Erratum: Odd elasticity and topological waves in active surfaces [Phys. Rev. E 109, 024608 (2024)].

This corrects the article DOI: 10.1103/PhysRevE.109.024608.

Learning a conserved mechanism for early neuroectoderm morphogenesis.

Morphogenesis is the process whereby the body of an organism develops its target shape. The morphogen BMP is known to play a conserved role across bilaterian organisms in determining the dorsoventral (DV) axis. Yet, how BMP governs the spatio-temporal dynamics of cytoskeletal proteins driving morphogenetic flow remains an open question. Here, we use machine learning to mine a morphodynamic atlas of Drosophila development, and construct a mathematical model capable of predicting the coupled dynamics of myosin, E-cadherin, and morphogenetic flow. Mutant analysis shows that BMP sets the initial condition of this dynamical system according to the following signaling cascade: BMP establishes DV pair-rule-gene patterns that set-up an E-cadherin gradient which in turn creates a myosin gradient in the opposite direction through mechanochemical feedbacks. Using neural tube organoids, we argue that BMP, and the signaling cascade it triggers, prime the conserved dynamics of neuroectoderm morphogenesis from fly to humans.

Odd elasticity and topological waves in active surfaces.

Odd elasticity describes active elastic systems whose stress-strain relationship is not compatible with a potential energy. As the requirement of energy conservation is lifted from linear elasticity, new antisymmetric (odd) components appear in the elastic tensor. In this work we study the odd elasticity and non-Hermitian wave dynamics of active surfaces, specifically plates of moderate thickness. These odd moduli can endow the vibrational modes of the plate with a nonzero topological invariant known as the first Chern number. Within continuum elastic theory, we show that the Chern number is related to the presence of unidirectional shearing waves that are hosted at the plate's boundary. We show that the existence of these chiral edge waves hinges on a distinctive two-step mechanism. Unlike electronic Chern insulators where the magnetic field at the same time gaps the spectrum and imparts chirality, here the finite thickness of the sample gaps the shear modes, and the odd elasticity makes them chiral.

Pattern formation by turbulent cascades.

Fully developed turbulence is a universal and scale-invariant chaotic state characterized by an energy cascade from large to small scales at which the cascade is eventually arrested by dissipation1-6. Here we show how to harness these seemingly structureless turbulent cascades to generate patterns. Pattern formation entails a process of wavelength selection, which can usually be traced to the linear instability of a homogeneous state7. By contrast, the mechanism we propose here is fully nonlinear. It is triggered by the non-dissipative arrest of turbulent cascades: energy piles up at an intermediate scale, which is neither the system size nor the smallest scales at which energy is usually dissipated. Using a combination of theory and large-scale simulations, we show that the tunable wavelength of these cascade-induced patterns can be set by a non-dissipative transport coefficient called odd viscosity, ubiquitous in chiral fluids ranging from bioactive to quantum systems8-12. Odd viscosity, which acts as a scale-dependent Coriolis-like force, leads to a two-dimensionalization of the flow at small scales, in contrast with rotating fluids in which a two-dimensionalization occurs at large scales4. Apart from odd viscosity fluids, we discuss how cascade-induced patterns can arise in natural systems, including atmospheric flows13-19, stellar plasma such as the solar wind20-22, or the pulverization and coagulation of objects or droplets in which mass rather than energy cascades23-25.

Learning a conserved mechanism for early neuroectoderm morphogenesis.

Morphogenesis is the process whereby the body of an organism develops its target shape. The morphogen BMP is known to play a conserved role across bilaterian organisms in determining the dorsoventral (DV) axis. Yet, how BMP governs the spatio-temporal dynamics of cytoskeletal proteins driving morphogenetic flow remains an open question. Here, we use machine learning to mine a morphodynamic atlas of Drosophila development, and construct a mathematical model capable of predicting the coupled dynamics of myosin, E-cadherin, and morphogenetic flow. Mutant analysis shows that BMP sets the initial condition of this dynamical system according to the following signaling cascade: BMP establishes DV pair-rule-gene patterns that set-up an E-cadherin gradient which in turn creates a myosin gradient in the opposite direction through mechanochemical feedbacks. Using neural tube organoids, we argue that BMP, and the signaling cascade it triggers, prime the conserved dynamics of neuroectoderm morphogenesis from fly to humans.

Non-reciprocal phase transitions.

Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter1-6, non-equilibrium systems7-9, networks of neurons10,11, social groups with conformist and contrarian members12, directional interface growth phenomena13-15 and metamaterials16-20. Although wave propagation in non-reciprocal media has recently been closely studied1,16-20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points21. We describe the emergence of these phases using insights from bifurcation theory22,23 and non-Hermitian quantum mechanics24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.

Liquid-crystal-based topological photonics.

Liquid crystals are complex fluids that allow exquisite control of light propagation thanks to their orientational order and optical anisotropy. Inspired by recent advances in liquid-crystal photo-patterning technology, we propose a soft-matter platform for assembling topological photonic materials that holds promise for protected unidirectional waveguides, sensors, and lasers. Crucial to our approach is to use spatial variations in the orientation of the nematic liquid-crystal molecules to emulate the time modulations needed in a so-called Floquet topological insulator. The varying orientation of the nematic director introduces a geometric phase that rotates the local optical axes. In conjunction with suitably designed structural properties, this geometric phase leads to the creation of topologically protected states of light. We propose and analyze in detail soft photonic realizations of two iconic topological systems: a Su-Schrieffer-Heeger chain and a Chern insulator. The use of soft building blocks potentially allows for reconfigurable systems that exploit the interplay between topological states of light and the underlying responsive medium.

