Order Phase Transition Point Research Articles

Third and fourth order phase transitions: Exact solution for the Ising model on the Cayley tree

In this work we present the first exact solution of a system of interacting particles with phase transitions of order higher than two. The presented analytical derivation shows that the Ising model on the Cayley tree exhibits a line of third order phase transition points, between temperatures T 2 = 2 k B − 1 J ln ( 2 + 1 ) and T B P = k B − 1 J ln ( 3 ) , and a line of fourth order phase transitions between T B P and ∞ , where k B is the Boltzmann constant, and J is the nearest-neighbor interaction parameter.

Correlation functions of the energy-momentum tensor in SU(2) gauge theory at finite temperature

We calculate correlation functions of the energy-momentum tensor in the vicinity of the deconfinement phase transition of ($3+1$)-dimensional SU(2) gauge theory and discuss their critical behavior in the vicinity of the second order deconfinement transition. We show that correlation functions of the trace of the energy-momentum tensor diverge uniformly at the critical point in proportion to the specific heat singularity. Correlation functions of the pressure, on the other hand, stay finite at the critical point. We discuss the consequences of these findings for the analysis of transport coefficients, in particular, the bulk viscosity, in the vicinity of a second order phase transition point.

Quasinormal modes and second order thermodynamic phase transition for Reissner–Nordström black hole

The relation between the quasinormal modes (QNMs) and the second order thermodynamic phase transition (SOTPT) for the Reissner–Nordström (RN) black hole is studied. It is shown that the quasinormal frequencies of the RN black hole start to get a spiral-like shape in the complex ω plane and both the real and imaginary parts become the oscillatory functions of the charge if the real part of the quasinormal frequencies arrives at its maximum at the second order phase transition point of Davies for given overtone number and angular quantum number. That is to say, we can find out the SOTPT point from the QNMs of the RN black hole. The fact shows that the quasinormal frequencies carry the thermodynamical information of the RN black hole.

Quasinormal frequencies and thermodynamic instabilities for the stationary axisymmetric Einstein-Maxwell dilaton-axion black hole

The massless scalar quasinormal frequencies of a stationary axisymmetric Einstein-Maxwell dilaton-axion (EMDA) black hole are investigated by using Leaver's continued fraction method. It is shown that in the complex plane the frequencies move counterclockwise and get a spiral-like shape as the angular momentum per unit mass a increases to its extremal value or the dilaton D decreases to its extremal value for the rotating black hole. However, for the non-rotating Garfinkle-Horowitz-Strominger dilaton (GHSD) black hole, the dilaton parameter D, which is related to the electric charge of this EMDA black hole, cannot make the frequencies spire in the complex ω plane, which is qualitatively different from the charge of the Reissner-Nordstrom (RN) black hole. The so-call ``Spiral-like Criterion" is obtained and it points out that the frequencies won't spire in the complex ω plane if the heat capacity for the considered black hole is always negative and vice versa. The most interesting outcome of our calculation is that the critical point, at which the imaginary part of the wave function related to time-dependent part (e−iωt) begins to oscillate obviously for the given quantum number, is just the second order phase transition point of Davies. The fact seems to imply that there is some relation between the dynamical evolution and thermodynamic instabilities for the black hole.

On the nature of isotherms at first order phase transitions for classical lattice models

Two main results are presented in this paper: Isakov’s Theorem about the impossibility of an analytic continuation of the pressure at a first order phase transition point and Friedli’s work about the restoration of the analytic continuation of the pressure in the van der Waals limit at such a transition point. An exposition of Pirogov-Sinai theory for models with two periodic ground-states is given in section 3, which can be read independently.

Remarks on the multiparameter reweighting method for the study of lattice QCD at nonzero temperature and density

We comment on the reweighting method for the study of finite density lattice QCD. We discuss the applicable parameter range of the reweighting method for models that have more than one simulation parameter. The applicability range is determined by the fluctuations of the modification factor of the Boltzmann weight. In some models having a first order phase transition, the fluctuations are minimized along the phase transition line if we assume that the pressure in the hot and the cold phases is balanced at the first order phase transition point. This suggests that the reweighting method with two parameters is applicable in a wide range for the purpose of tracing out the phase transition line in the parameter space. To confirm the usefulness of the reweighting method for two-flavor QCD, the fluctuations of the reweighting factor are measured by numerical simulations for the cases of reweighting in the quark mass and chemical potential directions. The relation to the phase transition line is discussed. Moreover, the sign problem caused by the complex phase fluctuations is studied.

