Specific Heat Critical Exponent Research Articles

The Ising model in a Bak–Tang–Wiesenfeld sandpile

We study the spin-1 Ising model with non-local constraints imposed by the Bak–Tang–Wiesenfeld sandpile model of self-organized criticality (SOC). The model is constructed as if the sandpile is being built on a (honeycomb) lattice with Ising interactions. In this way we combine two models that exhibit power-law decay of correlation functions characterized by different exponents. We discuss the model properties through an order parameter and the mean energy per node, as well as the temperature dependence of their fourth-order Binder cumulants. We find: (i) a thermodynamic phase transition at a finite T c between paramagnetic and antiferromagnetic phases, and (ii) that above T c the correlation functions decay in a way typical of SOC. The usual thermodynamic criticality of the two-dimensional Ising model is not affected by SOC constraints (the specific heat critical exponent α ≈ 0 ), nor are SOC-induced correlations affected by the interactions of the Ising model. Even though the constraints imposed by the SOC model induce long-range correlations, as if at standard (thermodynamic) criticality, these SOC-induced correlations have no impact on the thermodynamic functions.

Coexistence curves of electrical conductivity in critical solution: isobutyric acid–water with added ions

Comprehensive results of temperature (T) studies of electrical conductivity (σ) in the one-and-two phase regions of the binary fluid mixture of isobutyric acid–water (IBAW) with X M[KCl] at various critical concentrations are presented. A strong asymmetry of the coexistence curves as determined causes a strong violation of the law of rectilinear diameter. The obtained critical anomalies in the homogeneous phase σcrit (T) or the diameter of the σd (T) are associated with the same critical exponent θ = 1 − α ≈ 0.88, where α is the specific heat critical exponent. For the mixture with critical composition, the two phase regions show a concentration behavior with σl − σu ≈ |−t|β. The phase transition region shifts as a function of the salt concentration. The effective critical exponent β* = (0.264–0.296) is neither compatible with the Ising value β = 0.325 nor with the Fisher renormalized value ∼0.365.

Bond dilution in the 3D Ising model: a Monte Carlo study

We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent $\alpha$ . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.

Complex dielectric relaxation in supercooling and superpressing liquid-crystalline chiral isopentylcyanobiphenyl.

Results of broadband dielectric studies in glass-forming liquid crystalline chiral isopentylcyanobiphenyl (5(*)CB) are presented. Tests conducted as a function of temperature and pressure revealed the coexistence of glassy and critical properties. The latter are associated with the isotropic-cholesteric phase transition at T(I-Ch) approximately 250 K under atmospheric pressure. Dielectric loss curves in the isotropic liquid and in the cholesteric phase are clearly broadened on cooling and pressuring towards the glass transition. Although in the isotropic phase there is a single stretched loss curve, in the mesophase an additional relaxation process can be distinguished. The evolution of relaxation times is non-Arrhenius and can be portrayed by the Vogel-Fulcher-Tamman relation or its pressure counterpart. The glassy dynamics coexists with the critical-like behavior for the static dielectric permittivity and for the maxima of the dielectric loss curves. Their temperature and pressure dependences are associated with the critical exponent phi=1-alpha approximately 1/2, where alpha approximately 1/2 is the specific heat critical exponent. This behavior is associated with the continuous phase transition placed at DeltaT approximately 1.5 K below the clearing temperature for P=0.1 MPa. It has been found that 5(*)CB shows a unique pressure-temperature phase diagram. Pressure and temperature changes which begin in the isotropic liquid below at ca. T approximately 265 K always result in the transition to the cholesteric phase which can be supercooled or superpressed. For T>265 K the phase transition to another phase, presumably a solid one, always occurs. However, a cholesteric-solid phase border seems to exist only in isothermal pressure tests. It does not appear in the temperature studies.

Reentrant Behavior in Binary Mixtures of (Octyloxy)cyanobiphenyl (8OCB) and the Shorter Chain Homologue (Heptyloxy)cyanobiphenyl (7OCB)

The complete stable phase diagram of the two-component system (heptyloxy)cyanobiphenyl (7OCB) + (octyloxy)cyanobiphenyl (8OCB) was determined by means of modulated differential scanning calorimetry (MDSC), optical microscopy, and X-ray diffraction measurements. It was experimentally established that the 7OCB + 8OCB two-component system exhibits a monotropic reentrant nematic behavior. A complete quantitative thermodynamic analysis, through Oonk's equal G analysis, was performed, including the calculation of the monotropic reentrant behavior and a discussion of the stable melting phase diagram. Moreover, the specific-heat critical exponents (α), through second-order SmA to N transition, was obtained. If these α-values, together with those corresponding to other systems sharing cyanobiphenyl (nCB and nOCB) compounds, are plotted against the normalized nematic ranges, a common and uniform crossover trend is found. This behavior allows one to predict, in a simple way, the order of the SmA to N transition.

