Signatures of Topological Phonons in Superisostatic Lattices.

Soft topological surface phonons in idealized ball-and-spring lattices with coordination number z=2d in d dimensions become finite-frequency surface phonons in physically realizable superisostatic lattices with z>2d. We study these finite-frequency modes in model lattices with added next-nearest-neighbor springs or bending forces at nodes with an eye to signatures of the topological surface modes that are retained in the physical lattices. Our results apply to metamaterial lattices, prepared with modern printing techniques, that closely approach isostaticity.

Jamming as a Multicritical Point.

The discontinuous jump in the bulk modulus B at the jamming transition is a consequence of the formation of a critical contact network of spheres that resists compression. We introduce lattice models with underlying undercoordinated compression-resistant spring lattices to which next-nearest-neighbor springs can be added. In these models, the jamming transition emerges as a kind of multicritical point terminating a line of rigidity-percolation transitions. Replacing the undercoordinated lattices with the critical network at jamming yields a faithful description of jamming and its relation to rigidity percolation.

Topological Phonons and Weyl Lines in Three Dimensions.

Topological mechanics and phononics have recently emerged as an exciting field of study. Here we introduce and study generalizations of the three-dimensional pyrochlore lattice that have topologically protected edge states and Weyl lines in their bulk phonon spectra, which lead to zero surface modes that flip from one edge to the opposite as a function of surface wave number.

Topological boundary modes in jammed matter.

Granular matter at the jamming transition is poised on the brink of mechanical stability, and hence it is possible that these random systems have topologically protected surface phonons. Studying two model systems for jammed matter, we find states that exhibit distinct mechanical topological classes, protected surface modes, and ubiquitous Weyl points. The detailed statistics of the boundary modes shed surprising light on the properties of the jamming critical point and help inform a common theoretical description of the detailed features of the transition.

Penrose tilings as jammed solids.

Penrose tilings form lattices, exhibiting fivefold symmetry and isotropic elasticity, with inhomogeneous coordination much like that of the force networks in jammed systems. Under periodic boundary conditions, their average coordination is exactly four. We study the elastic and vibrational properties of rational approximants to these lattices as a function of unit-cell size N(S) and find that they have of order sqrt[N(S)] zero modes and states of self-stress and yet all their elastic moduli vanish. In their generic form, obtained by randomizing site positions, their elastic and vibrational properties are similar to those of particulate systems at jamming with a nonzero bulk modulus, vanishing shear modulus, and a flat density of states.

Giant director fluctuations in liquid crystal drops.

We report the discovery and elucidation of giant spatiotemporal orientational fluctuations in nematic liquid crystal drops with radial orientation of the nematic anisotropy axis producing a central "hedgehog" defect. We study the spatial and temporal properties of the fluctuations experimentally using polarized optical microscopy, and theoretically, by calculating the eigenspectrum of the Frank elastic free energy of a nematic drop of radius R_{2}, containing a spherical central core of radius R_{1} and constrained by perpendicular boundary conditions on all surfaces. We find that the hedgehog defect with radial orientation has a complex excitation spectrum with a single critical mode whose energy vanishes at a critical value µ_{c} of the ratio µ=R_{2}/R_{1}. When µ<µ_{c}, the mode has positive energy, indicating that the radial hedgehog state is stable; when µ>µ_{c}, it has negative energy indicating that the radial state is unstable to the formation of a lower-energy state. This mode gives rise to the large-amplitude director fluctuations we observe near the core, for µ near µ_{c}. A collapse of the experimental data corroborates model predictions for µ<µ_{c} and provides an estimate of the defect core size.

Theory of director fluctuations about a hedgehog defect in a nematic drop.

We present calculations of eigenmode energies and wave functions of both azimuthal and polar distortions of the nematic director relative to a radial hedgehog trapped in a spherical drop with a smaller concentric spherical droplet at its core. All surfaces interior to the drop have perpendicular (homeotropic) boundary conditions. We also calculate director correlation functions and their relaxation times. Of particular interest is a critical mode whose energy, with fixed Frank constants, vanishes as the ratio µ=R_{2}/R_{1} increases toward a critical value µ_{c}, where R_{2} is the radius of the drop and R_{1} that of the inner droplet, and then becomes negative for µ>µ_{c}. Our calculations form a basis for interpreting experimental measurements of director fluctuations relative to a radial hedgehog state in a spherical drop. We compare results with those obtained by previous investigations, which use a calculational approach different from ours, and with our experimental observations.

Distribution functions in percolation problems.

Percolation clusters are random fractals whose geometrical and transport properties can be characterized with the help of probability distribution functions. Using renormalized field theory, we determine the asymptotic form of various such distribution functions in the limits where certain scaling variables become small or large. Our study includes the pair-connection probability, the distributions of the fractal masses of the backbone, the red bonds, and the shortest, the longest, and the average self-avoiding walk between any two points on a cluster, as well as the distribution of the total resistance in the random resistor network. Our analysis draws solely on general, structural features of the underlying diagrammatic perturbation theory, and hence our main results are valid to arbitrary loop order.

