Connected Green function approach to symmetry breaking inΦ 1+1 4 -theory

Using the cluster expansions for n-point Green functions we derive a closed set of dynamical equations of motion for connected equal-time Green functions by neglecting all connected functions higher than 4 ^th order for theλΦ ⁴-theory in 1 + 1 dimensions. We apply the equations to the investigation of spontaneous symmetry breaking, i.e. to the evaluation of the effective potential at temperatureT=0. Within our momentum space discretization we obtain a second order phase transition (in agreement with the Simon-Griffith theorem) and a critical coupling ofλ _crit /4m ²=2.446 as compared to a first order phase transition andλ _crit /4m ²=2.568 from the Gaussian effective potential approach.     

QCD at finite temperature

Quantum chromodynamics is studied at finite temperatures and densities using the temperature Green functions method. For the Green functions the renormalized Schwinger-Dyson equations are obtained and their qualitative properties are discussed. The equality of the renormalization constants for the equations obtained at T, μ ≠ 0 with those for quantum field theory is pointed out. General properties of the gluon polarization tensor are investigated at T, μ ≠ 0. The temperature Green functions are calculated within the one-loop approximation using both relativistic and axial gauges. The fulfilment of the Slavnov-Taylor identities is verified. The asymptotic behaviour of the polarization tensor at T, μ ≠ 0 is established and the excitation spectrum of quark-gluon plasma is found. Both Fermi and Bose excitations are considered and the gauge invariance of the spectra is demonstrated. The renormalization group extension of the dispersion laws into the regions of high temperatures and densities is presented. The exact representation of the thermodynamical potential in QCD is found in terms of the temperature Green functions. For the quark-gluon plasma the thermodynamical potential is calculated with the g³-term taken into account. The equation of state of the hot quark-gluon plasma is found and its properties are discussed. The complete evolutional diagram of the hadronic matter is outlined. The phase curve asymptotics, which put bounds on the quark-gluon plasma domain, are found for the two limiting cases (μ = 0, T → T₀; T = 0, μ → μ₀). The phase transition of the hot quark-gluon plasma placed in external Abelian field is studied. The instability of such plasma has been found and the program of its stabilization is indicated. The infrared behaviour of the non-Abelian gauge theory is studied for finite temperatures when power divergencies are essential. The propagator of transverse gluons is shown to be singular for momenta |p| ˜ g²T and this cannot be avoided by summing the simplest bubble chains. The infrared asymptotic behaviour of the tree-gluon vertex is found and the results obtained are checked using the Slavnov-Taylor identities. The Green functions asymptotics found indicate either an instability of the quark-gluon plasma in the infrared momentum domain or the inconsistency of the perturbational methods. A non-perturbative approach to the infrared problem in QCD is developed within the axial gauge. The closed equations for the structure functions that determine the gluon polarization tensor are obtained by using the Slavnov-Taylor identities to found approximately the three-gluon vertex. It is shown that the solution of the equations obtained by iterations does not remove the infrared singularity from the temperature Green functions. The nonperturbative solution of such equations is discussed.     

Green function theory of Curie and order—order transitions in ferromagnetic systems

Dipolar critical temperatures in ferromagnetic systems with isotropic bilinear and biquadratic exchange are investigated by means of the Green function technique. Expressions are found for both the familiar Curie temperature, T_c, and the less well known order-order transition temperature, T_o, at which, under appropriate conditions, the magnetic ordering undergoes a change between fully aligned and canted ferromagnetism. At T = 0, a fully aligned state has <s_i^z = s for spin s and all lattice sites i, while a canted state has 〈s_i^z〉<s. It is shown independently of the Green function analysis that the T = 0 ground state is fully aligned if α, the ratio of biquadratic to bilinear exchange integrals, obeys ?[2s(s?1)]^?1<α< [2s²?2s+1]^?1. The region below the lower limit is identified as the range in which canted ferromagnetism can occur and is a range that does not appear to have been considered previously via the Green function formalism.The temperature dependence of the magnetic ordering is investigated by means of the double-time temperature-dependent Green function formalism. A new decoupling scheme is derived and used to reduce higher order Green functions to lowest order. It is found that a canted state, occuring at low temperatures, undergoes a transition to a fully aligned state at a temperature T₀ and subsequently becomes disordered at temperature T_c. Transitions to paramagnetism are found to be second order for α<α_c and first order for α?α_c where α_c is a critical value that depends on the atomic spin and weakly on the lattice structure. A phase diagram is given to illustrate the results, and a comparison is made with the corresponding results found in mean field theory.     

