Static Universality Class Research Articles

Dynamical Renormalization Group for Mode-Coupling Field Theories with Solenoidal Constraint

The recent inflow of empirical data about the collective behaviour of strongly correlated biological systems has brought field theory and the renormalization group into the biophysical arena. Experiments on bird flocks and insect swarms show that social forces act on the particles’ velocity through the generator of its rotations, namely the spin, indicating that mode-coupling field theories are necessary to reproduce the correct dynamical behaviour. Unfortunately, a theory for three coupled fields—density, velocity and spin—has a prohibitive degree of intricacy. A simplifying path consists in getting rid of density fluctuations by studying incompressible systems. This requires imposing a solenoidal constraint on the primary field, an unsolved problem even for equilibrium mode-coupling theories. Here, we perform an equilibrium dynamic renormalization group analysis of a mode-coupling field theory subject to a solenoidal constraint; using the classification of Halperin and Hohenberg, we can dub this case as a solenoidal Model G. We demonstrate that the constraint produces a new vertex that mixes static and dynamical coupling constants, and that this vertex is essential to grant the closure of the renormalization group structure and the consistency of dynamics with statics. Interestingly, although the solenoidal constraint leads to a modification of the static universality class, we find that it does not change the dynamical universality class, a result that seems to represent an exception to the general rule that dynamical universality classes are narrower than static ones. Our results constitute a solid stepping stone in the admittedly large chasm towards developing an off-equilibrium mode-coupling theory of biological groups.

Magnetic field driven critical behavior in bulk Gd

Gadolinium (Gd) has been proven to be such a complex magnetic system that its static universality class varies as a function of the reduced temperature (|ɛ| = |T−TC|/TC). Though many works have been done, physical mechanisms of this phenomenon have not been fully elucidated. In this work, we analyzed the static magnetization data, M(H), around the ferromagnetic-paramagnetic phase transition temperature (TC) based on Banerjee's criteria, the modified Arrott plot and Kouvel–Fisher techniques, the critical isotherm analysis, and the scaling hypothesis to reconsider the phase transition, magnetic order, and isotropic and uniaxial behaviors of a polycrystalline Gd bulk. The M(H) data analyses with H variations up to 140 kOe indicated that Gd still keeps the nature of a second-order phase transition and short-range magnetic order. In the critical region of |ɛ| = 3.4 × 10−3–6.9 × 10−2, we observed the critical-behavior transition from a mixed isotropic and uniaxial ferromagnet for H = 3–10 kOe through an isotropic 3D Heisenberg ferromagnet for H = 10–100 kOe to a uniaxial 3D Ising ferromagnet for H = 100–140 kOe. The results demonstrate the possibility of using high fields to identify the universality class of Gd. The fluctuations and preferred reorientation of spins in the hexagonal close-packed lattice at temperatures T > 230 K are thought to cause these interesting phenomena.

Dynamic universality class of model H with frustrated diffusion: ε expansion

We study a variation of the dynamic universality class of model H in a spatial dimension of $d=4-\epsilon$, by frustrating charge diffusion and momentum density fluctuations along $d_T=1$ or $d_T=2$ dimensions, while keeping the same dynamics of model H in the other $d_L=d-d_T$ dimensions. The case of $d_T=2$ describes the QCD critical point in a background magnetic field. We find that these models belong to a different dynamical universality class due to extended conservation laws compared to the model H, although the static universality class remains the same as the 3-dimensional Ising model. We compute the dynamic critical exponents of these models in first order of $\epsilon$-expansion to find that $x_\lambda\approx 0.847\,\epsilon$, $x_{\bar\eta}\approx 0.153\,\epsilon$, and $z=4-x_\lambda\approx 3.15$ when $\epsilon=1$ and $d_T=2$. For $d_T=1$ the results are numerically similar to the model H values: $z\approx 3.08$.

Thermal conductivity and chiral critical point in heavy ion collisions

Background: Quantum chromodynamics is expected to have a phase transition in the same static universality class as the 3D Ising model and the liquid-gas phase transition. The properties of the equation of state, the transport coefficients, and especially the location of the critical point are under intense theoretical investigation. Some experiments are underway, and many more are planned, at high-energy heavy ion accelerators.Purpose: Develop a model of the thermal conductivity, which diverges at the critical point, and use it to study the impact of hydrodynamic fluctuations on observables in high-energy heavy ion collisions.Methods: We apply mode-coupling theory, together with a previously developed model of the free energy that incorporates the critical exponents and amplitudes, to construct a model of the thermal conductivity in the vicinity of the critical point. The effect of the thermal conductivity on correlation functions in heavy ion collisions is studied in a boost-invariant hydrodynamic model via fluctuations, or noise.Results: We find that the closer a thermodynamic trajectory comes to the critical point the greater is the magnitude of the fluctuations in thermodynamic variables and in the 2-particle correlation functions in momentum space.Conclusions: It may be possible to discern the existence of a critical point, its location, and thermodynamic and transport properties near to it in heavy ion collisions using the methods developed here.

