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Motivated by the collective behaviour of biological swarms, we study thecritical dynamics of field theories with coupling between order parameter andconjugate momentum in the presence of dissipation. By performing a dynamicalrenormalization group calculation at one loop, we show that the violation ofmomentum conservation generates a crossover between a conservative yetIR-unstable fixed point, characterized by a dynamic critical exponent $z=d/2$,and a dissipative IR-stable fixed point with $z=2$. Interestingly, the twofixed points have different upper critical dimensions. The interplay betweenthese two fixed points gives rise to a crossover in the critical dynamics ofthe system, characterized by a crossover exponent $\kappa=4/d$. Such crossoveris regulated by a conservation length scale, $\mathcal R_0$, which is largerthe smaller the dissipation: beyond $\mathcal R_0$ the dissipative fixed pointdominates, while at shorter distances dynamics is ruled by the conservativefixed point and critical exponent, a behaviour which is all the more relevantin finite-size systems with weak dissipation. We run numerical simulations inthree dimensions and find a crossover between the exponents $z=3/2$ and $z=2$in the critical slowing down of the system, confirming the renormalizationgroup results. From the biophysical point of view, our calculation indicatesthat in finite-size biological groups mode-coupling terms in the equation ofmotion can significantly change the dynamical critical exponents even in thepresence of dissipation, a step towards reconciling theory with experiments innatural swarms. Moreover, our result provides the scale within which fullyconservative Bose-Einstein condensation is a good approximation in systems withweak symmetry-breaking terms violating number conservation, as quantum magnetsor photon gases.

Renormalization group crossover in the critical dynamics of field theories with mode coupling terms