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arXiv:1905.01228 · Physical review. E

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

Andrea Cavagna, Luca Di Carlo, Irene Giardina, Luca Grandinetti, Tomás S. Grigera, Giulia Pisegna

Published Dec 23, 2019• 13 citations
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Abstract

Motivated by the collective behavior of biological swarms, we study the critical dynamics of field theories with 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 crossover between an unstable fixed point, characterized by a dynamic critical exponent z=d/2, and a stable fixed point with z=2. Interestingly, the two fixed points have different upper critical dimensions. The interplay between these two fixed points gives rise to a crossover in the critical dynamics of the system, characterized by a crossover exponent κ=4/d. The crossover is regulated by a conservation length scale R_{0}, given by the ratio between the transport coefficient and the effective friction, which is larger as the dissipation is smaller: Beyond R_{0}, the stable fixed point dominates, while at shorter distances dynamics is ruled by the unstable fixed point and critical exponent, a behavior which is all the more relevant in finite-size systems with weak dissipation. We run numerical simulations in three dimensions and find a crossover between the exponents z=3/2 and z=2 in the critical slowdown 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 motion can significantly change the dynamical critical exponents even in the presence of dissipation, a step toward reconciling theory with experiments in natural swarms. Moreover, our result provides the scale within which fully conservative Bose-Einstein condensation is a good approximation in systems with weak symmetry-breaking terms violating number conservation, as quantum magnets or photon gases.

DOI
10.1103/physreve.100.062130

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Motivated by the collective behavior of biological swarms, we study the critical dynamics of field theories with 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 crossover between an unstable fixed point, characterized by a dynamic critical exponent z=d/2, and a stable fixed point with z=2. Interestingly, the two fixed points have different upper critical dimensions. The interplay between these two fixed points gives rise to a crossover in the critical dynamics of the system, characterized by a crossover exponent κ=4/d. The crossover is regulated by a conservation length scale R_{0}, given by the ratio between the transport coefficient and the effective friction, which is larger as the dissipation is smaller: Beyond R_{0}, the stable fixed point dominates, while at shorter distances dynamics is ruled by the unstable fixed point and critical exponent, a behavior which is all the more relevant in finite-size systems with weak dissipation. We run numerical simulations in three dimensions and find a crossover between the exponents z=3/2 and z=2 in the critical slowdown 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 motion can significantly change the dynamical critical exponents even in the presence of dissipation, a step toward reconciling theory with experiments in natural swarms. Moreover, our result provides the scale within which fully conservative Bose-Einstein condensation is a good approximation in systems with weak symmetry-breaking terms violating number conservation, as quantum magnets or photon gases.
We calculate the relaxational dynamical critical behavior of systems of O(n_{ parallel}) plus sign in circleO(n_{ perpendicular}) symmetry including conservation of magnetization by renormalization group theory within the minimal subtraction scheme in two-loop order. Within the stability region of the Heisenberg fixed point and the biconical fixed point, strong dynamical scaling holds, with the asymptotic dynamical critical exponent z=2varphinu-1 , where varphi is the crossover exponent and nu the exponent of the correlation length. The critical dynamics at n_{ parallel}=1 and n_{ perpendicular}=2 is governed by a small dynamical transient exponent leading to nonuniversal nonasymptotic dynamical behavior. This may be seen, e.g., in the temperature dependence of the magnetic transport coefficients.
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The critical dynamics of a non-conserved order parameter is generally expected to become overdamped at long distances, even when propagating modes occur at microscopic or intermediate scales. We investigate the critical dynamics of a scalar field theory in thermal equilibrium which, in addition to local friction and noise, also contains a time-dependent second-order kinetic term. We show how to build a supersymmetric field-theory formulation. Using a two-loop expansion about four spatial dimensions, we show that the propagating and strictly overdamped limits share the same static Gaussian and Wilson-Fisher fixed points but realize distinct dynamical scaling regimes. The overdamped limit reproduces Model A. On the surface where local friction and noise vanish, the theory instead supports an interacting propagating fixed point whose dynamic exponent receives corrections at two loops. We demonstrate that coarse-graining does not generate a local dissipative operator on this surface, which therefore remains invariant under the RG flow. Local dissipation is nevertheless relevant at the propagating fixed point: an arbitrarily small equilibrium friction-noise perturbation drives the flow away from propagating scaling. Propagating critical dynamics thus defines a consistent but fine-tuned regime that is unstable to local equilibrium dissipation.
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A semi-microscopic scaling-field theory is developed for crossover phenomena near critical and tricritical points. The theory is based on a renormalization-group description of a model with two competing fixed points (such as a critical and a tricritical fixed point) in terms of scaling fields. The coupled nonlinear differential equations for scaling fields are truncated such as to preserve the physics essential for crossover phenomena. The approach allows the explicit calculation of thermodynamic functions for (i) tricritical systems and (ii) critical systems with an irrelevant scaling field. We obtain, for example, an explicit expression for the scaling function of the susceptibility, which describes the crossover from the tricritical to the critical region. The idea of "flow diagrams" in the scaling-field space is used to characterize crossover phenomena globally in the whole critical region. The concept of asymptotic critical exponents is generalized and effective critical exponents are introduced as logarithmic derivatives of thermodynamic quantities with respect to experimental fields and scaling fields, respectively. By using the method of effective exponents the size of the crossover region between regions of different asymptotic critical behavior is estimated. For the susceptibility, the width of the crossover region in decades of the effective temperature variable is roughly equal to the inverse of the crossover exponent. In the case of a critical system with a slow transient the asymptotic critical exponent is only reached extremely close to the critical point (unless the amplitude of the transient vanishes). It might then be impossible to determine the asymptotic exponent experimentally or by conventional series-expansion techniques, and an analysis of the data in terms of effective exponents is the alternative. The scaling-field approach is applied to three systems with crossover phenomena: (i) the model for tricritical systems with molecular-field tricritical exponents, (ii) the Ashkin-Teller model in three dimensions, and (iii) a model for phase transitions with Fisher exponent renormalization due to a constraint.
The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length $ln\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{\psi}}/T$, with $\ensuremath{\psi}$ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions ddd less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For d>dDRd>{d}_{\text{DR}}d>dDR​, on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.