Isostaticity, auxetic response, surface modes, 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 and 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 the long-wavelength limit, the elastic theory of all
these lattices is a conformally invariant field theory with holographic
properties, 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.

Full text

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arXiv:1112.1109v1 [cond-mat.soft] 5 Dec 2011
Isostaticity, auxetic response, surface modes, and conformal invariance in twisted
kagome lattices
Kai Sun,1Anton Souslov,2Xiaoming Mao,2and T.C. Lubensky2
1Condensed Matter Theory Center and Joint Quantum Institute,
Department of Physics, University of Maryland, College Park, MD 20742, USA
2Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
Model lattices consisting of balls connected by central-force springs provide much of our under-
standing 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 struc-
tures, 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 and 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 sur-
face modes are not. In the the long-wavelength limit, the elastic theory of all these lattices is a
conformally invariant field theory with holographic properties, 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.
PACS numbers:
I.INTRODUCTION
Networks of balls and springs or frames of nodes con-
nected by compressible struts provide realistic models for
physical systems from bridges to condensed solids. Their
elastic properties depend on their coordination number z
– the average number of nodes each node is connected to.
If z is large enough, the networks are elastic solids whose
long-wavelength mechanical properties are described by a
continuum elastic energy with non-vanishing elastic mod-
uli. If z is small enough, the networks have deformation
modes of zero energy – they are floppy. As z is increased
from the floppy side, a critical value, zc, is reached at
which springs provide just enough constraints that the
system has no zero-energy “floppy” modes [1] (or mech-
anisms [2] in the engineering literature), and the sys-
tem is isostatic. The phenomenon of rigidity percolation
[3, 4] whereby a sample spanning rigid cluster develops
upon the addition of springs is one version of this floppy-
to-rigid transition. The coordination numbers of whole
classes of systems, including engineering structures [5, 6]
(bridges and buildings), randomly packed spheres near
jamming [7–10], network glasses [1, 11], cristobalites [12],
zeolites [13, 14], and biopolymer networks [15–18] are
close enough to zcthat their elasticity and mode struc-
ture is strongly influenced by those of the isostatic lattice.
Though the isostatic point always separates rigid from
floppy behavior, the properties of isostatic lattices are
not universal; rather they depend on lattice architecture.
Here we explore the the unusual properties of a partic-
ular class of periodic isostatic lattices derived from the
two-dimensional kagome lattice by rigidly rotating trian-
gles through an angle α without changing bond lengths
as shown in Fig. 1. The bulk modulus B of these lattices
is rigorously zero for all α ?= 0. As a result, their Poisson
ratio acquires its limit value of −1; when stretched in
one direction, they expand by an equal amount in the or-
thogonal direction: they are maximally auxetic [19–22].
These modes represent collapse pathways [23, 24] of the
kagome lattice. Modes of isostatic systems are generally
very sensitive to boundary conditions [25–27], but the de-
gree of sensitivity depends on the details of lattice struc-
ture. For reasons we will discuss more fully below, modes
of the square lattice, which is isostatic, are in fact insen-
sitive to changes from free boundary conditions (FBCs)
to periodic boundary conditions (PBCs), whereas those
of the undistorted kagome lattice are only mildly so. The
modes of both, however, change significantly when rigid
boundary conditions (RGBs) are applied. We show here
that in all families of the twisted kagome lattice, modes
depend sensitively on whether FBCs, PCBs or RGBs are
applied: finite lattices with free boundaries have floppy
surface modes that are not present in their periodic or
rigid spectrum or in that of finite undistorted kagome
lattices. In the long wavelength limit, the surface floppy
modes, which are present in any 2d material with B = 0,
reduce to surface Rayleigh waves [28] described by a con-
formally invariant energy whose analytic eigenfunctions
are fully determined by boundary conditions. At shorter
wavelengths, the surface waves become sensitive to lattice
structure and remain confined to within a distance of the
surface that diverges as the undistorted kagome lattice is

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2
(a)(b)
2a
a
Nx
Ny
α
(c)
(d)
3
3*
2
2*
1
aL
e1
e2
e3
1
1
1
1
1
1
2
3
3
3
3
3
3
4
4
4
4
4
4
4
2
2
2
2
2
FIG. 1: (a) Section of a kagome lattice with Nx = Ny = 4 and
Nc = NxNy three-site unit cells. Nearest-neighbor bonds, oc-
cupied 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 4 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 aL = 2acosα,
derived from the undistorted lattice by rigidly rotating tri-
angles 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 con-
ditions. Sites 1, 2, and 3 undergo no collective rotation about
their center of mass whereas sites 1, 2∗, and 3∗do. (d) Su-
perposed snapshots of the twisted lattice showing decreasing
areas with increasing α.
