Surface phonons, elastic response, and conformalinvariance in twisted kagome latticesKai Suna, Anton Souslovb,1, Xiaoming Maob, and T. C. Lubenskyb,2

aCondensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, MD 20742; andbDepartment of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104

Edited by Nigel Goldenfeld, University of Illinois at Urbana-Champaign, Urbana, IL, and approved May 29, 2012 (received for review December 7, 2011)

Model lattices consisting of balls connected by central-force springsprovide much of our understanding of mechanical response andphonon structure of real materials. Their stability depends criticallyon their coordination number z. d-dimensional lattices with z ¼ 2dare at the threshold of mechanical stability and are isostatic. Lat-tices with z < 2d exhibit zero-frequency “floppy” modes that pro-vide avenues for lattice collapse. The physics of systems as diverseas architectural structures, network glasses, randomly packedspheres, and biopolymer networks is strongly influenced by a near-by isostatic lattice. We explore elasticity and phonons of a specialclass of two-dimensional isostatic lattices constructed by distortingthe kagome lattice. We show that the phonon structure of theselattices, characterized by vanishing bulk moduli and thus negativePoisson ratios (equivalently, auxetic elasticity), depends sensitivelyon boundary conditions and on the nature of the kagome distor-tions. We construct lattices that under free boundary conditionsexhibit surface floppy modes only or a combination of both surfaceand bulk floppy modes; and we show that bulk floppy modes pre-sent under free boundary conditions are also present under peri-odic boundary conditions but that surface modes are not. In thelong-wavelength limit, the elastic theory of all these lattices is aconformally 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 resultsin relation to recent work on jammed systems. Our results highlightthe importance of network architecture in determining floppy-mode structure.

auxetic response ∣ self stress ∣ conformal field theory ∣ Cosserat elasticity

Networks of balls and springs or frames of nodes connected bycompressible struts provide realistic models for physical sys-

tems from bridges to condensed solids. Their elastic propertiesdepend on their coordination number z—the average numberof nodes each node is connected to. If z is small enough, the net-works have deformation modes of zero energy—they are floppy(1–6). As z is increased, a critical value, zc, is reached at whichsprings provide just enough constraints that the system has nozero-energy floppy modes (7) (or mechanisms, ref. 2, in the en-gineering literature), and the system is isostatic. For z > zc, net-works with appropriate geometry (see below) are rigid in thesense that they have no zero modes other than those associatedwith trivial rigid translations and rotations. If a network with z >zc is homogeneously distributed in space, it can be viewed as anelastic solid whose long-wavelength mechanical properties aredescribed by a continuum elastic energy with nonvanishing elasticmoduli (7–11). The phenomenon of rigidity percolation (12, 13)whereby a sample spanning rigid cluster develops upon the addi-tion of springs is one version of this floppy-to-rigid transition. Thecoordination numbers of whole classes of systems, including en-gineering structures (14, 15) (bridges and buildings), randomlypacked spheres near jamming (10, 11, 16, 17), network glasses(7, 8), cristobalites (18), zeolites (19, 20), and biopolymer net-works (21–24) are close enough to zc that their elasticity andmode structure is strongly influenced by those of the isostaticlattice.

