Critical Slowing-down Research Articles

Anisotropic broadening of the brillouin scattering spectra of hydrogenbonded ferroelectric PbHPo4

Abstract Brillouin scattering spectra of longitudinal acoustic phonons in ferroelectric lead monohydrogen phosphate, PbHPO4 (LHP) have been observed as a function of temperature in a region near the ferroelectric phase transition. The spectra of the c 22 mode shows an anomalous broadening just below the transition temperature in contrast to the temperature independent a* mode propagating nearly parallel to the spontaneous polarization. A critical slowing-down of the polarization is revealed by the analysis using the Debye type susceptibility function based on an order-disorder mechanism. The relaxation time follows well the Curie-Weiss law irrespective of the one-dimensional network of the hydrogen bonds and PO4 ions.

Multi-grid Monte Carlo (III). Two-dimensional O(4)-symmetric non-linear σ-model

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm with piecewise-constant interpolation applied to the two-dimensional O(4)-symmetric nonlinear $\sigma$-model [= SU(2) principal chiral model], on lattices up to $256 \times 256$. We find a dynamic critical exponent $z_{int,{\cal M}^2} = 0.60 \pm 0.07$ for the W-cycle and $z_{int,{\cal M}^2} = 1.13 \pm 0.11$ for the V-cycle, compared to $z_{int,{\cal M}^2} = 2.0 \pm 0.15$ for the single-site heat-bath algorithm (subjective 68% confidence intervals). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated. For a $256 \times 256$ lattice, W-cycle MGMC is about 35 times as efficient as a single-site heat-bath algorithm.

Dielectric Critical Slowing-Down in Ferroelectric Ca2Pb(C2H5CO2)6Crystal

Measurement of the complex dielectric constant of Ca 2 Pb(C 2 H 5 CO 2 ) 6 (DLP) has been made around the ferroelectric phase transition temperature, T c . The dielectric dispersion of relaxational type of which the relaxation time shows the critical slowing-down has been observed. The results indicate that the ferroelectric phase transition of DLP is essentially of the order-disorder type. No essential differences are found between DLP and isomorphic Ca 2 Sr(C 2 H 5 CO 2 ) 6 (DSP) in the static and dynamic dielectric properties around T c .

How to beat critical slowing-down: 1990 update

We discuss the problem of critical slowing-down (CSD) in Monte Carlo simulations, and review recent progress in devising collective-mode algorithms that reduce or eliminate CSD. Emphasis is on cluster and embedding algorithms; multigrid and Fourier acceleration are also discussed briefly.

Multi-grid Monte Carlo (II). Two-dimensional XY model

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm applied to the two-dimensional XY model, on lattices up to 128 × 128. In the low-temperature (spin-wave) phase, we find that critical slowing-down is completely eliminated ( z = 0). On the high-temperature side of critically, we find that the dynamic critical exponent is z = 1.4±0.3, compared to z = 2.1±0.3 for the heat-bath algorithm. We attribute the remaining critical slowing-down on the high-temperature side of criticality to the limited effectiveness of the MGMC updates in creating and destroying widely separated vortex-antivortex pairs.

Monte Carlo test of a hyperscaling relation for the two-dimensional self-avoiding walk. II

For pt. I see ibid., vol.60, p.1, (1990). By using a novel Monte Carlo algorithm which uses non-local moves to decrease the critical slowing-down, the authors simulate two-dimensional self-avoiding walks (SAWs) in a variable-length fixed-endpoint ensemble. This allows one to determine with reasonable accuracy the critical exponent alpha sing. As a byproduct, they obtain also accurate measurements of the exponent nu and the connective constant mu . They thus get a direct check of the hyperscaling relation dv=2- alpha sing. Estimates of alpha sing and mu are obtained by a maximum-likelihood fit which combines data generated at different fugacities.

Critical acceleration of Monte-Carlo simulations of Z2 gauge theory

Critical Slowing Down for Z 2 lattice gauge theory in 3 dimensions is reduced drastically by a stochastic cluster algorithm. It reduces the dynamical exponent from z > 2 (standard local algorithms) to z ≈ 0.73. 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 3-D Ising model and φ 4 model supports the conjecture of universality classes for stochastic cluster algorithms.

Nonlocal Monte Carlo algorithm for self-avoiding walks with fixed endpoints

We study a new Monte Carlo algorithm for generating self-avoiding walks with variable length (controlled by a fugacityβ) and fixed endpoints. The algorithm is a hybrid of local (BFACF) and nonlocal (cut-and-paste) moves. We find that the critical slowing-down, measured in units of computer time, is reduced compared to the pure BFACF algorithm:τ CPU ∼ 〈N〉≈2.3 versus 〈N〉≈3.0. We also prove some rigorous bounds on the autocorrelation time for these and related Monte Carlo algorithms.

