International Journal of Modern Physics C

The phase transition associated with the standard electroweak model is very weakly first order. The weakness of the transition means that around the critical temperature the finite-temperature Higgs mass is much less than the critical temperature. This leads to infrared problems in the calculation of the parameters of the potential. Therefore, theories of electroweak baryogenesis, which depend on the details of the transition, must be calculated with care.

In the first part of the talk, a short review of baryon number violation in the electroweak standard model is given. I concentrate on perturbative estimates for the electroweak phase transition. A strong first order phase transition could be relevant both for a possible generation of the baryon asymmetry of the universe, and the survival of this asymmetry afterwards. In the second part of the talk, some lattice results for the electroweak phase transition are presented. They tend to indicate a transition more strongly first order than predicted by perturbation theory. A definite condusion cannot be drawn, however, because of severe finite size effects.

We discuss several aspects of the electroweak phase transition, which is generally believed to be first order. Starting from Langer’s theory of metastable states we derive an approximate formula for the decay rate. We then investigate the uncertainties in present evaluations of size and surface tension of critical bubbles and comment on the stumbling blocks on the way towards a quantitative description of the electroweak phase transition.

Using a variation of Lueschers geometric charge definition for SU(2) lattice gauge theory, we have managed to give a geometric expression for it’s Chern-Simons term. From this definition we have checked the periodic structure. We determined the Chern-Simons density for lattices L⁴ and L³×2, 4 with L=4, 6, and 8 near the critical region in the SU(2) Higgs model. The data indicate that tunneling is increased at high temperature.

We briefly review some of the results on metastable and unstable states as observed by Monte Carlo simulations of short- and long-range Ising models.

Recent developments in the theory of pattern formation in diffusional growth are reviewed. Asymptotic analysis and scaling arguments are used to construct a morphology diagram in parameter space. The relevant variables spanning this space are the anisotropy ∈ of the capillary length and the dimensionless undercooling Δ. Structures are classified according to whether they are compact or fractal and whether they possess orientational order or not. Explicit expressions for selected velocities and typical length scales are given in the scaling regime. For compact structures the theory provides a prescription how to go beyond scaling results.

A discrete model for the late stages of order parameter conserving coarsening is studied in one and in two dimensions. Starting from stationary, unstable solutions, the growth dynamics is related to the evolution of relevant perturbations near these solutions.

We present an introduction to modern theories of interfacial fluctuations and the associated interfacial parameters: surface tension and surface stiffness, as well as their interpretation within the capillary wave model. Transfer matrix spectrum properties due to fluctuation of an interface in a long-cylinder geometry are reviewed. The roughening transition and properties of rigid interfaces below the roughening temperature in 3d lattice models are surveyed with emphasis on differences in fluctuations and transfer matrix spectral properties of rigid vs. rough interfaces.

In liquid mixtures or analogous binary systems at low temperatures the pure phases may coexist, separated by an interface. The interface tension vanishes according to σ=σ₀(1−T/T_c)^µ as the temperature T approaches the critical point from below. Similarly the correlation length diverges as ξ=ξ₀(T/T_c−1)^−ν. For three-dimensional systems the dimensionless product R=σ₀(ξ₀)² is universal. This and related universal quantities are investigated by means of a semiclassical expansion in the framework of Φ⁴ theory, as well as by Monte Carlo calculations in the three-dimensional Ising model.

The partition funtion of a fluctuating interface is argued to approach a universal form at large area, including the numerical coefficient, provided that the interface has periodic boundary conditions und continuum behavior. This is demonstrated by explicit calculations, using effective actions with different regularisations. The result is applied to an analysis of the vortex free energy in SU(2) lattice gauge theory.

In this paper, I give an overview over a recently developed rigorous theory of finite-size scaling near first order transitions. Leaving out details of the mathematical proofs, the main emphasis is put on the underlying physical ideas and the discussion of the validity of the results for regions which are, in the sense of mathematical rigour, not covered by the original papers. I present both the finite-size scaling for cubic systems and for long cylinders, discussing also a recent controversy which stems from our work on the finite-size scaling of the mass gap in long cylinders.

I present reaults of simulations of the q=10 and q=20 2-d Potts models in the transition region. The asymptotic finite size behavior sets in only for extremely large lattices. We learn from this simulation that finite size scaling cannot be used to decide that a transition is first order.

