Critical dynamics: a field-theoretical approach

Simulations of Stochastic Fluid Dynamics near a Critical Point in the Phase Diagram

We present simulations of stochastic fluid dynamics in the vicinity of a critical endpoint belonging to the universality class of the Ising model. This study is motivated by the challenge of modeling the dynamics of critical fluctuations near a conjectured critical endpoint in the phase diagram of quantum chromodynamics (QCD). We focus on the interaction of shear modes with a conserved scalar density, which is known as model H. We show that the observed dynamical scaling behavior depends on the correlation length and the shear viscosity of the fluid. As the correlation length is increased or the viscosity is decreased we observe a crossover from the dynamical exponent of critical diffusion, z≃4, to the expected scaling exponent of model H, z≃3. We use our method to investigate the time-dependent correlation function of non-Gaussian moments M^{n}(t) of the order parameter. We find that the relaxation time depends in a nontrivial manner on the power n.

Dynamical critical behavior of the two-dimensional three-state Potts model

We investigate the dynamical critical behavior of the two-dimensional three-state Potts model with single spin-flip dynamics in equilibrium. We focus on the mean-squared deviation of the magnetization M (MSD_{M}) as a function of time, as well as on the autocorrelation function of M. Our simulations reveal the existence of two crossover behaviors at times τ_{1}∼L^{z_{1}} and τ_{2}∼L^{z_{2}}, separating three dynamical regimes. MSD_{M} appears to shift from ordinary diffusion in the first regime, to anomalous diffusion in the second, and finally to be constant in the third regime. The magnetization autocorrelation function, on the other hand, is found to fluctuate between exponential decay, stretched-exponential decay, and then again exponential decay along these three regimes. This behavior is in agreement with the one reported recently for the two-dimensional Ising ferromagnet [Phys. Rev. E 108, 034118 (2023)2470-004510.1103/PhysRevE.108.034118], indicating that the existence of two dynamic critical exponents is not a peculiarity of the Ising model itself. A comparison of both MSD_{M} and the magnetization's autocorrelation function suggests that within our numerical accuracy the exponents z_{1} and z_{2} are shared between the Ising and three-state Potts models at least for the particular case of single spin-flip dynamics studied here, even though their equilibrium universality classes are clearly distinct. Continuity of MSD_{M} requires that α(z_{2}-z_{1})=γ/ν-z_{1}, in which α is the anomalous exponent in the intermediate regime. Since the ratio γ/ν is not shared between the two models, it follows that α is not shared either, an aspect well verified in our simulations. Finally, we also discuss the relevance of our main findings using another useful observable, namely the line magnetization M_{l}.

Thermo-osmosis of a near-critical binary fluid mixture: A general formulation and universal flow direction

We consider a binary fluid mixture, which lies in the one-phase region near the demixing critical point, and study its transport through a capillary tube linking two large reservoirs. We assume that short-range interactions cause preferential adsorption of one component onto the tube's wall. The adsorption layer can become much thicker than the molecular size, which enables us to apply hydrodynamics based on a coarse-grained free-energy functional. For transport processes induced by gradients of the pressure, composition, and temperature along a cylindrical tube, we obtain the formulas of the Onsager coefficients to extend our previous results on isothermal transport, assuming the critical composition in the middle of each reservoir in the reference equilibrium state. Among the processes, we focus on thermo-osmosis-mass flow due to a temperature gradient. We explicitly derive a formula for the thermal force density, which is nonvanishing in the adsorption layer and causes thermo-osmosis. This formula for a near-critical binary fluid mixture is an extension of the conventional formula for a one-component fluid, expressed in terms of local excess enthalpy. We predict that the direction of thermo-osmotic flow of a mixture near the upper (lower) consolute point is the same as (opposite to) that of the temperature gradient, irrespective of which component is adsorbed on the wall. Our procedure would also be applied to dynamics of a soft material, whose mesoscopic inhomogeneity can be described by a coarse-grained free-energy functional.