Employing the ising representation to implement nonlocal Monte Carlo updating inO(N) models

O(N) ferromagnets, anisotropic or not, with standard nearestneighbour interaction are expressed asN Ising models coupled with certain auxiliary fields. A hybrid Monte Carlo algorithm is then developed. It consists of global updates of the Ising variables using the Fortuin-Kasteleyn representation, combined with ordinary heat-bath or Metropolis updates of the remaining variables. The method generalizes toZ(2N) models. A numerical test forZ(8) at β=1.1 is reported. It reveals a dramatic reduction in critical slowing-down.

Dielectric Critical Slowing-Down in Ferroelectric Li2Ge7O15

The dielectric dispersion associated with the ferroelectric phase transition in Li 2 Ge 7 O 15 has been measured in the frequency range from 21 MHz to 7 GHz around the Curie point, T c =10.4°C. The Debye type dispersion which showed a critical slowing-down of dielectric response was observed, its relaxation frequency being about 1 GHz at T c . The relation between the observed critical slowing-down and the softening of the optical phonon reported previously is discussed.

Overcoming Critical Slowing Down

First Page

Dielectric dispersion in Li2Ge7O15

Abstract The complex dielectric constant of Li2Ge7O15 crystal has been measured at frequencies from 21 MHz to 7 GHz around the Curie temperature Tc. The dielectric dispersion of the Debye type has been found. A critical slowing-down of dielectric response has been observed, and the relaxation frequency decreases to about 1 GHz at Tc. The origin of the Debye dispersion observed in the vicinity of Tc is discussed in connection with the resonant type dispersion of soft optical phonon mode.

Dielectric relaxation near the antiferroelectric phase transition of squaric acid

The dielectric response of single crystals of squaric acid is studied around the antiferroelectric phase transition atT c ≈100°C. The dielectric data between 64 and 400 GHz can be explained by a simple Debye-relaxation with a critical slowing-down of the relaxation frequency to about 55 Ghz on approachingT c . The results in the paraelectric phase are discussed on the basis of weakly coupled one-dimensional Ising chains. Within the scope of this model we determine the intra and interchain coupling constants, the molecular dipolar moments and the noninteracting proton intrabond-jump time.

SCALING AND METASTABILITY OF THE RANDOM-FIELD SYSTEM Fe0.7Mg0.3Cl2

The field dependence of the critical divergence of (OM / dT), of zero-field cooled Fe0.7Mg0.3C12 agrees with scaling predictions for Fishman-Aharony realizations of the d = 3 random-field Ising model. The time dependence of the thermoremanent magnetization, due to cooling in a field Ho, obeys a generalized decay law, p (t) = a [In ( t / T)]-+ $ b. It accounts for the spin relaxation crossing over from a domain wall mechanism to pure domain growth on increasing T and/or Ho. Important aspects of the random-field Ising model (RFIM) problem [I] are stilI unclear: (i) the asymp totic critical behavior is a matter of controversy; (ii) the predicted relaxational behavior of the random-field (RF) induced metastable domain state has not yet convincingly been verified. Much of the problematics is closely tied to the unusual critical slowing-down, which nucleates the domain state upon RF cooling and gives rise to roundings of the critical divergences on usual laboratory time scales (7 5.10~ s) . Most of the experimental knowledge of the RFIM is based on the properties of diluted uniaxial antiferromagnets in a uniform external field (DAFF) [I]. They belong to the same universality class as the ferromagnetic RFIM, albeit showing differences in both the crossover [2] and the domain state behavior being subject to additional random-bond (RB) pinning [I, 31. In this paper we present new results on the d = 3 DAFF system Feo..i.Mgo.,C12 [4] concerning (i) critical behavior in the zero-fields cooled (ZFC) ground state and (ii) the zero-field relaxation of the field-cooled (FC) domain state. (i) The amplitude exponent y of the critical part of (aM / aT), cc H Y IEI-' [5], E = (T Tc) / T N , is difficult to measure owing to ambiguities in correcting for the non-critical background and for the peak suppression due to dynamical rounding [I]. However, since 6 11 0, the slope S of a plot of (bM /aT), vs. loglo Is1 (Fig. 1) is expected to vary as HY, neglecting the small T dependence of the regular background at T, (H) . Hence, loglo (S / So) vs. loglo (H / T) should yield a slope y = 2(1+6-a-q5/2) /q5 0.54 [5] inserting the widely accepted crossover, RF and random-exchange (RE) exponents q5 = 1.42, & = 0 and a = -0.09, respectively [I]. Satisfactory agreement is, indeed, found from Faraday rotation (0 cc M) measurements on Feo.7Mgo.,Clz. They reveal logarithmic divergences between the =to-RF crossover and the dynamical rounding temperatures (arrows in Fig. I), and y = 0.56 f 0.05 is obtained from a log-log plot of S us. H (Fig. 2). Fig. 1. (dB /dT)H vs. loglo 1 . ~ 1 after ZFC for H = 0.07 (I), 0.26 (2) and 0.39 T (3), below (0 ) and above (o) Tc, respectively. The solid best-fit lines characterize the RFIM regime undistorted by dynamical rounding (arrows; see text).