Critical behavior of dielectric permittivity and electric conductivity in temperature and pressure studies above and below the critical consolute point

Results of comprehensive temperature (T) and pressure (P) studies of static dielectric permittivity (ε′) and electric conductivity (σ) in the one- and two-phase regions of critical nitrobenzene–dodecane mixture are presented. A strong asymmetry of determined coexistence curves causes a strong violation of the law of rectilinear diameter. The obtained critical anomalies in the homogeneous phase [εhomo′(T),εhomo′(P),σhomo(P)] or the diameter of the binodal [εmean′(T),εmean′(P),σmean(T),σmean(P)] are associated with the same critical exponent φ=1−α ≈0.88, where α is the specific heat critical exponent. Critical anomalies for the isothermal, pressure path exhibit a set of favorite in comparison with results obtained in σ(T) and ε(T) tests. They are: the negligible influence of the critical Maxwell–Wagner effect, the hardly visible appearance of the correction-to-scaling term, a more pronounced manifestation of critical anomalies, and a reduced number of fitted parameters. Particularly noteworthy is the evidence for the σhomo(P) anomaly in the homogeneous phase, hardly obtained up to now in a mixture of a low electric conductivity. Results presented suggest the isomorphic behavior of ε′(T), σ(T) and ε′(P), σ(P) critical anomalies for the homogeneous phase and for the diameter of the binodal.

Ground-state numerical study of the three-dimensional random-field Ising model

The random field Ising model in three dimensions with Gaussian random fields is studied at zero temperature for system sizes up to 60^3. For each realization of the normalized random fields, the strength of the random field, Delta and a uniform external, H is adjusted to find the finite-size critical point. The finite-size critical point is identified as the point in the H-Delta plane where three degenerate ground states have the largest discontinuities in the magnetization. The discontinuities in the magnetization and bond energy between these ground states are used to calculate the magnetization and specific heat critical exponents and both exponents are found to be near zero.

Binary mixtures of nCB and nOCB liquid crystals. Two experimental evidences for a smectic A–nematic tricritical point

Modulated differential scanning calorimetry (MDSC) has been used to obtain specific heat measurements on octyloxycyanobiphenyl (8OCB), octylcyanobiphenyl (8CB) and nonyloxycyanobiphenyl (9OCB) liquid crystals and on several binary mixtures of 8OCB + 9OCB and 8CB + 9OCB. The order of the mesophase transitions, smectic A(SmA) to nematic(N) and N to isotropic (I) on pure components and on binary mixtures has been studied and, in addition, both binary mesophase diagrams have been built. For each set of mixtures, there exists a tricritical point (TCP) composition at the SmA–N transition. The TCP compositions have been found to be X9OCB = 0.63 for the system 8OCB + 9OCB and X9OCB = 0.42 for the system 8CB + 9OCB. In addition, the specific heat critical exponents (α) through second order SmA to N transition have been obtained between 8OCB(or 8CB) and the TCP composition for the two sets of binary mixtures. When these α-values were plotted against the normalised nematic range, a common and uniform crossover trend was found. Indeed, this fact suggests a uniform behaviour in terms of the McMillan ratio for the nCB and nOCB series of liquid crystals. Moreover, for these compounds there exists a common value of (TAN/TNI)TCP of 0.99 and also a value of (TAN/TNI)3D-XY of about 0.96.

Glassy and fluidlike behavior of the isotropic phase of n-cyanobiphenyls in broad-band dielectric relaxation studies.

It is shown that the temperature behavior of peaks f p, of dielectric loss curves in the isotropic phase of n-cyanobiphenyls n = 8, 9, 10 with isotropic-nematic and isotropic-smectic A transitions exhibits features characterisic for both supercooled, glass-forming liquids and critical, binary mixtures. The behavior of f p T can be portrayed by the Vogel-Fulcher-Tamman relation and the "critical-like", mode-coupling theory (MCT) equation. The latter is supported by the novel analysis of electric conductivity σ T . The obtained f p T and σT dependencies can be related by using the fractional Debye-Einstein-Stokes law. For all tested mesogens the static dielectric permittivities ɛ(') T and T are described by dependencies resembling those applied in the homogeneous phase of critical mixturesbut with specific-heat critical exponent α≈ 0.5. This behavior agrees with the novel fluidlike description for the isotropic-nematic transition (P.K. Mukherjee, Phys. Rev. E 51, 5745 (1995); A. Drozd-Rzoska, Phys. Rev. E 59, 5556 (1999)). The obtained glassy features of dielectric relaxation support the recent simulation analysis carried out by M. Letz et al.Phys. Rev. E 62, 5173 (2000)).

