Mechanical Graphene

We present a model of a mechanical system with a vibrational mode spectrum identical to the spectrum of electronic excitations in a tight-binding model of graphene. The model consists of point masses connected by elastic couplings, called"tri-bonds,"that implement certain three-body interactions, which can be tuned by varying parameters that correspond to the relative hopping amplitudes on the different bond directions in graphene. In the mechanical model, this is accomplished by varying the location of a pivot point that determines the allowed rigid rotations of a single tri-bond. The infinite system constitutes a Maxwell lattice, with the number of degrees of freedom equal to the number of constraints imposed by the tri-bonds. We construct the equilibrium and compatibility matrices and analyze the model's phase diagram, which includes spectra with Weyl points for some placements of the pivot and topologically polarized phases for others. We then discuss the edge modes and associated states of self stress for strips cut from the periodic lattice. Finally, we suggest a physical realization of the tri-bond, which allows access to parameter regimes not available to experiments on (strained) graphene and may be used to create other two-dimensional mechanical metamaterials with different spectral features.


I. INTRODUCTION
Topology [1][2][3] has become an important tool in advancing our understanding of electronic properties of solids. It plays an important role, for example, in determining the nature of surface states in a wide variety of systems including polyacetylene [4,5], quantum Hall systems [6,7], and topological insulators [8][9][10][11][12][13]. The success in applying topological ideas to electronic systems has recently inspired their generalization to certain classes of classical mechanical systems [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] by establishing a correspondence between quantum electronic Hamiltonians and the "square root" of the mechanical dynamical matrix. This correspondence is exact for a class of Maxwell lattices for which there is a balance between the number of degrees of freedom and the number of constraints per unit cell. A prototype example is the one-dimensional mechanical model [14,18,33] whose excitation spectrum precisely matches the spectrum of the Su-Schrieffer-Heeger (SSH) model for polyacetylene. In the onedimensional SSH model, electrons move on a bipartite lattice with different hopping matrix elements on alternating bonds that connect the sites on the A-and B-sublattice. The mechanical model consists of rigid rotors with pivot points fixed on the A-sublattice and connected by central-force springs residing on the the B-sublattice. There is a oneto-one correspondence between electrons on the Asublattice in the SSH model and the rotors in mechanical model and between electrons on the Bsublattice and the springs of the mechanical model. We note that in contrast to other models of topological mechanics [23][24][25][26][27][28][29][30][31] Maxwell lattices exhibit topologically protected zero frequency modes, along with an intrinsic particle-hole symmetry in the analog quantum Hamiltonian.
Here, we introduce and explore the properties of a model mechanical system whose bulk vibrational excitations are in correspondence with the electronic excitations of the particle-hole symmetric two-band tight-binding model of graphene. We consider a generalized graphene model describing nearest neighbor hopping on a honeycomb lattice in which the hopping matrix elements for the three bonds emanating from a given site are in general different. This model can represent strained graphene [34] for modest variations in the hopping amplitudes, and also arises in the analysis of Kitaev's honeycomb lattice model [35]. Our mechanical analog is constructed by closely following the paradigm used in the construction of the SSH analog. The honeycomb lattice of graphene, like the 1D SSH lattice, is bipartite with A-and B-sublattices, shown as red and blue disks in Fig. 1. Since there is only one degree of freedom per site in the graphene model, we assign a scalar variable z(R A ), which we identify as vertical height displacement, to each site on the A-sublattice, and we assign a kinetic energyż 2 (R A )/2 to that site. On each site in the B-sublattice, we need to assign the analog of a bond connecting sites on the A-sublattice, but this "bond" is connected to three rather than the usual two sites. To each of these triangular bonds, which we call tri-bonds, we assign a kind of "stretching" energy that depends quadrat- The tri-bond lattice. Each shaded triangle represents a tri-bond centered on a B site of the graphene lattice (blue disks). The tri-bonds are connected at their vertices, which lie on A-sublattice (red disks). The yellow disk in each tri-bond marks the pivot point. Dashed tri-bonds are those that are cut to create free surfaces. Bottom: A single tri-bond. The pivot point rp is at (x1d1 + x2d2 + x3d3).
ically on a certain linear combination of the heights at the three sites. Figure 1 shows our model mechanical graphene lattice, with red sites representing the locations of the height variables and gray triangles representing the tri-bonds.