Though the isostatic point always separates rigid from floppybehavior, the properties of isostatic lattices are not universal;rather they depend on lattice architecture. Here we explorethe unusual properties of a particular class of periodic isostaticlattices derived from the two-dimensional kagome lattice by ri-gidly rotating triangles through an angle α without changing bondlengths as shown in Fig. 1. The bulk modulus B of these lattices isrigorously zero for all α ≠ 0. As a result, their Poisson ratio ac-quires its limit value of −1; when stretched in one direction, theyexpand by an equal amount in the orthogonal direction: They aremaximally auxetic (25–28). These modes represent collapse path-ways (29, 30) of the kagome lattice. Modes of isostatic systems aregenerally very sensitive to boundary conditions (9, 31, 32), but thedegree of sensitivity depends on the details of lattice structure.For reasons we will discuss more fully below, modes of the squarelattice, which is isostatic, are in fact insensitive to changes fromfree boundary conditions (FBCs) to periodic boundary conditions(PBCs), whereas those of the undistorted kagome lattice are onlymildly so. The modes of both, however, change significantly whenrigid boundary conditions (RBCs) are applied. We show herethat, in all families of the twisted kagome lattice, modes dependsensitively on whether FBCs, PCBs, or RBCs are applied: Finitelattices with free boundaries have floppy surface modes that arenot present in their periodic or rigid spectrum or in that of finiteundistorted kagome lattices. In the long-wavelength limit, thesurface floppy modes, which are present in any 2d material withB ¼ 0, reduce to surface Rayleigh waves (33) described by a con-formally invariant energy whose analytic eigenfunctions are fullydetermined by boundary conditions. At shorter wavelengths, thesurface waves become sensitive to lattice structure and remainconfined to within a distance of the surface that diverges as theundistorted kagome lattice is approached. In the simplest twistedkagome lattice, all floppy modes are surface modes, but in morecomplicated lattices, including ones with uniaxial symmetry thatwe construct, there are both surface and bulk floppy modes.

Arguments due to Maxwell (1) provide a criterion for networkstability: Networks in d dimensions consisting of N nodes, eachconnected with central-force springs to an average of z neighbors,have N0 ¼ dN − 1

2zN zero-energy modes when z < 2d (in the

absence of redundant bonds—see below). Of these, a number,Ntr, which depends on boundary conditions, are trivial rigid trans-lations and rotations, and the remainder are floppy modes of in-ternal structural rearrangement. Under FBCs and PBCs, Ntrequals dðdþ 1Þ∕2 and d, respectively. With increasing z, mechan-ical stability is reached at the isostatic point at which N0 ¼ Ntr.

Author contributions: K.S., A.S., and T.C.L. designed research; K.S., A.S., X.M., and T.C.L.performed research; and K.S., A.S., X.M., and T.C.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

See Commentary on page 12266.1Present address: School of Physics, Georgia Institute of Technology, Atlanta, GA 30332.2To whom correspondence should be addressed. E-mail: tom@physics.upenn.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1119941109/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1119941109 PNAS ∣ July 31, 2012 ∣ vol. 109 ∣ no. 31 ∣ 12369–12374

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The Maxwell argument is a global one; it does not provide infor-mation about the nature of the floppy modes and does not dis-tinguish between bulk or surface modes.

Kagome Zero Modes and ElasticityThe kagome lattice of central-force springs shown in Fig. 1A is oneof many locally isostatic lattices, including the familiar square lat-tice in two dimensions (Fig. 1B) and the cubic and pyrochlore lat-tices in three dimensions, with exactly z ¼ 2d nearest-neighbor(NN) bonds connected to each site not at a boundary. UnderPBCs, there are no boundaries, and every site has exactly 2d neigh-bors. Finite,N-site sections of these lattices have surface sites withfewer than 2d neighbors and of order

ffiffiffiffiffiN

pzero modes. The free

kagome lattice with Nx and Ny unit cells along its sides (Fig. 1A)has N ¼ 3NxNy sites, NB ¼ 6NxNy − 2ðNx þNyÞ þ 1 bonds,and N0 ¼ 2ðNx þNyÞ − 1 zero modes, all but three of whichare floppy modes. These modes, depicted in Fig. 1A, consist ofcoordinated counterrotations of pairs of triangles along the sym-metry axes e1, e2, and e3 of the lattice. There are Nx modes asso-ciated with lines parallel to e1, Ny associated with lines parallel toe3, and Nx þNy − 1 modes associated with lines parallel to e2.