Cluster diffusion at the gelation point.

We use molecular dynamics simulations to study a model of the gelation transition with a dynamic bond forming procedure. After establishing evidence for three-dimensional (3D) percolation as the static universality class, we turn our attention to the dynamics of clusters at the gelation point, focusing in particular on the behavior of the diffusion constant D(s) as a function of cluster size s. We find a very clear power law behavior D(s) approximately s(-k) with a nontrivial exponent k approximately 0.69.

Relaxational dynamics study of the classical Heisenberg spinXYmodel in spherical coordinate representation

The two- and three-dimensional classical Heisenberg spin $\mathrm{XY}$ $(\mathrm{CHS}XY)$ models, with the spherical coordinates of spins taken as dynamic variables, are numerically investigated. We allow the polar \ensuremath{\theta} and azimuthal $\ensuremath{\varphi}$ angles to have uniform values in $[0,\ensuremath{\pi})$ and $[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi}),$ respectively, and the static universality class is shown to be identical to the classical $\mathrm{XY}$ model with two-component spins, as well as the $\mathrm{CHS}\mathrm{XY}$ model with a different choice of dynamic variables, conventionally used in the literature. The relaxational dynamic simulation reveals that the dynamic critical exponent z is found to have the value $z\ensuremath{\approx}2.0$ for both two and three dimensions, in contrast to $z\ensuremath{\approx}d/2$ $(d=$ spatial dimension) found previously with spin dynamics simulation of the conventional $\mathrm{CHS}\mathrm{XY}$ model. Comparisons with the usual two-component classical $\mathrm{XY}$ model are also made.

Static universality class for gadolinium

High-precision magnetization, $M(T,H),$ data have been taken along the c axis (easy direction of magnetization) of a high-purity Gd single crystal in the critical region near the ferromagnetic-paramagnetic phase transition. Elaborate data analyses demonstrate that the single power laws, by themselves, do not adequately describe the observed field dependence of M at the Curie point ${T}_{C},$ ${M(T}_{C},H),$ and the temperature variations of spontaneous magnetization, $M(T,0),$ and initial susceptibility, $\ensuremath{\chi}(T),$ in the asymptotic critical region $|\ensuremath{\epsilon}|=|(T\ensuremath{-}{T}_{C}{)/T}_{C}|l~2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3},$ but do so only when the multiplicative logarithmic corrections (LC), predicted by the renormalization group (RG) calculations for dipolar Ising (spin dimensionality $n=1)$ spin systems at the upper marginal space dimension ${d}^{*}=3,$ are taken into account. Such data analyses also permit the first accurate determination of LC exponents ${(x}^{\ensuremath{'}},x),$ the asymptotic critical exponents $\ensuremath{\beta},$ $\ensuremath{\gamma},$ and $\ensuremath{\delta},$ and critical amplitudes $\mathrm{B\ifmmode \hat{}\else \^{}\fi{}},$ $\stackrel{^}{\ensuremath{\Gamma}},$ and $\mathrm{D\ifmmode \hat{}\else \^{}\fi{}}$ for $M(T,0),$ $\ensuremath{\chi}(T),$ and ${M(T}_{C},H).$ The exponents ${x}^{\ensuremath{'}},$ x, $\ensuremath{\beta},$ $\ensuremath{\gamma},$ and $\ensuremath{\delta},$ as well as the universal amplitude ratio ${R}_{\ensuremath{\chi}}=\mathrm{D\ifmmode \hat{}\else \^{}\fi{}}{B}^{\ensuremath{\delta}\ensuremath{-}1}\stackrel{^}{\ensuremath{\Gamma}}$ possess the same (within the uncertainty limits) values as those yielded by the RG calculations for a $d=3$ uniaxial dipolar ferromagnet. Moreover, the presently determined values of $\ensuremath{\beta},$ $\ensuremath{\gamma},$ and $\ensuremath{\delta},$ together with the reported value of the specific heat critical exponent $\ensuremath{\alpha},$ obey the scaling relations $\ensuremath{\beta}+\ensuremath{\gamma}=\ensuremath{\beta}\ensuremath{\delta}$ and $\ensuremath{\alpha}+2\ensuremath{\beta}+\ensuremath{\gamma}=2$ accurately. By establishing that gadolinium belongs to the $d=3,$ $n=1$ dipolar static universality class, the present results resolve the long-standing controversy surrounding the nature of the asymptotic critical behavior of Gd.