Pathology of Schwinger boson mean-field theory for Heisenberg spin models

We have re-analyze the Schwinger boson mean field theory (SBMFT) for Heisenberg spin models on the cubic lattice. We find that the second order phase transition point for magnetic ordering previously reported corresponds to a local maximum of the free energy functional. For both ferromagnetic and antiferromagnetic Heisenberg models with spin $S \geq S_C$, where $S_C < 1/2$, the mean field transitions are first order from the magnetically long-ranged ordered phase to the completely uncorrelated phase. In addition to erroneously giving a first order transition for magnetic ordering, the mean field theory does not include a phase with finite short-range correlation, thus negating one of the prime advantages of SBMFT. The relevance of these pathologies to other situations beyond the cubic lattice is discussed.

Large- q expansion of the correlation length in the two-dimensional q-state Potts model

The large- q expansions of the exponential correlation length and the second moment correlation length for the q-state Potts model in two dimensions are calculated at the first order phase transition point both in the ordered and disordered phases. The expansion coefficients in the ordered and disordered phases coincide in lower orders for both of the two types of the correlation lengths, but they differ a little from each other in higher orders for the second moment correlation length. The second largest eigenvalues of the transfer matrix have the continuum spectrum both in the ordered and disordered phases in the large- q region, which is suggested to be maintained even in the limit of q → 4 from the analysis of the expansion series.

Large- q expansion of the two-dimensional q-state Potts model by the finite lattice method

We have calculated the large- q expansion for the energy and magnetization cumulants at the first order phase transition point in the two-dimensional q-state Potts model to the 21st or 23rd order in ▪ using the finite lattice method. The obtained series allow us to give highly convergent estimates of the cumulants for q > 4. The results confirm us the correctness of the conjecture by Bhattacharya et al. on the asymptotic behavior of the energy cumulants for q → 4 + and a similar new conjecture on the magnetization cumulants.

Deterministic Equations of Motion and Dynamic Critical Phenomena

Taking the two-dimensional $\phi^4$ theory as an example, we numerically solve the deterministic equations of motion with random initial states. Short-time behavior of the solutions is systematically investigated. Assuming that the solutions generate a microcanonical ensemble of the system, we demonstrate that the second order phase transition point can be determined already from the short-time dynamic behavior. Initial increase of the magnetization and critical slowing down are observed. The dynamic critical exponent z, the new exponent $\theta$ and the static exponents $\beta$ and $\nu$ are estimated. Interestingly, the deterministic dynamics with random initial states is in a same dynamic universality class of Monte Carlo dynamics.

Nonlinear Interaction of Light with Transversely Moving Medium.

Interaction of a light beam with matter reveals extreme sensitivity to transverse motions. Spatio-temporal delocalization of interaction and diffusive transport of light-induced perturbations is revealed through a new family of diffraction patterns. The response of the medium to the transverse motion of the light enhances near the second order phase transition point. The phenomenon has large application potential for the study of motions that are beyond the power of Doppler-effect-based methods, and suggests fundamental problems about critical processes with nonstationary influences.

Critical behavior of the three-dimensional random-field Ising model: Two-exponent scaling and discontinuous transition.

In extensive Monte Carlo simulations the phase transition of the random field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite size scaling. For a Gaussian distribution of the random fields it is found that the correlation length $\xi$ diverges with an exponent $\nu=1.1\pm0.2$ at the critical temperature and that $\chi\sim\xi^{2-\eta}$ with $\eta=0.50\pm0.05$ for the connected susceptibility and $\chi_{\rm dis}\sim\xi^{4-\bar{\eta}}$ with $\bar{\eta}=1.03\pm0.05$ for the disconnected susceptibility. Together with the amplitude ratio $A=\lim_{T\to T_c}\chi_{\rm dis}/\chi^2(h_r/T)^2$ being close to one this gives further support for a two exponent scaling scenario implying $\bar{\eta}=2\eta$. The magnetization behaves discontinuously at the transition, i.e. $\beta=0$, indicating a first order transition. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multi-peak structure that is characteristic for the phase-coexistence at first order phase transition points.

First order phase transitions of spin systems

I review some numerical ways to determine the parameters of systems close to a first order phase transition point: energy and specific heat of the coexisting phases and interface tension. Numerical examples are given for the 2-d $q$ states Potts model.

Large energy cumulants in the 2D Potts model and their effects in finite size analysis

We develop an ansatz for expressing the free energy of the two-dimensional q-states Potts model for q > 4 near its first order phase transition point. We notice that for the moderate values of q < 15, the energy profile at the phase transition is not expressible as a sum of gaussians. We discuss how this affects the traditional finite size analysis of this phase transition. In particular, the dominant length scale governing the finite size corrections turns out to be much (∼ 6 times) larger than the largest correlation length in the problem.

