Restoring the Fluctuation-Dissipation Theorem in Kardar-Parisi-Zhang Universality Class through a New Emergent Fractal Dimension.

The Kardar-Parisi-Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation-dissipation theorem (FDT) for spatial dimension d>1. Notably, these questions were answered exactly only for 1+1 dimensions. In this work, we propose a new FDT valid for the KPZ problem in d+1 dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension dn. We present relations between the KPZ exponents and two emergent fractal dimensions, namely df, of the rough interface, and dn. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent α, the surface fractal dimension df and, through our relations, the noise fractal dimension dn. Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, an FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class.

Jamming and unusual charge density fluctuations of strange metals.

The strange metallic regime across a number of high-temperature superconducting materials presents numerous challenges to the classic theory of Fermi liquid metals. Recent measurements of the dynamical charge response of strange metals, including optimally doped cuprates, have revealed a broad, featureless continuum of excitations, extending over much of the Brillouin zone. The collective density oscillations of this strange metal decay into the continuum in a manner that is at odds with the expectations of Fermi liquid theory. Inspired by these observations, we investigate the phenomenology of bosonic collective modes and the particle-hole excitations in a class of strange metals by making an analogy to the phonons of classical lattices falling apart across an unconventional jamming-like transition associated with the onset of rigidity. By making comparisons to the experimentally measured dynamical response functions, we reproduce many of the qualitative features using the above framework. We conjecture that the dynamics of electronic charge density over an intermediate range of energy scales in a class of strongly correlated metals can be at the brink of a jamming-like transition.

Universal scaling for disordered viscoelastic matter near the onset of rigidity.

The onset of rigidity in interacting liquids, as they undergo a transition to a disordered solid, is associated with a rearrangement of the low-frequency vibrational spectrum. In this Letter, we derive scaling forms for the singular dynamical response of disordered viscoelastic networks near both jamming and rigidity percolation. Using effective-medium theory, we extract critical exponents, invariant scaling combinations, and analytical formulas for universal scaling functions near these transitions. Our scaling forms describe the behavior in space and time near the various onsets of rigidity, for rigid and floppy phases and the crossover region, including diverging length scales and timescales at the transitions.

Multifunctional twisted kagome lattices: Tuning by pruning mechanical metamaterials.

This article investigates phonons and elastic response in randomly diluted lattices constructed by combining (via the addition of next-nearest bonds) a twisted kagome lattice, with bulk modulus B=0 and shear modulus G>0, with either a generalized untwisted kagome lattice with B>0 and G>0 or with a honeycomb lattice with B>0 and G=0. These lattices exhibit jamming-like critical endpoints at which B, G, or both B and G jump discontinuously from zero while the remaining moduli (if any) begin to grow continuously from zero. Pairs of these jamming points are joined by lines of continuous rigidity percolation transitions at which both B and G begin to grow continuously from zero. The Poisson ratio and G/B can be continuously tuned throughout their physical range via random dilution in a manner analogous to "tuning by pruning" in random jammed lattices. These lattices can be produced with modern techniques, such as three-dimensional printing, for constructing metamaterials.

Yield Precursor Dislocation Avalanches in Small Crystals: The Irreversibility Transition.

The transition from elastic to plastic deformation in crystalline metals shares history dependence and scale-invariant avalanche signature with other nonequilibrium systems under external loading such as colloidal suspensions. These other systems exhibit transitions with clear analogies to work hardening and yield stress, with many typically undergoing purely elastic behavior only after "training" through repeated cyclic loading; studies in these other systems show a power-law scaling of the hysteresis loop extent and of the training time as the peak load approaches a so-called reversible-to-irreversible transition (RIT). We discover here that deformation of small crystals shares these key characteristics: yielding and hysteresis in uniaxial compression experiments of single-crystalline Cu nano- and micropillars decay under repeated cyclic loading. The amplitude and decay time of the yield precursor avalanches diverge as the peak stress approaches failure stress for each pillar, with a power-law scaling virtually equivalent to RITs in other nonequilibrium systems.

