Specific Heat Critical Exponent Research Articles

Out-of-equilibrium scaling of the energy density along the critical relaxational flow after a quench of the temperature.

We study the out-of-equilibrium behavior of statistical systems along critical relaxational flows arising from instantaneous quenches of the temperature T to the critical point T_{c}, starting from equilibrium conditions at time t=0. In the case of soft quenches, i.e., when the initial temperature T is assumed sufficiently close to T_{c} (to keep the system within the critical regime), the critical modes develop an out-of-equilibrium finite-size-scaling (FSS) behavior in terms of the rescaled time variable Θ=t/L^{z}, where t is the time interval after quenching, L is the size of the system, and z is the dynamic exponent associated with the dynamics. However, the realization of this picture is less clear when considering the energy density, whose equilibrium scaling behavior (corresponding to the starting point of the relaxational flow) is generally dominated by a temperature-dependent regular background term or mixing with the identity operator. These issues are investigated by numerical analyses within the three-dimensional lattice N-vector models, for N=3 and 4, which provide examples of critical behaviors with negative values of the specific-heat critical exponent α, implying that also the critical behavior of the specific heat gets hidden by the background term. The results show that, after subtraction of its asymptotic critical value at T_{c}, the energy density develops an asymptotic out-of-equilibrium FSS in terms of Θ as well, whose scaling function appears singular in the small-Θ limit.

Finite-size scaling and bulk critical behavior of a quantum spherical model with a long-range interaction: entropy, internal energy and specific heat

The entropy, the internal energy and the specific heat of a d-dimensional quantum spherical model with a long-range interaction, the decreasing of which is controlled by a parameter σ (0 < σ ≤ 2), are studied close to its quantum critical point in the context of the finite-size scaling (FSS) theory for d = σ. The obtained specific heat critical exponent σ does not depend on σ and satisfies the quantum hyperscaling relation. Based on the derived FSS forms and asymptotes of the universal scaling functions of the considered quantities, the leading temperature dependencies of the entropy, the internal energy and the specific heat are obtained in both the renormalized classical and the quantum disordered regions of the phase diagram. It has been shown that when the temperature T → +0, the entropy and the specific heat in the renormalized classical region tend to zero as powers of T, while in the quantum disordered region they tend to zero exponentially.

Critical exponents of the O(N)-symmetric phi ^4 model from the varepsilon ^7 hypergeometric-Meijer resummation

We extract the varepsilon -expansion from the recently obtained seven-loop g-expansion for the renormalization group functions of the O(N)-symmetric model. The different series obtained for the critical exponents nu , omega and eta have been resummed using our recently introduced hypergeometric-Meijer resummation algorithm. In three dimensions, very precise results have been obtained for all the critical exponents for N=0,1,2,3 and 4. To shed light on the obvious improvement of the predictions at this order, we obtained the divergence of the specific heat critical exponent alpha for the XY model. We found the result -0.0123(11) which is compatible with the famous experimental result of -0.0127(3) from the specific heat of zero gravity liquid helium superfluid transition while the six-loop Borel with conformal mapping resummation result in literature gives the value -0.007(3). For the challenging case of resummation of the varepsilon -expansion series in two dimensions, we showed that our resummation results reflect a significant improvement to the previous six-loop resummation predictions.

Exact analysis of phase transitions in mean-field Potts models.

We construct the exact partition function of the Potts model on a complete graph subject to external fields with linear and nematic type couplings. The partition function is obtained as a solution to a linear diffusion equation and the free energy, in the thermodynamic limit, follows from its semiclassical limit. Analysis of singularities of the equations of state reveals the occurrence of phase transitions of nematic type at not zero external fields and allows for an interpretation of the phase transitions in terms of shock dynamics in the space of thermodynamic variables. The approach is shown at work in the case of a q-state model for q=3 but the method generalizes to arbitrary q. Critical asymptotics of magnetization, susceptibility, specific heat and relative critical exponents β,γ, and α are also provided.

Zeros of the 3-state Potts model partition function for the square lattice revisited**This paper is dedicated to the memory of Vladimir. What PM knows of mentoring he learned from Vladimir, his mentor. Hopefully FZ will become another link in the chain.

We compute the exact partition function for the 3-state Potts model on square lattices of several sizes larger than previously accessible. Making comparison with the exactly solved Ising model we show that, for aspects of the analytic structure close to the ferromagnetic transition point, these lattices are large enough to approach the thermodynamic limit. Subject to certain assumptions this allows for computation of estimates for the specific heat critical exponent. We thus obtain an estimate for this exponent. The estimate is consistent with the known result, thus demonstrating the potential use of this method for other models. We also discuss the antiferromagnetic transition.

