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Physical Review Letters

Theory of the Dynamics of First-Order Phase Transitions: Unstable Fixed Points, Exponents, and Dynamical Scaling

Fan Zhong, Qizhou Chen

Published Oct 19, 2005• 41 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.95.175701

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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.
Matter exhibits phases and their transitions. These transitions are classified as first-order phase transitions (FOPTs) and continuous ones. While the latter has a well-established theory of the renormalization group, the former is only qualitatively accounted for by classical theories of nucleation, since their predictions often disagree with experiments by orders of magnitude. A theory to integrate FOPTs into the framework of the renormalization-group theory has been proposed but seems to contradict with extant wisdom and lacks numerical evidence. Here we show that universal hysteresis scaling as predicted by the renormalization-group theory emerges unambiguously when the theory is combined intimately with the theory of nucleation and growth in the FOPTs of the paradigmatic two-dimensional Ising model driven by a linearly varying externally applied field below its critical point. This not only provides a new method to rectify the nucleation theories, but also unifies the theories for both classes of transitions and FOPTs can be studied using universality and scaling similar to their continuous counterpart.
Matter exhibits phases and their transitions. These transitions are classified as first-order phase transitions (FOPTs) and continuous ones. While the latter has a well-established theory of the renormalization group, the former is only qualitatively accounted for by classical theories of nucleation, since their predictions often disagree with experiments by orders of magnitude. A theory to integrate FOPTs into the framework of the renormalization-group theory has been proposed but seems to contradict with extant wisdom. Here we show first that classical nucleation and growth theories alone cannot explain the FOPTs of the paradigmatic two-dimensional Ising model driven by linearly varying an externally applied field. Then we offer compelling evidence that the transitions agree well with the renormalization-group theory when logarithmic corrections are properly considered. This unifies the theories for both classes of transitions and FOPTs can be studied using universality and scaling similar to their continuous counterpart.
Matter exhibits phases and their transitions. These transitions are classified as first-order phase transitions (FOPTs) and continuous ones. While the latter has a well-established theory of the renormalization group, the former is only qualitatively accounted for by classical theories of nucleation, since their predictions often disagree with experiments by orders of magnitude. A theory to integrate FOPTs into the framework of the renormalization-group theory has been proposed but seems to contradict with extant wisdom and lacks numerical evidence. Here we show that universal hysteresis scaling as predicted by the renormalization-group theory emerges unambiguously when the theory is combined intimately with the theory of nucleation and growth in the FOPTs of the paradigmatic two-dimensional Ising model driven by a linearly varying externally applied field below its critical point. This not only provides a new method to rectify nucleation theories, but also unifies the theories for both classes of transitions and FOPTs can be studied using universality and scaling similar to their continuous counterpart when nucleation and growth are properly taken into account.
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 φ3φ^{3}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 ϕ4ϕ^4ϕ4 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.
Abstract Phase transitions and critical phenomena are among the most intriguing phenomena in nature and society. They are classified into first-order phase transitions (FOPTs) and continuous ones. While the latter shows marvelous phenomena of scaling and universality, whether the former behaves similarly is a long-standing controversial issue. Here we definitely demonstrate complete universal scaling in field driven FOPTs for Langevin equations in both zero and two spatial dimensions by rescaling all parameters and subtracting nonuniversal contributions with singular dimensions from an effective temperature and a special field according to an effective theory. This offers a perspective different from the usual nucleation and growth but conforming to continuous phase transitions to study FOPTs.