Ferromagnetic and antiferromagnetic order in bacterial vortex lattices

Abstract

Despite their inherent non-equilibrium nature1, living systems can self-organize in highly ordered collective states2,3 that share striking similarities with the thermodynamic equilibrium phases4,5 of conventional condensed matter and fluid systems. Examples range from the liquid-crystal-like arrangements of bacterial colonies6,7, microbial suspensions8,9 and tissues10 to the coherent macro-scale dynamics in schools of fish11 and flocks of birds12. Yet, the generic mathematical principles that govern the emergence of structure in such artificial13 and biological6-9,14 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 phases of ferro- and antiferromagnetic order. The preferred phase can be controlled by tuning the vortex coupling through changes of the inter-cavity gap widths. The emergence of opposing order regimes is tightly linked to the existence of geometry-induced edge currents15,16, reminiscent of those in quantum systems17-19. 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.

Figure 1. Edge currents determine antiferromagnetic and ferromagnetic order in a square lattice of bacterial vortices.

a, Three domains of antiferromagnetic order highlighted by dashed white lines (gap width w = 6 μm). Scale bar: 50 μm. Overlaid false colour shows spin magnitude (see Supplementary Video 1 for raw data). b, Bacterial flow PIV field within an antiferromagnetic domain (Supplementary Video 1). For clarity, not all velocity vectors are shown. Largest arrows correspond to speed 40 μm/s. Scale bar: 20 μm. c, Schematic of bacterial flow circulation in the vicinity of a gap. For small gaps w < w_crit, bacteria forming the edge currents (blue arrows) swim across the gap, remaining in their original cavity. Bulk flow (red) is directed opposite to the edge current, (Supplementary Video 3). d, Graph of the Union Jack double-lattice model in an antiferromagnetic state with zero net pillar circulation. Solid and dashed lines depict vortex–vortex and vortex–pillar interactions of respective strengths J_v and J_p. Vortices and pillars are colour-coded according to their spin. e, For supercritical gap widths w > w_crit, extended domains of ferromagnetic order predominate (Supplementary Video 2; w = 11 μm). Scale bar: 50 μm. f, PIV field within a ferromagnetic domain (Supplementary Video 2). Largest arrows: 36 μm/s. Scale bar: 20 μm. g, For w > w_crit, bacteria forming the edge current (blue arrows) swim along the PDMS boundary through the gap, driving bulk flows (red) in the opposite directions, thereby aligning neighbouring vortex spins. h, Ferromagnetic state of the Union Jack lattice induced by edge current loops around the pillars. i, Trajectories of neighbouring spins (*-symbols in a,e) fluctuate over time, signalling exploration of an equilibrium under a non-zero effective temperature (top: antiferromagnetic; bottom: ferromagnetic). j, The zero of the spin–spin correlation χ at w_crit ≈ 8 μm marks the phase transition. The best-fit Union Jack model (solid line) is consistent with the experimental data. k, RMS vortex spin ${\langle V_i^2\rangle }^{1/2}$ decreases with the gap size w, showing weakening of the circulation. RMS pillar spin ${\langle P_j^2\rangle }^{1/2}$ increases with w, reflecting enhanced bacterial circulation around pillars. Each point in j,k represents an average over ≥ 5 movies in 3 μm bins at 1.5 μm intervals; vertical bars indicate standard errors (Methods).

Figure 2. Best-fit mean-field LFT model captures the phase transition in the square lattice.

a, A sign change of the effective interaction βJ signals the transition from antiferro- to ferromagnetic states. b, The effective energy barrier, βa²/(4b) when a < 0 and zero when a > 0 (Supplementary Sec. 5), decreases with the gap size w, reflecting increased susceptibility to fluctuations. c, The spin V_min minimizing the single-spin potential (Supplementary Sec. 5) decreases with w in agreement with the decrease in the RMS vortex spin (Fig. 1k). Each point in a–c represents an average over ≥ 5 movies in 3 μm bins at 1.5 μm intervals; blue circles are from distribution fitting, red diamonds are from SDE regression, and vertical bars indicate standard errors (Methods). d-f, Examples of the effective single-spin potential ${\mathcal{V}}^{\text{eff}}$ conditional on the mean spin of adjacent vortices [V]_V. Data (points) and estimated potential (surface) for three movies with gap widths 6, 10 and 17 μm.

Figure 3. Frustration in triangular lattices determines the preferred order.

a,b, Triangular lattices favour ferromagnetic states of either handedness (Supplementary Video 4). Vortices are colour-coded by spin. c, At the largest gap size, bacterial circulation becomes unstable. Scale bar: 50 μ m. d, The spin–spin correlation χ shows strongly enhanced ferromagnetic order compared with the square lattice (Fig. 1j). Each point represents an average over ≥ 5 movies in 3 μm bins at 1.5 μm intervals; vertical bars indicate standard errors (Methods).