approached. In the simplest twisted kagome lattice, all
floppy modes are surface modes, but in more complicated
lattices, including ones with uniaxial symmetry, we con-
struct, there are both surface and bulk floppy modes.
Arguments due to J.C. Maxwell [29] provide a criterion
for network stability: networks in d dimensions consist-
ing of N nodes, each connected with central-force springs
to an average of z neighbors, have N0= dN −1
energy modes when z < 2d (in the absence of redundant
bonds - see below). Of these a number, Ntr, which de-
2zN zero-
pends on boundary conditions, are trivial rigid transla-
tions and rotations, and the and the remainder are floppy
modes of internal structural rearrangement. Under FBCs
an PBCs, Ntrequals d(d+1)/2 and d, respectively. With
increasing z, mechanical stability is reached at the iso-
static point at which N0= Ntr. The Maxwell argument
is a global one; it does not provide information about
the nature of the floppy modes and does not distinguish
between bulk or surface modes.
II.KAGOME ZERO MODES AND ELASTICITY
The kagome lattice of central force springs shown in
Fig. 1(a) is one of many locally isostatic lattices, includ-
ing the familiar square lattice lattice in two dimensions
[Fig. 1(b)] and the cubic and pyrochlore lattices in three
dimensions, with exactly z = 2d nearest-neighbor (NN)
bonds connected to each site not at a boundary. Un-
der PBCs, there are no boundaries, and every site has
exactly 2d neighbors.Finite, N-site sections of these
lattices have surface sites with fewer than 2d neighbors
and of order
with Nxand Nyunit cells along its sides [Fig. 1(a)] has
N = 3NxNysites, NB= 6NxNy−2(Nx+Ny)+1 bonds,
and N0 = 2(Nx+ Ny) − 1 zero modes, all but three
of which are floppy modes. These modes, depicted in
Fig. 1(a), consist of coordinated counter rotations of pairs
of triangles along the symmetry axes e1, e2and e3of the
lattice. There are Nx modes associated with lines par-
allel to e1, Ny associated with lines parallel to e3, and
Nx+ Ny− 1 modes associated with lines parallel to e2.
In spite of the large number of floppy modes in the
kagome lattice, its longitudinal and shear Lam´ e coeffi-
cients, λ and µ, and its Bulk modulus B = λ + µ are
nonzero and proportional to the nearest neighbor (NN)
spring constant k: λ = µ =
√3k/4. The zero modes of this lattice can be used to
generate an infinite number of distorted lattices with un-
stretched springs and thus zero energy [24, 30]. We con-
sider only periodic lattices, the simplest of which are the
twisted kagome lattices obtained by rotating triangles of
the kagome unit cell through an angle α as shown in
Figs. 1(c) and (d) [24, 31]. These lattices have C3vrather
than C6vsymmetry and, like the undistorted kagome lat-
tice, three sites per unit cell. As Fig. 1(d) shows, the
lattice constant of these lattices is aL = 2acosα, and
their area Aα decreases as cos2α as α increases. The
maximum value that α can achieve without bond cross-
ings is π/3 so that the maximum relative area change
is Aπ/3/A0= 1/4. Since all springs maintain their rest
length, there is no energy cost for changing α, and as a
result, B is zero for every α ?= 0, whereas the shear mod-
ulus µ =√3k/8 remains nonzero and unchanged. Thus,
the Poisson ratio σ = (B − µ)/(B + µ) attains its small-
est possible value of −1. For any α ?= 0, the addition of
next-nearest-neighbor (NNN) springs, with spring con-
stant k′(or of bending forces between springs) stabilizes
√N zero modes. The free kagome lattice
√3k/8, and B = λ + µ =

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3
12
π
6
π
4
π
3
π
/
kk
′
α
σ>0σ<0
0.1 0.00.2 0.30.4
FIG. 2: phase diagram in the α − k′plane showing region
with negative Poisson ratio σ.
zero-frequency modes and increases B and σ. Neverthe-
less, for sufficiently small k′, σ remains negative. Figure
2 shows the region in the k′− α plane with negative σ.
