Monte Carlo study of tricritical dynamics in two dimensions

The dynamical tricritical behavior for the spin-1 Ising model with single-ion interaction is investigated in two dimensions using Monte Carlo simulations. The nonlinear dynamical tricritical exponentz _t is determined from the asymptotic power-law relaxation of the magnetization. The valuez _t = 1.99 ± 0.04 reported here is the first estimate of the dynamical exponent at a multicritical point, in two dimensions.On leave from Departamento de Fisica, Universidade Federal de Pernambuco, Recife-PE, Brazil.     

Renormalization group transformation for the master equation of a kinetic Ising chain

We define and solve a dynamical real-space renormalization group transformation for the master equation of a kinetic Ising model in one dimension.The exactly solvable model we treat has the dynamical exponentz=2. We show that spin flip rates differing from Glauber's choice may have exponentsz differing from 2. A particular model which hasz=4 is presented.     

Duality relations for asymmetric exclusion processes

We derive duality relations for a class ofU _q [SU(2)]-symmetric stochastic processes, including among others the asymmetric exclusion process in one dimension. Like the known duality relations for symmetric hopping processes, these relations express certainm-point correlation functions inN-particle systems (Nm) in terms of sums of correlation functions of the same system but with onlym particles. For the totally asymmetric case we obtain exact expressions for some boundary density correlation functions. The dynamical exponent for these correlators isz=2, which is different from the dynamical exponent for bulk density correlations, which is known to bez=3/2.     

Single-cluster Monte Carlo dynamics for the Ising model

We present an extensive study of a new Monte Carlo acceleration algorithm introduced by Wolff for the Ising model. It differs from the Swendsen-Wang algorithm by growing and flipping single clusters at a random seed. In general, it is more efficient than Swendsen-Wang dynamics ford>2, giving zero critical slowing down in the upper critical dimension. Monte Carlo simulations give dynamical critical exponentsz _w=0.33±0.05 and 0.44+0.10 ind=2 and 3, respectively, and numbers consistent withz _w=0 ind=4 and mean-field theory. We present scaling arguments which indicate that the Wolff mechanism for decorrelation differs substantially from Swendsen-Wang despite the apparent similarities of the two methods.     

High-temperature series expansion for the relaxation times of the two dimensional Ising model

We derive the high temperature series expansions for the two relaxation times of the single spin-flip kinetic Ising model on the square lattice. The series for the linear relaxation time _l is obtained with 20 non-trivial terms, and the analysis yields 2.183±0.005 as the value of the critical exponent _l , which is equal to the dynamical critical exponentz in the two-dimensional case. For the non-linear relaxation time we obtain 15 non-trivial terms, and the analysis leads to the results _nl = 2.08 ± 0.07. The scaling relation _l – _nl = ( being the exponent of the order parameter) seems to be fulfilled, though the error bars of _nl are still quite substantial. In addition, we obtain the series expansion of the linear relaxation time on the honeycomb lattice with 22 non-trivial terms. The result for the critical exponent is close to the value obtained on the square lattice, which is expected from universality.     

Scaling factors associated withM-furcations of the 1−μ∥x∥z map

A numerical study is made of the scaling behavior associated withM-furcations (M=3, 4, 5) in the mapx _t+1 =1–x _t ^z (z>1). The scaling constants and are calculated as functions ofz, as well as the more general scaling functions andf(a).     

Stretching dynamics of semiflexible polymers

We analyze the nonequilibrium dynamics of single inextensible semiflexible biopolymers as stretching forces are applied at the ends. Based on different (contradicting) heuristic arguments, various scaling laws have been proposed for the propagation speed of the backbone tension which is induced in response to stretching. Here, we employ a newly developed unified theory to systematically substantiate, restrict, and extend these approaches. Introducing the practically relevant scenario of a chain equilibrated under some prestretching force f _pre that is suddenly exposed to a different external force f _ext at the ends, we give a concise physical explanation of the underlying relaxation processes by means of an intuitive blob picture. We discuss the corresponding intermediate asymptotics, derive results for experimentally relevant observables, and support our conclusions by numerical solutions of the coarse-grained equations of motion for the tension.     

Critical acceleration of lattice gauge simulations

We present a stochastic cluster algorithm that drastically reduces critical slowing down forZ ₂ lattice gauge theory in three dimensions. The dynamical exponentz is reduced fromz>2 (standard Metropolis algorithm) tozO.73. The Monte Carlo pseudodynamics acts on the gauge-invariant flux tubes that are known to be the relevant large-scale low-energy excitations. A comparison of our results with known results for the 3D Ising model and ⁴ model supports the conjecture of universality classes for stochastic cluster algorithms.     