The tri-bonds are the analogs of springs in a standard ball-and-spring lattice, and each imposes a constraint that leads to a generalization of the Maxwell-Calladine theorem [36] relating the numbers of zero modes and states of self stress (SSS) to the number of degrees of freedom and number of constraints. Under periodic boundary conditions, our mechanical graphene lattice is a generalized Maxwell lattice [15,36] in which the number of constraints equals the number of degrees of freedom, and it exhibits all of the properties of ball-and-spring Maxwell lattices: (1) Each zero mode in the bulk spectrum, which again matches the electronic spectrum of graphene, is accompanied by a SSS. (2) Each lattice is characterized by a topological polarization or by Weyl modes. (3) Periodic strips or finite lattices cut from a periodic lattice whose spectrum is fully gapped have a number of zero-frequency surface modes equal to the number of tri-bonds cut, i.e., at least one zero mode per surface wavenumber, residing on one or the other of the opposite free edges. (4) The number of zero modes at a given wavenumber on a free surface is determined by the topological polarization or by the positions of Weyl modes and by a local surface polarization. (5) Domain walls connecting lattices of different topological polarization harbor either zero modes or self stress for each wavenumber along the wall; those connecting different Weyl lattices have zero modes or states of self stress for some wavenumbers but not for others.
Though the bulk spectrum of our model and graphene are identical, their surfaces modes are not. The top and bottom edges of a horizontal strip of the mechanical lattice, which can be created by removing the row of horizontal of dashed tri-bonds in Figure 1 from a periodic lattice, are different: the top surface exposes tri-bond vertices and the bottom a straight continuous line of tri-bond edges. These two surfaces correspond, respectively, to the bearded (with dangling bonds) and zigzag edges of graphene [37]. It is not possible to create a strip in the mechanical lattice like that in graphene in which both edges are zigzagged. Both edges of vertical strips exhibit a two-tri-bond zigzag pattern. The corresponding graphene edges correspond to armchair edges with an extra dangling bond at every second row. This paper is divided into five sections of which this is the first. Section II presents details of our model and defines the equilibrium and compatibility matrices that establish the Maxwell-Calladine theorem. Section III treats the excitation spectrum and establishes a phase diagram with a region harboring Weyl points and regions that carry a topological polarization. Section IV discusses zero modes at free edges or in domain walls. Section V presents physical models for the tri-bonds. Figure 1 provides a visual image of our model. We take the Bravais lattice constant to be a and define primitive reciprocal lattice vectors

II. MECHANICAL GRAPHENE MODEL
We also define the three vectors from a tri-bond centroid to its vertices: The centroids of the tri-bonds are located on B sites of the honeycomb and their vertices lie at A sites. Each site on the A-sublattice is occupied by a unit mass that is constrained (say by a frictionless rod) to move in the vertical direction. We assume these vertical displacements z(R A ) are small enough that linear approximations can be used. Each tribond is pinned at a pivot point that is displaced from its centroid by and the three tri-bonds meeting at a given site are connected in a way that allows each to freely rotate about an axis passing through its pivot point and the site in question. Any location of the pivot point (within or outside the triangle spanned by the tri-bond vertices) can be specified by a unique triple x = (x 1 , x 2 , x 3 ) with x 1 + x 2 + x 3 = 1. With this parametrization, the condition is satisfied for any rigid rotation of the tri-bond, where R B is the B-sublattice location of the tribond centroid and R B + d i are the A-sublattice positions of the vertices of the tri-bond. Violations of this condition necessarily cause a distortion of the tri-bond that costs some energy, which to lowest order in displacements must be quadratic in e(R B ), giving us an elastic energy where the sum is over all sites R B in the Bsublattice. This energy has exactly the same form as that of a lattice of harmonic springs where the sum is over bonds and e is the elongation of a bond. Following this analogy, we call e(R B ) the stretch of the tri-bond at R B . Of course, with the aid of Eq. (8) we can express U as a function of the displacements z(R A ) instead of stretches e(R B ). Given U , we can now construct expressions for the N B tri-bond tensions t(R B ) conjugate to stretches of the tri-bonds and the N site forces f (R A ) conjugate to vertical displacements of the masses: The tension t is one that induces "stretching" of the tri-bond in a manner exactly analogous to the tension in a spring bond inducing stretching of the spring. The tri-bond stretch is a measure of the deviation from coplanarity of its three vertices and pivot point.
