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

CAVAGNA, ANDREA; DI CARLO, LUCA; GIARDINA, IRENE; GRANDINETTI, LUCA; GRIGERA, TOMAS S.; PISEGNA, GIULIA

Physical Review E

american physical society

Año: 2019 vol. 100 p. 62130 - 62130

2470-0045

Motivated by the collective behavior of biological swarms, we study the critical dynamics of field theorieswith coupling between order parameter and conjugate momentum in the presence of dissipation. Under a fixed-network approximation, we perform a dynamical renormalization group calculation at one loop in the near-critical disordered region, and we show that the violation of momentum conservation generates a crossoverbetween an unstable fixed point, characterized by a dynamic critical exponent z = d/2, and a stable fixed pointwith z = 2. Interestingly, the two fixed points have different upper critical dimensions. The interplay betweenthese two fixed points gives rise to a crossover in the critical dynamics of the system, characterized by a crossoverexponent κ = 4/d. The crossover is regulated by a conservation length scale R 0 , given by the ratio between thetransport coefficient and the effective friction, which is larger as the dissipation is smaller: Beyond R 0 , thestable fixed point dominates, while at shorter distances dynamics is ruled by the unstable fixed point and criticalexponent, a behavior which is all the more relevant in finite-size systems with weak dissipation. We run numericalsimulations in three dimensions and find a crossover between the exponents z = 3/2 and z = 2 in the criticalslowdown of the system, confirming the renormalization group results. From the biophysical point of view,our calculation indicates that in finite-size biological groups mode coupling terms in the equation of motioncan significantly change the dynamical critical exponents even in the presence of dissipation, a step towardreconciling theory with experiments in natural swarms. Moreover, our result provides the scale within whichfully conservative Bose-Einstein condensation is a good approximation in systems with weak symmetry-breakingterms violating number conservation, as quantum magnets or photon gases.