In spite of the large number of floppy modes in the kagomelattice, its longitudinal and shear Lamé coefficients, λ and μ,and its bulk modulus B ¼ λþ μ are nonzero and proportionalto the NN spring constant k: λ ¼ μ ¼ ffiffiffi

3p

k∕8 and B ¼ λþ μ ¼ffiffiffi3

pk∕4. The zero modes of this lattice can be used to generate

an infinite number of distorted lattices with unstretched springsand thus zero energy (30). We consider only periodic lattices, thesimplest of which are the twisted kagome lattices obtained by ro-tating triangles of the kagome unit cell through an angle α asshown in Fig. 1 C and D (30, 34). These lattices have C3v ratherthanC6v symmetry and, like the undistorted kagome lattice, threesites per unit cell. As Fig. 1D shows, the lattice constant of theselattices is aL ¼ 2a cos α, and their areaAα decreases as cos2 α as αincreases. The maximum value that α can achieve without bondcrossings is π∕3, so that the maximum relative area change isAπ∕3∕A0 ¼ 1∕4. Because all springs maintain their rest length,there is no energy cost for changing α and, as a result, B is zerofor every α ≠ 0, whereas the shear modulus μ ¼ ffiffiffi

3p

k∕8 remainsnonzero and unchanged. Thus, the Poisson ratio σ ¼ ðB − μÞ∕ðBþ μÞ attains its smallest possible value of −1. For anyα ≠ 0, the addition of next-nearest-neighbor (NNN) springs, withspring constant k 0 (or of bending forces between springs) stabi-lizes zero-frequency modes and increases B and σ. Nevertheless,for sufficiently small k 0, σ remains negative. Fig. 2 shows the re-gion in the k 0 − α plane with negative σ.

Kagome Phonon SpectrumWe now turn to the linearized phonon spectrum of the kagomeand twisted kagome lattices subjected to PBCs. These conditionsrequire displacements at opposite ends of the sample to be iden-tical and thus prohibit distortions of the shape and size of the unitcell and rotations but not uniform translations, leaving two ratherthan three trivial zero modes. The spectrum (35) of the three low-est frequency modes along symmetry directions of the undis-torted kagome lattice with and without NNN springs is shownin Fig. 3A. When k 0 ¼ 0, there is a floppy mode for each wave-number q ≠ 0 running along the entire length of the three sym-metry-equivalent straight lines running from M to Γ to M in theBrillouin zone (see Fig. 3, Inset). When Nx ¼ Ny, there are ex-actlyNx − 1 wavenumbers with q ≠ 0 along each of these lines fora total of 3ðNx − 1Þ floppy modes. In addition, there are threezero modes at q ¼ 0 corresponding to two rigid translationsand one floppy mode that changes unit cell area at second butnot first order in displacements, yielding a total of 3Nx zeromodes rather than the 4Nx − 1modes expected from theMaxwellcount under FBCs. This discrepancy is our first indication of theimportance of boundary conditions. The addition of NNN springsendows the floppy modes at k 0 ¼ 0 with a characteristic fre-quency ω� ∼

ffiffiffiffiffik 0p

and causes them to hybridize with the acousticphonon modes (Fig. 3A) (35). The result is an isotropic phononspectrum up to wavenumber q� ¼ 1∕l� ∼

ffiffiffiffiffik 0p

and gaps at Γ andM of order ω�. Remarkably, at nonzero α and k 0 ¼ 0, the modestructure is almost identical to that at α ¼ 0 and k 0 > 0 with char-acteristic frequency ωα ∼

ffiffiffik

p j sin αj and length lα ∼ 1∕ωα. In otherwords, twisting the kagome lattice through an angle α has essen-tially the same effect on the spectrum as adding NNN springs withspring constant j sin αj2k. Thus, under PBCs, the twisted kagome

A

C

D

B

Fig. 1. (A) Section of a kagome lattice with Nx ¼ Ny ¼ 4 and Nc ¼ NxNy

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 flop-py mode. Next-nearest-neighbor bonds are shown as dotted lines in the low-er left hexagon. The vectors e1, e2, and e3 indicate symmetry directions of thelattice. The numbers in the triangles indicate those that twist together underPBCs in zero modes along the three symmetry direction. Note that there areonly four of these modes. (B) Section of a square lattice depicting a floppymode in which all sites along a line are displaced uniformly. (C) Twisted ka-gome lattice, with lattice constant aL ¼ 2a cos α, derived from the undis-torted lattice by rigidly rotating triangles through an angle α. A unit cell,bounded by dashed lines, is shown in violet. Arrows depict site displacementsfor the zone-center (i.e., zero wavenumber) ϕ mode which has zero (non-zero) frequency under free (periodic) boundary conditions. Sites 1, 2, and3 undergo no collective rotation about their center of mass, whereas sites1, 2�, and 3� do. (D) Superposed snapshots of the twisted lattice showingdecreasing areas with increasing α.