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.

Weirdest Martensite: Smectic Liquid Crystal Microstructure and Weyl-Poincaré Invariance.

Smectic liquid crystals are remarkable, beautiful examples of materials microstructure, with ordered patterns of geometrically perfect ellipses and hyperbolas. The solution of the complex problem of filling three-dimensional space with domains of focal conics under constraining boundary conditions yields a set of strict rules, which are similar to the compatibility conditions in a martensitic crystal. Here we present the rules giving compatible conditions for the concentric circle domains found at two-dimensional smectic interfaces with planar boundary conditions. Using configurations generated by numerical simulations, we develop a clustering algorithm to decompose the planar boundaries into domains. The interfaces between different domains agree well with the smectic compatibility conditions. We also discuss generalizations of our approach to describe the full three-dimensional smectic domains, where the variant symmetry group is the Weyl-Poincaré group of Lorentz boosts, translations, rotations, and dilatations.

Elasticity of randomly diluted honeycomb and diamond lattices with bending forces.

We use numerical simulations and an effective-medium theory to study the rigidity percolation transition of the honeycomb and diamond lattices when weak bond-bending forces are included. We use a rotationally invariant bond-bending potential, which, in contrast to the Keating potential, does not involve any stretching. As a result, the bulk modulus does not depend on the bending stiffness κ. We obtain scaling functions for the behavior of some elastic moduli in the limits of small ΔP = 1-P, and small δP = P-Pc, where P is an occupation probability of each bond, and Pc is the critical probability at which rigidity percolation occurs. We find good quantitative agreement between effective-medium theory and simulations for both lattices for P close to one.

Visualization, coarsening, and flow dynamics of focal conic domains in simulated smectic-A liquid crystals.

Smectic liquid crystals vividly illustrate the subtle interplay of broken translational and orientational symmetries, by exhibiting defect structures forming geometrically perfect confocal ellipses and hyperbolas. Here, we develop and numerically implement an effective theory to study the dynamics of focal conic domains in smectic-A liquid crystals. We use the information about the smectic's structure and energy density provided by our simulations to develop several novel visualization tools for the focal conics. Our simulations accurately describe both simple and extensional shear, which we compare to experiments, and provide additional insight into the coarsening dynamics of focal conic domains.

Tricritical behavior of soft nematic elastomers.

We propose a lattice statistical model to investigate the phase diagrams and the soft responses of nematic liquid-crystal elastomers. Using suitably scaled infinite-range interactions, we obtain exact self-consistent equations for the tensor components of the nematic order parameter in terms of temperature, the distortion and stress tensors, and the initial nematic order. These equations are amenable to simple numerical calculations, which are used to characterize the low-temperature soft regime. We find a peculiar phase diagram, in terms of temperature and the diagonal component of the distortion tensor along the stretching direction, with first- and second-order transitions to the soft phase, and the prediction of tricritical points. This behavior is not qualitatively changed if we use different values of the initial nematic order parameter.

Elastic Maier-Saupe-Zwanzig model and some properties of nematic elastomers.

We introduce a simple mean-field lattice model to describe the behavior of nematic elastomers. This model combines the Maier-Saupe-Zwanzig approach to liquid crystals and an extension to lattice systems of the Warner-Terentjev theory of elasticity, with the addition of quenched random fields. We use standard techniques of statistical mechanics to obtain analytic solutions for the full range of parameters. Among other results, we show the existence of a stress-strain coexistence curve below a freezing temperature, analogous to the P-V diagram of a simple fluid, with the disorder strength playing the role of temperature. Below a critical value of disorder, the tie lines in this diagram resemble the experimental stress-strain plateau and may be interpreted as signatures of the characteristic polydomain-monodomain transition. Also, in the monodomain case, we show that random fields may soften the first-order transition between nematic and isotropic phases, provided the samples are formed in the nematic state.