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Physical Review Letters

Renormalization Group Theory of Hysteresis

Fan Zhong, Jinxiu Zhang

Published Sep 4, 1995• 55 citations
Publisher

Abstract

We apply renormalization group theory directly to the first-order phase transition of the large-N model driven linearly by an external magnetic field $H\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}t$, where $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}$ is the sweeping rate. Novel dynamic scaling forms for the magnetization, the structure factor, and the area of hysteresis loop are $M(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H},t,T)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}f({\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{1/2}t,{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{(d\ensuremath{-}2)/4}T)$, $C(k,\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H},t,T)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{\ensuremath{-}d/4}{f}^{\ensuremath{'}}({\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{\ensuremath{-}1/4}k,{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{1/2}t,{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{\left(d\ensuremath{-}2\right)/4}T)$, and $A\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{1/2}g({\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{\left(d\ensuremath{-}2\right)/4}T)$, respectively, where $T$ is the temperature of the system, $d$ the spatial dimensionality, $k$ the wave number, and $f$, ${f}^{\ensuremath{'}}$, and $g$ scaling functions. These results show that the rate of the external driving field can serve as a scaling parameter to study hysteresis.

DOI
10.1103/physrevlett.75.2027

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We apply renormalization group theory directly to the first-order phase transition of the large-N model driven linearly by an external magnetic field H=H\ifmmode˙\else˙\fitH\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}tH=H\ifmmode˙\else˙\fi​t, where H\ifmmode˙\else˙\fi\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}H\ifmmode˙\else˙\fi​ is the sweeping rate. Novel dynamic scaling forms for the magnetization, the structure factor, and the area of hysteresis loop are M(H\ifmmode˙\else˙\fi,t,T)=f(H\ifmmode˙\else˙\fi1/2t,H\ifmmode˙\else˙\fi(d\ensuremath−2)/4T)M(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H},t,T)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}f({\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{1/2}t,{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{(d\ensuremath{-}2)/4}T)M(H\ifmmode˙\else˙\fi​,t,T)=f(H\ifmmode˙\else˙\fi​1/2t,H\ifmmode˙\else˙\fi​(d\ensuremath−2)/4T), C(k,H\ifmmode˙\else˙\fi,t,T)=H\ifmmode˙\else˙\fi\ensuremath−d/4f\ensuremath′(H\ifmmode˙\else˙\fi\ensuremath−1/4k,H\ifmmode˙\else˙\fi1/2t,H\ifmmode˙\else˙\fi(d\ensuremath−2)/4T)C(k,\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H},t,T)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{\ensuremath{-}d/4}{f}^{\ensuremath{'}}({\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{\ensuremath{-}1/4}k,{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{1/2}t,{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{\left(d\ensuremath{-}2\right)/4}T)C(k,H\ifmmode˙\else˙\fi​,t,T)=H\ifmmode˙\else˙\fi​\ensuremath−d/4f\ensuremath′(H\ifmmode˙\else˙\fi​\ensuremath−1/4k,H\ifmmode˙\else˙\fi​1/2t,H\ifmmode˙\else˙\fi​(d\ensuremath−2)/4T), and A=H\ifmmode˙\else˙\fi1/2g(H\ifmmode˙\else˙\fi(d\ensuremath−2)/4T)A\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{1/2}g({\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}}^{\left(d\ensuremath{-}2\right)/4}T)A=H\ifmmode˙\else˙\fi​1/2g(H\ifmmode˙\else˙\fi​(d\ensuremath−2)/4T), respectively, where TTT is the temperature of the system, ddd the spatial dimensionality, kkk the wave number, and fff, f\ensuremath′{f}^{\ensuremath{'}}f\ensuremath′, and ggg scaling functions. These results show that the rate of the external driving field can serve as a scaling parameter to study hysteresis.
Dynamical phase transitions in the Ising model on hypercubic lattices are considered. Under a linearly swept magnetic field, the hysteresis loop that characterizes the field-driven first-order phase transition is studied carefully. Using the Glauber dynamics, we find that, in the mean-field approximation, the energy dissipation of this phase transition or the hysteresis loop area A of the M-H curve can be scaled with respect to the sweep rate h of magnetic field in the form with a = 2 and b = 2/3. However, b varies when fluctuations and spin correlations are taken into account. Monte Carlo simulation is used to obtain the scaling relation for A in two-, three- and four-dimensional Ising models and we obtain the exponents and respectively. These exponents are obviously different from those obtained by scaling A as for any temperatures in Ising models under a sinusoidal field. Finally we point out that, in the concept of universality, field-driven first-order phase transitions in the Ising model in different dimensions belong to different universal classes due to the spin fluctuation and correlation below the Curie temperature.
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.
Abstract We investigate the off-equilibrium dynamics of a classical spin system with symmetry in 2 < D < 4 spatial dimensions and in the limit . The system is set up in an ordered equilibrium state and is subsequently driven out of equilibrium by slowly varying the external magnetic field h across the transition line at fixed temperature . We distinguish the cases where the magnetic transition is continuous and where the transition is discontinuous. In the former case, we apply a standard Kibble–Zurek approach to describe the non-equilibrium scaling and formally compute the correlation functions and scaling relations. For the discontinuous transition we develop a scaling theory which builds on the coherence length rather than the correlation length since the latter remains finite for all times. Finally, we derive the off-equilibrium scaling relations for the hysteresis loop area during a round-trip protocol that takes the system across its phase transition and back. Remarkably, our results are valid beyond the large- n limit.