Out-of-equilibrium scaling of the energy density along the critical relaxational flow after a quench of the temperature

We study the out-of-equilibrium behavior of statistical systems along critical relaxational flows arising from instantaneous quenches of the temperature T to the critical point T_{c}, starting from equilibrium conditions at time t=0. In the case of soft quenches, i.e., when the initial temperature T is assumed sufficiently close to T_{c} (to keep the system within the critical regime), the critical modes develop an out-of-equilibrium finite-size-scaling (FSS) behavior in terms of the rescaled time variable Θ=t/L^{z}, where t is the time interval after quenching, L is the size of the system, and z is the dynamic exponent associated with the dynamics. However, the realization of this picture is less clear when considering the energy density, whose equilibrium scaling behavior (corresponding to the starting point of the relaxational flow) is generally dominated by a temperature-dependent regular background term or mixing with the identity operator. These issues are investigated by numerical analyses within the three-dimensional lattice N-vector models, for N=3 and 4, which provide examples of critical behaviors with negative values of the specific-heat critical exponent α, implying that also the critical behavior of the specific heat gets hidden by the background term. The results show that, after subtraction of its asymptotic critical value at T_{c}, the energy density develops an asymptotic out-of-equilibrium FSS in terms of Θ as well, whose scaling function appears singular in the small-Θ limit.

Critical dynamical behavior of the Ising model

We investigate the dynamical critical behavior of the two- and three-dimensional Ising models with Glauber dynamics in equilibrium. In contrast to the usual standing, we focus on the mean-squared deviation of the magnetization M, MSD_{M}, as a function of time, as well as on the autocorrelation function of M. These two functions are distinct but closely related. We find that MSD_{M} features a first crossover at time τ_{1}∼L^{z_{1}}, from ordinary diffusion with MSD_{M}∼t, to anomalous diffusion with MSD_{M}∼t^{α}. Purely on numerical grounds, we obtain the values z_{1}=0.45(5) and α=0.752(5) for the two-dimensional Ising ferromagnet. Related to this, the magnetization autocorrelation function crosses over from an exponential decay to a stretched-exponential decay. At later times, we find a second crossover at time τ_{2}∼L^{z_{2}}. Here, MSD_{M} saturates to its late-time value ∼L^{2+γ/ν}, while the autocorrelation function crosses over from stretched-exponential decay to simple exponential one. We also confirm numerically the value z_{2}=2.1665(12), earlier reported as the single dynamic exponent. Continuity of MSD_{M} requires that α(z_{2}-z_{1})=γ/ν-z_{1}. We speculate that z_{1}=1/2 and α=3/4, values that indeed lead to the expected z_{2}=13/6 result. A complementary analysis for the three-dimensional Ising model provides the estimates z_{1}=1.35(2), α=0.90(2), and z_{2}=2.032(3). While z_{2} has attracted significant attention in the literature, we argue that for all practical purposes z_{1} is more important, as it determines the number of statistically independent measurements during a long simulation.

Critical dynamics of the superfluid phase transition: Multiloop calculation of the microscopic model

Results and method of a three-loop renormalization-group calculation in the model of a Bose gas with a local density-density interaction in the formalism of time-dependent Green functions at finite temperature are presented. The results provide support to the recent conjecture [J. Honkonen, M. V. Komarova, Y. G. Molotkov, and M. Y. Nalimov, Nucl. Phys. B 939, 105 (2019)0550-321310.1016/j.nuclphysb.2018.12.015; Y. A. Zhavoronkov, M. V. Komarova, Y. G. Molotkov, M. Y. Nalimov, and J. Honkonen, Theor. Math. Phys. 200, 1237 (2019)0040-577910.1134/S0040577919080142] that the dynamics of the superfluid phase transition is described by a model which belongs to the same universality class as the stochastic model A.

Critical dynamics of the antiferromagnetic O(3) nonlinear sigma model with conserved magnetization

We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equations of motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d=2+ɛ expansion about its lower critical dimension d_{lc}=2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d_{c}=4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε=4-d expansion near d_{c}. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z_{Γ}=d-2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values z_{c}=d/2 in an intermediate crossover region.

