Despite their inherently non-equilibrium nature(1), living systems can self-organize in highly ordered collective states(2,3) that share striking similarities with the thermodynamic equilibrium phases(4,5) of conventional condensed-matter and fluid systems. Examples range from the liquid-crystal-like arrangements of bacterial colonies(6,7), microbial suspensions(8,9) and tissues(10) to the coherent macro-scale dynamics in schools of fish(11) and flocks of birds(12). Yet, the generic mathematical principles that govern the emergence of structure in such artificial(13) and biological(6-9,14) systems are elusive. It is not clear when, or even whether, well-established theoretical concepts describing universal thermostatistics of equilibrium systems can capture and classify ordered states of living matter. Here, we connect these two previously disparate regimes: through microfluidic experiments and mathematical modelling, we demonstrate that lattices of hydrodynamically coupled bacterial vortices can spontaneously organize into distinct patterns characterized by ferro-and antiferromagnetic order. The coupling between adjacent vortices can be controlled by tuning the inter-cavity gap widths. The emergence of opposing order regimes is tightly linked to the existence of geometry-induced edge currents(15,16), reminiscent of those in quantum systems(17-19). Our experimental observations can be rationalized in terms of a generic lattice field theory, suggesting that bacterial spin networks belong to the same universality class as a wide range of equilibrium systems.
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