Determination of susceptibility and specific heat critical exponents for weak itinerant-electron ferromagnets from vibrating reed experiments

We report the observation of a linear relationship between the magnetic contribution to Young's modulus, $\ensuremath{\Delta}{E/E}_{0},$ and inverse magnetic susceptibility ${\ensuremath{\chi}}^{\ensuremath{-}1}$ for amorphous weak itinerant-electron ferromagnets ${\mathrm{Fe}}_{90}{\mathrm{Zr}}_{10}$ and ${\mathrm{Fe}}_{91}{\mathrm{Zr}}_{9}$ in the asymptotic critical region near the ferromagnetic-paramagnetic phase transition. The proportionality $\ensuremath{\Delta}{E(T)/E}_{0}\ensuremath{\propto}{\ensuremath{\chi}}^{\ensuremath{-}1}(T)$ is shown to provide as accurate a means of determining the asymptotic critical exponent $\ensuremath{\gamma}$ and the leading ``correction-to-scaling'' amplitudes for susceptibilty from the $\ensuremath{\Delta}{E/E}_{0}$ data as a direct measurement of magnetic susceptibilty does. Similarly, the well-known relation between the magnetic contributions to sound velocity and specific heat is fully exploited to extract accurate estimates for the universal critical amplitude ratio ${A}^{+}{/A}^{\ensuremath{-}}$ and the asymptotic critical exponents ${\ensuremath{\alpha}}^{\ifmmode\pm\else\textpm\fi{}}$ for the specific heat from the sound velocity data. The presently determined values of ${\ensuremath{\alpha}}^{\ifmmode\pm\else\textpm\fi{}}$ and $\ensuremath{\gamma},$ together with the reported value for spontaneous magnetization critical exponent $\ensuremath{\beta},$ not only obey the scaling equalities ${\ensuremath{\alpha}}^{+}={\ensuremath{\alpha}}^{\ensuremath{-}}$ and $\ensuremath{\alpha}+2\ensuremath{\beta}+\ensuremath{\gamma}=2$ but also assert that the atomic magnetic moments in the alloys in question interact with one another through an attractive interaction which decays faster than ${1/r}^{5}$ with the interatomic spacing, r.

Pretransitional behavior of dielectric permittivity on approaching a clearing point in a mixture of nematogens with antagonistic configurations of dipoles.

The results of studies of static dielectric permittivity epsilon(T) in the isotropic phase of 4-cyano-4-pentylalkylbiphenyl and n-p-methoxybenzylidene-p'-butylaniline (5CB-MBBA) mixtures are presented. 5CB and MBBA are nematogens with antagonistic permanent dipole moments configurations. An increase in MBBA concentration strongly decreases the pretransitional effect. However, a derivative analysis detected the existence of a pretransitional anomaly even if its weakness made a straightforward epsilon(T) fit impossible. The obtained anomalies can be well portrayed by fluidlike equations, isomorphic to that applied in critical binary mixtures. Every time the same value of the specific heat critical exponent alpha approximately 0.5 was obtained. A preliminary discussion of the influence of the permanent dipole configuration on the pretransitional behavior of dielectric permittivity was also possible.

CsMn(BrxI1−x)3:Crossover from anXYto an Ising chiral antiferromagnet

We report on high-resolution specific-heat and magnetocaloric-effect measurements of the triangular-lattice antiferromagnets CsMn(Br_xI_{1-x})_3 with different x. The evolution of the magnetic phase diagrams from the easy-axis system for x = 0 to the easy-plane system for x = 1 was studied in detail. The specific-heat critical exponent alpha of the almost isotropic x = 0.19 system agrees with the value predicted for a chiral Heisenberg scenario. In an applied magnetic field (B = 6 T) a crossover to a weak first-order transition is detected.

Critical behavior of electrical resistivity in amorphous Fe-Zr alloys

Electrical resistivity (ρ) of the amorphous (a-)Fe100−c Zr c (c=8.5, 9.5 and 10) alloys has been measured in the temperature range 77 to 300 K, which embraces the second-order magnetic phase transition at the Curie temperature point T c. Analysis of the resistivity data particularly in the critical region reveals that these systems have a much wider range of critical region compared to other crystalline ferromagnetic materials. The value of T c and specific heat critical exponent, α has the same values as those determined from our earlier magnetic measurements. The value of α for all the present investigated alloys are in close agreement with the values predicted for three-dimensional (3D) Heisenberg ferromagnet systems, which gives contradiction to the earlier results on similar alloys. It is observed from the analysis that the presence of quenched disorder does not have any influence on critical behavior.

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.

