Topologically protected pseudospins in 2D 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 springmass 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 also 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), while coupled pendula [32], and a mechanical granular graphene system [33] may mimic QSHE-like phenomena.
In this paper, we propose a two-dimensional (2-D) 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: Fig. 1 (a). TRS protected edge modes, incorporating the propagation of such pseudospins, are shown to exist. This structure 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 " symmetry. The and $ 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 spring-mass constituted lattice by the zone-folding method [8].

II. THE SPRING-MASS SYSTEM MODEL AND COMPUTATIONAL METHODS
We consider a hexagonal lattice with equal masses connected by linear springs , as shown in Brillouin zone (BZ) of the hexagonal lattice, twice, to form a new BZ with 1/3 of its original area, as shown in Fig. 1(b). Consequently, the Κ (Κ ′ ) 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 ' and ) ( ' = ) = 3 ), as indicated in Fig. 1 (a). The band structure based on the expanded unit cell is plotted in Fig. 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 ' from the inter unit-cell spring constant: ) . Such distinction still preserves the " symmetry of the unit cell. It was found that when ' ≠ ) , the band degeneracy at the Γ point is lifted and yields a band gap, as indicated in Fig. 2 (b) and (c). With ) and constant, we continuously change the value of ' from ' > ) to ' < ) , through which the band gap at Γ point first closes and then reopens. When ' = ) , there is no band gap[ Fig. 2 (a) (b) and (c)]. We study the modes related to this transition for (i) ' > ) and (ii) ' < ) . respectively. The constituent and direction displacements are plotted successively below. We find that the / direction displacements fields at Γ are of odd and even spatial paritiesof the 9 (/ : ) and 9 E F: E (/ 9: ) variety, as inferred both from the sense of the displacements and stated relationships in the C 6 character table [34]. For instance, the 9 (/ : ) character is inferred through Hybridizing the ' and ) modes in a symmetric and antisymmetric manner yields pseudospins [8], ± = ( ' ± ) )/ 2 , and ± = ( ' ± ) )/ 2.
(1) Fig. 3 (e) -(h) illustrates the related phase distribution of C , F , C and F in the range of -π to π. Clearly seen from the phase relationship that harmonic wave propagation in C / C and F / F  We find that for the case of ' < ) , the modal displacement fields have exactly the same odd and even spatial parities, but ' and ) are now associated with the higher two degenerate bands, while ' and ) corresponds to the lower two bands (Fig. 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 C , C , F , F can be obtained (see Supplementary Material) to be of the following form: where ± = 9 ± : , and ) = 9 ) + : ) . = ) is imaginary ( >0), and < 0. = p q Fp r ) indicates the relative energy of and bands, which is positive in the lattice of ' > ) , and negative in the lattice of ' < ) , respectively. ℎ spin Chern number can be calculated from Since is negative, v depends on the sign of , which leads to v = 0 when > 0, and v = ±1 when < 0 . This means that for the lattice with ' > ) , v = 0 , and the band gap is topologically trivial (Fig. 2 (b)). When we decrease ' to ' < ) , the band gap becomes topologically nontrivial (Fig. 2 (c)) and v = ±1. 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 nontrivial lattices.

B. 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: Figure 5 gap as related to the frequency ranges indicated in Fig. 2(b) and (c). The band structure of the ribbon supper cell is shown in Fig. 5 (b)  ). Compared to the band structures in Fig. 2 (b) 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 Fig. 5 (b). It was noted that these two new modes propagate with a group    Pseudospin-up modes are supported to propagate towards the right, but splits into 2 waves at the cross, that exit from port 1 and 2 respectively. But it is forbidden to exit from port 3, as right-

IV. 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 time reversal symmetry. Through varying the inter-and inter-unit cell spring constants of such a lattice, for a given mass, a clear and