Susceptibility of the Ising Model on a Kagomé Lattice by Using Wang-Landau Sampling

The susceptibility of the Ising model on a kagome lattice has never been obtained. We investigate the properties of the kagome-lattice Ising model by using the Wang-Landau sampling method. We estimate both the magnetic scaling exponent yh = 1.90(1) and the thermal scaling exponent yt = 1.04(2) only from the susceptibility. From the estimated values of yh and yt, we obtain all the critical exponents, the specific-heat critical exponent α = 0.08(4), the spontaneous-magnetization critical exponent β = 0.10(1), the susceptibility critical exponent γ = 1.73(5), the isothermalmagnetization critical exponent δ = 16(4), the correlation-length critical exponent ν = 0.96(2), and the correlation-function critical exponent η = 0.20(4), without using any other thermodynamic function, such as the specific heat, magnetization, correlation length, and correlation function. One should note that the evaluation of all the critical exponents only from information on the susceptibility is an innovative approach.

Critical behavior in the vicinity of nematic to smectic A (N–SmA) phase transition of two polar–polar binary liquid crystalline systems

ABSTRACTPhase diagram, critical behavior and order of the nematic (N)–smectic A (SmA) phase transition of two polar–polar binary systems (i) 4-n-heptyloxy-4′-cyanobiphenyl (7OCB) and 4-n-octyloxy-4′-cyanobiphenyl (8OCB); (ii) 4-n-octyloxy-4′-cyanobiphenyl (8OCB) and 4-n-nonyloxy-4′-cyanobiphenyl (9OCB) by means of a high-resolution temperature scanning measurement of birefringence have been reported in this work. A simple power law analysis has been adopted to extract the specific heat critical exponent (α′) at N–SmA transition from birefringence data. The α′ for N–SmA transition indicates a uniform crossover behavior and has appeared to be non-universal in nature. With increasing concentration of the higher homologues for both the binary systems, the N–SmA transition reveals a strong tendency to be driven towards the tricritical nature. The 3D-XY limit (i.e. α′ = −0.007) for N–SmA transition reaches at the concentration x8OCB = 0.28 corresponding to the McMillan ratio 0.914, whereas the tricritical point has been found to appear near x9OCB = 1.0 corresponding to McMillan ratio 0.992.

Theory of the anomalous critical behavior for the smectic-A-hexatic transition.

We propose a theoretical explanation for the long-standing problem of the anomalous critical behavior of the heat capacity near the smectic-A-hexatic phase transition. Experiments find a large specific heat critical exponent α=0.5-0.7, which is inconsistent with a small negative value α≈-0.01 expected for the three-dimensional XY universality class. We show that most of the observed features can be explained by treating simultaneously fluctuations of the hexatic orientational and translational (positional) order parameters. Assuming that the translational correlation length ξ_{tr} is much larger than the hexatic correlation length ξ_{h}, we calculate the temperature dependence of the heat capacity in the critical region near the smectic-A-hexatic phase transition. Our results are in quantitative agreement with the calorimetric experimental data.

Dimensional Reduction in Quantum Dipolar Antiferromagnets.

We report ac susceptibility, specific heat, and neutron scattering measurements on a dipolar-coupled antiferromagnet LiYbF_{4}. For the thermal transition, the order-parameter critical exponent is found to be 0.20(1) and the specific-heat critical exponent -0.25(1). The exponents agree with the 2D XY/h_{4} universality class despite the lack of apparent two-dimensionality in the structure. The order-parameter exponent for the quantum phase transitions is found to be 0.35(1) corresponding to (2+1)D. These results are in line with those found for LiErF_{4} which has the same crystal structure, but largely different T_{N}, crystal field environment and hyperfine interactions. Our results therefore experimentally establish that the dimensional reduction is universal to quantum dipolar antiferromagnets on a distorted diamond lattice.

On the Phase Diagram of the 2d Ising Model with Frustrating Dipole Interaction

Due to intrinsic frustrations of interaction, the 2d Ising model with competing ferromagnetic short-range nearest-neighbour and antiferromagnetic long-range dipole interactions possesses a rich phase diagram. The order of the phase transition from the striped h = 1 phase to the tetragonal phase that is observed in this model has been a subject of recent controversy. We address this question by using the partition function density analysis in the complex temperature plane. Our results support the second-order phase transition scenario. To measure the strength of the phase transition, we calculate the values of specific heat critical exponent a. Along with the space dimension, it appears to depend on the ratio of strengths of the short-range and long-range interactions.

