Through a detailed study of the mean-field approximation, the Gaussian approximation, the perturbation expansion, and the field-theoretic renormalization-group analysis of a φ 3 theory, we show that the instability fixed points of the theory, together with their associated instability exponents, are quite probably relevant to the scaling and universality behavior exhibited by the first-order phase transitions in a field-driven scalar ϕ 4 model, below its critical temperature and near the instability points. Finitetime scaling and leading corrections to the scaling are considered. We also show that the instability exponents of the first-order phase transitions are equivalent to those of the Yang–Lee edge singularity, and employ the latter to improve our estimates of the former. The outcomes agree well with existing numerical results.
Through a detailed study of the mean-field approximation, the Gaussian approximation, the perturbation expansion, and the field-theoretic renormalization-group analysis of a φ 3 theory, we show that the instability fixed points of the theory, together with their associated instability exponents, are quite probably relevant to the scaling and universality behavior exhibited by the first-order phase transitions in a field-driven scalar ϕ 4 model, below its critical temperature and near the instability points. Finitetime scaling and leading corrections to the scaling are considered. We also show that the instability exponents of the first-order phase transitions are equivalent to those of the Yang–Lee edge singularity, and employ the latter to improve our estimates of the former. The outcomes agree well with existing numerical results.
We show by a detailed study of the mean-field approximation, the Gaussian approximation, the perturbation expansion, and the field-theoretic renormalization-group analysis of a φ3 theory that its instability fixed points with their associated instability exponents are quite probably relevant to the scaling and universality behavior exhibited by the first-order phase transitions in a field-driven scalar ϕ4 model below its critical temperature near their instability points. Finite-time scaling and leading corrections to scaling are considered. We also show that the instability exponents of the first-order phase transitions equal those of the Yang--Lee edge singularity and employ the latter to improve our estimates of the former. The outcomes agree well with existent numerical results.
We study the scaling and universal behavior of temperature-driven first-order phase transitions in scalar models. These transitions are found to exhibit rich phenomena, though they are controlled by a single complex-conjugate pair of imaginary fixed points of ϕ 3 theory. Scaling theories and renormalization group theories are developed to account for the phenomena, and three universality classes with their own hysteresis exponents are found: a field-like thermal class, a partly thermal class, and a purely thermal class, designated, respectively, as Thermal Classes I, II, and III. The first two classes arise from the opposite limits of the scaling forms proposed and may cross over to each other depending on the temperature sweep rate. They are both described by a massless model and a purely massive model, both of which are equivalent and are derived from ϕ 3 theory via symmetry. Thermal Class III characterizes the cooling transitions in the absence of applied external fields and is described by purely thermal models, which include cases in which the order parameters possess different symmetries and thus exhibit different universality classes. For the purely thermal models whose free energies contain odd-symmetry terms, Thermal Class III emerges only at the mean-field level and is identical to Thermal Class II. Fluctuations change the model into the other two models. Using the extant three- and two-loop results for the static and dynamic exponents for the Yang–Lee edge singularity, respectively, which falls into the same universality class as ϕ 3 theory, we estimate the thermal hysteresis exponents of the various classes to the same precision. Comparisons with numerical results and experiments are briefly discussed.
Phase transitions are of great importance in a diversity of fields. They are usually classified into continuous phase transitions and first-order phase transitions (FOPTs). Whereas the former has a well-developed theoretical framework of the renormalization-group (RG) theory, no general theory has yet been developed for the latter that appear far more frequently. Focusing on the dynamics of a generic FOPT in the phi4 model below its critical point, we show by a field-theoretic RG method that it is governed by an unexpected unstable fixed point of the corresponding phi3 model. Accordingly, it exhibits a distinct scaling and universality behavior with unstable exponents different from the critical ones.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.
We apply the functional renormalization group theory to the dynamics of first-order phase transitions and show that a potential with all odd-order terms can describe spinodal decomposition phenomena. We derive a momentum-dependent dynamic flow equation which is decoupled from the static flow equation. We find the expected instability fixed points; and their associated exponents agree remarkably with the existent theoretical and numerical results. The complex renormalization group flows are found and their properties are shown. Both the exponents and the complex flows show that the spinodal decomposition possesses singularity with consequent scaling and universality.