Anomalous elasticity in nematic and smectic elastomer tubule phases.

We study anomalous elasticity in the tubule phases of nematic and smectic elastomer membranes, which are flat in one direction and crumpled in another. These phases share the same macroscopic symmetry properties including spontaneously broken in-plane isotropy and hence belong to the same universality class. Below an upper critical value D_{c}=3 of the membranes' intrinsic dimension D , thermal fluctuations renormalize the elasticity with respect to elastic displacements along the tubule axis so that elastic moduli for compression along the tubule axis and for bending the tubule axis become length-scale dependent. This anomalous elasticity belongs to the same universality class as that of d -dimensional conventional smectic liquid crystals with D taking on the role of d . For physical tubule phases, D=2 , this anomaly is of power-law type and thus might by easier to detect experimentally than the logarithmic anomaly in conventional smectics.

Field theory of directed percolation with long-range spreading.

It is well established that the phase transition between survival and extinction in spreading models with short-range interactions is generically associated with the directed percolation (DP) universality class. In many realistic spreading processes, however, interactions are long ranged and well described by Lévy flights-i.e., by a probability distribution that decays in d dimensions with distance r as r;{-d-sigma} . We employ the powerful methods of renormalized field theory to study DP with such long-range Lévy-flight spreading in some depth. Our results unambiguously corroborate earlier findings that there are four renormalization group fixed points corresponding to, respectively, short-range Gaussian, Lévy Gaussian, short-range, and Lévy DP and that there are four lines in the (sigma,d) plane which separate the stability regions of these fixed points. When the stability line between short-range DP and Lévy DP is crossed, all critical exponents change continuously. We calculate the exponents describing Lévy DP to second order in an epsilon expansion, and we compare our analytical results to the results of existing numerical simulations. Furthermore, we calculate the leading logarithmic corrections for several dynamical observables.

Smectic-A elastomers with weak director anchoring.

Experimentally it is possible to manipulate the director in a (chiral) smectic- A elastomer using an electric field. This suggests that the director is not necessarily locked to the layer normal, as described in earlier papers that extended rubber elasticity theory to smectics. Here, we consider the case that the director is weakly anchored to the layer normal assuming that there is a free energy penalty associated with relative tilt between the two. We use a recently developed weak-anchoring generalization of rubber elastic approaches to smectic elastomers and study shearing in the plane of the layers, stretching in the plane of the layers, and compression and elongation parallel to the layer normal. We calculate, inter alia, the engineering stress and the tilt angle between director and layer normal as functions of the applied deformation. For the latter three deformations, our results predict the existence of an instability towards the development of shear accompanied by smectic- C-like order.

Smectic-C tilt under shear in smectic-A elastomers.

Stenull and Lubensky [Phys. Rev. E 76, 011706 (2007)] have argued that shear strain and tilt of the director relative to the layer normal are coupled in smectic elastomers and that the imposition of one necessarily leads to the development of the other. This means, in particular, that a smectic-A elastomer subjected to a simple shear will develop smectic-C-like tilt of the director. Recently, Kramer and Finkelmann [e-print arXiv:0708.2024; Phys. Rev. E 78, 021704 (2008)], performed shear experiments on smectic-A elastomers using two different shear geometries. One of the experiments, which implements simple shear, produces clear evidence for the development of smectic-C-like tilt. Here, we generalize a model for smectic elastomers introduced by Adams and Warner [Phys. Rev. E 71, 021708 (2005)] and use it to study the magnitude of SmC-like tilt under shear for the two geometries investigated by Kramer and Finkelmann. Using reasonable estimates of model parameters, we estimate the tilt angle for both geometries, and we compare our estimates to the experimental results. The other shear geometry is problematic since it introduces additional in-plane compressions in a sheetlike sample, thus inducing instabilities that we discuss.

Smectic elastomer membranes.

We present a model for smectic elastomer membranes which includes elastic and liquid-crystalline degrees of freedom. Based on our model, we determined the qualitative phase diagram of a smectic elastomer membrane using mean-field theory. This phase diagram is found to comprise five phases, viz., smectic- A -flat, smectic- A -crumpled, smectic- C -flat, smectic- C -crumpled, and smectic- C -tubule phases, where in the latter phase, the membrane is flat in the direction of mesogenic tilt and crumpled in the perpendicular direction. The transitions between adjacent phases are second-order phase transitions. We study in some detail the elasticity of the smectic- C -flat and the smectic- C -tubule phases which are associated with a spontaneous breaking of in-plane rotational symmetry. As a consequence of the Goldstone theorem, these phases exhibit soft elasticity characterized by the vanishing of in-plane shear moduli.

Unconventional elasticity in smectic-A elastomers.