Coupled two-species model for the pair contact process with diffusion

The contact process with diffusion (PCPD) defined by the binary reactions B+B→B+B+B, B+B→∅ and diffusive particle spreading exhibits an unusual active to absorbing phase transition whose universality class has long been disputed. Multiple studies have indicated that an explicit account of particle pair degrees of freedom may be required to properly capture this system's effective long-time, large-scale behavior. We introduce a two-species representation for the PCPD in which single particles B and particle pairs A are dynamically coupled according to the stochastic reaction processes B+B→A, A→A+B, A→∅, and A→B+B, with each particle type diffusing independently. Mean-field analysis reveals that the phase transition of this model is driven by competition and balance between the two species. We employ Monte Carlo simulations in one, two, and three dimensions to demonstrate that this model consistently captures the pertinent features of the PCPD. In the inactive phase, A particles rapidly go extinct, effectively leaving the B species to undergo pure diffusion-limited pair annihilation kinetics B+B→∅. At criticality, both A and B densities decay with the same exponents (within numerical errors) as the corresponding order parameters of the original PCPD, and display mean-field scaling above the upper critical dimension d_{c}=2. In one dimension, the critical exponents for the B species obtained from seed simulations also agree well with previously reported exponent value ranges. We demonstrate that the scaling properties of consecutive particle pairs in the PCPD are identical with that of the A species in the coupled model. This two-species picture resolves the conceptual difficulty for seed simulations in the original PCPD and naturally introduces multiple length scales and timescales to the system, which are also the origin of strong corrections to scaling. The extracted moment ratios from our simulations indicate that our model displays the same temporal crossover behavior as the PCPD, which further corroborates its full dynamical equivalence with our coupled model.

Critical dynamics of anisotropic antiferromagnets in an external field

We numerically investigate the nonequilibrium critical dynamics in three-dimensional anisotropic antiferromagnets in the presence of an external magnetic field. The phase diagram of this system exhibits two critical lines that meet at a bicritical point. The nonconserved components of the staggered magnetization order parameter couple dynamically to the conserved component of the magnetization density along the direction of the external field. Employing a hybrid computational algorithm that combines reversible spin precession with relaxational Monte Carlo updates, we study the aging scaling dynamics for the model C critical line, identifying the critical initial slip, autocorrelation, and aging exponents for both the order parameter and the conserved field, thus also verifying the dynamic critical exponent. We further probe the model F critical line by investigating the system size dependence of the characteristic spin wave frequencies near criticality and measure the dynamic critical exponents for the order parameter including its aging scaling at the bicritical point.

Renormalization group study of superfluid phase transition: Effect of compressibility

Dynamic critical behavior in superfluid systems is considered in the presence of external stirring and advecting processes. The latter are generated by means of the Gaussian random velocity ensemble with white-noise character in time variable and self-similar spatial dependence. The main focus of this work is to analyze an effect of compressible modes on the critical behavior. The model is formulated through stochastic Langevin equations, which are then recast into the Janssen-De Dominicis response formalism. Employing the field-theoretic perturbative renormalization group method we analyze large-scale properties of the model. Explicit calculations are performed to the leading one-loop approximation in the double (ɛ,y) expansion scheme, where ɛ is a deviation from the upper critical dimension d_{c}=4 and y describes a scaling property of the velocity ensemble. Altogether five distinct universality classes are expected to be macroscopically observable. In contrast to the incompressible case, we find that compressibility leads to an enhancement and stabilization of nontrivial asymptotic regimes.

Monte Carlo static and dynamic simulations of a three-dimensional Ising critical model