III. KAGOME PHONON SPECTRUM
We now turn to the linearized phonon spectrum of the
kagome and twisted kagome lattices subjected to PBCs.
These conditions require displacements at opposite ends
of the sample to be identical and thus prohibit distortions
of the shape and size of the unit cell and rotations but
not uniform translations, leaving two rather than three
trivial zero modes. The spectrum [32] of the three lowest
frequency modes along symmetry directions of the undis-
torted kagome lattice with and without NNN springs
is shown in Fig. 3(a). When k′= 0, there is a floppy
mode for each wavenumber q ?= 0 running along the
entire length of the three symmetry-equivalent straight
lines running from M to Γ to M in the Brillouin zone
[See inset to Fig. 3]. When Nx= Ny, there are exactly
Nx− 1 wavenumbers with q ?= 0 along each of these
lines for a total of 3(Nx− 1) floppy modes. In addition,
there are three zero modes at q = 0 corresponding to
two rigid translations, and one floppy mode that changes
unit cell area at second but not first order in displace-
ments, yielding a total of 3Nx zero modes rather than
the 4Nx−1 modes expected from the Maxwell count un-
der FBCs. This is our first indication of the importance
of boundary conditions. The addition of NNN springs
endows the floppy modes at k′= 0 with a characteris-
tic frequency ω∗∼
with the acoustic phonon modes [Fig. 3(a)] [32]. The
result is an isotropic phonon spectrum up to wavenum-
ber q∗= 1/l∗∼
Remarkably, at nonzero α and k′= 0, the mode struc-
ture is almost identical to that at α = 0 and k′> 0
with characteristic frequency ωα∼
lα∼ 1/ωα. In other words, twisting the kagome lattice
through an angle α has essentially the same effect on the
spectrum as adding NNN springs with spring constant
|sinα|2k. Thus under PBCs, the twisted kagome lattice
has no zero modes other than the trivial ones: it is “col-
√k′and causes them to hybridize
√k′and gaps at Γ and M of order ω∗.
√k|sinα| and length
KMK
Γ
KMK
Γ
(a)(b)
M K
Γ
qq
α=0α>0
( )
q
ω
FIG. 3: (a)Phonon spectrum for the undistorted kagome lat-
tice. Dashed lines depict frequencies at k′= 0 and full lines
at k′> 0. The inset shows the Brillouin zone with symme-
try 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 ω∗∼
a plateau beginning at q ≈ q∗∼
l∗= 1/q∗. (b) Phonon spectrum for α > 0 and k′= 0. The
plateau along ΓM defines ωα ∼
qα ∼ ωα defines a length lα ∼ 1/|sinα|.
√k′on
√k′defining a length scale
√k|sinα| and its onset at
lectively” jammed in the language of references [27, 33],
but because it is not rigid with respect to changing the
unit cell size, it is not strictly jammed.
IV.MODE COUNTING AND STATES OF SELF
STRESS
To understand the origin of the differences in the zero-
mode count for different boundary conditions, we turn to
an elegant formulation [2] of the Maxwell rule that takes
into account the existence of redundant bonds (i.e., bonds
whose removal does not increase the number of floppy
modes [4]) and states in which springs can be under states
of self-stress. Consider a ring network in two dimensions
shown in Fig. 4 with N = 4 nodes and Nb= 4 springs
with three springs of length a and one spring of length
b. The Maxwell count yields N0= 4 = 3+1 zero modes:
two rigid translations, one rigid rotation, and one internal
floppy mode – all of which are “finite-amplitude” modes
with zero energy even for finite-amplitude displacements.