The tension of the interface between the high and low temperature phases of finite temperature Quantum Chromodynamics is an important characteristic of the quarkhadron phase transition. We discuss its cosmological effects and its determination, for planar and spherical interfaces, with lattice Monte Carlo techniques for the case of pure glue, N_t=2 SU(3) gauge theory.

We present the results of a computation of the surface tension on hadronic and gluonic droplets in finite temperature quenched lattice QCD. The surface free energy was obtained by integrating the average action along a path in coupling space corresponding to growing and melting a spherical droplet of a given radius. So far droplets of radius 4, 5 and 8 (in lattice units) have been considered. After discussing ambiguities in the definition of the surface tension due to a droplet’s finite size and thick interface, we compare our results with the Bag model study of Mardor and Svetitsky. Our calculations appear to confirm qualitatively their prediction of a negative surface tension for hadronic droplets with small radii. A similar result is found for gluonic droplets, in contrast however with the Bag model. We also compare our results with the recent lattice simulation of Kajantie, Karkkainen and Rummukainen.

In the SU(3) pure gauge theory the high temperature deconfinement phase transition is first order. At the transition the confined phase coexists with three distinct deconfined phases. The interfaces separating the various phases show universal critical behavior when the confined phase completely wets the deconfined-deconfined interfaces. The critical exponents of complete wetting are derived from an interface solution of a Z(3) symmetric effective theory for the Polyakov loop. When complete wetting occurs in the pure glue theory the deconfinement transition proceeds via domain wall splitting. When quarks are present wetting disappears and the standard bubble nucleation scenario applies.

The transfer matrix method to describe finite size effects due to tunneling are worked out for Z(2)- and Z(3)-symmetric models. We used this method to extract the surface tension σ in the SU(3) gauge theory at the finite temperature phase transition on lattices with an extent T=2 in the euclidean time direction. We also discuss if the confined phase completely wets the deconfined phase at this first order phase transition.

The free energy distribution across the interface of coexisting quark and hadron mat-ter is studied in the framework of lattice QCD. We calculate the interface tension α with the “differential method” for pure SU(3) gauge theory and in the presence of dynamical quarks with four flavors. Using lattices with spatial volume 8²×16 it turns out that α is very small and compatible with zero. The chiral condensate indicates the same width of the domain wall as the Polyakov loop distribution and the other thermodynamical observables.

The thermal phase transition in lattice QCD at finite quark mass is studied from the distribution of the zeros of the partition function in the complex quark mass plane. Results from simulations at strong and intermediate coupling are presented. Evidence for a chirally related transition at non-zero quark mass in the intermediate coupling region for β<5.2. The method exhibits remarkable stability in the evaluation of the critical mass as the lattice volume increases. Conclusions regarding the possible phase structure of QCD in the mass-temperature plane are discussed.

The shrinking of a chirally symmetric droplet of finite baryonic charge, immersed into the broken symmetry medium is described. The treatment is parallel to the methods applied in the Skyrme model of the nucleon.

We report on results about the mass sensitivity of chiral symmetry restoration in the linear sigma model. For masses of the pseudoscalar meson octet which are close to the experimental values, we observed only a gradual change in the order parameters, when the temperature was changed. To estimate the size of the first order transition region around the chiral limit, we have varied the mass input for the tree level parametrization in several ways. The point with realistic meson masses turned out to lie well inside the crossover region.

The extension of dynamical scaling to local, space-time dependent rescaling factors is investigated. For a dynamical exponent z=2, the corresponding invariance group is the Schrödinger group. Schrödinger invariance is shown to determine completely the two-point correlation function. The result is checked in two exactly solvable models.

We present a numerical method for the study of the stability-capacity diagram of the Ising perceptron that is readily scalable, thus providing the opportunity to resolve some open issues on this problem. We also discuss the possibility of studying multi-layer systems.

This brief review discusses methods to locate and characterize first order phase transitions, paying particular attention to finite size effects. In the first part, the order parameter probability distribution and its fourth-order cumulant is discussed for thermally driven first-order transitions (the 3-state Potts model in d=3 dimensions is treated as an example). First-order transitions are characterized by a minimum of the cumulant, which gets very deep for large enough systems. In the second part, we discuss how to locate first order phase boundaries ending in a critical point in a large parameter space. As an example, the study of the unmixing transition of asymmetric polymer mixtures by a combination of histogram techniques and finite size scaling is described.