The critical dynamics of 'model A" of Hohenberg and Halperin has been studied by the Monte Carlo method. Simulations have been carried out in the three-dimensional (3d) simple cubic Ising model for lattices of sizes L=16 to L=512. Using Wolff's cluster algorithm, the critical temperature is precisely found as β_{c}=0.22165468(5). By Fourier transform of the lattice configurations, the critical scattering intensities I(q[over ⃗]) can be obtained. After circular averaging, the static simulations with L=256 and L=512 provide an estimate of the critical exponent γ/ν=2-η=1.9640(5). The |q[over ⃗]|-dependent distribution of I(q[over ⃗]) showed an exponential distribution, corresponding to a Gaussian distribution of the scattering amplitudes for a large q domain. The time-dependent intensities were then used for the study of the critical dynamics of 3d lattices at the critical point. To simulate results of an x-ray photon correlation spectroscopy experiment, the time-dependent correlation function of the intensities was studied for each |q[over ⃗]|-value. In the q region where I(q[over ⃗]) had an exponential distribution, the time correlations can be fit to a stretched exponential, where the exponent μ=γ/νz≃0.975 provides an estimate of the dynamic exponent z. This corresponds to z=2.0145, in agreement with the observed variations of the characteristic fluctuation time of the intensity: τ(q)∝q^{-z}, which gives z=2.015. These results agree with the ε expansion of field-theoretical methods (2.017). In this paper, the need to take account of the anomalous time behavior (μ<1) in the dynamics is exemplified. This dynamics reflects a nonlinear time behavior of model A, and its large time extension is discussed in detail.

Dynamic critical exponent z of the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model

We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain z=2.0245(15) for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our estimate of z is fully consistent with recent field theoretic results.

Computation of the dynamic critical exponent of the three-dimensional Heisenberg model

Working in and out of equilibrium and using state-of-the-art techniques we have computed the dynamic critical exponent of the three-dimensional Heisenberg model. By computing the integrated autocorrelation time at equilibrium, for lattice sizes L≤64, we have obtained z=2.033(5). In the out-of-equilibrium regime we have run very large lattices (L≤250) obtaining z=2.04(2) from the growth of the correlation length. We compare our values with that previously computed at equilibrium with relatively small lattices (L≤24), with that provided by means a three-loops calculation using perturbation theory and with experiments. Finally we have checked previous estimates of the static critical exponents, η and ν, in the out-of-equilibrium regime.

Renormalization Group in Non-Relativistic Quantum Statistics

Dynamic behaviour of a boson gas near the condensation transition in the symmetric phase is analyzed with the use of an effective large-scale model derived from time-dependent Green functions at finite temperature. A renormalization-group analysis shows that the scaling exponents of critical dynamics of the effective multi-charge model coincide with those of the standard model A. The departure of this result from the description of the superfluid transition by either model E or F of the standard phenomenological stochastic models is corroborated by the analysis of a generalization of model F, which takes into account the effect of compressible fluid velocity. It is also shown that, contrary to the single-charge model A, there are several correction exponents in the effective model, which are calculated at the leading order of the ɛ= 4 − d expansion.

Symmetry Breaking in Stochastic Dynamics and Turbulence

Symmetries play paramount roles in dynamics of physical systems. All theories of quantum physics and microworld including the fundamental Standard Model are constructed on the basis of symmetry principles. In classical physics, the importance and weight of these principles are the same as in quantum physics: dynamics of complex nonlinear statistical systems is straightforwardly dictated by their symmetry or its breaking, as we demonstrate on the example of developed (magneto)hydrodynamic turbulence and the related theoretical models. To simplify the problem, unbounded models are commonly used. However, turbulence is a mesoscopic phenomenon and the size of the system must be taken into account. It turns out that influence of outer length of turbulence is significant and can lead to intermittency. More precisely, we analyze the connection of phenomena such as behavior of statistical correlations of observable quantities, anomalous scaling, and generation of magnetic field by hydrodynamic fluctuations with symmetries such as Galilean symmetry, isotropy, spatial parity and their violation and finite size of the system.

Bacterial range expansions on a growing front: Roughness, fixation, and directed percolation

Directed percolation (DP) is a classic model for nonequilibrium phase transitions into a single absorbing state (fixation). It has been extensively studied by analytical and numerical techniques in diverse contexts. Recently, DP has appeared as a generic model for the evolutionary and ecological dynamics of competing bacterial populations. Range expansion-the stochastic reproduction of bacteria competing for space to be occupied by their progeny-leads to a fluctuating and rough growth front, which is known from experiment and simulation to affect the underlying critical behavior of the DP transition. In this work, we employ symmetry arguments to construct a pair of nonlinear stochastic partial differential equations describing the coevolution of surface roughness with the composition field of DP. Macroscopic manifestations (phenomenology) of these equations on growth patterns and genealogical tracks of range expansion are discussed; followed by a renormalization group analysis of possible scaling behaviors at the DP transition.