When b = 3a, the Maxwell rule breaks down. In the zero-
energy configuration, the long spring and the three short
ones are colinear, and a prestressed state in which the b-
spring is under compression and the three a-springs are
under tension (or vice versa) but the total force on each
node remains zero becomes possible. This is called a state
of self-stress. The system still has three finite amplitude
zero modes corresponding to arbitrary rigid translations
and rotations, but the finite-amplitude floppy mode has
disappeared. In the absence of prestress, it is replaced
by two “infinitesimal” floppy modes of displacements of
the two internal nodes perpendicular of the now linear
network. In the presence of prestress, these two modes
have a frequency proportional to the square root of the
tension in the springs. Thus, the system now has one
state of self stress and one extra zero mode in the absence

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4
of prestress, implying N0= 2N −NB+S, where S is the
number of states of self stress.
This simple count is more generally valid as can be
shown with the aid of the equilibrium and compatibility
matrices [2], denoted, respectively, as H and C ≡ HT. H
relates the vector t of NB spring tensions to the vector
f of dN forces at nodes via H · t = f, and C relates
the the vector d of dN node displacements to the vector
e of NB spring stretches via C · d = e. The dynamical
matrix determining the phonon spectrum is D = kH·HT.
Vectors t0in the null space of H, (H · t0= 0), describe
states of self-stress whereas vectors d0in the null space of
C represent displacements with no stretch e, i.e., modes
of zero energy. Thus the nullspace dimensions of H and
C are, respectively, S and N0. The rank-nullity theorem
of linear algebra [34] states that the rank r of a matrix
plus the dimension of its null space equals its column
number. Since the rank of a matrix and its transpose
are equal, the H and C matrices, respectively, yield the
relations r + S = NBand r + N0= dN, implying N0=
dN − NB+ S.
have z = 2d exactly, and the Maxwell rule yields N0=
0: there should be no zero modes at all. But we have
just seen that both the square and undistorted kagome
lattices under PBCs have of order
calculated from the dynamical matrix, which, because it
is derived from a harmonic theory, does not distinguish
between infinitesimal and finite-amplitude zero modes.
Thus, in order for there to be zero modes, there must be
states of self-stress, in fact one state of self-stress for each
zero mode.
In the square lattice under FBCs, N = NxNy and
NB= 2NxNy−Nx−Ny, there are no states of self stress,
and N0 = Nx+ Ny zero modes depicted in Fig. 1(b).
Under PBCs, the dimension of the nullspace of H is
S = Nx+ Ny, and there are also N0 = S = Nx+ Ny
zero modes that are identical to those under FBCs. We
have already seen that there are N0= 2(Nx+ Ny) − 1
zero modes in the free undistorted kagome lattice. Di-
rect evaluations [23] (See Text S1) of the dimension of
the null spaces of H and C for the undistorted kagome
lattice with PBCs yields S = N0= 3Nxwhen Nx= Ny.
The zero modes under PBCs are identical to those un-
der FBCs except that the 2Nx−1 modes associated with
lines parallel to e2under FBCs get reduced to Nxmodes
because of the identification of apposite sides of the lat-
tice required by the PBCs as shown in Fig. 1(a). Thus
the modes of both the square and kagome lattices do
not depend strongly on whether FCBs or PBCs are ap-
plied. Under RBC’s, however, the floppy modes of both
disappear. The situation for the twisted kagome lattice
is different. There are still 2(Nx+ Ny) − 1 zero modes
under FBCs, but there are only two states of self stress
under PBCs and thus only N0= S = 2 zero modes, as
a direct evaluation of the null spaces of H and C verifies
(See Appendix for details), in agreement with the results
obtained via direct evaluation of the eigenvalues of the
dynamical matrix [32, 35]. All of the floppy modes under
Under PBCs, locally isostatic lattices
√N zero modes as
a
a
b
b
(a)(b)
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.
FBCs have disappeared.
