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Journal of Physics A Mathematical and General

Hysteresis and self-organized criticality in the O(N) model in the limit N to infinity

Deepak Dhar, P B Thomas

Published Oct 7, 1992• 91 citations
Publisher

Abstract

The authors consider the response of the ferromagnetic N-vector model to a sinusoidally varying external magnetic field in the large-N limit. In all dimensions d>2, they show that at low frequencies omega , and small amplitudes H 0 of the field, the area of the hysteresis loop scales as (H 0 omega ) 1/2 with logarithmic corrections. At very high frequencies, the area varies as H 0 2 / omega . They find that for any H 0 there is a dynamical phase transition separating these two frequency regimes. They determine numerically the critical frequency as a function of the field strength. In the high-frequency phase the magnetization is predominantly transverse to the external magnetic field.

DOI
10.1088/0305-4470/25/19/012

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Physical Review Letters•Sep 4, 1995•55 citations
The authors consider the response of the ferromagnetic N-vector model to a sinusoidally varying external magnetic field in the large-N limit. In all dimensions d>2, they show that at low frequencies omega , and small amplitudes H 0 of the field, the area of the hysteresis loop scales as (H 0 omega ) 1/2 with logarithmic corrections. At very high frequencies, the area varies as H 0 2 / omega . They find that for any H 0 there is a dynamical phase transition separating these two frequency regimes. They determine numerically the critical frequency as a function of the field strength. In the high-frequency phase the magnetization is predominantly transverse to the external magnetic field.
We have studied in this paper,by performing the Monte Carlo numerical simulation,both the hysteretic scaling and the dynamical phase transition of a three-dimen sional,(3D) classical X-Y model driven by an sinusoidally oscillating external m agnetic field.A scaling formula has been worked out which relates the hysteresis loop area with the amplitude h0 and frequency ω of the external fie ld as wel l as the reduced temperature T/Tc of the system in the form:Area～hα0ωβ(1-T/Tc)γ.The best-fit expo nents are α=0.57,β=0.34 and γ=0.9.The 3D X- Y model also characterizes a distinctive discrepancy in dynamical transition fea ture after short and long term evolution of magnetization,respectively.Our simul ation disclosed that the short-term evolution of magnetization (period number≤1 0) attains the symmetry-breaking of system with a nonzero dynamical order parame ter (Q≠0) at a either critical amplitude h0c or frequency ωc.The symme try-breaking in short term,however,evolves steadily into a symmetric disorder st ate (Q=0) after a longer term relaxation of system.The specific relaxation times at which the Q value becomes zero from nonzero increase evidently as the temper ature of system drops.
We investigate the off-equilibrium dynamics of a spin system with O(NNN) symmetry in 2<d<42 < d < 42<d<4 spatial dimensions arising by the presence of a slowly varying time-dependent magnetic field h(t,ts)∼t/tsh(t,t_s) \sim t/t_sh(t,ts​)∼t/ts​, tst_sts​ is a time scale, at the critical temperature T=TcT = T_cT=Tc​ and below it T<TcT < T_cT<Tc​. After showing the general theory, we demonstrate the off-equilibrium scaling and we formally compute the correlation functions in the limit of large NNN. We derive the off-equilibrium scaling relations for the hysteresis loop area and for the magnetic work done by the system when the magnetic field h(t,ts)h(t,t_s)h(t,ts​) is varied across the phase transitions cyclically in time. We also investigate the first deviations from the equilibrium behavior in the correlation functions checking the consistence for an exponential approach.
We have studied, using Monte Carlo (MC) simulation for ferromagnetic Ising systems in one to four dimensions and solving numerically the mean-field (MF) equation of motion, the nature of the response magnetization m(t) of an Ising system in the presence of a periodically varying external field [h(t)=h0{\mathit{h}}_{0}h0​cos(\ensuremath{\omega}t)]. From these studies, we determine the m-h loop or hysteresis loop area A(=\ensuremath{\oint}mdh) and the dynamic order parameter Q(=\ensuremath{\oint}mdt) and investigate their variations with the frequency (\ensuremath{\omega}) and amplitude (h0{\mathit{h}}_{0}h0​) of the applied external magnetic field and the temperature (T) of the system. The variations in A are fitted to a scaling form, assumed to be valid over a wide range of parameter (\ensuremath{\omega},h0{\mathit{h}}_{0}h0​,T) values, and the best-fit exponents are obtained in all three dimensions (D=2,3,4). The scaling function is Lorentzian in the MF case and is exponentially decaying, with an initial power law, for the MC cases. The dynamic phase boundary (in the h0{\mathit{h}}_{0}h0​-T plane) is found to be frequency dependent and the transition (from Q\ensuremath{\ne}0 for low T and h0{\mathit{h}}_{0}h0​ to Q=0 for high T and h0{\mathit{h}}_{0}h0​) across the boundary crosses over from a discontinuous to a continuous one at a tricritical point. These boundaries are determined in various cases.We find that the response can be generally expressed as m(t)=P(\ensuremath{\omega}(t-${\mathrm{\ensuremath{\tau}}}_{\mathrm{eff}}$)) where P denotes a periodic function with the same frequency \ensuremath{\omega} of the perturbing field and ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$(h0{\mathit{h}}_{0}h0​,\ensuremath{\omega},T) denotes the effective delay. We established that this effective delay ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ of the response is the crucial term and it practically determines all the above observations for A, Q, etc. Investigating the nature of the in-phase (\ensuremath{\chi}\ensuremath{'}) and the out-of-phase (\ensuremath{\chi}\ensuremath{'}\ensuremath{'}) susceptibility, defined as \ensuremath{\chi}\ensuremath{'}=(m0{\mathit{m}}_{0}m0​/h0{\mathit{h}}_{0}h0​)cos(\ensuremath{\varphi}) and \ensuremath{\chi}\ensuremath{'}\ensuremath{'}=(m0{\mathit{m}}_{0}m0​/h0{\mathit{h}}_{0}h0​)sin(\ensuremath{\varphi}); \ensuremath{\varphi}=\ensuremath{\omega}${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ [and m0{\mathit{m}}_{0}m0​ is the amplitude of m(t)], we find that the loop area A is directly given by \ensuremath{\chi}\ensuremath{'}\ensuremath{'} and also the temperature variation of \ensuremath{\chi}\ensuremath{'}\ensuremath{'}, at fixed \ensuremath{\omega} and h0{\mathit{h}}_{0}h0​, gives a prominent peak at the dynamic transition point. We have also studied the behavior of the response magnetization by the application of a short-duration (compared with the relaxation time) pulsed magnetic field.Here, we observed (both in the MC and MF cases) that the width ratio (of the half-width and the width of the response magnetization and of the pulsed field, respectively) and the susceptibility (the ratio of excess magnetization over its equilibrium value and the height of the pulsed field) both show sharp peaks at the order-disorder (ferromagnetic-paramagnetic) transition point. We have also studied similar response behavior of Ising systems in the presence of time-varying longitudinal and transverse fields, solving numerically the mean-field equation of motion. We have again studied here the nature of the dynamic phase transition and the behavior of the ac susceptibility (both longitudinal and transverse) across the dynamic phase boundary. For a short-duration pulse of the transverse field, the width ratio and the pulse susceptibility are again seen to diverge at the order-disorder transition point.
We consider the dynamical off-equilibrium behavior of the three-dimensional O(N) vector model in the presence of a slowly varying time-dependent spatially uniform magnetic field H(t)=h(t)e, where e is an N-dimensional constant unit vector, h(t)=t/t(s), and t(s) is a time scale, at fixed temperature T≤T(c), where T(c) corresponds to the continuous order-disorder transition. The dynamic evolutions start from equilibrium configurations at h(i)<0, correspondingly t(i)<0, and end at time t(f)>0 with h(t(f))>0, or vice versa. We show that the magnetization displays an off-equilibrium scaling behavior close to the transition line H(t)=0. It arises from the interplay among the time t, the time scale t(s), and the finite size L. The scaling behavior can be parametrized in terms of the scaling variables t(s)(κ)/L and t/t(s)(κ(t)), where κ>0 and κ(t)>0 are appropriate universal exponents, which differ at the critical point and for T<T(c). In the latter case, κ and κ(t) also depend on the shape of the lattice and on the boundary conditions. We present numerical results for the Heisenberg (N=3) model under a purely relaxational dynamics. They confirm the predicted off-equilibrium scaling behaviors at and below T(c). We also discuss hysteresis phenomena in round-trip protocols for the time dependence of the external field. We define a scaling function for the hysteresis loop area of the magnetization that can be used to quantify how far the system is from equilibrium.
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 TTT is the temperature of the system, ddd the spatial dimensionality, kkk the wave number, and fff, ${f}^{\ensuremath{'}}$, and ggg scaling functions. These results show that the rate of the external driving field can serve as a scaling parameter to study hysteresis.