Fig. 2. Phase diagram in the α − k 0 plane showing region with negativePoisson ratio σ.

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lattice has no zero modes other than the trivial ones: It is“collectively” jammed in the language of refs. 32 and 36, but be-cause it is not rigid with respect to changing the unit cell size, it isnot strictly jammed.

Mode Counting and States of Self-StressTo understand the origin of the differences in the zero-modecount for different boundary conditions, we turn to an elegantformulation (2) of the Maxwell rule that takes into accountthe existence of redundant bonds (i.e., bonds whose removal doesnot increase the number of floppy modes; ref. 13) and states inwhich springs can be under states of self-stress (3–6). Consider aring network in two dimensions shown in Fig. 4 withN ¼ 4 nodesand Nb ¼ 4 springs with three springs of length a and one springof length b. The Maxwell count yields N0 ¼ 4 ¼ 3þ 1 zeromodes: two rigid translations, one rigid rotation, and one internalfloppy mode—all of which are “finite-amplitude” modes withzero energy even for finite-amplitude displacements. Whenb ¼ 3a, the Maxwell rule breaks down. In the zero-energy con-figuration, the long spring and the three short ones are colinear,and a prestressed state in which the b spring is under compressionand the three a springs are under tension (or vice versa) but thetotal force on each node remains zero becomes possible. This iscalled a state of self-stress. The system still has three finite-am-plitude zero modes corresponding to arbitrary rigid translationsand rotations, but the finite-amplitude floppy mode has disap-peared. In the absence of prestress, it is replaced by two “infini-tesimal” floppy modes of displacements of the two internal nodesperpendicular of the now linear network. In the presence of pres-tress, these two modes have a frequency proportional to thesquare root of the tension in the springs. Thus, the systemnow has one state of self-stress and one extra zero mode inthe absence of prestress, implying N0 ¼ 2N −NB þ S, whereS is the number of states of self-stress.

This simple count is more generally valid, as can be shown withthe aid of the equilibrium and compatibility matrices (2), de-noted, respectively, as H and C ≡ HT . H relates the vector t ofNB spring tensions to the vector f of dN forces at nodes viaH · t ¼ f , and C relates the vector d of dN node displacementsto the vector e of NB spring stretches via C · d ¼ e. The dynami-cal matrix determining the phonon spectrum is D ¼ kH · HT .

Vectors t0 in the null space of H, (H · t0 ¼ 0), describe statesof self-stress, whereas vectors d0 in the null space of C representdisplacements with no stretch e—i.e., modes of zero energy. Thusthe null-space dimensions of H and C are, respectively, S andN0.The rank-nullity theorem of linear algebra (37) states that therank r of a matrix plus the dimension of its null space equalsits column number. Because the rank of a matrix and its transposeare equal, the H and C matrices, respectively, yield the relationsr þ S ¼ NB and r þN0 ¼ dN, implying N0 ¼ dN −NB þ S.Under PBCs, locally isostatic lattices have z ¼ 2d exactly, andthe Maxwell rule yields N0 ¼ 0: There should be no zero modesat all. But we have just seen that both the square and undistortedkagome lattices under PBCs have of order

ffiffiffiffiffiN

pzero modes as

calculated from the dynamical matrix, which, because it is derivedfrom a harmonic theory, does not distinguish between infinitesi-mal and finite-amplitude zero modes. Thus, in order for there tobe zero modes, there must be states of self-stress, in fact, one stateof self-stress for each zero mode.