V. EFFECTIVE THEORY AND EDGE MODES
An effective long-wavelength energy Eeff for the low-
energy acoustic phonons and nearly floppy distortions
provides insight into the nature of the modes of the
twisted kagome lattice. The variables in this theory are
the vector displacement field u(x) of nodes at undistorted
positions x and the scalar field φ(x) describing nearly
floppy distortions within a unit cell. The detailed form
of Eeff depends on which three lattice sites are assigned
to a unit cell. Figure 1(c) depicts the lattice distortion
φ for the nearly floppy mode at Γ (with energy propor-
tional to |sinα|2) along with a particular representation
of a unit cell, consisting of a central asymmetric hexagon
and two equilateral triangles, with 8 sites on its bound-
ary. If sites 1, 2, and 3 are assigned to the unit cell,
then the distortion φ involves no rotations of these sites
relative to their center of mass, and the harmonic limit
of Eeff depends only on the symmetrized and linearized
strain uij= (∂iuj+ ∂jui)/2 and on φ:
E =
1
2
?
d2x
?
2µ˜ u2
ij
+K(φ + ξuii)2+ V (∂iφ)2− WΓijkuij∂kφ
?
, (1)
where ˜ uij = uij −1
stain tensor, µ =
acscα/(2√3), W =√3k/4 + O(α2), and V =√3k/8 +
O(α2). The last term in which Γijkis a third-rank ten-
sor, whose only non-vanishing components are Γxxx=
−Γxyy= −Γyyx= −Γxyx= 1, is invariant under op-
erations of the group C6v but not under arbitrary rota-
tions. The Kξφuiiterm is the only one that reflects the
C3v (rather than C6v) symmetry of the lattice. There
are several comments to make about this energy. The
gauge-like coupling in which the isotropic strain uiiap-
pears only in the combination (φ + ξuii)2guarantees
that the bulk modulus vanishes: φ will simply relax to
−ξuii to reduce to zero the energy of any state with
nonvanishing uii. The coefficient K can be calculated
directly from the observation that under φ alone, the
length of every spring changes by δa = −√3φsinα, and
this length change is reversed by a homogenous volume
change uii = δAα/Aα = −2δa/a. In the α → 0 limit,
2δijukk is the symmetric-traceless
√3k/8, K = 3√3tan2α/a2, ξ =

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5
K → 0, and the energy reduces to that of an isotropic
solid with bulk modulus B0 = limα→0Kξ2=
the V and W terms, which are higher order in gradients,
are ignored. The W term gives rise to a term, singular in
gradients of u, when φ is integrated out that is respon-
sible for the deviations of the finite-wavenumber elastic
energy from isotropy. At small α, the length scale lαap-
pears in several places in this energy: in the length ξ and
in the ratios
?µ/K,
scales much larger than lα, the V and W terms can be
ignored, and φ relaxes to −ξuii leaving only the shear
elastic energy of an elastic solid proportional µ˜ u2
length scales shorter than lα, φ deviates from −ξuiiand
contributes significantly to the form of the energy spec-
trum. If 1, 2∗and 3∗in Fig. 1(d) are assigned to the unit
cell, then φ involves rotations relative to the lattice axes,
and the energy develops a Cosserat-like form [36, 37] that
is a function of φ − a(∇ × u)z/2 rather than φ.
The modes of our elastic energy in the long-wavelength
limit (qlα ≪ 1) are simply those of an elastic medium
with B = 0. In this limit, there are transverse and lon-
gitudinal bulk sound modes with equal sound velocities
cT =
?µ/ρ = (a/2)?k/m and cL=
where m is the particle mass at each node and ρ is the
mass density. In addition there are Rayleigh surface
waves [28] in which there is a single decay length (rather
than the two at B > 0), and displacements are propor-
tional to e−qyycos[qxx] with qy = qx for a semi-infinite
sample in the right half plane so that the penetration
depth into the interior is 1/qx. These waves have zero
frequency in two dimensions when B = 0, and they do
not appear in the spectrum with PBCs. Thus this sim-
ple continuum limit provides us with an explanation for
the difference between the spectrum of the free and pe-
riodic twisted kagome lattices. Under FBCs, there are
zero-frequency surface modes not present under PBCs.
Further insight into how boundary conditions affect
spectrum follows from the observation that the contin-
uum elastic theory with B = 0 depends only on ˜ uij.