Study of the antiferromagnetic Blume-Capel model by using the partition function zeros in the complex temperature plane

The critical temperature and the thermal scaling exponent of the square-lattice Blume-Capel model are obtained as functions of fugacity by calculating the partition function zeros in the complex temperature plane of the square-lattice Blume-Capel antiferromagnet for the first time. The thermal scaling exponent yt determines the specific-heat critical exponent and the correlation-length critical exponent. The value of yt = 2 indicates a strong first-order phase transition in two dimensions whereas yt = 1 corresponds to an Ising-class second-order phase transition. The full spectrum (from yt = 2 to yt = 1) of the thermal scaling exponent for the square-lattice Blume-Capel model is evaluated for the first time. The phase diagram of the square-lattice Blume-Capel model, including the tricritical point, is also obtained, and it is compared with other results in the literature.

Fluctuations and criticality in the random-field Ising model

We investigate the critical properties of the $d=3$ random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength $h$ and system sizes $\mathcal{V}=L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$, with $L\ensuremath{\le}156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength ${h}_{\mathrm{c}}$ and the correlation length exponent $\ensuremath{\nu}$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\ensuremath{\alpha}$ and the violation of the hyperscaling exponent $\ensuremath{\theta}$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\ensuremath{\beta}/\ensuremath{\nu}$ and $\overline{\ensuremath{\gamma}}/\ensuremath{\nu}$, respectively.

Positive specific-heat critical exponent of a three-dimensional three-state random-bond Potts model

Whereas the stability of a pure critical system is determined by the sign of its specific-heat critical exponent α according to Harrisʼ criterion, whether a d-dimensional dirty system should satisfy ν⩾2/d and α<0 or not has been a controversial issue for several decades, where ν is its correlation-length critical exponent. Here, contrary to recent analytical and numerical results, we find for the three-dimensional three-state random-bond Potts model whose pure version exhibits a first-order phase transition a random fixed point whose ν<2/d and α>0 using a finite-time scaling combining with extended dynamic Monte Carlo renormalization-group method. This suggests further studies are still needed to clarify the issue in three-dimensional systems.

Monte Carlo analysis of the critical properties of the two-dimensional randomly bond-diluted Ising model via Wang–Landau algorithm

The influence of random bond-dilution on the critical properties of the two-dimensional Ising model on a square lattice with periodic boundary conditions is presented using the Monte Carlo process with Wang–Landau sampling. The lattice linear size is L = 20 – 140 and the concentration of the diluted bonds spans a wide range from weak to strong dilution, namely, q = 0.05 , 0.1 , 0.2 , 0.3 , 0.4 ; the respective percolation limit for the square lattice is q c P E R C = 0.5 . Its pure version ( q = 0 ) has a second-order phase transition with vanishing specific heat critical exponent, an example of inapplicability of the Harris criterion. The main effort is focused on the temperature dependence of the specific heat and magnetic susceptibility to estimate the respective maximum values and subsequent pseudocritical temperatures for extracting the relative critical exponents. We have also looked at the probability distribution of the susceptibility, pseudocritical temperature and specific heat for assessing self-averaging. The study is carried out in the appropriately restricted but dominant energy subspaces. By applying the finite-size scaling analysis, the critical exponents are estimated; the specific heat exponent vanishes, the correlation-length exponent ν is equal to one and the critical exponents’ ratio ( γ / ν ) retains its pure-Ising-model value, thus supporting the strong universality hypothesis.

The isotropic-to-nematic transition in a two-dimensional fluid of hard needles: a finite-size scaling study

The isotropic-to-nematic transition in a two-dimensional fluid of hard needles is studied using grand canonical Monte Carlo simulations, multiple histogram reweighting, and finite size scaling. The transition is shown to be of the Kosterlitz-Thouless type, via a direct measurement of the critical exponents eta and beta, of the susceptibility and order parameter, respectively. At the transition, eta=1/4 and beta=1/8 are observed, in excellent agreement with Kosterlitz-Thouless theory. Also the shift in the chemical potential of the nematic susceptibility maximum with system size is in good agreement with theoretical expectations. Some evidence of singular behavior in the density fluctuations is observed, but no divergence, consistent with a negative specific heat critical exponent. At the transition, a scaling analysis assuming a conventional critical point also gives reasonable results. However, the apparent critical exponent beta_eff obtained in this case is not consistent with theoretical predictions.

