Being Discrete about Yang and Mills: Basic Techniques of Euclidean Lattice Gauge Theory

We discuss the formulation of gauge-invariant quantum field theories (without dynamical matter fields) as statistical mechanics systems on four-dimensional Euclidean lattices. Approximation methods including strong- and weak-coupling expansions, mean-field theory and Monte Carlo simulations are reviewed in detail, and Abelian duality transformations are derived. New models are discussed. An action is defined on 2 × 1 rectangular loops of links and its properties are investigated. It is found to result in phase transitions in 2, 3 and 4 dimensions with Z(2) and SU(2) gauge groups. A large class of models with Z(N) symmetry realised on plaquettes is investigated, and several phase diagrams are presented. A mixed model with interactions through both plaquettes and rectangles is found to have a line of phase transitions and a critical point associated with the crossover region in the Wilson SU(2) model.     

Generalised actions for lattice gauge models

We consider simple modifications of the conventional Wilson action for lattice gauge theory. An SU(2) action is defined on “plaquettes” of 2×1 links. It is found to possess phase transitions in three- and four-dimensional realisations of the model. A similar model with gauge group Z(2) is also studied, and found to have two phases in three and four dimensions. We discuss the phase structure of Z(N) gauge models in four dimensions with several coupling constants and present phase diagrams for Z(4), Z(5) and Z(6).     

Comparative analysis of the twin and heterophase structures in (1−x)Pb(Mg1/3Nb2/3)O3−xPbTiO3 crystals

The specific features of the coexistence of phases in heavily twinned crystals of the (1?x)Pb(Mg_1/3Nb_2/3)O₃?xPbTiO₃ system in the vicinity of the morphotropic phase boundary (0.30≤x≤0.40) are investigated. The phase transformations in crystals at x = const during cooling are considered at electric field strengths E=0 and 0.1 MV/m. The conditions of the formation of interphase boundaries (zero net strain planes) are determined for different first-order phase transitions. The results of calculating the tetragonal-monoclinic M _C and monoclinic-monoclinic (M _C –M _A ) phase transitions are represented in the form of “twin state-interphase boundary” diagrams. The effect of a 90° domain (twin) structure of the tetragonal phase on the heterophase state associated with the presence of monoclinic phases is analyzed.     

On Z → γγ decay and cancellation of axial anomaly in Z → γγ transition amplitude for massive fermions

Z → γγ decay amplitude is considered and proven to be zero due to properties of polarization vectors and Bose statistics. Triangular diagrams for a pseudoscalar → γγ and Z → γγ processes with massive fermions in the loop are explicitely calculated. In the Standard Model axial anomaly vanishes in the sum of these diagrams as Z boson is mixed with one of the Goldstone bosons.     

The non-perturbative renormalization group in the ordered phase

We study some analytical properties of the solutions of the non-perturbative renormalization group flow equations for a scalar field theory with Z₂ symmetry in the ordered phase, i.e. at temperatures below the critical temperature. The study is made in the framework of the local potential approximation. We show that the required physical discontinuity of the magnetic susceptibility χ(M) at M=±M₀ (M₀ spontaneous magnetization) is reproduced only if the cut-off function which separates high and low energy modes satisfies to some restrictive explicit mathematical conditions; we stress that these conditions are not satisfied by a sharp cut-off in dimensions of space d<4.By generalizing a method proposed earlier by Bonanno and Lacagnina [Nucl. Phys. B 693 (2004) 36] to any kind of cut-off we propose to solve numerically the renormalization group flow equations for the threshold functions rather than for the local potential. It yields an algorithm sufficiently robust and precise to extract universal as well as non-universal quantities from numerical experiments at any temperature, in particular at sub-critical temperatures in the ordered phase. Numerical results obtained for the φ⁴ potential with three different cut-off functions are reported and compared. The data confirm our theoretical predictions concerning the analytical behavior of χ(M) at M=±M₀.Fixed point solutions of the adimensioned renormalization group flow equations are also obtained in the same vein, that is by solving the fixed points equations and the associated eigenvalue problem for the threshold functions rather than for the potential. We report high precision data for the odd and even spectra of critical exponents for different cut-offs obtained in this way.     

One-Dimensional Models for Atoms in Strong Magnetic Fields,II: Anti-Symmetry in the Landau Levels

Electrons in strong magnetic fields can be described by one-dimensional models in which the Coulomb potential and interactions are replaced by regularizations associated with the lowest Landau band. For a large class of models of this type, we show that the maximum number of electrons that can be bound is less than aZ+Zf(Z). The function f(Z) represents a small non-linear growth which reduces to A _p Z(logZ)²when the magnetic field B=O(Z ^p ) grows polynomially with the nuclear charge Z. In contrast to earlier work, the models considered here include those arising from realistic cases in which the full trial wave function for N-electrons is the product of an N-electron trial function in one-dimension and an antisymmetric product of states in the lowest Landau level.     