Magnetic-field dependence of the critical dynamics of a superconductingYBa2Cu3O7−δdetwinned single crystal

The in-plane longitudinal resistivities ${\ensuremath{\rho}}_{\mathrm{aa}}$ and ${\ensuremath{\rho}}_{\mathrm{bb}}$ of a single-crystal of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ have been measured as a function of magnetic field $H$ (less than 6.75 T) and temperature $T$ near ${T}_{c}.$ We found that the ${\ensuremath{\rho}}_{\mathrm{aa}}(T,H)$ data in the mixed state collapse into a single curve as a function of ${\ensuremath{\rho}}_{\mathrm{bb}}(T,H).$The extracted fluctuation conductivity exhibits dynamic scaling over a wide range of $T$ and $H:$ assuming that the static universality class is the three-dimensional $\mathrm{XY}$ model yields the observed dynamic critical exponent $z=1.5\ifmmode\pm\else\textpm\fi{}0.1(H>0),$ apparently consistent with the dynamic universality class of superfluid ${}^{4}\mathrm{He}.$ In zero field, the data are inconclusive, and a reliable determination of $z$ is not possible.

Dynamic critical phenomena in aqueous protein solutions.

An aqueous protein solution is a binary mixture of large particles (protein molecules) and small particles (water molecules). In contrast, a binary liquid mixture is a mixture of two different kinds of particles that are approximately the same size. Thus, a comparison of the critical behavior in these two systems allows one to determine the effect on critical behavior of a relative difference of size in the two kinds of particles in a binary mixture. The correlation length j, osmotic compressibility kT , and coexistence curve of aqueous protein solutions in the vicinity of the critical point for liquid-liquid phase separation have been reported recently [1‐3]. These experimental results established that the static critical properties of these solutions are consistent with the behavior of members of the static universality class of three-dimensional systems with short range interactions and scalar order parameters. This universality class includes binary liquid mixtures. In contrast to the equilibrium properties, the dynamic properties of aqueous protein solutions in the vicinity of the critical point have not been investigated as thoroughly. In this Letter, we report our use of quasielastic light scattering (QLS) to investigate the time decay of spontaneous concentration fluctuations, as a function of both wave number q of the fluctuations and temperature T, along the critical isochore of aqueous solutions of the bovine eye lens protein gII-crystallin. The only previous study of critical dynamics in a protein solution of which we are aware is the work of Ishimoto and Tanaka who made QLS measurements at a single fixed wave number of the fluctuations using a 19 channel digital correlator on aqueous solutions of lysozyme [4] along the critical isochore for sT 2 TcdyTc $ 5 3 10 23 (q j, 0.1), where Tc is the critical temperature. The present study is a significant experimental advance over the work of Ishimoto and Tanaka in that (i) we obtain data for fluctuations at up to 12 different wave numbers, (ii) we obtain data using a 144 channel digital correlator, and (iii) we obtain data much closer to the critical point [ sT 2 TcdyTc $ 1.56 3 10 24 and q j# 3.18]. The more thorough investigation of critical dynamics presented here has revealed new and unexpected behavior. Specifically, we find that the wave number and temperature dependence of the average rate of decay of the concentration fluctuations are consistent with the theory for critical dynamics in binary liquid mixtures [5] only if we allow both the background viscosity and the background contribution to the decay rate to be unusually large. Furthermore, in contrast to the behavior seen in binary liquid mixtures we find that the concentration fluctuations exhibit very significant deviation from exponential decay with time. Of course, these protein solutions may be viewed as

Static universality class implied by the critical exponents of Gd.

Structural characteristics and available critical exponent measurements fail to distinguish clearly between Ising and Heisenberg critical behavior in Gd. In this context we present a new perturbed \ensuremath{\gamma}\ensuremath{\gamma}-angular correlation measurement of the exponent \ensuremath{\beta} which appears to resolve the ambiguity. From our result, \ensuremath{\beta}=0.399(16), we conclude that Gd is described by Heisenberg critical behavior for reduced temperatures \ensuremath{\epsilon}${>10}^{\mathrm{\ensuremath{-}}3}$. This does not exclude crossover to another value for smaller \ensuremath{\epsilon}.