Monte Carlo analysis of critical properties of the two-dimensional randomly site-diluted Ising model via Wang–Landau algorithm

The influence of random site dilution on the critical properties of the two-dimensional Ising model on a square lattice was explored by Monte Carlo simulations with the Wang–Landau sampling. The lattice linear size was L = 20 – 120 and the concentration of diluted sites q = 0.1 , 0.2 , 0.3 . Its pure version displays a second-order phase transition with a vanishing specific heat critical exponent α , thus, the Harris criterion is inconclusive, in that disorder is a relevant or irrelevant perturbation for the critical behaviour of the pure system. The main effort was focused on the specific heat and magnetic susceptibility. We have also looked at the probability distribution of susceptibility, pseudocritical temperatures and specific heat for assessing self-averaging. The study was carried out in appropriate restricted but dominant energy subspaces. By applying the finite-size scaling analysis, the correlation length exponent ν was found to be greater than one, whereas the ratio of the critical exponents ( α / ν ) is negative and ( γ / ν ) retains its pure Ising model value supporting weak universality.

Thermodynamics of a heavy-ion-irradiated superconductor: The zero-field transition

Specific heat measurements show that the introduction of amorphous columnar defects considerably affects the transition from the normal to the superconducting state in zero magnetic field. Experimental results are compared to numerical simulations of the three-dimensional $XY$ model for both the pure system and the system containing random columnar disorder. The numerics reproduce the salient features of experiment, showing in particular that the specific heat peak changes from cusplike to smoothly rounded when columnar defects are added. By considering the specific heat critical exponent $\ensuremath{\alpha}$, we argue that such behavior is consistent with recent numerical work [A. Vestergren et al., Phys. Rev. B 70, 054508 (2004)] showing that the introduction of columnar defects changes the universality class of the transition.

Re-entrant Nematic Behavior in the 7OCB+9OCB Mixtures: Evidence for Multiple Nematic−Smectic Tricritical Points

The metastable phase diagram of the two-component system heptyloxycyanobiphenyl (7OCB)+nonyloxycyanobiphenyl (9OCB) was determined by means of modulated differential scanning calorimetry (MDSC) and optical microscopy measurements. It was experimentally established that the 7OCB+9OCB two-component system exhibits a monotropic re-entrant nematic behavior. A complete quantitative thermodynamic analysis, through Oonk's equal G analysis, was performed, allowing for the calculation of the monotropic re-entrant behavior and the prediction of two tricritical points, one of them experimentally accessible for the SmAd-to-N transition and the other non-experimentally accessible for the RN-to-SmAd transition. From specific-heat measurements, latent heats were obtained for those mixtures displaying a first-order SmAd-to-N transition. Additionally, for some mixtures, the specific-heat critical exponents (alpha), through the second-order SmAd-to-N transition, were obtained. Both batches of data enable us to access to the experimental tricritical temperature for the SmAd-to-N transition.

Strain effects at the hexatic-B–smectic-Atransition in the 65OBC liquid crystal

We have studied the hexatic-B-smectic-A (HexB-SmA) transition in n -hexyl- 4'-n -pentyloxybiphenyl-4-carboxylate (65OBC) by means of a high-resolution ac photopyroelectric (PPE) calorimetric technique. A procedure for the interpretation of the PPE data, which allows the detection of an internal heating source due to strain annealing and/or latent heat, has been applied. We have found that the strain present in the sample depends on the kinetics of formation of the smectic phase once the sample is cooled from the isotropic one. The strain field keeps memory of this kinetics and can be only partially annealed on decreasing the temperature or cycling it around the Hex-SmA transition. A reversible ordering-disordering process has been found at T(c) and has been explained in terms of the competition between the order parameter variation with temperature and the constraints imposed by the disorder. The results confirm that the transition has a weakly first-order character with a specific heat critical exponent that disagrees with the available theoretical predictions. Our data show the importance of the disorder in 65OBC and we tried to clarify what would be the consequence of this result in theoretical modeling devoted to solving the puzzle of the HexB-SmA transition in this compound.

Self-Consistent Scaling Theory for Logarithmic-Correction Exponents

Multiplicative logarithmic corrections frequently characterize critical behavior in statistical physics. Here, a recently proposed theory relating the exponents of such terms is extended to account for circumstances which often occur when the leading specific-heat critical exponent vanishes. Also, the theory is widened to encompass the correlation function. The new relations are then confronted with results from the literature, and some new predictions for logarithmic corrections in certain models are made.