Motile bacteria in a critical fluid mixture

We studied the swimming of Escherichia coli bacteria in the vicinity of the critical point in a solution of the nonionic surfactant C_{12}E_{5} in buffer solution. In phase-contrast microscopy, each swimming cell produces a transient trail behind itself lasting several seconds. Comparing quantitative image analysis with simulations show that these trails are due to local phase reorganization triggered by differential adsorption. This contrasts with similar trails seen in bacteria swimming in liquid crystals, which are due to shear effects. We show how our trails are controlled, and use them to probe the structure and dynamics of critical fluctuations in the fluid medium.

Solvent coarsening around colloids driven by temperature gradients

Using mesoscopic numerical simulations and analytical theory, we investigate the coarsening of the solvent structure around a colloidal particle emerging after a temperature quench of the colloid surface. Qualitative differences in the coarsening mechanisms are found, depending on the composition of the binary liquid mixture forming the solvent and on the adsorption preferences of the colloid. For an adsorptionwise neutral colloid, the phase next to its surface alternates as a function of time. This behavior sets in on the scale of the relaxation time of the solvent and is absent for colloids with strong adsorption preferences. A Janus colloid, with a small temperature difference between its two hemispheres, reveals an asymmetric structure formation and surface enrichment around it, even if the solvent is within its one-phase region and if the temperature of the colloid is above the critical demixing temperature T_{c} of the solvent. Our phenomenological model turns out to capture recent experimental findings according to which, upon laser illumination of a Janus colloid and due to the ensuing temperature gradient between its two hemispheres, the surrounding binary liquid mixture develops a concentration gradient.

Mutual diffusion governed by kinetics and thermodynamics in the partially miscible mixture methanol + cyclohexane

To gain an understanding of the transport and thermodynamic behavior of the highly non-ideal mixture methanol + cyclohexane, three complementary approaches, i.e. experiment, molecular simulation and predictive equations, are employed. The temperature and composition dependence of different diffusion coefficients is studied around the miscibility gap at ambient pressure. On the one hand Fick diffusion coefficients are measured experimentally by interferometric probing and on the other hand Maxwell-Stefan diffusion coefficients and intradiffusion coefficients are sampled by equilibrium molecular dynamics simulation at five temperatures below the upper critical temperature of ∼319 K. The spinodal curve is determined from extrapolation of the experimental Fick diffusion coefficient data and compared to predictions from excess Gibbs energy models. It is found that these models are not capable to correctly describe the activity coefficients over the whole composition range of the studied mixture. Thus, different parameter sets for a modified Wilson model are used for calculations of the thermodynamic factor, which is needed to transform Maxwell-Stefan into Fick diffusion coefficients and vice versa. Further, predictive equations for the Maxwell-Stefan diffusion coefficient, which are based on intradiffusion coefficients, are compared to simulation results. Using different approaches provides a clearer understanding of the relations between kinetic and thermodynamic properties contributing to the diffusion behavior of partially miscible mixtures.

Renormalization-group theory for cooling first-order phase transitions in Potts models

We develop a dynamic field-theoretic renormalization-group (RG) theory for cooling first-order phase transitions in the Potts model. It is suggested that the well-known imaginary fixed points of the q-state Potts model for q>10/3 in the RG theory are the origin of the dynamic scaling found recently from numerical simulations, apart from logarithmic corrections. This indicates that the real and imaginary fixed points of the Potts model are both physical and control the scalings of the continuous and discontinuous phase transitions, respectively, of the model. Our one-loop results for the scaling exponents are already not far away from the numerical results. Further, the scaling exponents depend on q only slightly, consistent with the numerical results. Therefore, the theory is believed to provide a natural explanation of the dynamic scaling including the scaling exponents and their scaling laws for various observables in the cooling first-order phase transition of the Potts model.