In the square lattice under FBCs, N ¼ NxNy and NB ¼2NxNy −Nx −Ny, there are no states of self-stress, and N0 ¼Nx þNy zero modes depicted in Fig. 1B. Under PBCs, the di-mension of the null space of H is S ¼ Nx þNy, and there arealso N0 ¼ S ¼ Nx þNy zero modes that are identical to thoseunder FBCs. We have already seen that there are N0 ¼ 2ðNx þNyÞ − 1 zero modes in the free undistorted kagome lattice. Directevaluations (29) (see SI Text) of the dimension of the null spacesof H and C for the undistorted kagome lattice with PBCs yieldsS ¼ N0 ¼ 3Nx when Nx ¼ Ny. The zero modes under PBCs areidentical to those under FBCs except that the 2Nx − 1 modes as-sociated with lines parallel to e2 under FBCs get reduced to Nxmodes because of the identification of apposite sides of the latticerequired by the PBCs, as shown in Fig. 1A. Thus the modes ofboth the square and kagome lattices do not depend stronglyon whether FBCs or PBCs are applied. Under RBCs, however,the floppy modes of both disappear. The situation for the twistedkagome lattice is different. There are still 2ðNx þNyÞ − 1 zeromodes under FBCs, but there are only two states of self-stressunder PBCs and thus only N0 ¼ S ¼ 2 zero modes, as a directevaluation of the null spaces of H and C verifies (SI Text), inagreement with the results obtained via direct evaluation ofthe eigenvalues of the dynamical matrix (35, 38). All of the floppymodes under FBCs have disappeared.

Effective Theory and Edge ModesAn effective long-wavelength energy Eeff for the low-energyacoustic phonons and nearly floppy distortions provides insightinto the nature of the modes of the twisted kagome lattice. Thevariables in this theory are the vector displacement field uðxÞ ofnodes at undistorted positions x and the scalar field ϕðxÞ describ-ing nearly floppy distortions within a unit cell. The detailed formof Eeff depends on which three lattice sites are assigned to a unitcell. Fig. 1C depicts the lattice distortion ϕ for the nearly floppymode at Γ (with energy proportional to j sin αj2) along with a par-ticular representation of a unit cell, consisting of a central asym-metric hexagon and two equilateral triangles, with eight sites onits boundary. If sites 1, 2, and 3 are assigned to the unit cell, thenthe distortion ϕ involves no rotations of these sites relative totheir center of mass, and the harmonic limit of Eeff depends onlyon the symmetrized and linearized strain uij ¼ ð∂iuj þ ∂juiÞ∕2and on ϕ:

E ¼ 1

2

Zd2x½2μ ~u2

ij þKðϕþ ξuiiÞ2 þ V ð∂iϕÞ2 −WΓ ijkuij∂kϕ�;[1]

where ~uij ¼ uij − 12δijukk is the symmetric-traceless stain

tensor, μ ¼ ffiffiffi3

pk∕8,K ¼ 3

ffiffiffi3

ptan2 α∕a2, ξ ¼ a csc α∕ð2 ffiffiffi

3p Þ,W ¼ffiffiffi

3p

k∕4þOðα2Þ, and V ¼ ffiffiffi3

pk∕8þOðα2Þ. The last term in

A B

Fig. 3. (A) Phonon spectrum for the undistorted kagome lattice. Dashedlines depict frequencies at k 0 ¼ 0 and full lines at k 0 > 0. The inset showsthe Brillouin zone with symmetry points Γ, M, and K. Note the line of zeromodes along ΓM when k 0 ¼ 0, all of which develop nonzero frequencies forwavenumber q > 0when k 0 > 0 reaching ω� ∼

ffiffiffiffiffik 0p

on a plateau beginning atq ≈ q� ∼

ffiffiffiffiffik 0p

defining a length scale l � ¼ 1∕q�. (B) Phonon spectrum for α > 0

and k 0 ¼ 0. The plateau along ΓM defines ωα ∼ffiffiffik

pj sin αj and its onset at qα ∼

ωα defines a length lα ∼ 1∕j sin αj.

A B

Fig. 4. (A) Ring network with b > 3a showing internal floppy mode.(B) Ring-network with b ¼ 3a showing one of the two infinitesimal modes.