Symmetries and Dualities in the Theory of Elasticity.

Microscopic symmetries impose strong constraints on the elasticity of a crystalline solid. In addition to the usual spatial symmetries captured by the tensorial character of the elastic tensor, hidden nonspatial symmetries can occur microscopically in special classes of mechanical structures. Examples of such nonspatial symmetries occur in families of mechanical metamaterials where a duality transformation relates pairs of different configurations. We show on general grounds how the existence of nonspatial symmetries further constrains the elastic tensor, reducing the number of independent moduli. In systems exhibiting a duality transformation, the resulting constraints on the number of moduli are particularly stringent at the self-dual point but persist even away from it, in a way reminiscent of critical phenomena.

Dualities and non-Abelian mechanics.

Dualities are mathematical mappings that reveal links between apparently unrelated systems in virtually every branch of physics1-8. Systems mapped onto themselves by a duality transformation are called self-dual and exhibit remarkable properties, as exemplified by the scale invariance of an Ising magnet at the critical point. Here we show how dualities can enhance the symmetries of a dynamical matrix (or Hamiltonian), enabling the design of metamaterials with emergent properties that escape a standard group theory analysis. As an illustration, we consider twisted kagome lattices9-15, reconfigurable mechanical structures that change shape by means of a collapse mechanism9. We observe that pairs of distinct configurations along the mechanism exhibit the same vibrational spectrum and related elastic moduli. We show that these puzzling properties arise from a duality between pairs of configurations on either side of a mechanical critical point. The critical point corresponds to a self-dual structure with isotropic elasticity even in the absence of spatial symmetries and a twofold-degenerate spectrum over the entire Brillouin zone. The spectral degeneracy originates from a version of Kramers' theorem16,17 in which fermionic time-reversal invariance is replaced by a hidden symmetry emerging at the self-dual point. The normal modes of the self-dual systems exhibit non-Abelian geometric phases18,19 that affect the semiclassical propagation of wavepackets20, leading to non-commuting mechanical responses. Our results hold promise for holonomic computation21 and mechanical spintronics by allowing on-the-fly manipulation of synthetic spins carried by phonons.

A mechanism for anomalous transport in chiral active liquids.

Chiral active fluids are known to have anomalous transport properties such as the so-called odd viscosity. In this paper, we provide a microscopic mechanism for how such anomalous transport coefficients can emerge. We construct an Irving-Kirkwood-type stress tensor for chiral liquids and express the transport coefficients in terms of orientation-averaged intermolecular forces and distortions of the pair correlation function induced by a flow field. We then show how anomalous transport properties can be expected naturally due to the presence of a transverse component in the orientation-averaged intermolecular forces and anomalous distortion modes of the pair correlation function between chiral active particles. We anticipate that our work can provide a microscopic framework to explain the transport properties of nonequilibrium chiral systems.

Topological Waves in Fluids with Odd Viscosity.

Fluids in which both time reversal and parity are broken can display a dissipationless viscosity that is odd under each of these symmetries. Here, we show how this odd viscosity has a dramatic effect on topological sound waves in fluids, including the number and spatial profile of topological edge modes. Odd viscosity provides a short-distance cutoff that allows us to define a bulk topological invariant on a compact momentum space. As the sign of odd viscosity changes, a topological phase transition occurs without closing the bulk gap. Instead, at the transition point, the topological invariant becomes ill defined because momentum space cannot be compactified. This mechanism is unique to continuum models and can describe fluids ranging from electronic to chiral active systems.

Soft self-assembly of Weyl materials for light and sound.

Soft materials can self-assemble into highly structured phases that replicate at the mesoscopic scale the symmetry of atomic crystals. As such, they offer an unparalleled platform to design mesostructured materials for light and sound. Here, we present a bottom-up approach based on self-assembly to engineer 3D photonic and phononic crystals with topologically protected Weyl points. In addition to angular and frequency selectivity of their bulk optical response, Weyl materials are endowed with topological surface states, which allow for the existence of one-way channels, even in the presence of time-reversal invariance. Using a combination of group-theoretical methods and numerical simulations, we identify the general symmetry constraints that a self-assembled structure has to satisfy to host Weyl points and describe how to achieve such constraints using a symmetry-driven pipeline for self-assembled material design and discovery. We illustrate our general approach using block copolymer self-assembly as a model system.

Topological index for periodically driven time-reversal invariant 2D systems.

We define a new Z2-valued index to characterize the topological properties of periodically driven two dimensional crystals when the time-reversal symmetry is enforced. This index is associated with a spectral gap of the evolution operator over one period of time. When two such gaps are present, the Kane-Mele index of the eigenstates with eigenvalues between the gaps is recovered as the difference of the gap indices. This leads to an expression for the Kane-Mele invariant in terms of the Wess-Zumino amplitude. We illustrate the relation of the new index to the edge states in finite geometries by numerically solving an explicit model on the square lattice that is periodically driven in a time-reversal invariant way.