Equilibrium and non-equilibrium concentration fluctuations in a critical binary mixture

When a macroscopic concentration gradient is present across a binary mixture, long-ranged non-equilibrium concentration fluctuations (NCF) appear as a consequence of the coupling between the gradient and spontaneous equilibrium velocity fluctuations. Long-ranged equilibrium concentration fluctuations (ECF) may be also observed when the mixture is close to a critical point. Here we study the interplay between NCF and critical ECF in a near-critical mixture aniline/cyclohexane in the presence of a vertical concentration gradient. To this aim, we exploit a commercial optical microscope and a simple, custom-made, temperature-controlled cell to obtain simultaneous static and dynamic scattering information on the fluctuations. We first characterise the critical ECF at fixed temperature T above the upper critical solution temperature T_c, in the wide temperature range [Formula: see text] ^°C. In this range, we observe the expected critical scaling behaviour for both the scattering intensity and the mass diffusion coefficient and we determine the critical exponents [Formula: see text], [Formula: see text] and [Formula: see text], which are found in agreement with the 3D Ising values. We then study the system in the two-phase region (T < T _c). In particular, we characterise the interplay between ECF and NCF when the mixture, initially at a temperature T_i, is rapidly brought to a temperature T _f > T _i. During the transient, a vertical diffusive mass flux is present that causes the onset of NCF, whose amplitude vanishes with time, as the flux goes to zero. We also study the time dependence of the equilibrium scattering intensity I _eq, of the crossover wave vector q _co and of the diffusion coefficient D during diffusion and find that all these quantities exhibit an exponential relaxation enslaved to the diffusive kinetics.

Off-equilibrium scaling behaviors driven by time-dependent external fields in three-dimensional O(N) vector models

We consider the dynamical off-equilibrium behavior of the three-dimensional O(N) vector model in the presence of a slowly varying time-dependent spatially uniform magnetic field H(t)=h(t)e, where e is an N-dimensional constant unit vector, h(t)=t/t(s), and t(s) is a time scale, at fixed temperature T≤T(c), where T(c) corresponds to the continuous order-disorder transition. The dynamic evolutions start from equilibrium configurations at h(i)<0, correspondingly t(i)<0, and end at time t(f)>0 with h(t(f))>0, or vice versa. We show that the magnetization displays an off-equilibrium scaling behavior close to the transition line H(t)=0. It arises from the interplay among the time t, the time scale t(s), and the finite size L. The scaling behavior can be parametrized in terms of the scaling variables t(s)(κ)/L and t/t(s)(κ(t)), where κ>0 and κ(t)>0 are appropriate universal exponents, which differ at the critical point and for T<T(c). In the latter case, κ and κ(t) also depend on the shape of the lattice and on the boundary conditions. We present numerical results for the Heisenberg (N=3) model under a purely relaxational dynamics. They confirm the predicted off-equilibrium scaling behaviors at and below T(c). We also discuss hysteresis phenomena in round-trip protocols for the time dependence of the external field. We define a scaling function for the hysteresis loop area of the magnetization that can be used to quantify how far the system is from equilibrium.

Splitting of the Universality Class of Anomalous Transport in Crowded Media

We investigate the emergence of subdiffusive transport by obstruction in continuum models for molecular crowding. While the underlying percolation transition for the accessible space displays universal behavior, the dynamic properties depend in a subtle nonuniversal way on the transport through narrow channels. At the same time, the different universality classes are robust with respect to introducing correlations in the obstacle matrix as we demonstrate for quenched hard-sphere liquids as underlying structures. Our results confirm that the microscopic dynamics can dominate the relaxational behavior even at long times, in striking contrast to glassy dynamics.

Superfluid phase transition with activated velocity fluctuations: Renormalization group approach

A quantum field model that incorporates Bose-condensed systems near their phase transition into a superfluid phase and velocity fluctuations is proposed. The stochastic Navier-Stokes equation is used for a generation of the velocity fluctuations. As such this model generalizes model F of critical dynamics. The field-theoretic action is derived using the Martin-Siggia-Rose formalism and path integral approach. The regime of equilibrium fluctuations is analyzed within the perturbative renormalization group method. The double (ε,δ)-expansion scheme is employed, where ε is a deviation from space dimension 4 and δ describes scaling of velocity fluctuations. The renormalization procedure is performed to the leading order. The main corollary gained from the analysis of the thermal equilibrium regime suggests that one-loop calculations of the presented models are not sufficient to make a definite conclusion about the stability of fixed points. We also show that critical exponents are drastically changed as a result of the turbulent background and critical fluctuations are in fact destroyed by the developed turbulence fluctuations. The scaling exponent of effective viscosity is calculated and agrees with expected value 4/3.