Sun et al. PNAS ∣ July 31, 2012 ∣ vol. 109 ∣ no. 31 ∣ 12371

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which Γ ijk is a third-rank tensor, whose only nonvanishing com-ponents are Γxxx ¼ −Γxyy ¼ −Γyyx ¼ −Γxyx ¼ 1, is invariant underoperations of the group C6v but not under arbitrary rotations.The Kξϕuii term is the only one that reflects the C3v (rather thanC6v) symmetry of the lattice. There are several comments tomake about this energy. The gauge-like coupling in which the iso-tropic strain uii appears only in the combination ðϕþ ξuiiÞ2 guar-antees that the bulk modulus vanishes: ϕ will simply relax to −ξuiito reduce to zero the energy of any state with nonvanishing uii.The coefficient K can be calculated directly from the observationthat, under ϕ alone, the length of every spring changes byδa ¼ −

ffiffiffi3

pϕ sin α, and this length change is reversed by a homo-

genous volume change uii ¼ δAα∕Aα ¼ −2δa∕a. In the α → 0limit, K → 0, and the energy reduces to that of an isotropic solidwith bulk modulus B0 ¼ limα→0Kξ2 ¼ ffiffiffi

3p

k∕4 if the V and Wterms, which are higher order in gradients, are ignored. TheW term gives rise to a term, singular in gradients of u, whenϕ is integrated out that is responsible for the deviations of thefinite-wavenumber elastic energy from isotropy. At small α, thelength scale lα appears in several places in this energy: in thelength ξ and in the ratios

ffiffiffiffiffiffiffiffiffiffiμ∕K

p,

ffiffiffiffiffiffiffiffiffiffiffiV∕K

p, and

ffiffiffiffiffiffiffiffiffiffiffiffiW∕K

p. At length

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 anelastic solid proportional μ ~u2

ij. At length scales shorter than lα, ϕdeviates from −ξuii and contributes significantly to the form ofthe energy spectrum. If 1, 2�, and 3� in Fig. 1D are assignedto the unit cell, then ϕ involves rotations relative to the latticeaxes, and the energy develops a Cosserat-like form (39, 40) thatis a function of ϕ − að∇ × uÞz∕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 ¼ 0. Inthis limit, there are transverse and longitudinal bulk sound modeswith equal sound velocities cT ¼ ffiffiffiffiffiffiffiffiffi

μ∕ρp ¼ ða∕2Þ ffiffiffiffiffiffiffiffiffiffi

k∕mp

andcL ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðBþ μÞ∕ρp

→ cT , where m is the particle mass at eachnode and ρ is the mass density. In addition there are Rayleighsurface waves (33) in which there is a single decay length (ratherthan the two at B > 0), and displacements are proportional toe−qyy cos½qxx� with qy ¼ qx for a semiinfinite sample in the upperhalf-plane so that the penetration depth into the interior is 1∕qx.These waves have zero frequency in two dimensions when B ¼ 0,and they do not appear in the spectrum with PBCs. Thus this sim-ple continuum limit provides us with an explanation for the dif-ference between the spectrum of the free and periodic twistedkagome lattices. Under FBCs, there are zero-frequency surfacemodes not present under PBCs.

Further insight into how boundary conditions affect spectrumfollows from the observation that the continuum elastic theorywith B ¼ 0 depends only on ~uij. The metric tensor gijðxÞ ofthe distorted lattice is related to the strain uijðxÞ via the simplerelation gijðxÞ ¼ δij þ 2uijðxÞ; and ~uij ¼ ½gijðxÞ − 1

2δijgkkðxÞ�∕2,

which is zero for gij ¼ δij, is invariant, and thus remains equalto zero, under conformal transformations that take the metrictensor from its reference form δij to hðxÞδij for any continuousfunction hðxÞ. The zero modes of the theory thus correspond sim-ply to conformal transformations, which in two dimensions arebest represented by the complex position and displacement vari-ables z ¼ xþ iy and wðzÞ ¼ uxðzÞ þ iuyðzÞ. All conformal trans-formations are described by an analytic displacement fieldwðzÞ. Because, by Cauchy’s theorem, analytic functions in the in-terior of a domain are determined entirely by their values on thedomain’s boundary (the “holographic” property; ref. 41), the zeromodes of a given sample are simply those analytic functions thatsatisfy its boundary conditions. For example, a disc with fixededges (u ¼ 0) has no zero modes because the only analytic func-tion satisfying this FBC is the trivial one wðzÞ ¼ 0; but a disc withfree edges (stress and thus strain equal to zero) has one zeromode for each of the analytic functions wðzÞ ¼ anzn for integer

