The properties of the absorbing states of non-equilibrium models belonging to the conserved directed percolation universality class are studied. We find that at the critical point the absorbing states are hyperuniform, exhibiting anomolously small density fluctuations. The exponent characterizing the fluctuations is measured numerically, a scaling relation to other known exponents is suggested, and a new correlation length relating to this ordering is proposed. These results may have relevance to photonic band-gap materials.PACS numbers: 05.65.+b, 47.57.E-, 05.20.-y Dynamical systems with absorbing states [1][2][3] are intrinsically out of equilibrium, as they necessarily violate detailed balance -by definition, once they arrive at an absorbing state, they can never get out. There are many such systems, and they have an extensive literature [3]. Recently [4,5], an experimental system of non-Brownian colloids displaying a transition to an absorbing state has been studied, and a model, called "random organization" [5], has been proposed as a description of the experimental phenomena.Absorbing states satisfy some model-dependent condition, the specifics of which do not appear to change the behavior qualitatively. The control-parameter space is divided into two regions -one where absorbing states are achieved for any typical initial state, the other where absorbing states, although possible, are not attained, resulting in a non-equilibrium steady state with a non-zero average number of "active" particles which violate the stipulated condition. These regions are separated by a critical line, and the transition between the two phases displays characteristics of a continuous phase transition, with a diverging correlation length. This phenomenology is common to all absorbing state models.In this Letter we shall study the "order" that develops in several absorbing state models as the system approaches the critical point from within the absorbing phase. Remarkably, even though the initial states are random, the dynamics are random, and there is an enormous number of random absorbing states, those absorbing states actually attained by the system are special, with greatly diminshed density fluctuations. As we approach criticality, these fluctuations scale differently from the usual random system -they are, in fact, hyperuniform [6]. Hyperuniform systems have recently attracted interest due to their relation to photonic bandgap materials [7,8]. Thus, the setup in [4,5] may provide an efficient method to create such materials in bulk.For a system in d dimensions, we characterize the density fluctuations in a region of volume d as a function of byRandom systems, such as a Poisson process, have the standard " √ N " fluctuations; this corresponds to λ = d. When λ > d, the particles are distributed more uniformly; such distributions are termed hyperuniform [6]. For example, a crystal whose atoms are displaced by small random amounts from their lattice sites will have λ = d + 1. We find that for the models we study, λ > d for c...
The discovery that the band structure of electronic insulators may be topologically non-trivial has revealed distinct phases of electronic matter with novel properties 1,2 . Recently, mechanical lattices have been found to have similarly rich structure in their phononic excitations 3,4 , giving rise to protected unidirectional edge modes [5][6][7] . In all of these cases, however, as well as in other topological metamaterials 3,8 , the underlying structure was finely tuned, be it through periodicity, quasi-periodicity or isostaticity. Here we show that amorphous Chern insulators can be readily constructed from arbitrary underlying structures, including hyperuniform, jammed, quasi-crystalline and uniformly random point sets. While our findings apply to mechanical and electronic systems alike, we focus on networks of interacting gyroscopes as a model system. Local decorations control the topology of the vibrational spectrum, endowing amorphous structures with protected edge modes-with a chirality of choice. Using a real-space generalization of the Chern number, we investigate the topology of our structures numerically, analytically and experimentally. The robustness of our approach enables the topological design and self-assembly of non-crystalline topological metamaterials on the micro and macro scale.Condensed-matter science has traditionally focused on systems with underlying spatial order, as many natural systems spontaneously aggregate into crystals. The behaviour of amorphous materials, such as glasses, has remained more challenging 9 . In particular, our understanding of common concepts such as bandgaps and topological behaviour in amorphous materials is still in its infancy when compared with crystalline counterparts. This is not only a fundamental problem; advances in modern engineering, both of metamaterials and of quantum systems, have opened the door for the creation of materials with arbitrary structure, including amorphous materials. This prompts a search for principles that can apply to a wide range of amorphous systems, from interacting atoms to mechanical metamaterials.In the exploration of topological insulators, conceptual advances have proven to carry across between disparate physical realizations, from quantum systems 10 , to photonic waveguides 11 , to acoustical resonators 12,13 , to hinged or geared mechanical structures 3,14 . One promising model system is a class of mechanical insulators consisting of gyroscopes suspended from a plate. Appropriate crystalline arrangements of such gyroscopes break time-reversal symmetry, opening topological phononic bandgaps and supporting robust chiral edge modes 5,6 . Unlike trivial insulators, whose electronic states can be thought of as a sum of independent local insulating states, topological insulators require the existence of delocalized states in each non-trivial band and prevent a description in terms of a basis of localized Wannier states [15][16][17] . It is natural, therefore, to assume that some regularity over long distances may be key to top...
We consider driven many-particle models which have a phase transition between an active and an absorbing phase. Like previously studied models, we have particle conservation, but here we introduce an additional symmetry -when two particles interact, we give them stochastic kicks which conserve center of mass. We find that the density fluctuations in the active phase decay in the fastest manner possible for a disordered isotropic system, and we present arguments that the large scale fluctuations are determined by a competition between a noise term which generates fluctuations, and a deterministic term which reduces them. Our results may be relevant to shear experiments and may further the understanding of hyperuniformity which occurs at the critical point.
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