Critical slowing down near the multiferroic phase transition in MnWO4

By using broadband dielectric spectroscopy in the radio frequency and microwave range, we studied the magnetoelectric dynamics in the multiferroic chiral antiferromagnet MnWO_{4}. Above the multiferroic phase transition at T_{N2}≈12.6  K we observe a critical slowing of the corresponding magnetoelectric fluctuations resembling the soft-mode behavior in canonical ferroelectrics. This electric-field-driven excitation carries much less spectral weight than ordinary phonon modes. Also, the critical slowing down of this mode scales with an exponent larger than 1, which is expected for magnetic second-order phase transition scenarios. Therefore, the investigated dynamics have to be interpreted as the softening of an electrically active magnetic excitation, an electromagnon.

Non-equilibrium Casimir force between vibrating plates

We study the fluctuation-induced, time-dependent force between two plates confining a correlated fluid which is driven out of equilibrium mechanically by harmonic vibrations of one of the plates. For a purely relaxational dynamics of the fluid we calculate the fluctuation-induced force generated by the vibrating plate on the plate at rest. The time-dependence of this force is characterized by a positive lag time with respect to the driving. We obtain two distinctive contributions to the force, one generated by diffusion of stress in the fluid and another related to resonant dissipation in the cavity. The relation to the dynamic Casimir effect of the electromagnetic field and possible experiments to measure the time-dependent Casimir force are discussed.

Critical dynamics of an isothermal compressible nonideal fluid

A pure fluid at its critical point shows a dramatic slow-down in its dynamics, due to a divergence of the order-parameter susceptibility and the coefficient of heat transport. Under isothermal conditions, however, sound waves provide the only possible relaxation mechanism for order-parameter fluctuations. Here we study the critical dynamics of an isothermal, compressible nonideal fluid via scaling arguments and computer simulations of the corresponding fluctuating hydrodynamics equations. We show that, below a critical dimension of 4, the order-parameter dynamics of an isothermal fluid effectively reduces to "model A," characterized by overdamped sound waves and a divergent bulk viscosity. In contrast, the shear viscosity remains finite above two dimensions. Possible applications of the model are discussed.

Simulation of static critical phenomena in nonideal fluids with the lattice Boltzmann method

A fluctuating nonideal fluid at its critical point is simulated with the lattice Boltzmann method. It is demonstrated that the method, employing a Ginzburg-Landau free energy functional, correctly reproduces the static critical behavior associated with the Ising universality class. A finite-size scaling analysis is applied to determine the critical exponents related to the order parameter, compressibility and specific heat. A particular focus is put on finite-size effects and issues related to the global conservation of the order parameter.

Field theory of bicritical and tetracritical points. IV. Critical dynamics including reversible terms

This article concludes a series of papers [Folk, Holovatch, and Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)] where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here we give the complete two-loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described, in distances from the critical point accessible to experiments, by the critical behavior qualitatively found in one-loop order. Although one may conclude from the effective dynamic exponents (taking almost their one-loop values) that weak scaling for the order parameter components is valid, the flow of the time-scale ratios is quite different, and they do not reach their asymptotic values.

Dynamic scaling functions and amplitude ratios of stochastic models with energy conservation above Tc

Dynamical scaling functions above Tc for the characteristic frequencies and the dynamical correlation functions of the order parameter and the conserved density of model C are calculated in one loop order. By a proper exponentiation procedure these results can be extended in order to consider the changes in these functions using the fixed point values and exponents in two loop order. The dynamical amplitude ratio R of the characteristic frequencies is generalized to the critical region. Surprisingly the decay of the shape functions at large scaled frequency does not behave as expected from applying scaling arguments. The exponent upsilon of the decay does not change when going from the critical to the hydrodynamic region although the shape functions change. The value of upsilon for the order parameter is in agreement with its value in the critical region, whereas for the conserved density it is equal to 2, the value in the hydrodynamic region.