n ≥ 0. The boundary conditions limx→∞uðx; yÞ ¼ 0 and uðx; yÞ ¼uðxþ L; yÞ on a semiinfinite cylinder with axis along x are satis-fied by the function wðzÞ ¼ eiqxz ¼ eiqxxe−qxy when qx ¼ 2nπ∕L,where n is an integer. This solution is identical to that for classicalRayleigh waves on the same cylinder. Like the Rayleigh theory,the conformal theory puts no restriction on the value of n(or equivalently qx). Both theories break down, however, atqx ¼ qc ≈minðl−1α ; a−1Þ, beyond which the full lattice theory,which yields a complex value of qy ¼ q 0

y þ iq 0 0y , is needed.

Fig. 5A shows an example of a surface wave. At the bottom ofthis figure, uyðxÞ is an almost perfect sinusoid. As y decreases to-ward the surface, the amplitude grows, and in this picture reachesthe nonlinear regime by the time the surface at y ¼ 0 is reached.Fig. 5B plots q 0

y as a function of qx obtained both by direct nu-merical evaluation and by an analytic transfer matrix procedure(42) for different values of α (SI Text). The Rayleigh limit q 0

y ¼ qxis reached for all α as qx → 0. Interestingly, the Rayleigh limitremains a good approximation up to values of qx that increasewith increasing α. The inset to Fig. 5 plots q 0

y lα as a functionof η ¼ qxlα and shows that in the limit α → 0 (lα∕a → ∞),q 0y obeys an α-independent scaling law of the form q 0

y ¼l−1α f ðqxlαÞ. The full complex qy obeys a similar equation. This typeof behavior is familiar in critical phenomena where scaling occurswhen correlation lengths become much larger than microscopiclengths. The function f ðηÞ approaches η as η → 0 and asymptotesto 4∕3 for η → ∞. Thus for qxlα ≪ 1, q 0

y ¼ qx and for qxlα ≫ 1,q 0y ¼ ð4∕3Þl−1α . As α increases, lα∕a is no longer much larger than

one, and deviations from the scaling law result. The situation for

A

B

Fig. 5. (A) Lattice distortions for a surface wave on a cylinder, showing ex-ponential decay of the surface displacements into the bulk. This figure wasconstructed by specifying a small sinusoidal modulation on the bottomboundary and propagating lattice-site positions upward to the free bound-ary at the top under the constraint of constant lengths and periodic bound-ary conditions around the cylinder. Distortions near and at the top boundary,which have become nonlinear, are not described by our linearized treatment.(B) q 0

yaL as a function of qxaL for lattice Rayleigh surface waves for α ¼ π∕20,π∕10, 3π∕20, π∕5, π∕4, in order from bottom to top. Smooth curves are theanalytic results from a transfer matrix calculation, and dots are from directnumerical calculations. The dashed line is the continuum Rayleigh limitq 0y ¼ qx . Curves at smaller α break away from this curve at smaller values

of qy than do those at large α. At α ¼ π∕4, q 0x diverges at qyaL ¼ π. The inset

plots q 0y lα as a function of qxlα for different α. The lower curve in the inset

(black) is the α-independent scaling function of qy lα reached in the α → 0 lim-it. The other curves from top to bottom are for α ¼ π∕25, π∕12, π∕9, and π∕6(chosen to best present results rather than to match the curves in the mainfigure). Curves for α < π∕15 are essentially indistinguishable from the scalinglimit. The curve at α ¼ π∕6 stops because qy < π∕aL.

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surfaces along different directions (e.g., along x ¼ 0 rather thany ¼ 0) is more complicated and is beyond the scope of this paper.

