Universal behavior at the Lifshitz Points of an active Malthusian Ising model

TL;DR

This work presents a minimal active matter model, the Active Malthusian Ising Model, to explore Lifshitz points in nonequilibrium settings with motility and birth–death dynamics. Using dynamical renormalization group analysis with one-loop epsilon expansions, it identifies two distinct Lifshitz points, longitudinal and transverse, and characterizes their universal behavior. The longitudinal LP exhibits a generic divergent flow indicative of a possible strong-coupling fixed point or fluctuation-induced first order, while the transverse LP remains in an anisotropic equilibrium LP universality class, with another fixed point emerging when the active coupling is tuned away from zero. Extending to the all-direction Lifshitz point, the analysis again finds a divergent flow, signaling a first-order-like transition; these results connect LP physics in active matter to known driven-diffusive models and provide concrete predictions for simulations and experiments.

Abstract

Lifshitz points (LPs) are multicritical points where ordered, disordered, and patterned phases meet. Originally studied in equilibrium magnetic systems, LPs have since been identified in soft matter and even cosmological settings. Their role in active, living matter, however, remains entirely unexplored. Here we address this gap by introducing and analyzing LPs in the Active Malthusian Ising Model (AMIM) -- a minimal model of living matter that incorporates motility together with birth-death dynamics. Despite its simplicity, the AMIM provides direct experimental relevance. We show that the system generically exhibits two distinct LPs and elucidate their universal behavior using a dynamic renormalization group analysis with the $ε$-expansion method at one loop. Our results yield testable predictions for future simulations and experiments, establishing LPs as a fertile testing ground for novel physics in active matter.

Figures (2)

• Figure 1: A microscopic active Malthusian Ising model (MAIM). (a) & (b) In this MAIM, spins' directions preferentially align with the vertical $x$-axis and they dictate the spins' direction motion (i). However, fluctuations can modify spins directions, leading to the spins moving sideways (ii), and spin-flips (iii). Further, we allow for the appearance (or birth) of particle (iv) and disappearance (or death) of particle (v), thus leading to the fact that the particle number is not conserved (Malthusian dynamics).
• Figure 2: RG flow diagrams of the two distinct Lifshitz Points (LPs) (a) Longitudinal LP: A generically divergent RG flow is observed for nonzero $\lambda$. The RG flow is generated for $d=4.4$, with the Gaussian fixed point (FP) depicted by the blue triangle, and the equilibrium anisotropic Ising LP FP depicted by the purple hexagon. (b) Transverse LP: The LP multicritical LP behavior is generically described by the equilibrium anisotropic Ising LP FP (red circle). Upon further fine tuning $\beta$ to zero, a new FP emerges (green square). The RG flow is generated for $d=6.9$ and the Gaussian FP is depicted by the yellow pentagon.