Critical Slowing Down of the Spin Relaxation and Center Shift Anomaly in NiFe2O4 Observed with Mössbauer Spectroscopy

Detailed Mössbauer-effect studies of NiFe2O4 were carried out in the temperature range from 13.4 to 890 K. The respective temperature dependences of the center shifts, the hyperfine fields, and the longitudinal relaxation rates of the spin fluctuation were determined for both A- and B-site ferric ions. An anomalous change in the slope of the center shift of the B-site ferric ion was clearly observed near 475 K. The Néel temperatures and the critical exponents of the average spins were determined for the A- and B-site ions. It was also observed that the spin relaxation rates decreased remarkably in the critical region. The exponents which characterized the critical slowing down of spin relaxation were also determined.

Dynamical studies of highly deuterated betaine phosphate near the antiferroelectric phase transition

The dielectric response and the Raman spectra of single crystals of deuterated betaine phosphate are studied around the antiferroelectric phase transition. The dielectric data between 10 MHz and 11 GHz can be explained on the basis of a simple Debye-relaxation with a critical slowing-down of the relaxation rate on approachingT C . Using the Cole-Davidson form of the dielectric function we succeeded in fitting the data in the whole frequency range from 10 MHz to 11 GHz and from 64–400 GHz over a temperature range from 145–280 K. Raman spectra clearly indicate that the doubling of the unit cell does not take place at the antiferroelectric transition temperature, but some degrees below.

Critical Raman scattering in orthorhombic KNbO3

Abstract The Raman spectrum of a single-domain orthorhombic KNbO3 sample exhibits the appearance of a central peak beside the soft phonon when the transition to the tetragonal phase is approached. The low-frequency Raman profile is well reproduced by a coupled relaxation-phonon model. It is pointed out that the orthorhombic-tetragonal phase transition is related to a critical slowing-down of the phonon renormalized by the interaction with the relaxation, whereas the orthorhombic-rhombohedral transition is associated solely with a smooth softening of the decoupled phonon.

Dielectric Critical Slowing-Down in Hydrogen-Bonded Ferroelectrics: PbHPO4and PbDPO4

Complex dielectric constants of PbHPO 4 and PbDPO 4 have been studied at frequencies from 10 3 Hz to 10 9 Hz in the Curie temperature region. The Curie-Weiss behavior of the low-frequency dielectric constant is confirmed. The dielectric dispersion of the Debye type which shows a critical slowing-down has been observed in both materials. The phase transition is considered to be of an order-disorder type. The effects of deuteration in the static and dynamic dielectric properties are discussed.

Incommensurate phase transition in sodium nitrite: A study of the dynamics by neutron spin echo technique

NaNO2 is a model system for crystals undergoing an incommensurate order-disorder phase transition. In this case, the incommensurate phase transition is charaterized by a relaxational mode which exhibits critical slowing-down. We have performed neutron experiments to study the pretransitional fluctuations and the dynamics in the modulated phase. The order of magnitude of relaxation times being 109 s, we used a high resolution technique: neutron spin-echo. Surprisingly, it has been found that the relaxation process contains a fast and a "central peak"-type slow component. It appears that previous macroscopic measurements, interpreted in terms of a single relaxation time model, produced intermediate effective relaxation rates.

Dielectric relaxation mechanism in the incommensurate phase of K2SeO4-type crystals

Abstract The dynamic susceptibility x(w) in the incommensurate phase of K2SeO4-type crystals has been derived from the phenomenological kinetic equations of the Landau-Khalatnikov type using phase soliton model. The critical slowing-down of the polarization fluctuation which has been observed around the incommensurate-commensurate transition temperature T c can be explained as the softening of the phase mode of the discommensuration lattice. In the case of K2SeO4 crystals, the frequency of the soft phase mode is estimated as w1/2° = 0.7(T - Tc)1/2 (cm−1) near T c.