Introducing the N -dimensional vectors z and f of site-displacements and forces and the N B dimensional vectors of e and t of stretches and tensions, we can write Eqs. (8), (10), and (11) as where C is the compatibility matrix with components and Q is the equilibrium matrix with components Q = C T as required. In Fourier space, where C(q) and Q(q) in this case are 1 × 1 matrices with As in the case of central-force springs, the null space of C consists of zero modes and that of Q of SSSs. The global and wave-number specific Maxwell-Calladine index theorems [15] follow immediately: and where N 0 and S are, respectively, the total number of zero modes and the total number of SSSs, n 0 (q) and s(q) are the numbers of zero modes and states of self stress at wavenumber q, and n and n B are the number of sites (1 under PBC) and number of tri-bonds (1 under PBC) per unit cell. The energy can be written in various ways in terms of these variables: where D(q) = k|C(q)| 2 is the (one-dimensional) dynamical matrix. For the system with PBCs, the corresponding quantum Hamiltonian is block diagonal with 2 × 2 blocks of the form where we define the normal mode frequency scale ω 0 = √ k. The square of this Hamiltonian is diagonal with lower entry kCC * and upper entry kC * C = D (which are identical as C is one-dimensional). The eigenvalues of H(q), given by ±ω(q), are simply the two square roots of D and thus specify the normal mode dispersion ω(q), with C(0) = 1 implying ω 0 = ω(q = 0).
If we identify the bond hopping amplitudes t i = ω 0 x i , then H(q) is formally identical to the Hamiltonian of the nearest neighbor tight binding model of graphene [34]. Equivalently, the constraint x i = 1 corresponds to setting ω 0 = i t i .

III. WEYL POINTS AND PHASE DIAGRAM
The spectrum and topological properties of mechanical graphene depend on the location of the pivot point in the tri-bond. The features of the different models can thus be represented on a ternary phase diagram in which each point (x 1 , x 2 , x 3 ) corresponds precisely to the placement of the pivot point. Unlike a typical ternary phase diagram, however, negative values of x i (corresponding to pivot points outside the triangle) are allowed here.
In this section, we will derive the phase diagram shown in Fig. 2(a), in which there are three gapped regions with different topological polarizations R T and four regions, W 0,1,2,3 , with Weyl points. The point x = (1/3, 1/3, 1/3) corresponds to undistorted graphene. Other points correspond to strained graphene, though the regions outside W 0 correspond to degrees of strain that are probably too large to be physically realizable.
Weyl points arise when C(q) = 0, which generically occurs for pairs of points ±q * , where q * is a solution of C(q * ) = x 1 e iq * ·d1 + x 2 e iq * ·d2 + x 3 e iq * ·d3 = 0. (23) When x = (1/3, 1/3, 1/3) (at the middle of W 0 in Fig. 2(a)) Eq. (23) is satisfied at the ±K points q * = ±q K , where This corresponds to the well known Weyl point at the Brillouin zone corner in unstrained graphene. In Fig. 3(a) we show the displacements of one of the zero frequency modes at q K . It is straightforward to see in Fig. 3(a) that this is indeed a "floppy mode" in which, to linear order, each tri-bond undergoes a rigid rotation about its pivot point and hence is not stretched.
Because the phase of C(q) advances by 2π as q wraps around q * , the Weyl points are locally pro- tected and cannot be removed by a smooth deformation. Therefore, there must be a Weyl phase in a finite region around (1/3, 1/3, 1/3). However, it is also clear from (23) that for x = (1, 0, 0), (0, 1, 0) or (0, 0, 1) there are no solutions to C(q * ) = 0. Thus, there must be a gapped phase in the vicinity of the three corners of the dashed triangle in Fig. 2(a).
To determine the phase boundaries, we note that Weyl points can only only disappear if they meet, which must occur at a time-reversal-invariant point q * = −q * +G, where G is a reciprocal lattice vector. q * = 0 is ruled out for finite x because C(0) = x 1 + x 2 + x 3 = 1, so this must occur at one of the three M points which satisfy Then Eq. (23) requires where the subscript is defined cyclically. Together with x i = 1, this implies x k = 1/2 and x k−1 + x k+1 = 1/2, which define the three red lines bounding the triangle inscribed in the dashed region of Fig. 2(a). The Weyl phase in the vicinity of x = (1/3, 1/3, 1/3) thus corresponds to the region W 0 . On the boundary of W 0 the Weyl points meet and annihilate.
