Surface phonons, elastic response, and conformal invariance in twisted kagome lattices

Abstract

Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number z. d-dimensional lattices with z = 2d are at the threshold of mechanical stability and are isostatic. Lattices with z < 2d exhibit zero-frequency "floppy" modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios (equivalently, auxetic elasticity), depends sensitively on boundary conditions and on the nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties (characteristics of the bulk are encoded on the sample boundary), and the surface waves are Rayleigh waves. We discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.

Fig. 1.

(A) Section of a kagome lattice with N_x = N_y = 4 and N_c = N_xN_y three-site unit cells. Nearest-neighbor bonds, occupied by harmonic springs, are of length a. The rotated row (second row from the top) represents a floppy mode. Next-nearest-neighbor bonds are shown as dotted lines in the lower left hexagon. The vectors e₁, e₂, and e₃ indicate symmetry directions of the lattice. The numbers in the triangles indicate those that twist together under PBCs in zero modes along the three symmetry direction. Note that there are only four of these modes. (B) Section of a square lattice depicting a floppy mode in which all sites along a line are displaced uniformly. (C) Twisted kagome lattice, with lattice constant a_L = 2a cos α, derived from the undistorted lattice by rigidly rotating triangles through an angle α. A unit cell, bounded by dashed lines, is shown in violet. Arrows depict site displacements for the zone-center (i.e., zero wavenumber) ϕ mode which has zero (nonzero) frequency under free (periodic) boundary conditions. Sites 1, 2, and 3 undergo no collective rotation about their center of mass, whereas sites 1, 2^∗, and 3^∗ do. (D) Superposed snapshots of the twisted lattice showing decreasing areas with increasing α.

Fig. 2.

Phase diagram in the α - k^′ plane showing region with negative Poisson ratio σ.

Fig. 3.

(A) Phonon spectrum for the undistorted kagome lattice. Dashed lines depict frequencies at k^′ = 0 and full lines at k^′ > 0. The inset shows the Brillouin zone with symmetry points Γ, M, and K. Note the line of zero modes along ΓM when k^′ = 0, all of which develop nonzero frequencies for wavenumber q > 0 when k^′ > 0 reaching on a plateau beginning at defining a length scale l^∗ = 1/q^∗. (B) Phonon spectrum for α > 0 and k^′ = 0. The plateau along ΓM defines and its onset at q_α ∼ ω_α defines a length l_α ∼ 1/| sin α|.

Fig. 4.

(A) Ring network with b > 3a showing internal floppy mode. (B) Ring-network with b = 3a showing one of the two infinitesimal modes.

Fig. 5.

(A) Lattice distortions for a surface wave on a cylinder, showing exponential decay of the surface displacements into the bulk. This figure was constructed by specifying a small sinusoidal modulation on the bottom boundary and propagating lattice-site positions upward to the free boundary at the top under the constraint of constant lengths and periodic boundary conditions around the cylinder. Distortions near and at the top boundary, which have become nonlinear, are not described by our linearized treatment. (B) as a function of q_xa_L for lattice Rayleigh surface waves for α = π/20, π/10, 3π/20, π/5, π/4, in order from bottom to top. Smooth curves are the analytic results from a transfer matrix calculation, and dots are from direct numerical calculations. The dashed line is the continuum Rayleigh limit . Curves at smaller α break away from this curve at smaller values of q_y than do those at large α. At α = π/4, diverges at q_ya_L = π. The inset plots as a function of q_xl_α for different α. The lower curve in the inset (black) is the α-independent scaling function of q_yl_α reached in the α → 0 limit. The other curves from top to bottom are for α = π/25, π/12, π/9, and π/6 (chosen to best present results rather than to match the curves in the main figure). Curves for α < π/15 are essentially indistinguishable from the scaling limit. The curve at α = π/6 stops because q_y < π/a_L.

Fig. 6.

(A) Kagome-based lattice with pgg space group symmetry and uniaxial C_2v point group symmetry. (B) Lattice with p6 space symmetry but global C₆ point-group symmetry. (C) Density plot of the spectrum of the lowest frequency branch of the pgg uniaxial kagome lattice. The spectrum is absolutely isotropic near the origin point Γ, but it has a zero modes on two symmetry related continuous curves at large values of wavenumber.