As a final problem, we discuss the shift of the gas-liquid condensation in thin-film geometry confined between two parallel plates due to boundary fields (“capillary condensation”). Being interested in temperatures far below bulk criticality (e. g. near the wetting transition), special thermodynamic integration techniques are the method of choice, rather than the use of finite sizes scaling to map out the (asymmetric) phase diagram.

The standard Ising model in two and three dimensions was simulated with Metropolis-Glauber kinetics on the Intel Hypercube with 32 MIMD processors of i860 type, each with 16 megabytes of distributed memory, and on 8 such processors of the Alliant FX with shared memory. The nucleation time in reverse magnetic fields h. was found to increase as 1/h² in three dimensions, for times up to millions of iterations. The relaxation towards the equilibrium spontaneous magnetization was a simple exponential in three dimensions and a stretched exponential in two below the critical temperature; its power-law decay at the critical point lead to z=2.18(d=2) and =2.09 (d=3). The Becker-Döring equation for the growth of microdroplets leads at the critical temperature to a growth rate vanishing with some power of the time.

Problems associated with analyticity of thermodynamic functions close to first-order phase transitions are briefly reviewed. The bubble model for correlation functions is then applied to planar Ising-like models at subcritical temperatures (T<T_c) with a bulk magnetic field h. The fluctuation sum is used to calculate the susceptibility χ(h) from the bubble correlation function. We show that χ(h), calculated this way, must contain an essential singularity at h=0 i.e. at the first-order phase boundary. This has important implications to metastability, where we demonstrate that if the ensemble is restricted such that the magnetization stays positive when h goes negative, χ(h) has an infinite number of poles along the negative real axis with a limit-point at h=0. For an unrestricted ensemble, a Yang-Lee circle theorem is derived.

The multicanonical ensemble is reviewed. Simulations of this ensemble promise improvements for a wide range of systems. Applications to first order phase transitions and spin glasses are summarized.

Describing asymmetric first-order phase transitions by an order parameter distribution, there is a controversy in the literature concerning the relative normalization of the two occurring peaks. There are two rules for normalizing the distribution, called the “equal weight”-rule and the “equal height”-rule, which lead to different predictions of the finite-size scaling. We tested these predictions for an asymmetric model with two co-existing phases by means of a Monte Carlo simulation. We find overwhelming numerical evidence in favour of the “equal weight”-rule, showing at the same time that the shift of the susceptibility maximum with respect to the infinite volume transition point h_t is proportional to L^−2d, and not to L^−d. In addition we tested a new method to determine the transition point h_t which was recently proposed by Borgs and Janke.

The presence of strong metastabilities in computer simulations of models showing a first order phase transition hinders a reliable determination of the weight ratio between the two phases. We discuss a new phenomenological method which allows an accurate fixing of the weight ratio using the standard multihistogram procedure.

We report tests of finite-size scaling ansatzes in the low temperature phase of the two-dimensional Ising model. For moments of the magnetisation density, we find good agreement with the new ansatz of Borgs and Kotecký, and clear evidence of violations of the double Gaussian ansatz. We note that certain consequences of the convexity of the free energy are not adequately treated in either of these approaches.

At low temperature, the Potts models can have interfaces between domains ordered to different directions. We use the difference of periodic and twisted boundary conditions to study planar order-order interfaces. We show that the interface tension, energy and width display critical properties as one approaches the critical point. We measure the critical exponents connected to their power like singular behaviour at the critical point.

Using multicanonical Monte Carlo simulations, we have calculated the interface free energy between the coexisting ordered and disordered phase at the temperature driven first-order phase transition of the two-dimensional 7-state Potts model. Our result 2f³= 0.0241±0.0010 is about an order of magnitude smaller than other estimates in the recent literature. We discuss the differences in analysis and give a possible explanation for the discrepancy. Finally, we briefly mention similar investigations at the deconfining phase transition of SU(3) lattice gauge theory.

We present a multicanonical algorithm for the SU(3) pure gauge theory at the deconfinement phase transition. We measure the tunneling times for lattices of size L²×2 for L=8, 10, and 12. In contrast to the canonical algorithm the tunneling time increases only moderately with L. Finally, we determine the interfacial free energy applying the multicanonical algorithm.

To investigate order-order interfaces, we perform multimagnetical Monte Carlo simulations of the 2D and 3D Ising model. Stringent tests of the numerical methods are performed by reproducing with high precision exact 2D results. In the physically more interesting 3D case we estimate the amplitude of the critical interfacial tension.