Dynamical Correlation Functions in the Two-Dimensional Kinetic Ising Model: A Real-Space Renormalization-Group Approach

We calculate the static and dynamic single-particle correlation functions in one-dimensional (1D) trapped Bose gases and discuss experimental measurements that can directly probe such correlation functions. Using a quantized hydrodynamic theory for the low energy excitations, we calculate both the static and dynamic single-particle correlation functions for a 1D Bose gas that is a phase-fluctuating quasi-condensate. For the static (equal-time) correlation function, our approximations and results are equivalent to those of Petrov, Shlyapnikov and Walraven. The Fourier transform of the static single-particle correlation function gives the momentum distribution, which can be measured using Doppler-sensitive Bragg scattering experiments on a highly elongated Bose gas. We show how a two-photon Raman out-coupling experiment can measure the characteristic features of the dynamic or time-dependent single-particle correlation function of a 1D Bose quasi-condensate.

We make use of the calculation of Achiam and Kosterlitz and investigate the generalization of the Migdal-type renormalization-group calculation to critical dynamics, by looking at the one- and two-dimensional kinetic Ising model with no conserved magnetization. In the two-dimensional case the dynamical critical index ${Z}_{M}(\mathrm{magnetic}\mathrm{perturbation})=2.064$ and ${Z}_{E}(\mathrm{energylike}\mathrm{perturbation})=1.819$ for scale factor $\ensuremath{\lambda}=1$. ${Z}_{M}$ involves the static $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ exponent, while ${Z}_{E}$ involves $\frac{1}{\ensuremath{\nu}}$, and therefore, it is not surprising that ${Z}_{M}$ is closer to the high-temperature expansion results, since $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ in the Migdal approximation for statics is much closer to the exact value than $\frac{1}{\ensuremath{\nu}}$. In one dimension we obtain ${Z}_{M}={Z}_{E}=2$, which is the exact result of Glauber.

Renormalization-group (RG) methods developed previously for the study of the growth of order in unstable systems are extended to treat the spinodal decomposition of the two-dimensional spin-exchange kinetic Ising model. The conservation of the order parameter and fixed-length sum rule are properly preserved in the theory. Various correlation functions in both coordinate and momentum space are calculated as functions of time. The scaling function for the structure factor is extracted. We compare our results with direct Monte Carlo (MC) simulations and find them in good agreement. The time rescaling parameter entering the RG analysis is temperature dependent, as was determined in previous work through a RG analysis of MC simulations. The results exhibit a long-time logarithmic growth law for the typical domain size, both analytically and numerically. In the time region where MC simulations have previously been performed, the logarithmic growth law can be fitted to a power law with an effective exponent. This exponent is found to be in excellent agreement with the result of MC simulations. The logarithmic growth law agrees with a physical model of interfacial motion which involves an interplay between the local curvature and an activated jump across the interface.

We develop a general approach to calculating ``nonuniversal'' prefactors in static and dynamic correlation functions of one-dimensional (1D) quantum liquids at zero temperature by relating them to the finite-size scaling of certain matrix elements (form factors). This represents a powerful tool for extracting data valid in the thermodynamic limit from finite-size effects. As the main application, we consider weakly interacting spinless fermions with an arbitrary pair interaction potential, for which we perturbatively calculate certain prefactors in static and dynamic correlation functions. We also evaluate prefactors of the long-distance behavior of correlation functions nonperturbatively for the exactly solvable Lieb-Liniger model of 1D bosons.

We study the dynamical response of a two-dimensional Ising model subject to a square-wave oscillating external field. In contrast to earlier studies, the system evolves under a so-called soft Glauber dynamic [Rikvold and Kolesik, J. Phys. A 35, L117 (2002)], for which both nucleation and interface propagation are slower and the interfaces smoother than for the standard Glauber dynamic. We choose the temperature and magnitude of the external field such that the metastable decay of the system following field reversal occurs through nucleation and growth of many droplets of the stable phase, i.e., the multidroplet regime. Using kinetic Monte Carlo simulations, we find that the system undergoes a nonequilibrium phase transition, in which the symmetry-broken dynamic phase corresponds to an asymmetric stationary limit cycle for the time-dependent magnetization. The critical point is located where the half period of the external field is approximately equal to the metastable lifetime of the system. We employ finite-size scaling analysis to investigate the characteristics of this dynamical phase transition. The critical exponents and the fixed-point value of the fourth-order cumulant are found to be consistent with the universality class of the two-dimensional equilibrium Ising model. This universality class has previously been established for the same nonequilibrium model evolving under the standard Glauber dynamic, as well as in a different nonequilibrium model of CO oxidation. The results reported in the present paper support the hypothesis that this far-from-equilibrium phase transition is universal with respect to the choice of the stochastic dynamics.