Magnetization processes of CuMn below the freezing temperature Tf

A simple classical model [5] that is based on a blocking of magnetic clusters in a uniaxial anisotropy field explains characteristic properties of magnetization processes of Cu(5–15 at.%) Mn as transition states between the “zero point magnetization” M₀ = M(T → OK) and the thermal equilibrium magnetization M_∞ = M(t → ∞, T). Further it shows two facts the experimental confirmations of which we report in this paper: (a) a magnetization decrease with increasing temperature in high fields and (b) a superposition rule for the magnetization processes in small fields and at low temperatures.     

Dynamic magnetic hysteresis behavior and dynamic phase transition in the spin-1 Blume-Capel model

The nature (time variation) of response magnetization m(wt) of the spin-1 Blume-Capel model in the presence of a periodically varying external magnetic field h(wt) is studied by employing the effective-field theory (EFT) with correlations as well as the Glauber-type stochastic dynamics. We determine the time variations of m(wt) and h(wt) for various temperatures, and investigate the dynamic magnetic hysteresis behavior. We also investigate the temperature dependence of the dynamic magnetization, hysteresis loop area and correlation near the transition point in order to characterize the nature (first- or second-order) of the dynamic transitions as well as obtain the dynamic phase transition temperatures. The hysteresis loops are obtained for different reduced temperatures and we find that the areas of the loops are decreasing with the increasing of the reduced temperatures. We also present the dynamic phase diagrams and compare the results of the EFT with the results of the dynamic mean-field approximation. The phase diagrams exhibit many dynamic critical points, such as tricritical (•), zero-temperature critical (Z), triple (TP) and multicritical (A) points. According to values of Hamiltonian parameters, besides the paramagnetic (P), ferromagnetic (F) fundamental phases, one coexistence or mixed phase region, (F+P) and the reentrant behavior exist in the system. The results are in good agreement with some experimental and theoretical results.     

A variational approach for the classical anisotropic Heisenberg model in a crystal field

The variational approach based on the Bogoliubov inequality for the free energy is used to study the three-dimensional anisotropic Heisenberg XXZ model with a crystal field. The magnetization and the phase diagrams are obtained as a function of the parameters of the Hamiltonian. Limiting cases, such as isotropic Heisenberg, XY, and planar rotator models in two and three dimensions, are analyzed and compared to previous results obtained from analytical approximations as well as to those obtained from more reliable approaches such as series expansion and Monte Carlo simulations. A parametric procedure has been used in order to simplify the solutions of the self-consistent coupled equations.     

Role of heavy fermions and heavy Higgs scalars in the electroweak correction to the MW − MZ relation

The role of heavy fermions and heavy Higgs-scalars in the M_W ? M_Z relation resulting from the one-loop-corrected μ decay width Γ⁽¹⁾ and its experimental data Γ^exp is studied in the framework of the standard electroweak theory. Exact and approximate formulae are both given for these heavy particle effects. The quadratic dependence of Γ⁽¹⁾ on large fermion mass m_heavy gives a positive contribution to the calculation of M_W from Γ⁽¹⁾ = Γ^exp for a given M_Z, and cancels the light fermion contributions of the form ～ αln(m_light/M_W) at the value of m_heavy ～ 200 GeV. On the other hand, the Higgs mass dependence of the calculation is, at best, logarithmic, and does not produce visible effects. Applications for deriving constraints for the top-quark mass (or heavier fermion mass) are discussed, and a concrete example is given of the relation between experimental uncertainties in measurements of M_W,Z and the corresponding allowed region for m_t.     

The phase diagrams and compensation behaviors of mixed spin Blume-Capel model in a trimodal magnetic field

The phase diagrams and compensation behaviors of mixed spin-1/2 and spin-1 Blume-Capel model in a trimodal magnetic field are investigated in the framework of the effective field theory on simple cubic lattice. The change of negative crystal field and trimodal concentration can affect the TCP, the second-order phase and the magnetic field degeneration at ground state in T-H space. In T-D space, the trajectory of the TCP takes on the acre curve and there exist the two TCPs under certain condition. In addition to giving one or two compensation temperature points in M-T space, the mixed spin Blume-Capel model also provides one or two novel compensation magnetic field points in M-H space. Some results are not revealed in previous works.     

Light thresholds in Grand Unified Theories

In a generic Grand Unified Theory with a relatively small dispersion of the spectrum around the Z-boson and the unification masses, a connection is established, exact at one loop level, between M_Z, G_F, α(M_Z and the strong coupling constant α₃(M_Z. At this level of precision, this avoids the logical and phenomenological inconsistency of predicting α₃(M_Z) by means of the electroweak couplings as extracted from the data in the Standard Model rather than in the complete theory. Attention is paid to the independence of the physical results from regularization and/or renormalization schemes.As a particularly relevant example, the analysis is specialized to the case of the Minimal Supersymmetric Standard Model, with emphasis on light charginos and neutralinos.     