The points outside the dashed triangle of Fig. 2(a) have the pivot outside the tri-bond and have one or two of the x i negative. This corresponds in the graphene model to having bond(s) with a negative hopping amplitude. Systems with negative hopping amplitudes are closely related to systems with positive hopping amplitudes. The sign of the hopping amplitude on one of the three bonds (say along d 1 ) can be changed by a non-uniform gauge transformation that changes the signs of all sites on every other horizontal (zig-zag) line of bonds on the honeycomb lattice. This transformation takes bond hopping amplitudes (t 1 , t 2 , t 3 ) → (−t 1 , t 2 , t 3 ), which in our mechanical model takes . This gauge transformation does not change the normal mode frequencies except for an overall constant factor due to the relation between x i and t i . However, since the gauge transformation is at a nonzero wavevector q M1 , it leads to a shift in the wavevector of the normal modes and hence a transformation of the dispersion relation: This transformation maps x = (1/3, 1/3, 1/3) to x ′ = (−1, 1, 1), so the normal mode spectra at these two pivot point locations are related by Eq. (28). In particular, for x ′ there are Weyl points at q K + q M1 . Indeed, Eq. (23) is satisfied for q * = ±q L , where (29) It can be checked that q L = −q K +q M1 up to a reciprocal lattice vector. More generally, the Weyl phase W 0 maps to the region W 1 in Fig. 2(a). Similar transformations identify the Weyl phases W 2 and W 3 , whose boundaries are again given by the red lines defined by Eq. (27). When x is in W 0 in Fig. 2(a), the Weyl points are at ±q * , which reside in the dark gray regions of Fig. 2(b). When x is in W 1 , the Weyl points are at ±q * residing in the light tan regions of Fig. 2(b), and there are symmetry related regions corresponding to W 2 and W 3 . When a path through W 0 is traversed, beginning on one edge and ending another, a pair of Weyl points created at M j pass through the diametrically opposite dark gray regions of Fig. 2(b) and annihilate at M j±1 . Similarly, for a path that begins on right boundary of W 1 , passes through W 1 , and terminates on the left boundary, the Weyl points are born at ±M 2 , pass through the opposite tan regions of Fig. 2(b), and annihilate at ±M 3 . For x = (−α, (α + 1)/2, (α + 1)/2) with α ≫ 1, Weyl points occur at q * ≈ ±(2 2/α /a)x, converging to the Γ point as α → ∞.
Outside the W n regions there is a gap everywhere in the Brillouin zone. There are three disconnected phases that are topologically distinct. The topological polarization can be most easily evaluated at the three simple points (indexed by j = 1, 2, 3) x i = δ ij . Then, The topological polarization R T is determined by the winding numbers n i of this phase over the independent cycles of the Brillouin zone: where C i is the cycle along reciprocal lattice generator b i , which satisfies b i · a j = 2πδ ij for Bravais lattice generators a j . (Note the minus sign in this equation, which appears because it is defined as an integral over C(q) rather than Q(q) as in [14].) Writing then gives which follows from the "completeness" relation 2 i=1 a i b i = 2πI, where I is the unit matrix. The value of R T depends on the real-space positions assigned to the A and B lattice sites, i.e., on our gauge choice. In our current symmetric choice, in which the B sites do not sit at a center of inversion, R T is not a lattice vector, but the differences between of R T 's in different phases are. Arrows indicating R T are indicated in Fig. 2(a). (In the gauge where the origin lies at an A site, we subtract a particular d k from each d j in Eq. (30), and the n i are all 0 or ±1.) Finally, we note that the line in the phase diagram corresponding to x 1 = 0 (where the pivot point is along the bottom edge of the tri-bond) corresponds to a one dimensional limit, in which the system consists of decoupled horizontal lines that are similar to the SSH model. In this case there is a direct transition between topologically distinct gapped phases, which occurs at the blue point x = (0, 1/2, 1/2). Here so that there is a line of zero modes at q x = ±π/a along the vertical line that connects M 2 and M 3 , indicated by a blue line in Fig. 2(b). This situation is analogous to that in critical kagome lattices when there are parallel straight lines of bonds [38]. The zero modes are easy to visualize: as we have discussed, rotation of a tri-bond about any axis passing through its pivot point produces no stretch and costs no energy. Consider a horizontal line of edges containing the d 2 and d 3 vertices of the row of tri-bonds above it and the d 1 vertex of the row of tri-bonds below it as shown in Fig. 3(b). Rotating neighboring tri-bonds in the upper row by δθ in opposite directions about the axis passing through the pivot point and d 1 (the top vertex) while rotating neighboring tri-bonds in the lower row in opposite directions by δθ/ √ 3 about the axis along their bottom edges produces a zero mode. This operation only affects the given rows, and there is a zero mode for each line of bonds.