Metastability and Spinodal Points for a Random Walker on a Triangle

We investigate time-dependent properties of a single-particle model in which a random walker moves on a triangle and is subjected to nonfocal boundary conditions. This model exhibits spontaneous breaking of a Z ₂ symmetry. The reduced size of the configuration space (compared to related many-particle models that also show spontaneous symmetry breaking) allows us to study the spectrum of the time evolution operator. We break the symmetry explicitly and find a stable phase, and a metastable phase which vanishes at a spinodal point. At this point, the spectrum of the time evolution operator has a gapless and universal band of excitations with a dynamical critical exponent z=1. Surprisingly, the imaginary parts of the eigenvalues E _j(L) are equally spaced, following the rule . Away from the spinodal point, we find two time scales in the spectrum. These results are related to scaling functions for the mean path of the random walker and to first passage times. For the spinodal point, we find universal scaling behavior. A simplified version of the model which can be handled analytically is also presented.     

The effects of random fields on the critical and bicritical behavior of Mn x Zn1−x F2

We have carried out a comprehensive study of the static and dynamic spin-spin correlations of Mn _x Zn_1–x F₂ in a magnetic field. Samples withx=0.75 andx=0.5 have been studied. This system exhibits behavior closely related, if not identical, to that of the Random Field Ising Model (RFIM). An additional feature of Mn _x Zn_1–x F₂ is that it exhibits an easily accessible bicritical point; thus one can study the changeover from the RFIM to the uniformXY model with a transverse random field. Quite generally, the instantaneous spin-spin correlations in a field are described by a combination of Lorentzian, Lorentzian-squared and delta function terms the latter corresponds to the long range order (LRO) component. In the Ising phase one finds history dependent behavior as discussed previously. In theXY phase, except very near the spin-flop boundary, one finds ergodic behavior withXY LRO and Lorentzian squared Ising fluctuations. Rather complicated instability effects are found all along the spin-flop boundary. Further, when one establishes LRO in theXY phase and lowers the field through the spin-flop value, one obtains a LRO Ising state in thex=0.75 sample whereas one obtains the field-cooled domain state in thex=0.50 sample. This dramatic difference in behavior is not understood. Our results on the RFIM aspects of the problem are consistent with our previous studies. The transition is dominated by the metastability effects with an underlying equilibrium transition which is either first order or weakly second order (0). The underlying transition manifests itself directly in measurements of the dynamic response nearT _N (H). From the data above the metastability boundary we deduce for the static correlation length exponentv=1.4±0.3 in good agreement with theory. We find for the RFIM crossover exponent _RF=1.5±0.2 where the errors represent the spread in values obtained from different techniques. Finally, we have determined in detail the field-temperature phase diagram of thex=0.5 sample including the critical behavior along the spin-flop line; the latter transition appears to be second order for an extended region.     

Dynamical correlations of two-dimensional vortex-like defects

High-order dynamical correlations of defects (quantum vortices, disclinations, etc.) in thin films are examined by starting from the Langevin equation for the defect motion. It is demonstrated that the dynamical correlation functions F _2n of the vorticity or disclinicity behave as F _2n ∼y ²/r ⁴ⁿ , where r is the characteristic scale and y is the renormalized fugacity. Therefore below the Berezinskii-Kosterlitz-Thouless transition temperature the F _2n are characterized by anomalous scaling exponents. The behavior differs strongly from the normal law F _2n ∼F ₂ ⁿ obeyed by equal-time correlation functions; the unequal-time correlation functions appear to be much larger. The phenomenon resembles intermittency in turbulence. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 10, 675–679 (25 November 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Partition function zeros and the three-dimensional Ising spin glass

We have computed the exact partition function of the 3D Ising spin glass on lattices of effective size 3×3×L_z, 4×4×L_z, and 5×5×L_z forL _z up to 9, and several random bond configurations. Studying the distribution of zeros of the associated partition functions, we find further evidence that these systems have a singularity in the thermodynamic limit.     

Testing “microscopic” theories of glass-forming liquids

We assess the validity of “microscopic” approaches of glass-forming liquids based on the sole knowledge of the static pair density correlations. To do so, we apply them to a benchmark provided by two liquid models that share very similar static pair density correlation functions while displaying distinct temperature evolutions of their relaxation times. We find that the approaches are unsuccessful in describing the difference in the dynamical behavior of the two models. Our study is not exhaustive, and we have not tested the effect of adding corrections by including, for instance, three-body density correlations. Yet, our results appear strong enough to challenge the claim that the slowdown of relaxation in glass-forming liquids, for which it is well established that the changes of the static structure factor with temperature are small, can be explained by “microscopic” approaches only requiring the static pair density correlations as nontrivial input.     

