(b). It was noted that these two new To verify the unidirectional propagation of the topological edge modes, we conducted time-domain numerical simulations on finite spring-mass lattices consists of both T and NT units. The governing equation for the spring-mass lattice takes the form u Au F t , = + ( ) where Au is the restoring/displacement-dependent force due to spring deformations, and F t ( ) is a time-dependent excitation. We solve the equivalent ODE:  propagate into the bulk, while a force (with g w w = ) will only excite states that propagate at the edge of the T and NT domains. A sharp discontinuity turning boundary between T and NT as indicated in figure 6(d) demonstrates that the edge states were immune to backscattering figure 6(e).
As the indicated pseudospins are symmetrized configurations of modal displacement fields, they are not prone to selective and individual excitation. However, in another application of the T-NT unit arrangement shown in figure 7(a), it may be able to separate out the counter-propagating states, as broadly constructed in figures 3(e)-(h). With F F e t 0 i = w it was seen that when a left-moving state (say, with positive group velocity) reaches the crossing, it will propagate up to port 1 and down to port 2 along the edges but will not propagate right to the port 3. Consequently, the trajectory of wave propagation (figures 7(b)-(f)) forms a 'T' shape. It

Conclusions
In summary, we have shown that a mass-spring based lattice system may have attributes related to that of a topological insulator, in the presence of TRS. Through varying the inter-and inter-unit cell spring constants of such a lattice, for a given mass, a clear and distinct variation of the band structure was seen. A concomitant change in the modal displacement fields, corresponding to a band inversion, may be generated. The deconvolution of the fields as well as their hybridization in a symmetric and antisymmetric manner yields a basis for the creation of pseudo-spins, corresponding to clockwise/counter-clockwise rotation of the modal displacement vector. Both pseudo spin-up and pseudo spin-down modalities, corresponding to the positive or negative group velocity are proposed. The existence of polarized edge states as well as corresponding modes was demonstrated through both frequency domain analysis and time domain simulations. These edge modes are topologically protected, as they are immune to backscattering when encountering sharp edges. Considering that harmonic oscillators (which are direct manifestations of spring-mass units) form the basis for many physical systems, ranging from acoustics to electromagnetics, this work yields a general foundational framework and related methodology, i.e. modulating band structure and constituent modes through varying the respective spring constants of the physical system.

Acknowledgments
This work was supported by ARO grant No. W911NF-17-1-0453. The authors acknowledge discussions with D Bisharat and X Kong.

Appendix A. Dynamical matrix for spring-mass lattice
To get the dispersion relations, we evaluate where D is the dynamical matrix, and u is the displacements of the masses.
The elements of D were obtained through assuming a Bloch wave solution of form u Ue .
q l qa la t , i 1 1 , , which are elements of the first raw of equation (5). Other entries of D can be obtained in a similar manner.  The dynamical matrix D for 6 masses with 12 constituent modal displacements (i.e.    ) Each element in H can be approximated to the second order using Taylor expansion.
= Neglect second-order off-diagonal terms, the effective Hamiltonian is (here 1 2   = w w which is negative when k k, 1 2 < and B , which is also negative. Since H s NT has a similar formula as the Bernevig-Hughes-Zhang model [35], the spin Chern number can be calculated from equation (3). Since M and B are both negative, the spin Chern number for lattice with k k 1 2 < is 1,  which indicates it is topologically non-trivial.  is the quantum spin Hall conductivity [1]. The calculated spin Chern numbers C s  and C s  give n 1, s = implying Z 2 is unity. Similarly, for a lattice with k k, 1 2 > the effective spin Hamiltonian takes the same form as equation (16), but with M 0, > and B 0. < According to equation (3), C s =0, which proves that the lattice with k k 1 2 > is topologically trivial. The projections of pseudo spin eigenvectors with k 1.2, 1 = k 1 2 = and m 1 = are plotted in figure 12, which shows eigenvectors of the lower bands are more p-like, while eigenvectors to the higher bands tend to be d-like, as expected for an ordinary/trivial insulator.
Appendix D. Mini band gap due to C 6 symmetry breaking at the T-NT boundary There would indeed be level repulsion/band anti-crossings (mini bandgap) when levels/bands of similar symmetry intersect, as would be relevant to the slight perturbation from C 6 symmetry at the boundary between the T and the NT regions. The magnitude of the gap could be related to the extent of asymmetry and could, in principle, be reduced, e.g. through minimizing the effect of C 6 symmetry breaking at the T-NT interface [37]. We indicate such influences in figure 13. The figures have differing relative mass ratios, and ratio of the intercell spring constant (T): intra-cell spring constant: inter-cell spring constant (NT). It can then be seen that as the asymmetry in mass and spring constants between the T and NT lattices increases, the mini band gap at the 'crossing' becomes larger as well.