Classical Critical Behavior Research Articles

Quantum version of a spherical model: Crossover from quantum to classical critical behavior.

We investigate a quantum version of the spherical model which is obtained from the classical Berlin-Kac spherical model by a simple canonical quantization scheme. We find a complete solution of the model for short-range as well as for long-range interactions. At finite temperatures the critical behavior is the same as in the classical spherical model whereas at zero temperature we find a quantum phase transition characterized by new critical exponents. Based on a functional-integral representation of the partition function the free energy of the model is shown to be equivalent to that of the nonlinear \ensuremath{\sigma} model in the limit of infinite order-parameter dimensionality. \textcopyright{} 1996 The American Physical Society.

First- or second-order transition in the melting of repeat sequence DNA

Both theoretical analysis and observation of the continuity of the melted fraction of base pairs indicate that the melting transition in DNA is second order. Analysis of the salt dependence of the transition by polyelectrolyte limiting laws, however, has first-order dynamics imbedded in the analysis. This paper proposes that the observation taken to be a latent heat of melting in the limiting law analysis could instead be a specific heat anomaly associated with a second-order transition. The limiting laws can be reconstructed based on a second-order transition with a specific heat anomaly. The T2M dependence of this excess heat is also consistent with its being a specific heat anomaly of a system displaying classical critical behavior. Classical critical behavior indicates that theoretical mean field approaches such as MSPA should be particularly appropriate to helix melting studies.

Chemical reaction driven phase transitions and critical points

Two simple examples of model (mean-field) equations of state for phase equilibrium in chemically reactive systems are examined for ‘‘unexpected’’ phase equilibria. They are, essentially, exactly soluble and give classical critical behavior. One of these leads to a lower critical solution temperature that results in closed-loop coexistence curves similar to those seen in hydrogen-bonding mixtures. The second leads to less familiar, but interesting phase diagrams that exhibit a phenomenon analogous to critical azeotropy. The same phenomena occur in two examples of lattice gas models the partition functions of which can be mapped exactly to that of the Ising model thus resulting in nonclassical critical behavior. These models demonstrate how a chemical reaction can provide a mechanism leading to interesting phase equilibria and critical phenomena.

The Random-Field Ising Model: A Free Energy Study

Thermodynamic properties of the random-field Ising model are reviewed and compared with published experimental results on diluted antiferromagnets in a uniform field. The Landau-Ginzburg free energy is calculated in the self-consistent approximation for 3-dimensional systems treating the random field induced fluctuations and the thermodynamic fluctuations of the order parameter separately. At low temperatures and in the presence of random fields two states exist, the stable, long-range ordered state, and the metastable disordered microdomain state. In a heating process, the ordered state changes to the high-temperature phase at the well defined transition temperature. The transition is first order for arbitrarily weak random field. Critical properties are briefly examined and according to the Ginzburg criterion, the classical critical behaviour is predicted for large random fields. For weak fields, nonclassical critical behaviour is expected.

Classical critical behaviour in crystal surfaces near smooth and sharp edges

Theoretical approaches suggested that the crystal shape near a smooth edge should show a non-classical critical behaviour. A generalized mean field analysis of the available experimental data in small Pb crystals is given. The same framework is used to analyse sharp edges in the surface of small gold crystals. It is shown that the experiments are consistent with a classical critical behaviour in contrast with recent interpretations.

Classical critical behaviour in crystal surfaces near smooth and sharp edges

Classical critical behaviour in crystal surfaces near smooth and sharp edges

Random field effects in classical and quantum critical phenomena

The effect of quenched random fields on classical and quantum critical behaviour is studied by means of the ϵ-analysis for a number of systems. The investigation is performed in terms of a generalized random-field correlation function. The interplay of short-range correlations as well as of a parameter-dependent variety of long-range correlations with thermal and quantum fluctuations is revealed.