We study two aspects of the elasticity of smectic- A elastomers that make these materials genuinely and qualitatively different from conventional uniaxial rubbers. Under strain applied parallel to the layer normal, monodomain smectic- A elastomers exhibit a drastic change in Young's modulus above a threshold strain value of about 3%, as has been measured in experiments by [Nishikawa and Finkelmann, Macromol. Chem. Phys. 200, 312 (1999)]. Our theory predicts that such strains induce a transition to a smectic-C-like state and that it is this transition that causes the change in elastic modulus. We calculate the stress-strain behavior as well as the tilt of the smectic layers and the molecular orientation for strain along the layer normal, and we compare our findings with the experimental data. We also study the electroclinic effect in chiral smectic-A* elastomers. According to experiments by [Lehmann, Nature (London) 410, 447 (2001)] and [Köhler, Appl. Phys. A 80, 381 (2003)], this effect leads in smectic-A* elastomers to a giant or, respectively, at least very large lateral electrostriction. Incorporating polarization into our theory, we calculate the height change of smectic-A* elastomer films in response to a lateral external electric field, and we compare this result to the experimental findings.

Finite-size scaling of directed percolation in the steady state.

Recently, considerable progress has been made in understanding finite-size scaling in equilibrium systems. Here, we study finite-size scaling in nonequilibrium systems at the instance of directed percolation (DP), which has become the paradigm of nonequilibrium phase transitions into absorbing states, above, at, and below the upper critical dimension. We investigate the finite-size scaling behavior of DP analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d-dimensional hypercube of finite edge length L with periodic boundary conditions near the bulk critical point. In particular, we study the order parameter and its higher moments using renormalized field theory. We derive finite-size scaling forms of the moments in a one-loop calculation. Moreover, we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous Binder cumulant in equilibrium systems and that, in particular, provides a signature of the DP universality class. To complement our analytical work, we perform Monte Carlo simulations which confirm our analytical results.

Scaling behavior of linear polymers in disordered media.

It has long been known that the universal scaling properties of linear polymers in disordered media are well described by the statistics of self-avoiding walks (SAWs) on percolation clusters and their critical exponent nu(SAW), with the SAW implicitly referring to the average SAW. Hitherto, static averaging has been commonly used, e.g., in numerical simulations, to determine what the average SAW is. We assert that only kinetic, rather than static, averaging can lead to asymptotic scaling behavior and corroborate our assertion by heuristic arguments and a renormalizable field theory. Moreover, we calculate to two-loop order nu(SAW), the exponent nu(max) for the longest SAW, and a family of multifractal exponents nu(alpha).

Dynamics of smectic elastomers.

We study the low-frequency, long-wavelength dynamics of liquid crystal elastomers, crosslinked in the smectic-A phase, in their smectic-A, biaxial smectic and smectic-C phases. Two different yet related formulations are employed. One formulation describes the pure hydrodynamics and does not explicitly involve the Frank director, which relaxes to its local equilibrium value in a nonhydrodynamic time. The other formulation explicitly treats the director and applies beyond the hydrodynamic limit. We compare the low-frequency, long-wavelength dynamics of smectic-A elastomers to that of nematics and show that the two are closely related. For the biaxial smectic and the smectic-C phases, we calculate sound velocities and the mode structure in certain symmetry directions. For the smectic-C elastomers, in addition, we discuss in some detail their possible behavior in rheology experiments.

Dynamics, dynamic soft elasticity, and rheology of smectic-C elastomers.

We present a theory for the low-frequency, long-wavelength dynamics of soft smectic-C elastomers with locked-in smectic layers. Our theory, which goes beyond pure hydrodynamics, predicts a dynamic soft elasticity of these elastomers and allows us to calculate the storage and loss moduli relevant for rheology experiments as well as the mode structure.

Soft elasticity in biaxial smectic and smectic-C elastomers.

Ideal (monodomain) smectic-A elastomers cross-linked in the smectic-A phase are simply uniaxial rubbers, provided deformations are small. From these materials smectic-C elastomers are produced by a cooling through the smectic-A to smectic-C phase transition. At least in principle, biaxial smectic elastomers could also be produced via cooling from the smectic-A to a biaxial smectic phase. These phase transitions, respectively, from Dinfinityh to C2h and from Dinfinityh to D2h symmetry, spontaneously break the rotational symmetry in the smectic planes. We study the above transitions and the elasticity of the smectic-C and biaxial phases in three different but related models: Landau-like phenomenological models as functions of the Cauchy-Saint-Laurent strain tensor for both the biaxial and the smectic-C phases and a detailed model, including contributions from the elastic network, smectic layer compression, and smectic-C tilt for the smectic-C phase as a function of both strain and the c-director. We show that the emergent phases exhibit soft elasticity characterized by the vanishing of certain elastic moduli. We analyze in some detail the role of spontaneous symmetry breaking as the origin of soft elasticity and we discuss different manifestations of softness like the absence of restoring forces under certain shears and extensional strains.