Radiatively induced spontaneous symmetry breaking and phase transitions in curved spacetime

Local gauge symmetries which are spontaneously broken in flat spacetime are shown to be restored for large spacetime curvatures. The case of symmetry breaking due to radiative quantum corrections in gauge theories with elementary scalar fields is considered explicitly. In spacetimes with a positive Ricci curvature scalar R and a cosmological event horizon, the critical curvature R_C is of O(m_H²) or O(m_W²), depending on whether the theory is formulated with conformal or minimal scalar fields. In Ricci flat spacetimes with a conventional event horizon the symmetry is expected to be restored when the temperature of the Hawking thermal radiation is of O(m_W). This phenomenon is described in detail, using functional integral methods and dimensional renormalization, for massless scalar electro-dynamics in de-Sitter spacetime. For conformal scalars, the symmetry restoring phase transition is first order, the critical curvature being R_C = 0.910 m_H². For minimal scalars, an anomalous, curvature dependent mass counterterm is required. The phase transition in this case is second order, and occurs at R_C = 83.57 m_W². Symmetry restoration at finite temperature in flat spacetime is considered in an appendix. The critical temperature at which a first-order phase transition occurs in the Weinberg-Salam model is found to be T_C = 0.329 m_W.     

An X-Y antiferromagnet

A two-sublattice X-Y antiferromagnet is treated using the second-order Green function theory. The higher-order Green functions are decoupled by the random phase approximation (RPA), and the transition temperature T_N, where the paramagnetic staggered susceptibility diverges, and the critical values of the spin correlation functions are calculated for systems having simple cubic and body centred cubic lattice structures and for all spin values. Comparing the results with the experimental Néel temperatures of Co(C₅H₅NO)₆(ClO₄)₂ and Co(C₅H₅NO)₆(BF₄)₂ we get J/k_B=0.258 K and 0.216 K for these compounds, respectively.     

Application of the Green's function method to ferrimagnetism

The Green function theory, which has been used hitherto for ferro- and antiferromagnetism, has been extended for the case of a two-sublattice ferrimagnet. Two parametrized Green functions are used corresponding to the two sublattices. The equations of motion are set up and the higher order Green functions are decoupled according toCallen's approximation. The functions are obtained by solving the simultaneous equations. The quasi-particle energies are evaluated from the singularities of the Green functions and the magnetization at low temperatures is found to obey theT ^3/2-law. These results obtained for the energy and magnetization agree well with those obtained by the conventional spin wave method.     

Vacuum stability conditions from copositivity criteria

In this paper, we introduce a non-minimally conformally coupled scalar field and dark matter in F(T) cosmology and study their dynamics. We investigate the stability and phase space behavior of the parameters of the scalar field by choosing an exponential potential and cosmologically viable form of F(T). We found that the dynamical system of equations admits two unstable critical points; thus no attractor solutions exist in this cosmology. Furthermore, taking into account the scalar field mimicking quintessence and phantom energy, we discuss the corresponding cosmic evolution for both small and large times. We investigate the cosmological implications of the model via the equation of state and deceleration parameters of our model and show that the late-time Universe will be dominated by phantom energy and, moreover, phantom crossing is possible. Our results do not lead to explicit predictions for inflation and the early Universe era.     

Thermal conductivity of two binary mixtures of gases of equal molecular weight

The paper reports new measurements of the thermal conductivity of two binary mixtures at a nominal temperature of 27.5°C but over a pressure range. The two systems, N₂-CO (three compositions up to 12MPa) and N₂O-CO₂ (four mixtures up to 4.2 MPa) have been chosen because they consist of molecules of similar structure and equal molecular weight (28.01 and 44.01, respectively). Moreover, the zero-density viscosity is identical within each mixture.The results have been extrapolated and fitted to equations in the usual way. The zero-density thermal conductivity in each system varies systematically with composition and differs by about 4% for the pure components in each case. This is a measure of the effect of the internal degrees of freedom on thermal conductivity.In the absence of a reliable theory, it is shown that the equations of kinetic theory can be used to represent the composition-dependence of zero-density thermal conductivity with a judicious choice of two quantities which cannot be calculated independently.In both systems, both the excess thermal conductivity λ(T,?) - λ(T, 0) and the ratio λ(T, ?)?λ(T, 0) are sole functions of density with a remarkable degree of precision.The variation of the thermal conductivity with density could be predicted quit accurately with the aid of the theory due to Mason.     