Kosterlitz-thouless transition and self-duality in three dimensions

A class of self-dual globally symmetric Z_N models in three dimensions is presented. The limit N → ∞ is a type of anisotropic U(1) model (XY model) dual to a gas of integer point charges, interacting via a logarithmic potential in three dimensions. The latter is, at low temperature, nothing but a sine-Gordon theory with an anisotropic, logarithmic propagator. It therefore has a low-temperature Kosterlitz-Thouless phase and KT phase transition to a massive phase.The relation of the U(1) model to the thermodynamics of a helical magnet along the ferromagnetic-helical phase boundary in zero applied field (or to the smectic A to amectic C phase boundary in a liquid crystal) is indicated.     

Electromagnetic corrections to neutrino processes in the leading logarithmic approximation,the value of sin θ and the W and Z masses

The parameters of the Weinberg-Salam model can be defined by amplitudes at a momentum scale M = O(M_W, M_Z). We derive the leading logarithmic e.m. correction to the relations giving the neutrino amplitudes at a momentum scale μ ? M in terms of sin²θ(M), α(M), M_W and M_Z. For leptonic processes, the Fermi constant is not corrected, but a running, universal, sin²θ(μ) > sin²θ(M) should be used. The Fermi constant for semileptonic processes is renormalized by a factor $\text{?(μ) > 1}$, for charged currents, and is not renormalized, for neutral current processes. The latter are described by the same sin²θ(μ) as the leptonic ones. We estimate that sin²θ(M) is about 0.013 smaller than the value of sin²θ obtained from semileptonic data with no correction, thereby improving the agreement with grand unified theories. The prediction for W (Z) masses and widths in terms of the low energy parameters are discussed. Using previous calculations at order α, we obtain predictions for the masses which are accurate up to and including terms of order (αlnM²)².     

Summation of ring diagrams of nuclear thermodynamic potential

Using an imaginary-time linked diagram expansion of the thermodynamic potential Ω, we derive methods by which the particle-particle (hole-hole) and the particle-hole ring diagrams of Ω can each be summed up to all orders in a rather convenient way. A model space P and a corresponding finite temperature reaction matrix K are introduced in order to take care of the strong short-range correlations between nucleons. Methods of the finite temperature Green functions are employed. The contribution from the particle-particle (hole-hole) ring diagrams to Ω is given in terms of simple integrals of the form ∝₀¹dλ tr_p {M(λ)K} where λ is a strength parameter, and M a thermal transition matrix which is calculated from a finite-temperature RPA-type secular equation. Solutions of this equation are derived, and their possible connections with phase transitions are discussed. The particle-hole ring diagrams are treated in a similar way. Applications of the present approach and prospects for formulating a Brueckner-type theory of nuclear matter at finite temperature are discussed.     

Dynamic phase transitions in the kinetic spin-1 Blume-Capel model on the Bethe lattice

The stationary states of the kinetic spin-1 Blume-Capel (BC) model on the Bethe lattice are analyzed in detail in terms of recursion relations. The model is described using a Glauber-type stochastic dynamics in the presence of a time-dependent oscillating external magnetic field (h) and crystal field (D) interactions. The dynamic order parameter, the hysteresis loop area and the dynamic correlation are calculated. It is found that the magnetization oscillates around nonzero values at low temperatures (T) for the ferromagnetic (F) phase while it only oscillates around zero values at high temperatures for the paramagnetic (P) phase. There are regions of the phase space where the two solutions coexist. The dynamic phase diagrams are obtained on the (kT/J,h/J) and (kT/J,D/J) planes for the coordination number q=4. In addition to second-order and first-order phase transitions, dynamical tricritical points and triple points are also observed.     

Higher order field equations II; Limit properties of green's functions

In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions G^M_R(x) and G^M_A(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for |M| → ∞ the Green's functions G^M_R(x) and G^M_A(x) approach the Green's functions Δ_R(x) and Δ_A(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of G^M_A(x) and G^M_A(x) is the same as of Δ_R(x) and Δ_A(x) - and also the same as for D_R(x) and D_A(x) for t→ ± ∞, where D_R and D_A are the Green n's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense.     

Phase transitions and optical properties in 〈001〉 (1−<Emphasis Type="Italic">x</Emphasis>)PbZn<Subscript>1/3</Subscript>Nb<Subscript>2/3</Subscript>O<Subscript>3</Subscript>−<Emphasis Type="Italic">x</Emphasis>PbTiO<Subscript>3</Subscript> single crystals

The effect of a constant electric field (0 < E < 5 kV/cm) on the optical properties of PbZn_1/3Nb_2/3O₃?xPbTiO₃ (PZN-xPT) crystals with x = 7 and 9% has been analyzed. It has been shown that, at temperatures close to the temperature of the transition from the rhombohedral [R(X)] phase to the tetragonal (T) phase, two induced phase transitions are observed in the electric field, which are associated with the appearance of new intermediate phases M_a and M_c [R(X)-M_a-M_c-T]. Correlation between these two transitions and the PbTiO₃ content x has been found. The E-T phase diagrams have been obtained. The M_c phase in PZN-9PT crystals is found to remain the ground state after the removal of the electric field, whereas the M_c phase in PZN-7PT crystals is metastable and is transformed into the M_a phase after the removal of the electric field.