Interpretation of electric susceptibility and order-parameter measurements near the first- and second-order Jahn-Teller phase transitions in DyAsO4and DyVO4using correlated-effective-field theory

Precise measurements of the order parameter and the c axis electric susceptibility have been made near the Jahn-Teller phase transitions in DyAsO4 and DyVO4. Both compounds show large departures from simple mean-field (MF) theory. The phase transition in DyAsO4 is of first order while that of DyVO4 is continuous. The data are analysed using a correlated-effective-field (CEF) model in which a simple approximation, consistent with classical critical behaviour, is made for the dependence on q of the effective exchange constant J(q). The CEF model gives an excellent description of the data over a wide range of temperatures and magnetic fields, both below and above TD. This agreement establishes that the competing effects of short- and long-range interactions are responsible for the deviations from MF behaviour; the model accounts for the first- and second-order nature of the phase transitions in these compounds. The large c axis dielectric anomalies imply the existence of appreciable electric dipole moments and are interpreted in terms of coupling to odd-symmetry q=0 modes. From the measurements they find that the odd-symmetry (electric dipolar) couplings are in order of magnitude larger in DyVO4 than in DyAsO4. These interactions of relatively long range appear to be strong enough in DyVO4 to ensure a continuous phase transition, but are too weak to do so in DyAsO4.

Analysis of electric susceptibility data and critical behaviour in the uniaxial ferromagnet LiHoF4using correlated-effective-field theory

Measurements of the electric susceptibility (which is orthogonal to the magnetic susceptibility) in LiHoF4 show a sharp cusp at Tc that cannot be explained by simple mean-field theory. Correlated-effective-field (CEF) theory is applied to this system and is found to provide an excellent fit to the data over a wide range of temperatures. In addition the critical behaviour of LiHoF4 is re-examined by the application of the authors' CEF model to the precise order-parameter data of Griffin and co-workers (1980). Although the authors' CEF model ultimately shows classical critical behaviour, over the temperature range of the data it yields a fit of quality equivalent to that obtained by Griffin and co-workers using renormalisation-group theory. Hence the experiments of Griffin and co-workers do not unequivocally establish a marginal dimensionality of three for LiHoF4.

Evidence for classical critical behavior in long-range site percolation

A new large-cell Monte Carlo renormalization group following the approach of Reynolds et al. is developed to study site percolation with long-range interaction. The results show that site percolation with a long, but finite, range of interactions has the same correlation-length exponent ${\ensuremath{\nu}}_{P}$ as ordinary percolation. However, as the interaction range tends to infinity, ${\ensuremath{\nu}}_{P}$ approaches its mean-field value of $\frac{1}{2}$ indicating that infinite-range percolation exhibits classical critical behavior.

Dielectric relaxation in pseudo-one-dimensional ferroelectric CsD2PO4

Dielectric dispersion in pseudo-one-dimensional hydrogen-bonded ferroelectric Cs${\mathrm{D}}_{2}$P${\mathrm{O}}_{4}$ was measured in the frequency range from 100 kHz to 2 GHz for temperatures between -13 and 30\ifmmode^\circ\else\textdegree\fi{}C. The observed relaxation is monodispersive in the paraelectric phase, except in the vicinity of the transition. The static dielectric constant ${\ensuremath{\epsilon}}_{s}$ shows classical critical behavior, with the critical coefficient $\ensuremath{\gamma}=1$. The relaxation frequency ${f}_{\ensuremath{\epsilon}}$ is much lower than in corresponding three-dimensional systems and exhibits a pronounced critical slowing down with a nonclassical exponent of 1.3. Outside the critical region in the paraelectric phase, $\ensuremath{\epsilon}$ and ${f}_{\ensuremath{\epsilon}}$ can be described reasonably well by a pseudo-one-dimensional Ising model, treating the interchain interactions in the mean-field approximation. The intrachain and interchain interactions obtained agree well with previously determined values. The noninteracting deuteron intrabond-jump time is much smaller than in K${\mathrm{D}}_{2}$P${\mathrm{O}}_{4}$ but has the same activation energy.

Evidence for classical critical behavior at the nematic-isotropic phase transition

Magnetically induced birefringence measurements were performed in the isotropic phase of methoxybenzylidene butylaniline (MBBA). Over the entire temperature range studied, $\ensuremath{\Delta}n$ was found to be linear in ${H}^{2}$ to fields over 10 T. This result is consistent with the Landau model of the nematic-isotropic transition, and strongly inconsistent with the results of Keyes and Shane [Phys. Rev. Lett. 42, 722 (1979)] in a similar experiment. Their results indicate both the approach of a field-induced critical point and tricritical behavior near the phase transition.