Critical slowing down of fluctuations in p-terphenyl and p-quaterphenyl observed by proton spin-lattice relaxation

The temperature evolution of the proton spin-lattice relaxation time T₁ in p-terphenyl and in p-quaterphenyl around their order-disorder phase transition has been measured. In both cases pretransitional collective fluctuations destroy the high temperature Arrhenius behaviour of the relaxation rate corresponding to a single reorientational jump motion. The spin-lattice relaxation times present then a drastic decrease until the transition temperature (T₀ = 193 K in p-terphenyl, T₀ = 238 K in p-quaterphenyl). This decrease is associated to the critical slowing down of fluctuations. In the low temperature phase the ordering phenomena lead to a sharp drop of the spin-lattice relaxation rate.     

Temperature dependence of coupling constants

We study the temperature-dependence of coupling constants at the one-loop level for massive ?⁴ theory and massive scalar electrodynamics (SED). It is found that the scalar coupling constant λ for m² > 0 decreases with temperature leading to a phase transition to a non-interacting phase. In a model with m² < 0, λ increases as 1n T. The gauge coupling constant of SED increases uniformly with temperature.     

The Non-Perturbative Approach to the Infrared Problem in QCD at Finite Temperature

A non-perturbative approach is developed for investigation of the infrared problem in QCD at T ≠ 0 in the ghost-free axial gauge. The problem is solved by using a 3-dimensional theory within the exact Slavnov-Taylor identities and Schwinger-Dyson equations. The system of two non-linear integral equations for the structural functions of the gluon polarization tensor is obtained whose solution determines the infrared behavior of the temperature Green functions in the 4-dimensional QCD. The simplest solution of these equations which is the same as the first term of the perturbation expansion shows the presence of singularities in the gluon propagator at momenta p ∼ g²T, that cannot be eliminated by any choice of the gauge. The infrared instability of QCD at T ≠ 0 caused by these singularities is discussed.     

First order β, γ phase transition in NH4Br

The course of the order parameter around the β, γ phase transition (235 K) in the NH₄Br has been closely observed by means of linear optical birefringence. The experimental data were fitted in the critical region T > 0.9T₀ by the Landau theory and a simple power law. Both relations reveal a first order transition, with the jump height in the birefringence amounting to Δn(T₀)/?Δn(120 K) = 0.145.     

On heat diffusion mode of structural phase transition by two-fluid model

With the help of the two-fluid model developed by Götze and Michel for phonons it is shown for a simple model Hamiltonian that in the low temperature phase the optical soft mode becomes isothermal, the heat diffusion mode is dominant near the transition temperatureT _c and the quasiparticle interaction is of great importance in determining the thermodynamic quantities nearT _c. Green function techniques are applied to describe the two-fluid model functions in a microscopic way. The simplest approximations are discussed for the model equations describing nonequilibrium phenomena of the soft optical phonon mode in the low temperature phase. The quasiparticle interaction operator can be related to the interaction operator between quasiparticles and the condensed mode. This relation enables one to understand the behaviour of the thermodynamic quantities near the transition temperature on a microscopic way. The first order displacive phase transition is also discussed.     

Effects of charge on dynamical instability of spherical collapse in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) gravity

The effects of charge on stable structure of spherically symmetric collapsing model comprising anisotropic matter distribution are studied in f(R, T) gravity, where R and T correspond to scalar curvature and trace of the energy-momentum tensor, respectively. We construct the field equations, Maxwell equations and dynamical equations in this scenario. We employ linear perturbation scheme on physical variables, metric functions as well as modified terms to establish the evolution or collapse equation for a consistent functional form of f(R, T) gravity. We investigate the limit of instability in Newtonian as well as post Newtonian regimes. It is found that charge plays a fundamental role to slow down the collapse and form a more stable system.     

Exploiting the flexibility of a family of models for taxation and redistribution

In this paper we present and discuss results of Monte Carlo numerical simulations of the two-dimensional Ising ferromagnet in contact with a heat bath that intrinsically has a thermal gradient. The extremes of the magnet are at temperatures T ₁?<?T _c ?<?T ₂, where T _c is the Onsager critical temperature. In this way one can observe a phase transition between an ordered phase (T?<?T _c ) and a disordered one (T?>?T _c ) by means of a single simulation. By starting the simulations with fully disordered initial configurations with magnetization m????0 corresponding to T?=???, which are then suddenly annealed to a preset thermal gradient, we study the short-time critical dynamic behavior of the system. Also, by setting a small initial magnetization m?=?m ₀, we study the critical initial increase of the order parameter. Furthermore, by starting the simulations from fully ordered configurations, which correspond to the ground state at T?=?0 and are subsequently quenched to a preset gradient, we study the critical relaxation dynamics of the system. Additionally, we perform stationary measurements (t??????) that are discussed in terms of the standard finite-size scaling theory. We conclude that our numerical simulation results of the Ising magnet in a thermal gradient, which are rationalized in terms of both dynamic and standard scaling arguments, are fully consistent with well established results obtained under equilibrium conditions.     

