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A passive solid cannot do work on its surroundings through any quasistatic cycle of deformations. This property places strong constraints on the allowed elastic moduli. In this talk, we show that static elastic moduli altogether absent in passive elasticity can arise from active, non-conservative microscopic interactions. These active moduli enter the antisymmetric (or odd) part of the static elastic modulus tensor and quantify the amount of work extracted along quasistatic strain cycles. In two-dimensional isotropic media, two chiral odd-elastic moduli emerge in addition to the bulk and shear moduli. We discuss microscopic realizations that include networks of Hookean springs augmented with active transverse forces and non-reciprocal active hinges. Using coarse-grained microscopic models, numerical simulations and continuum equations, we uncover phenomena ranging from auxetic behavior induced by odd moduli to elastic wave propagation in overdamped media enabled by self-sustained active strain cycles. We discuss active metamaterial realizations that conserve linear momentum but exhibit a non-reciprocal linear response. Next we ask: what happens to the well-established framework of phase transitions in these non-reciprocal systems far from equilibrium? Simple demonstrations with robots will be presented along with naturally occurring phenomena from various domains of science. In all these cases, the emergence of unique time-dependent many-body phases can be captured by combining insights from non-Hermitian quantum mechanics and bifurcation theory. This mathematical approach lays the foundation for a general theory of critical phenomena in systems whose collective dynamics is not governed by an optimization principle.
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