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arXiv:1504.07268

From quantum to classical dynamics: Dynamic crossover in the relativistic $O(N)$ model

D. Mesterházy, Jan H. Stockemer, Yuya Tanizaki

Published Apr 27, 2015• 0 citations
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Abstract

We investigate the transition from unitary to dissipative dynamics in the relativistic $O(N)$ vector model with the $\lambda (\varphi^{2})^{2}$ interaction using the nonperturbative functional renormalization group in the real-time formalism. In thermal equilibrium, the theory is characterized by two scales, the interaction range for coherent scattering of particles and the mean free path determined by the rate of incoherent collisions with excitations in the thermal medium. Their competition determines the renormalization group flow and the effective dynamics of the model. Here we quantify the dynamic properties of the model in terms of the scale-dependent dynamic critical exponent $z$ in the limit of large temperatures and in $2 \leq d \leq 4$ spatial dimensions. We contrast our results to the behavior expected at vanishing temperature and address the question of the appropriate dynamic universality class for the given microscopic theory.

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From quantum to classical dynamics: Dynamic crossover in the relativistic O(N)O(N)O(N) model
We investigate the transition from unitary to dissipative dynamics in the relativistic O(N)O(N)O(N) vector model with the λ(φ2)2\lambda (\varphi^{2})^{2}λ(φ2)2 interaction using the nonperturbative functional renormalization group in the real-time formalism. In thermal equilibrium, the theory is characterized by two scales, the interaction range for coherent scattering of particles and the mean free path determined by the rate of incoherent collisions with excitations in the thermal medium. Their competition determines the renormalization group flow and the effective dynamics of the model. Here we quantify the dynamic properties of the model in terms of the scale-dependent dynamic critical exponent zzz in the limit of large temperatures and in 2≤d≤42 \leq d \leq 42≤d≤4 spatial dimensions. We contrast our results to the behavior expected at vanishing temperature and address the question of the appropriate dynamic universality class for the given microscopic theory.
We study the crossover between classical and nonclassical critical behaviors. The critical crossover limit is driven by the Ginzburg number G. The corresponding scaling functions are universal with respect to any possible microscopic mechanism which can vary G, such as changing the range or the strength of the interactions. The critical crossover describes the unique flow from the unstable Gaussian to the stable nonclassical fixed point. The scaling functions are related to the continuum renormalization-group functions. We show these features explicitly in the large-NNN limit of the $\mathrm{O}(N) {\ensuremath{\varphi}}^{4}$ model. We also show that the effective susceptibility exponent is nonmonotonic in the low-temperature phase of the three-dimensional Ising model.
The perturbative approach to the description of long-wavelength excitations at high temperature breaks down near the critical point of a second order phase transition. We study the dynamics of these excitations in a relativistic scalar field theory at and near the critical point via a renormalization group approach at high temperature and an $\ensuremath{\epsilon}$ expansion in $d=5\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions. The long-wavelength physics is determined by a nontrivial fixed point of the renormalization group. At the critical point we find that the dispersion relation and width of quasiparticles of momentum p are ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{p}^{z}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p},$ respectively, and the group velocity of quasiparticles ${v}_{g}\ensuremath{\sim}{p}^{z\ensuremath{-}1}$ vanishes in the long-wavelength limit at the critical point. Away from the critical point for $T\ensuremath{\gtrsim}{T}_{c}$ we find ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}z}[1+(p\ensuremath{\xi}{)}^{2z}{]}^{1/2}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p}(p\ensuremath{\xi}{)}^{2z}/[1+(p\ensuremath{\xi}{)}^{2z}]$ with $\ensuremath{\xi}$ the finite temperature correlation length $\ensuremath{\xi}\ensuremath{\propto}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}.$ The new dynamical exponent z results from anisotropic renormalization in the spatial and time directions. For a theory with O(N)O(N)O(N) symmetry we find $z=1+\ensuremath{\epsilon}{(N+2)/(N+8)}^{2}+\mathcal{O}({\ensuremath{\epsilon}}^{2}).$ This dynamical critical exponent describes a new universality class for dynamical critical phenomena in quantum field theory. Critical slowing down, i.e., a vanishing width in the long-wavelength limit, and the validity of the quasiparticle picture emerge naturally from this analysis.
Nonequilibrium dynamics play an important role in all contexts of physics, both classical and quantum as well as living and nonliving, so it is crucial to develop a foundational understanding of nonequilibrium phase transitions. In this work we investigate an important class of nonequilibrium dynamics in the form of nonreciprocal interactions. In particular we study how nonreciprocal coupling between two O(n_{i}) order parameters (with i=1,2) affects the universality at a multicritical point, extending the analysis of J. T. Young et al. [Phys. Rev. X 10, 011039 (2020)2160-330810.1103/PhysRevX.10.011039], which considered the case n_{1}=n_{2}=1, i.e., a Z_{2}×Z_{2} model. We show that nonequilibrium fixed points (NEFPs) emerge for a broad range of n_{1},n_{2} and exhibit intrinsically nonequilibrium critical phenomena, namely a violation of fluctuation-dissipation relations at all scales and underdamped oscillations near criticality in contrast to the overdamped relaxational dynamics of the corresponding equilibrium models. Furthermore, the NEFPs exhibit an emergent discrete scale invariance in certain physically relevant regimes of n_{1},n_{2}, but not others, depending on whether the critical exponent ν is real or complex. The boundary between these two regions is described by an exceptional point in the renormalization group (RG) flow, leading to distinctive features in correlation functions and the phase diagram. Another contrast with the previous work is the number and stability of the NEFPs as well as the underlying topology of the RG flow. Finally, we investigate an extreme form of nonreciprocity where one order parameter is independent of the other order parameter but not vice versa. Unlike the Z_{2}×Z_{2} model, which becomes nonperturbative in this case, we identify a distinct nonequilibrium universality class whose dependent field similarly violates fluctuation-dissipation relations but does not exhibit discrete scale invariance or underdamped oscillations near criticality.
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.
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