Palatini formulation of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) gravity theory,and its cosmological implications

We consider the Palatini formulation of f(R, T) gravity theory, in which a non-minimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similar to the metric case, the field equations impose the non-conservation of the energy-momentum tensor. We obtain the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra force, which is identical to the one obtained in the metric case. The thermodynamic interpretation of the theory is also briefly discussed. We investigate in detail the cosmological implications of the theory, and we obtain the generalized Friedmann equations of the f(R, T) gravity in the Palatini formulation. Cosmological models with Lagrangians of the type \(f=R-\alpha ^2/R+g(T)\) and \(f=R+\alpha ^2R^2+g(T)\) are investigated. These models lead to evolution equations whose solutions describe accelerating Universes at late times.     

The magnetic phase diagram of CsNiF3 as determined by neutron diffraction

By neutron diffraction we investigated the magnetic properties of CsNiF₃ around and below T_N with and without magnetic field. We measured the phase diagram for 1.9 K ≤ T ≤ T_N = 2.7 K in an external field. The phase transition in this temperature-range is always continuous. We were able to measure the critical scattering connected with the magnetic phase transition. This critical scattering is centered around the position of the magnetic Bragg reflections. The critical scattering was found to be similar everywhere on the phase-separation line (H ≠ )0 and at T_N(H = 0). When applying a small field only processes were found, which correspond to a growing or diminishing of certain domains.     

Critical changes in vibrational modes in a magnetic debye crystal with a magnon-phonon coupling

Using the Green function technique we calculate the mean square of the vibrational amplitude of an ion (or equivalently the Debye Waller factor W, or the Mössbauer recoil free fraction ?) in a magnetic Debye crystal near the magnetic phase transition temperature T_c. We find that the magnon-phonon coupling which leads to the phonon spectrum broadening and frequency shift leads to sharp changes in W near T_c. The frequency shift leads to an increase in W, the broadening leads to a decrease in W. These results can explain the anisotropic critical changes in W near T_c observed recently in the magnetostrictive materials RFe₂ and RCo₂ (R = rare earth). In addition we calculate the expected changes in the second order Doppler shift near T_c. We find that when T approaches T_c from above a positive discontinuity and decrease in absolute value of slope of the thermal shift should occur.     

Effect of irradiation at low doses and incubation of voids within the rate theory approach

The evolution of defects in a material under irradiation is studied at low doses (∼5 dpa or less) using rate equations. It is shown that as a function of temperature at a critical valueT _c a transition occurs in the behaviour of the solutions of the rate equations. BelowT _c the voids show incubation effects. An expression is derived for the critical dislocation density at which the void growth starts. This is related to the trapped vacancy fraction ε in vacancy dislocation loops. AboveT _c the incubation effects are shown to be related to the gas production rate which becomes the rate controlling parameter in determining the evolution of the defects. A gas-bubble to void transition occurs at a critical void radius and expressions are derived for the critical void size and dose at which the transition appears. It is shown that closely related to this is the incubation dose for interstitial loops. Finally, these features are corroborated by actual numerical integration of the rate equations.     

An apparent Landau-type second order transition in the spinel CdIn2S4

Experimental evidence is given of a second order transition in the spinel CdIn₂S₄. The specific heatc _p shows a step at the transition temperature (T _c=403°K), the thermal expansion coefficientα remains unaffected. An analysis yields a critical region of this transition considerably narrower (5°K) than expected from the critical regions of transitions of a similar type (quartz: 160°K,β-brass: 250°K) and therefore the critical region has not been observed experimentally, i.e. the transition appears to follow Landau’s theory. Landau’s group theoretical analysis of second order transitions in crystals leads to a model for the transition in CdIn₂S₄ on the atomic scale. If the temperature is decreased below the critical temperature a 1:1 ordering occurs between Cd and In on the tetrahedral sites. With this model the gross features of this transition can be qualitatively understood: The unaffected thermal expansion and the long relaxation times for the establishment of equilibrium approachingT _c. Also the narrow critical region can be explained in terms of a large correlation length in CdIn₂S₄. Finally it is shown that a generalized formulation of the thermodynamic part of Landau’s theory correlates thermodynamic quantities far above and far below the critical temperature, as illustrated for CdIn₂S₄. This generalization allows also a more satisfactory determination of the critical region.