Computer simulation of the nonequilibrium critical behavior of disordered two-dimensional Ising systems

We report the results of a computer simulation of the critical relaxation of the magnetization in the two-dimensional Ising model with nonmagnetic impurity atoms frozen at the lattice sites. We assume a square lattice of dimension 400² with spin concentrationsp=1.0, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7. The Monte Carlo and dynamic renormalization group methods are used to determine the dynamical critical indexz as a function ofp: z(p): z(1)=2.24±0.07,z(0.95)=2.24±0.06,z(0.85)=2.38±0.05,z(0.8)=2.51±0.06,z(0.75)=2.66±0.07,z(0.7)=2.88±0.06. It is shown thatz(p) obeys a singular scaling law of the formz=A | ln (p–p _c) |+B withA=0.56±0.07,B=1.62±0.07.Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 83–88, August, 1994.     

Zero temperature phase transitions in spin-ladders: Phase diagram and dynamical studies of Cu2(C5H12N2)2Cl4>

In a magnetic field, spin-ladders undergo two zero-temperature phase transitions at the critical fields H_c1 and H_c2. An experimental review of static and dynamical properties of spin-ladders close to these critical points is presented. The scaling functions, universal to all quantum critical points in one-dimension, are extracted from (a) the thermodynamic quantities (magnetization) and (b) the dynamical functions (NMR relaxation). A simple mapping of strongly coupled spin ladders in a magnetic field on the exactly solvable XXZ model enables to make detailed fits and gives an overall understanding of a broad class of quantum magnets in their gapless phase (between H_c1 and H_c2). In this phase, the low temperature divergence of the NMR relaxation demonstrates its Luttinger liquid nature as well as the novel quantum critical regime at higher temperature. The general behavior close these quantum critical points can be tied to known models of quantum magnetism. Received: 13 March 1998 / Received in final form and Accepted: 21 July 1998     

Kawasaki dynamics with infinite-range spin exchange

The essential feature of the Kawasaki model is the conserved order parameter, which places the model in class B of the Halperin, Hoheberg, and Ma classification. We have studied the energy relaxation of this model in one and two dimensions with the added feature that spin exchange may take place between any pair of sites within the system. Our results for the dynamic exponentz are indistinguishable from those for class A models, in which the order parameter is not conserved.     

A renormalization group analysis of correlation functions for the dipole gas

We develop a renormalization group method for analyzing the generating functional for charge correlations of a dilute classical dipole gas. It is based on and extends the renormalization group analysis introduced by Brydges and Yau for the dipole gas partition function. Our method leads to systematic formulas for the large-distance behavior of correlation functions of all orders. We prove that in any dimensiond2, at any value>0 of the inverse temperature, and at sufficiently small activityz, the correlation functions exhibit at large distances the same behavior as for a vacuum (z=0), but with a new dielectric constant 1+ over which we have good control. The results proved here extend existing results on the two-point correlations to all higher correlations, and constitute a general confirmation of the fact that dipoles do not screen.     

The irrelevance of detailed balance for the dynamical critical exponent

We construct the most general real space renormalization for the two-dimensional kinetic Ising model on the triangular lattice which, to second order in the high-temperature expansion, conserves detailed balance and avoids fast transients for the cell spins. We show the corresponding dynamical recursion relations (as well as the exponentz) to be unaltered with respect to the ones found, in a previous paper, for a completely different class of transformations. This finding resolves long-standing confusions and controversies.     

Effect of oxygen on the NMR relaxation properties of crude oils

Nuclear magnetic resonance relaxation measurements of bulk fluids provide a sensitive probe of the dynamics of molecular motion. Dissolved oxygen can interfere with this technique as its paramagnetic nature leads to a reduction of the paramagnetic relaxation times of the fluids. We studied this effect for the relaxation properties of crude oils that are in general characterized by a distribution of relaxation times. The samples were stock tank oils that have been exposed to air. We comparedT ₁ andT ₂ relaxation time distributions and their correlation functions of the initial (oxygenated) samples with those from the deoxygenated samples. Oxygen was removed from the oils with a freeze-thaw technique. As expected, the effect of oxygen is most apparent in oils with long relaxation times. In these oils the effect of oxygen can be described by an additional relaxation rate 1/T _1,2 ^ox to the transverse and longitudinal relaxation rates that is sample dependent but does not vary within the relaxation time distribution of the oil. Values of 1/T _1,2 ^ox for different crude oils were found to be in the range of 2.5 to 8.3 s. For crude oils that have components with relaxation times less than 100 ms, no significant oxygen effect is observed.