Other Lattices and Relation to JammingThe C3v twisted kagome lattice is the simplest of many latticesthat can be formed from the kagome and other periodic isostaticlattices. Fig. 6 A and B shows two other examples of isostaticlattices constructed from the kagome lattice. Most intriguing isthe lattice with pgg symmetry. Its geometry has uniaxial symme-try, yet its long-wavelength elastic energy is identical to that of theC3v twisted kagome lattice (i.e., it is isotropic with a vanishingbulk modulus), and its mode structure near q ¼ 0 is isotropicas shown in Fig. 6C. Thus, this system loses long-wavelengthzero-frequency bulk modes of the undistorted kagome latticeto surface modes. However, at large wavenumber, lattice aniso-tropy becomes apparent, and (infinitesimal) floppy bulk modesappear. Thus in this and related systems, a fraction of the zeromodes under FBCs are bulk modes that are visible under PBCs,and a fraction are surface modes that are not.

Randomly packed spheres above the jamming transition withaverage coordination number z ¼ 2dþ Δz exhibit a characteristicfrequency ω� ∼ Δz and length l� ∼ ðΔzÞ−1 and a transition from aDebye-like (∼ ωd−1) to a flat density of states at ω ≈ ω� (43, 44).The square and kagome lattices with randomly added NNNsprings have the same properties (45, 46). A general “cutting”argument (9, 31) provides a procedure for perturbing away from

the isostatic limit and an explanation for these properties. How-ever, it only applies provided a finite fraction of the of orderLd−1

floppy modes of a sample with sides of length L cut from an iso-static lattice with PBCs are extended—i.e., have wave functionsthat extend across the sample rather than remaining localizedeither in the interior or at the surface the sample. Clearly thetwisted kagome lattice, whose floppy modes are all surfacemodes, violates this criterion; and indeed, the density of statesof the lattice with Δz ¼ 0 shows Debye behavior crossing overto a flat plateau at ω ≈ ωα. Adding NNN bonds gives rise to alength lc, which is approximately the parallel combination,ðl−1α þ l�−1Þ−1, of the lengths lα and l� arising, respectively, fromtwisting the lattice and adding NNN springs, and cross-over to theplateau at ωc ∼ l−1c . The pgg lattice in Fig. 6A, however, has bothextended and surface floppy modes, so its cross-over to a flat pla-teau occurs at ω ≈ ω� rather than at ωα or ωc.

Connections to Other SystemsOur study highlights the rich and remarkable variety of physicalproperties that isostatic systems can exhibit. Under FBCs, floppymodes can adopt a variety of forms, from all being extended to allbeing localized near surfaces to a mixture of the two. UnderPBCs, the presence of floppy modes depends on whether the lat-tice can or cannot support states of self-stress. When a lattice ex-hibits a large number of zero-energy edge modes, its mechanical/dynamical properties become extremely sensitive to boundaryconditions, much as do the electronic properties of the topologi-cal states of matter studied in quantum systems (47–50). Thezero-energy edge modes observed in our isostatic lattices are col-lective modes whose amplitudes decay exponentially from theedge with a finite decay length, in direct contrast to the verylocalized and trivial floppy modes arising from dangling bonds.We focused primarily on high-symmetry lattices derived fromthe kagome lattice, but the properties they exhibit (namely, a def-icit of floppy modes in the bulk and the existence of floppy surfacemodes) are shared by any two-dimensional system with a vanish-ing bulk modulus (or the equivalent in anisotropic systems).Three-dimensional analogs of the twisted kagome lattice canbe constructed by rotating tetrahedra in pyrochlore and zeolitelattices (19, 20) and in cristobalites (18). These lattices are ani-sotropic. With NN forces only, they exhibit a vanishing modulusfor compression applied in particular planes rather than isotro-pically, but we expect them to exhibit many of the properties thetwo-dimensional lattices exhibit. Finally, we note that Maxwell’sideas can be applied to spin systems such as the Heisenberg anti-ferromagnet on the kagome lattice (51, 52), and the possibility ofunusual edge states in them is intriguing.

ACKNOWLEDGMENTS. We are grateful for informative conversationswith Randall Kamien, Andea Liu, and S.D. Guest. This work was supportedin part by the National Science Foundation under Grants DMR-0804900,DMR-1104707 (to T.C.L. and X.M.), Materials Research Science and Engineer-ing Center DMR-0520020 (to T.C.L. and A.S.), and JQI-NSF-PFC (to K.S.).

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