MCRG study of d-dimensional random field Ising models

A Monte Carlo Renormalization Group method has been used to study d-dimensional hypercubic Ising models in random fields (±H). For d = 4 our results are consistent with classical critical behavior. For d = 3 we find a sharp transition with critical exponents which are apparently modified by the random field. We find no sharp transition for d = 2.

Specific heat behaviour of the ferroelectric PbHPO 4

Specific heat measurements of PbHPO 4 confirm the second order character for the Ferro-paraelectric phase transition at T c = 307.75 ± 0.09 K. A comparison between the experimental results and phenomenological and statistical theories is carried out. The classical critical behaviour for both α + and α - are shown to be valid up to |T − T c | < 0.1K.

Tricriticality and the failure of scaling in the many-component limit

We analyze exactly in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, the $n$-component continuous-spin model with a cubic field term ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}}_{3}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}|}^{2}$. Full "four-field" tricritical behavior is exhibited for dimensionalities $d\ensuremath{\ge}3$. However, orthodox tricritical scaling is shown to be impossible for $3&lt;d&lt;4$: to obtain the correct nonclassical spherical-model exponents on the $\ensuremath{\lambda}$ line ($H={H}_{3}=0, T&gt;{T}_{t}$) it is essential to allow for a dangerous irrelevant scaling variable $p\ensuremath{\propto}\frac{1}{{R}_{0}^{d}}$, where ${R}_{0}$ is the range of the pair interactions. The appropriate crossover exponent is ${\ensuremath{\varphi}}_{p}=3\ensuremath{-}d$ so that $p$ is marginal for $d=3$: orthodox scaling is then possible but the scaling functions are nonuniversal. On the disordered symmetry plane ($H={H}_{3}=0$) only the corrections to scaling survive and describe crossover to Gaussian tricritical behavior. For $T&gt;{T}_{t}$ bicritical crossover from spherical to classical critical behavior occurs when ${H}_{3}$ varies but scaling is fully obeyed. Some inferences for systems with finite $n$ are drawn.

Quantum classical crossover critical behavior of the Ising model in a transverse field

Abstract The Ising model (spin 1 2 ) in a transverse field has many applications in the field of cooperative phenomena (magnetism, Jahn-Teller effects, ferroelectricity). The quantum classical crossover critical behavior is studied for a d-dimensional system with exact results when d = 1. The crossover behavior depends strongly on the relative magnitude of the crossover exponent φ and of the critical line shift exponent Ψ.

Phason Instability in Tetrathiafulvalene-Tetracyanoquinodimethane (TTF-TCNQ)

The 49-K phase transition in tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) is interpreted in terms of a phason instability of the charge-density-wave ground state. The order parameter couples linearly to an external strain, and the phase transition can be suppressed by clamping the crystal. Recent neutron scattering experiments show indications of this effect. Renormalization-group calculations yield pure classical critical behavior at this phase transition, and $\mathrm{XY}$ exponents for the 54-K transition.

A field-theoretic model for the critical behaviour of4He above the Lambda point

It is shown that the critical behaviour of4He aboveTλ can be described in terms of a peculiar solution of a set of approximate equations related to a field-theoretic model describing a system of bosons interacting via a « phononlike » field. The basic set of equations is derived by means of rather general physical arguments from an exact set of identities of the theory. The solution which describes the critical behaviour is only asymptotically valid if an « effective coupling constant », associated with the model, reaches a nontrivial « fixed point » as the correlation length goes to infinity. In the asymptotic region we derive the well-known scaling lawdν + α = 2. The nontrivial fixed pointyF, the rate of approach to the asymptotic regionb and the critical indexν are calculated with an approximate method. The results are functions of the dimensionality parameter e = 4 —d for e > 0. For dimensionality d>4 the expected classical critical behaviour is recovered. Our best estimate is yF 0.11,b 4e, υ = 1/(2[1-υ(I-yF)/4]). These results are not derived with an expansion in the parameter e but involve an approximation that presumably is equivalent to neglecting terms of the order of the critical indexrη. In three dimensions we getv 0.643. A peculiar feature of our results is that the fixed pointyF does not depend eithei on the physical parameters of the theory or on the dimensionality of the system, but it depends on the topological structure of the diagrams.