A two-dimensional lattice-gas model with square symmetry is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. It has been shown that the precision of the method depends strongly not only on the number of sites in the block but also on its symmetry. In general, the accuracy of the method increases with the number of sites in the block. The minimal relative error in determining the critical values of the interaction parameters is equal to \(\). Using the RSRG method, we have explored phase diagrams of both a two-dimensional Ising spin model and of a square lattice gas with lateral interactions between adparticles. We also have investigated the influence of the attractive and repulsive interactions on both the thermodynamic properties of the lattice gas and the diffusion of adsorbed particles over surface. We have calculated adsorption isotherms and coverage dependences of the pair correlation function, isothermal susceptibility and the chemical diffusion coefficient. In addition, we have included in our analysis the interaction of the activated particle in the saddle point with its nearest neighbors. We have also used Monte Carlo (MC) technique to calculate these dependences. Despite the fact that both methods constitute very different approaches, the correspondence of the numerical data is surprisingly good. Therefore, we conclude that the RSRG approach can be applied to characterize the thermodynamic and kinetic properties of systems of particles with strong lateral interactions.

The usual spin-wave theory, when applied to the calculation of rotationally invariant averages in the Heisenberg model, permits exact calculation of static and dynamical correlation functions in one dimension. The results are also believed to be exact in two dimensions, although this has not been show rigorously here. The one-dimensional ferromagnet in the hydrodynamic regime and the two-dimensional ferromagnet at all wave vectors do not satisfy the dynamical scaling hypothesis. There is no characteristic exponent in these cases. The present results contradict the prediction of the mode-coupling theory.

The static and dynamic properties of a dilute binary macroion mixture, where a small amount of large and highly charged spheres is immersed in a system of small spheres, are theoretically discussed. From light scattering experiments on such systems, the self-diffusion of the large spheres within the system of small spheres is essentially determined. Such systems have recently been studied by Phillies, using suspensions of highly charged polystyrene spheres. Modelling the elements of the memory function matrix by single exponentials, relating the k-dependent amplitudes and relaxation rates to the moments of the one-particle propagators, compact analytic expressions are derived for the normalized mean squared displacements and long-time self-diffusion coefficients. The moments are calculated from a generalized Smoluchowski equation and are given by integrals over the static correlation functions. Using the so-called two-component macrofluid model to calculate the static correlation functions, analytic expressions of the static partial total correlation functions in k-space are derived within the mean-spherical approximation for the binary mixture. Because of the strong dilution of the mixture, a rescaling procedure has to be used. This rescaling procedure is a simple generalization of that given by Hansen and Hayter for the one-component macroion fluid. Comparison of the theoretically calculated long-time self-diffusion of the large spheres with the experimental results of Phillies was quite satisfactory. Also, calculations of the two self-diffusion coefficients as functions of the diameter ratio and salt concentration have been performed, as well as of the normalized mean squared displacements.Key wordsColloidal suspensionself-diffusionBrownian motion

We explore the emergence of a discrete time crystalline (DTC) order and its stability against thermal fluctuations in a driven kinetic Ising model on a two-dimensional square lattice using the drive protocol invented in a recent work (Gambetta et al 2019 Phys. Rev. E 100 060105(R)). The DTC order is found to be quite robust in the presence of thermal fluctuations. We construct the resulting three-dimensional phase diagram for the DTC order, which manifests a striking resemblance to the jamming phase diagram proposed by Liu and Nagel. This finding may suggest a new way to view the DTC order as a new type of nonequilibrium soft matter. The quench dynamics exhibits a unique feature due to the nature of the employed drive protocol, namely, breakdown of the inverse relationship between the domain growth and defect relaxation, which holds in the usual quench dynamics of the kinetic Ising model.

The static and dynamic pair correlation functions of the Ising mixture were obtained and investigated on the basis of cluster expansion of the free energy and cluster approximation to Glauber equation for unary correlation function.