The metric tensor gij(x) of the distorted lattice is re-
lated to the strain uij(x) via the simple relation gij(x) =
δij+2uij(x); and ˜ uij= [gij(x) −1
zero for gij= δij, is invariant, and thus remains equal to
zero, under conformal transformations that take the met-
ric tensor from its reference form δij to h(x)δij for any
continuous function h(x). The zero modes of the theory
thus correspond simply to conformal transformations,
which in two dimensions are best represented by the com-
plex position and displacement variables z = x + iy and
w(z) = ux(z) + iuy(z). All conformal transformations
are described by an analytic displacement field w(z).
Since by Cauchy’s theorem, analytic functions in the in-
terior of a domain are determined entirely by their val-
ues on the domain’s boundary (the “holographic” prop-
erty [38]), the zero modes of a given sample are simply
those analytic functions that satisfy its boundary con-
ditions. For example, a disc with fixed edges (u = 0)
has no zero modes because the only analytic function
√3k/4 if
?V/K, and
?W/K. At length
ij. At
?(B + µ)/ρ → cT,
2δijgkk(x)]/2, which is
satisfying this FBC is the trivial one w(z) = 0; but
a disc with free edges (stress and thus strain equal to
zero) has one zero mode for each of the analytic functions
w(z) = anznfor integer n ≥ 0. The boundary conditions
limx→∞u(x,y) = 0 and u(x,y) = u(x+L,y) on a semi-
infinite cylinder with axis along x are satisfied by the
function w(z) = eiqxz= eiqxxe−qxywhen qx = 2nπ/L,
where n is an integer. This solution is identical to that
for classical Rayleigh waves on the same cylinder. Like
the Rayleigh theory, the conformal theory puts no restric-
tion on the value of n (or equivalently qx). Both theories
break down, however, at qx= qc≈ min(l−1
which the full lattice theory, which yields a complex value
of qy= q′
Figure 5(a) shows an example of a surface wave. At the
bottom of this figure, uy(x) is an almost perfect sinusoid.
As y decreases toward the surface, the amplitude grows,
and in this picture reaches the nonlinear regime by the
time the surface at y = 0 is reached. Figure 5(b) plots
q′
evaluation and by an analytic transfer matrix procedure
[39] for different values of α (Text S1). The Rayleigh
limit q′
y= qxis reached for all α as qx→ 0. Interestingly
the Rayleigh limit remains a good approximation up to
values of qx that increase with increasing α. The inset
to Fig. 5, plots q′
that in the limit α → 0 (lα/a → ∞), q′
independent scaling law of the form q′
full complex qy obeys a similar equation. This type of
behavior is familiar in critical phenomena where scaling
occurs when correlation lengths become much larger than
microscopic lengths. The function f(η) approaches η as
η → 0 and asymptotes to 4/3 for η → ∞. Thus for
qxlα ≪ 1, q′
As α increases, lα/a is no longer much larger than one,
and deviations from the scaling law result. The situation
for surfaces along different directions (e.g., along x =
0 rather than y = 0) is more complicated and will be
treated in a future publication [40].
α,a−1) beyond
y+ iq′′
y, is needed.
yas a function of qxobtained both by direct numerical
ylαas a function of η = qxlαand shows
yobeys an α-
y= l−1
αf(qxlα). The
y= qx and for qxlα ≫ 1, q′
y= (4/3)l−1
α.
VI.OTHER LATTICES AND RELATION TO
JAMMING
The C3vtwisted kagome lattice is the simplest of many
lattices that can be formed from the kagome and other
periodic isostatic lattices. Figures 6 (a) and (b) show
two other examples of isostatic lattices constructed from
the kagome lattice. Most intriguing is the lattice with
pgg symmetry. Its geometry has uniaxial symmetry, yet
its long-wavelength elastic energy is identical to that of
the C3vtwisted kagome lattice, i.e, it is isotropic with a
vanishing bulk modulus, and its mode structure near q =
0 is isotropic as shown in Fig. 6 (c). Thus, this system
loses long-wavelength zero-frequency bulk modes of the
undistorted kagome lattice to surface modes. However, at
large wavenumber, lattice anisotropy becomes apparent,
and (infinitesimal) floppy bulk modes appear. Thus in