Associated with each zero mode, there must be a state of self stress. Alternating equal-amplitude stresses on tri-bonds along any row produces the desired zero-force state. In the case with x 1 = 0 and x 2 = x 3 = 1/2, the stress tends to bend the tri-bonds symmetrically either upward or downward about a line passing through the pivot point and vertex 1. Alternation of the sign of the stresses causes vertices 2 and 3 to experience equal and opposite forces from the two tri-bonds each shares along the x axis. When x 2 = x 3 , neighboring rows are still decoupled, and each is equivalent to the SSH model, whose critical point occurs when x 2 = x 3 . Thus the lines defined by x i = 0 for some i, which include the perimeter of the region where all x i are positive, correspond to the 1D limit.

IV. EDGE STATES
Strips with periodic boundary conditions in one direction or samples with free sides on all boundaries can be produced by removing lines of tri-bonds from the system under full periodic boundary con-ditions. Each line of cut tri-bonds creates two free edges. Since the number of sites and tri-bonds are equal under periodic boundary conditions, the index theorem reduces to where ∆N B is the total number of tri-bonds cut to produce the free edges. A similar equation applies to each wavenumber q along the cut producing a strip: where ∆n B is the number of bonds cut per unit cell of one of the exposed edges. How zero modes are distributed on the free edges depends on the topological polarization R T and a local surface polarization R L [14,15] according to the same formula derived for central-force Maxwell lattices. The number of zero modes per edge unit cell (or equivalently per edge wavenumber q) for a given edge corresponding to a lattice "plane" indexed by the reciprocal lattice vector G pointing along the edge's outer normal is The local polarization R L is simply the electric polarization at the given edge that arises from assigning a charge +1 to sites on the A-sublattice and a charge −1 to sites on the B sublattice. Of course only components of R L parallel to G contribute to n 0 so we are free to add arbitrary components to R L parallel to the edge .
It is instructive to look at a couple of examples. Consider the strip with edges parallel to the x-axis as shown in Fig. 3(b). To produce this strip, one tri-bond per surface unit cell had to be removed, so there is a total of one zero mode per wavenumber on the two exposed edges. The local polarization R L L on the lower edge with outer surface normal, G = −4π/ √ 3aŷ, is equally well represented by −d 1 /2 , Insight into Eq. (37) can be gained by considering what happens to C when the sites and tri-bonds are indexed at different positions (without changing the lattice itself). Let and define the "gauge-transformed" compatibility matrix, Then Thus if we chose ∆R A − ∆R B = R L , we find that the total polarization of C ′ (q) is R ′ = R L + R T . Let q = (q ⊥ , q ), where q ⊥ and q are, respectively, the components of q perpendicular (positive toward the sample interior) and parallel to the lattice plane in question, and define η = exp(iq ⊥ a ⊥ ), where a ⊥ is the depth of the surface unit cell, and 0 < q ⊥ ≤ |G|, where G is the inner normal reciprocal lattice vector associated with the lattice plane. C ′ (η, q ) contains only positive powers of η, and thus no poles in η, and as a result, the integral (41) counts the number of zero modes at the surface determined by G. As particular examples, consider the bottom and left edges in Fig. 1. In the first case, R L = −d 1 /2 and where η = exp(iq y √ 3 a/2), has at most one zero and no poles, in agreement with our result that the top and bottom surfaces can have either one zero mode or none. In the second case, R L = d 2 , and where η = exp(iq x a/2). Again, there are no poles, but the highest power of η is 2, and according to Eq. (41), there can be 0, 1, or 2 zero modes in agreement with our previous results. When there are Weyl modes, the number of zero modes on a free edge of a strip will change when q || passes through the projection of a Weyl point onto that edge [32,39]. The total number of zero modes does not change at this transition, so there must be a change of the opposite sign in the number of zero modes on the opposite edge. In other words, zero modes move from one side of the sample to the opposite at a projection of a Weyl point. A similar phenomenon occurs at domain walls in systems under periodic boundary conditions, in which the number of zero modes equals the number of states of self-stress and is given by where ν T is equal to the number of zero modes per wavenumber if it is positive and minus the number of states of self stress if it is negative. Thus a change in the number of zero modes on a zeromode domain wall, which occurs at q equal to the projected Weyl wavenumber, must be accompanied by an equal change in the number of Weyl states of self-stress on a self-stress domain wall.