FERROMAGNETIC POTTS MODEL ON A HIERARCHICAL LATTICE WITH RANDOM LAYERED INTERACTIONS

We analyze the critical behavior of a q-state Potts model with correlated disordered ferromagnetic exchange interactions along the layers of a diamond hierarchical lattice. For a special class of weakly disordered distributions, we use the topological properties of the lattice to write a set of recursion relations for the moments of the probability distribution of the interaction parameters. We identify a small parameter, q-q0, where q0=0.537…, to expand and decouple the recursion relations. For q<q0, there is just a trivial stable fixed point, associated with the critical behavior of the uniform model. For q>q0 (that correponds, in a uniform case, to a specific heat critical exponent α>-2), the existence of a stable disordered fixed point indicates a change in the critical behavior. We make some remarks on the validity of the Harris criterion for hierarchical lattices.

Thermodynamic study of the magnetic phase transition in UNi 4B

We investigate the magnetic phase transition in UNi 4B by specific heat, c( T), thermal expansion and resistivity under pressure. These experiments reveal that the hexagonal symmetry is not broken at T N = 20 K, and predicts a hydrostatic pressure dependence of T N of − 29 mK/kbar, close to the result from resistivity. The anomalous upturn of c/ T to 470 mJ/mol K 2 at 0.4 K is reduced to 160 mJ/mol K 2 in 16 T ∥ b-axis. The specific heat critical exponent α = −0.15(2) suggests 3D Heisenberg universality in zero field.

Short-range Potts spin-glass model: Renormalization-group method.

We study the phase diagram of the short-range Potts spin glass within the Migdal-Kadanoff renormalization group of the hypercubic lattices. Using some analytical approaches and performing the real-space renormalization group of the problem we show that the model, for q\ensuremath{\gtrsim}2, freezes into a Potts spin-glass phase at finite temperature, ${\mathit{T}}_{\mathrm{SG}}$\ensuremath{\ne}0, in d=4. This indicates an algebraic power-law decay of the correlation functions in this low-temperature phase. We show that the Potts spin-glass phase does not occur in d=3 for q\ensuremath{\gtrsim}2 and calculate the lower critical dimension for all continuous values of q. The specific-heat critical exponent is found to be large and negative. \textcopyright{} 1996 The American Physical Society.

Critical behavior of superfluid 4He in aerogel.

We report on Monte Carlo studies of the critical behaviour of superfluid $^4$He in the presence of quenched disorder with long-range fractal correlations. According to the heuristic argument by Harris, uncorrelated disorder is irrelevant when the specific heat critical exponent $\alpha$ is negative, which is the case for the pure $^4$He. However, experiments on helium in aerogel have shown that the superfluid density critical exponent $\zeta$ changes. We hypothesize that this is a cross-over effect due to the fractal nature of aerogel. Modelling the aerogel as an incipient percolating cluster in 3D and weakening the bonds at the fractal sites, we perform XY-model simulations, which demonstrate an increase in $\zeta$ from $0.67 \pm 0.005 $ for the pure case to an apparent value of $0.722\pm 0.005$ in the presence of the fractal disorder, provided that the helium correlation length does not exceed the fractal correlation length.

Replica-symmetry breaking in the critical behaviour of the random ferromagnet

We study the critical properties of the weakly disordered $p$-component random Heisenberg ferromagnet. It is shown that if the specific heat critical exponent of the pure system is positive, the traditional renormalization group (RG) flows at dimensions $D=4-\e$, which are usually considered as describing the disorder-induced universal critical behavior, are {\it unstable} with respect to replica symmetry breaking (RSB) potentials as found in spin glasses. It is demonstrated that the RG flows involving RSB potentials lead to fixed points which have the structure known as the 1 step RSB, and there exists a whole spectrum of such fixed points. It is argued that spontaneous RSB can occur due to the interactions of the fluctuating fields with the local non-perturbative degrees of freedom coming from the multiple local minima solutions of the mean-field equations. However, it is not clear whether or not RSB occurs for infinitesimally weak disorder. Physical consequences of these conclusions are discussed.

Absence of a stiffness instability for a model critical-wetting transition in three dimensions

We have tested the generality of the stiffness instability mechanism recently proposed by Fisher and Jin (1993) for the critical wetting phase transition in three-dimensional systems with short-ranged forces. We extend the analysis of Fisher and Jin to a class of Aukrust-Hauge type models and find that the stiffness instability is specific to the case where the unrenormalized transition is precisely second order (i.e where the mean-field specific heat critical exponent is precisely zero). A linear functional renormalization-group analysis of this Aukrust-Hauge class of models yields exactly the same dramatic fluctuation-induced non-universal critical exponents that have previously been predicted. We conclude by considering possible implications of this result on future Ising model simulations and on the basis of this propose a numerical test for the validity of existing renormalization-group analyses of continuum effective interfacial Hamiltonians.