We characterize the steady-state static and dynamic properties in a broad class of mass transport processes on a periodic hypercubic lattice of volume Ld , where both the mass and center-of-mass (CoM) remain conserved and the detailed balance is violated in the bulk; we specifically consider the models in the d = 1 and 2 dimensions. Using a microscopic approach, we exactly determine the decay (or growth) exponents for various dynamic and static correlation functions. We show that despite constrained dynamics due to CoM conservation (CoMC), the density relaxation is indeed diffusive. However, the fluctuation properties are strikingly different from those in diffusive systems with a single (mass) conservation law as dynamic and static fluctuations are more suppressed in systems with CoMC, resulting in an extreme (‘class-I’) hyperuniformity in certain cases. In the thermodynamic limit, the steady-state variance ⟨ Q 2 ( T ) ⟩ c of time-integrated bond current Q ( T ) across a bond in the time interval T exhibits the following long-time behavior: ⟨ Q 2 ( T ) ⟩ c ≃ A 1 T + A 2 + A 3 T − d / 2 . The exponents governing the small-frequency behavior of the power spectra S J ( f ) ≃ A 1 + Const . f ψ J for bond current are exactly determined as ψ J = 3 / 2 and 2 in d = 1 and 2 dimensions, respectively; the corresponding unequal-time current-current correlation function decay as t − 5 / 2 and t −3 as a function of time t in d = 1 and 2 dimensions, respectively. Remarkably, depending on the dimensions and microscopic details, the prefactor A 1 can vanish, causing the variance to eventually saturate, implying a class-I ‘dynamic hyperuniformity’. We also compute the static structure factor S(q), which varies as the square of the wave number q in the small-q limit, i.e. S ( q ) ∼ q 2 , implying a class-I spatial hyperuniformity.

The Maier-Saupe-Zwanzig model for the nematic phase transitions in liquid crystals is investigated in a diamond hierarchical lattice. The model takes into account a parameter to describe the biaxiality of the microscopic units. Also, a suitably chosen external field is added to the Hamiltonian to allow the determination of critical parameters associated with the nematic phase transitions. Using the transfer-matrix technique, the free energy and its derivatives are obtained in terms of recursion relations between successive generations of the hierarchical lattice. In addition, a real-space renormalization-group approach is developed to obtain the critical parameters of the same model system. Results of both methods are in excellent agreement. There are indications of two continuous phase transitions. One of them corresponds to a uniaxial-isotropic transition, in the class of universality of the three-state Potts model on the diamond hierarchical lattice. The transition between the biaxial and the uniaxial phases is in the universality class of the Ising model on the same lattice.

We perform a numerical study of the kinetic Blume-Capel (BC) model to find if it exhibits the metamagnetic anomalies previously observed in the kinetic Ising model for supercritical periods [P. Riego et al., Phys. Rev. Lett. 118, 117202 (2017)0031-900710.1103/PhysRevLett.118.117202; G. M. Buendía et al., Phys. Rev. B 96, 134306 (2017)2469-995010.1103/PhysRevB.96.134306]. We employ a heat-bath Monte Carlo (MC) algorithm on a square lattice in which spins can take values of ±1,0, with a nonzero crystal field, subjected to a sinusoidal oscillating field in conjunction with a constant bias. In the ordered region, we find an equivalent hysteretic response of the order parameters with its respective conjugate fields between the kinetic and the equilibrium model. In the disordered region (supercritical periods), we observed two peaks, symmetrical with respect to zero bias, in the susceptibility and scaled variance curves, consistent with the numerical and experimental findings on the kinetic Ising model. This behavior does not have a counterpart in the equilibrium model. Furthermore, we find that the peaks occur at higher values of the bias field and become progressively smaller as the density of zeros, or the amplitude of the oscillating field, increases. Using nucleation theory, we demonstrate that these fluctuations, as in the Ising model, are not a critical phenomenon, but that they are associated with a crossover between a single-droplet (SD) and a multidroplet (MD) magnetization switching mechanism. For strong (weak) bias, the SD (MD) mechanism dominates. We also found that the zeros concentrate on the droplets' surfaces, which may cause a reduced interface tension in comparison with the Ising model [M. Schick et al., Phys. Rev. B 34, 1797 (1986)0163-182910.1103/PhysRevB.34.1797]. Our results suggest that metamagnetic anomalies are not particular to the kinetic Ising model, but rather are a general characteristic of spin kinetic models, and provide further evidence that the equivalence between dynamical phase transitions and equilibrium ones is only valid near the critical point.

First results of massive Monte Carlo simulations of the d=3 Ising spin glass with ± J bond distribution, performed on a fast special purpose computer, are discussed. Equilibrium static and dynamic correlation functions have been recorded for temperatures T/J=1.10 to 2.00. A qualitative change in the behavior of the system, and the best fits for the spin glass correlation length and relaxation time favor a continuous phase transition at Tc/J≊1.20±0.05. We find no evidence for the Tc=0 freezing picture.

The dynamical properties of liquid caesium near the melting point are investigated within the generalized collective-modes approach in a Markovian approximation. The generalized thermodynamic quantities and spectra of collective modes have been obtained for the five-variables description of longitudinal fluctuations. As the basis variables, three conserved and two nonconserved quantities have been used: the number density, longitudinal momentum, and energy density; and the first time derivatives of the longitudinal momentum and the energy density. All of the static and dynamic correlation functions were calculated directly from molecular dynamics simulations for 500 particles over 165 000 time steps, avoiding any fitting parameters. A comparison with the experimental data and results of previous molecular dynamics simulations is made.