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arXiv:1705.03198 · Physical review. E

Dynamic finite-size scaling at first-order transitions

Andrea Pelissetto C, Ettore Vicari

Published Jul 12, 2017• 14 citations
arXivHTMLPDF

Abstract

We investigate the dynamic behavior of finite-size systems close to a first-order transition (FOT). We develop a dynamic finite-size scaling (DFSS) theory for the dynamic behavior in the coexistence region where different phases coexist. This is characterized by an exponentially large time scale related to the tunneling between the two phases. We show that, when considering time scales of the order of the tunneling time, the dynamic behavior can be described by a two-state coarse-grained dynamics. This allows us to obtain exact predictions for the dynamical scaling functions. To test the general DFSS theory at FOTs, we consider the two-dimensional Ising model in the low-temperature phase, where the external magnetic field drives a FOT, and the 20-state Potts model, which undergoes a thermal FOT. Numerical results for a purely relaxational dynamics fully confirm the general theory.

DOI
10.1103/physreve.96.012125

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We investigate the dynamic behavior of finite-size systems close to a first-order transition (FOT). We develop a dynamic finite-size scaling (DFSS) theory for the dynamic behavior in the coexistence region where different phases coexist. This is characterized by an exponentially large time scale related to the tunneling between the two phases. We show that, when considering time scales of the order of the tunneling time, the dynamic behavior can be described by a two-state coarse-grained dynamics. This allows us to obtain exact predictions for the dynamical scaling functions. To test the general DFSS theory at FOTs, we consider the two-dimensional Ising model in the low-temperature phase, where the external magnetic field drives a FOT, and the 20-state Potts model, which undergoes a thermal FOT. Numerical results for a purely relaxational dynamics fully confirm the general theory.
Abstract We investigate how the scaling behavior of finite systems at magnetic first-order transitions (FOTs) with relaxational dynamics changes in correspondence of various boundary conditions. As a theoretical laboratory, we consider the two-dimensional Ising model in the low-temperature phase. When the boundary conditions do not favor any specific phase of the system, we show that a dynamic finite-size scaling (DFSS) theory can be developed to describe the dynamic behavior in the coexistence region, where different phases coexist. When the boundary conditions at two opposite sides of the system generate a planar interface separating the phases, we show that the autocorrelation times are characterized by a power-law behavior, related to the dynamics enforced by the interface. Numerical results for a purely relaxational dynamics confirm the general picture.
We study off-equilibrium behaviors at first-order transitions (FOTs) driven by a time dependence of the temperature across the transition point T(c), such as the linear behavior T(t)/T(c)=1±t/t(s) where t(s) is a time scale. In particular, we investigate the possibility of nontrivial off-equilibrium scaling behaviors in the regime of slow changes, corresponding to large t(s). We consider the two-dimensional Potts models, which provide an ideal theoretical laboratory to investigate issues related to FOTs driven by thermal fluctuations. We put forward general ansatzes for off-equilibrium scaling behaviors around the time t=0 corresponding to T(c). Then we present numerical results for the q=10 and 20 Potts models. We show that off-equilibrium scaling behaviors emerge at FOTs with relaxational dynamics, when appropriate boundary conditions are considered, such as mixed boundary conditions favoring different phases at the opposite sides of the system, which enforce an interface in the system.
We investigate finite-size effects at first-order quantum transitions. For this purpose we consider the one-dimensional q-state quantum Potts chain, in particular with q=10, which undergoes a first-order transition, separating the quantum disordered and ordered phases with a discontinuity in the energy density of the ground state. In agreement with the general theory, around the transition the low-energy properties show finite-size scaling with respect to appropriate scaling variables. Their size dependence is particularly sensitive to boundary conditions, which is a specific feature of first-order quantum transitions. Finally, we also discuss the finite-size behavior of the q-state Potts model (q≥2) at the first-order transitions driven by a parallel magnetic field, occurring in the ferromagnetic phase.
We investigate the dynamic properties of systems at thermal first-order transitions (FOT), when boundary conditions favor one of the two phases. In particular, we study the dynamic scaling properties arising from a slow heating of systems with disordered boundary conditions, starting in the ordered phase and moving across the FOT. As a theoretical laboratory we consider the two-dimensional Potts model. We show that a power-law dynamic scaling emerges, associated with a mixed regime where the two phases are spatially separated. We argue that these features generally apply to systems at FOTs, when boundary conditions favor one of the two phases. In particular, they should be relevant for the experimental search of FOTs of the quark-gluon plasma in heavy-ion collisions.
We consider a particular case of the two dimensional Blume-Emery-Griffiths model to study the finite-size scaling for a field driven first-order phase transition with two coexisting phases not related by a symmetry. For low temperatures we verify the asymptotic (large volume) predictions of the rigorous theory of Borgs and Kotecky. Near the critical temperature we show that all data fit onto a unique curve, even when the correlation length ξ becomes comparable to or larger than the size of the system, provided the linear dimension L of the system is rescaled by ξ.
We study finite-size effects at first-order quantum transitions (FOQTs). We show that the low-energy properties show a finite-size scaling (FSS) behavior, the relevant scaling variable being the ratio of the energy associated with the perturbation driving the transition and the finite-size energy gap at the FOQT point. The size dependence of the scaling variable is therefore essentially determined by the size dependence of the gap at the transition, which in turn depends on the boundary conditions. Our results have broad validity and, in particular, apply to any FOQT characterized by the degeneracy and crossing of the two lowest-energy states in the infinite-volume limit. In this case, a phenomenological two-level theory provides exact expressions for the scaling functions. Numerical results for the quantum Ising chain in transverse and parallel magnetic fields support the FSS Ansätzes.