Field theory of bicritical and tetracritical points. III. Relaxational dynamics including conservation of magnetization (model C)

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.

Field theory of bicritical and tetracritical points. I. Statics

We calculate the static critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by the renormalization group method within the minimal subtraction scheme in two-loop order. Summation methods lead to fixed points describing multicritical behavior. Their stability border lines in the space of the order parameter components n_|| and n_perpendicular and spatial dimension d are calculated. The essential features obtained already in two-loop order for the interesting case of an antiferromagnet in a magnetic field ( n_|| =1, n_perpendicular =2 ) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points, leading to very small transient exponents. We are also able to calculate the flow of static couplings, which allows us to consider the attraction region. Depending on the nonuniversal background parameters, the existence of different multicritical behavior (bicritical or tetracritical) is possible, including a triple point.

Field theory of bicritical and tetracritical points. II. Relaxational dynamics

We calculate the relaxational dynamical critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by renormalization group method within the minimal subtraction scheme in two-loop order. The three different bicritical static universality classes previously found for such systems correspond to three different dynamical universality classes within the static borderlines. The Heisenberg and the biconical fixed point lead to strong dynamic scaling whereas in the region of stability of the decoupled fixed point weak dynamic scaling holds. Due to the neighborhood of the stability border between the strong and the weak scaling dynamic fixed point to the dynamical stable fixed point a very small dynamic transient exponent of omega(Beta)_(v) =0.0044 is present in the dynamics for the physically important case n_|| =1 and n_perpendicular =2 in d=3 .

Static and dynamic structure factors in three-dimensional randomly diluted Ising models

We consider the three-dimensional randomly diluted Ising model and study the critical behavior of the static and dynamic spin-spin correlation functions (static and dynamic structure factors) at the paramagnetic-ferromagnetic transition in the high-temperature phase. We consider a purely relaxational dynamics without conservation laws, the so-called model A. We present Monte Carlo simulations and perturbative field-theoretical calculations. While the critical behavior of the static structure factor is quite similar to that occurring in pure Ising systems, the dynamic structure factor shows a substantially different critical behavior. In particular, the dynamic correlation function shows a large-time decay rate which is momentum independent. This effect is not related to the presence of the Griffiths tail, which is expected to be irrelevant in the critical limit, but rather to the breaking of translational invariance, which occurs for any sample and which, at the critical point, is not recovered even after the disorder average.

Determination of the dynamic and static critical exponents of the two-dimensional three-state Potts model using linearly varying temperature

We employ a successive Monte Carlo renormalization group procedure in the presence of a linearly varying temperature to study the three-state (q=3) Potts model on square lattices. By matching correlation functions of different lattice sizes at different renormalization levels, a rate exponent r associated with the temperature sweep rate R and the correlation length exponent nu are obtained. The dynamic critical exponent z is then obtained by the scaling law z=r-1/nu derived in our method. The results of z=2.171(62) for q=3 in this work and the previously obtained z=2.15(13) for q=2 seem to support the extension of the "weak universality" hypothesis to dynamic critical behavior. With these calculated exponents, the dynamic scaling forms of both specific heat and order parameter at various R are presented and all the other static critical exponents are determined, which verify our method. Discussions are made on how to improve the accuracy of the estimation of the dynamic critical exponent.

Spin-dynamics simulations of the XY vector Blume-Emery-Griffiths model in three dimensions

We present extensive Monte Carlo spin-dynamics simulations of the vector Blume-Emery-Griffiths model with three-dimensional spins on a simple cubic lattice and periodic boundary conditions. For a range of Hamiltonian parameters, this model reproduces the phase diagram topology of the one experimentally observed for bulk mixtures of 3He-4He . Highly efficient decomposition time integration algorithms have been used to obtain the evolution of the equations of motion from which we have computed the dynamic structure factor S(q,omega) close to critical and tricritical points. The dispersion relation and the corresponding dynamic exponents have been obtained in the critical and multicritical transitions. We find that the dynamic critical exponents for in-plane and out-of-plane fluctuations are the same at the tricritical point but are in fact different along the critical line.