Effect of hydrostatic pressure on the phase transitions and dielectric properties of {(CH3)2NH2}5CD3CL11

Abstract The effect of hydrostatic pressure has been studied for the phase transition and dielectric properties of {(CH3)2NH2}5Cd3Cl11. The maximum value of the dielectric constant peak, which exists at T 1 = 280 K at 0 pressure, decreases with increasing pressure obeying a Curie-Weiss like law with the Curie-Weiss pressure of about −860 MPa. This implies that the material becomes ferroelectric at this hypothetical negative pressure. As pressure increases the first order phase transition point Tc , which exist at 127 K at 0 MPa, first decreases with dTc /dp = − 0.064 K/MPa, and then it turns to increase with dTc /dp = + 0.25 K/MPa above about 100 MPa. This indicates that there is a pressure-induced low-temperature phase above about 100 MPa.

2S + 1 State Vertex Model at a First Order Transition Point

We present study on a multi-state vertex model solved by Sogo et al. at one of first order phase transition points. The origin of the level crossings is analyzed in terms of vertex weights and the Bethe ansatz equation. Singularity at this point is discussed in terms of quantum enveloping algebra

AC calorimetry at the first order phase transition point

A model is proposed for AC calorimetry (ACC) at the first order phase transition point. The model is compared with the results of ACC around the melting point of ann-paraffin (C20H42). The observed frequency dependence of ACC is consistent with the model. A harmonic component of the temperature modulation with a frequency equal to twice the heating frequency was observed at the phase transition point. It is shown that the harmonic component can be explained on the basis of the proposed model.

SELF-CONSISTENT SOLUTION OF THE SIMULTANEOUS SCHWINGER-DYSON EQUATION IN STRONG COUPLING QED

In the strong coupling region of QED, we have obtained a numerical solution to the simultaneous Schwinger-Dyson equation for the fermion and the photon propagators. It is shown that there is a critical point separating the spontaneous-chiral-symmetry-breaking (strong coupling) phase and the weak coupling phase. This critical point is consistently interpreted as the second order phase transition point at which the continuum limit of cutoff QED may be taken. The vacuum polarization function obtained in this framework exhibits essentially the same asymptotic uv behavior as that predicted from the one-loop calculation. The scaling behavior is not inconsistent with the mean-field result as predicted from the one-loop case, which favors the triviality of QED.

SIMULATIONS OF FOUR-DIMENSIONAL SIMPLICIAL QUANTUM GRAVITY AS DYNAMICAL TRIANGULATION

Four-Dimensional Simplicial Quantum Gravity is simulated using the dynamical triangulation approach. We studied simplicial manifolds of spherical topology and found the critical line for the cosmological constant as a function of the gravitational one, separating the phases of opened and closed Universe. When the bare cosmological constant approaches this line from above, the four-volume grows: we reached about 5×104 simplexes, which proved to be sufficient for the statistical limit of infinite volume. However, for the genuine continuum theory of gravity, the parameters of the lattice model should be further adjusted to reach the second order phase transition point, where the correlation length grows to infinity. We varied the gravitational constant, and we found the first order phase transition, similar to the one found in three-dimensional model, except in 4D the fluctuations are rather large at the transition point, so that this is close to the second order phase transition. The average curvature in cutoff units is large and positive in one phase (gravity), and small negative in another (antigravity). We studied the fractal geometry of both phases, using the heavy particle propagator to define the geodesic map, as well as with the old approach using the shortest lattice paths. The heavy propagator geodesic appeared to be much smoother, so that the scaling laws were found, corresponding to finite fractal dimensions: D+~2.3 in the gravity phase and D−~4.6 in the antigravity phase. Similar, but somewhat lower numbers were obtained from the heat kernel singularity. The influence of the αR2 terms in 2, 3 and 4 dimensions is discussed.

Schwinger-dyson equation approach to strong coupling QED

In the strong coupling region of QED, we have tried to obtain a solution to the simultaneous Schwinger-Dyson equation for the fermion and the photon propagators. It is shown that there is a critical point separating the spontaneous-chiral-symmetry-breaking (strong coupling) phase and the weak coupling phase. This critical point is consistently interpreted as the second order phase transition point at which the continuum limit of cutoff QED may be taken. The vacuum polarization function obtained numerically in this framework exhibits essentially the same asymptotic ultraviolet behavior as that predicted from the one-loop calculation. The scaling behavior is not inconsistent with the mean-field result as predicted from the one-loop case, which favors the triviality of QED.