V. PHYSICAL MODELS
Physical realization of the mechanical graphene model poses some technical challenges, but some straightforward approaches are possible. The simplest version of the tri-bond uses a spring that directly measures the stretch. The tri-bond consists of a rigid plate suspended at its pivot point on a vertical spring that is attached to a rigid ceiling. The plates are connected at each vertex to a ball of mass m through a mechanism that allows free rotations about the axes passing through the center of the ball and the pivot point on the plate. A possible design with three nested universal ball joints is shown in Fig. 4. By attaching a lever arm to each plate that extends arbitrarily far from its centroid and attaching the spring to the end of it as shown in Fig. 4, the pivot point can be placed at any location in the plane, thus allowing the realization of the complete phase diagram of Fig. 2.
The height of the pivot point is precisely x i z i , so the difference in energy in the spring (and gravity) from the equilibrium configuration in which all plates are horizontal is exactly the desired tri-bond energy. To mimic the spectrum of graphene, we need the kinetic energy matrix expressed in terms of the corner height variables z(R A ) to be a multiple of the identity: E K = (1/2)m Aż (R A ) 2 . Thus we take the mass of a plate to be negligible compared to the mass of a corner ball. Note that the deviation from coplanarity of the plate corners and pivot point, which defines the tri-bond stretch, is measured with respect to the fixed equilibrium position of the pivot point rather than the pivot point that moves with the rigid plate (i.e., the position where the spring is attached).
In the limit of small deviations from equilibrium, this model directly mimics the generic discussion of the tri-bond above. Because the tri-bond stretch is directly encoded in a single spring and the height variables are literally encoded as heights of the ball joints, both e and z are immediately visible. For example, the self-stress state of the one-dimensional row of Fig. 3 is simply an alternating pattern of tension and compression in the springs along that row.
Other physical models could exhibit the mechanical graphene phase diagram, if not the exact same spectrum. All that is required is a potential energy of the form where the effective stiffness k may depend on the location of the pivot. Because the features of greatest interest are zero modes, the form of the kinetic energy does not affect their existence or locations in the Brillouin zone, though it will affect the details of the spectrum at finite frequency if the kinetic energy is not simply a multiple ofż 2 1 +ż 2 2 +ż 2 3 . A natural example realizes the tri-bond energy as the bending energy of a triangular elastic plate that is pinned at the pivot point and attached at each vertex to two other plates. While the precise form of the lowest potential energy of the sheet for arbitrary (small) choices of the corner heights is difficult to calculate due to the boundary conditions at the corners and on the free edges, it must vanish for all rigid rotations of the plate, for which x 1 z 1 +x 2 z 2 +x 3 z 3 = 0, and thus cannot depend on any other linear combination of the corner heights.
Another possibility consists of rigid plates coupled pairwise by springs at their corners. In this model, there are two degrees of freedom per plate (the two rocking angles) and thus two vibrational bands. One band is fully gapped, consisting of modes in which the net displacement of the three plate corners at any given A site is zero. The other band exhibits the mechanical graphene phases, with each plate effectively acting as a tri-bond that couples the average displacements at each A site.

VI. CONCLUDING REMARKS
This paper has introduced a mechanical model that is a precise analog of the tight-binding model of graphene, and defines an appropriate two dimensional generalization of the SSH analog introduced in Ref. [14]. This model system exhibits a rich phase diagram of Weyl phases, along with gapped phases with distinct topological polarizations. Our pro-posed structures are amenable to physical implementation, and it will be interesting to construct them and to probe their mechanical mode structures.
In addition, our construction introduces the tribond, which opens a new avenue for studies of Maxwell lattices. It will be interesting to use this approach, and generalizations of it, to construct new classes of two and three dimensional mechanical systems.