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arXiv:1512.06201 · Physical review. E

Off-equilibrium scaling behaviors driven by time-dependent external fields in three-dimensional O(N) vector models

Andrea Pelissetto C, Ettore Vicari

Published Mar 24, 2016• 27 citations
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Abstract

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.

DOI
10.1103/physreve.93.032141

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We study the out-of-equilibrium scaling behavior of two-dimensional and three-dimensional Ising systems, when they are slowly driven across their magnetic first-order transitions at low temperature T<T_{c}, where T_{c} is the temperature of their continuous transition. We consider Kibble-Zurek (KZ) protocols in which a spatially homogenous magnetic field h varies as h(t)=t/t_{s} with a timescale t_{s}. The KZ dynamics starts from negatively magnetized configurations equilibrated at h_{i}<0 and stops at a positive value of h where the configurations acquire a positive average magnetization. We consider the Metropolis and the heat-bath dynamics, which are two specific examples of a purely relaxational dynamics. We focus on two different dynamic regimes. We consider the out-equilibrium finite-size scaling (OFSS) limit in which the system size L and the timescale t_{s} diverge simultaneously, keeping an appropriate combination fixed. Then, we analyze the KZ dynamics in the thermodynamic limit (TL), obtained by taking first the L→∞ limit at fixed t and t_{s}, and then considering the scaling behavior in the large-t_{s} limit. Our numerical results provide evidence of OFSS, as predicted by general scaling arguments. The results in the TL show the emergence of spinodal-like behaviors: The passage from the negatively magnetized phase to the positively magnetized one occurs at positive values h_{*}>0 of the magnetic field, which decrease as h_{*}∼1/(lnt_{s})^{κ}, with κ=2 and κ=1 in two and three dimensions, respectively, for t_{s}→∞. We identify σ≡t(lnt)^{κ}/t_{s} as the relevant scaling variable associated with the KZ dynamics in the TL.
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.
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 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 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 study the critical dynamics arising from a time-dependent periodic homogenous source coupled to the order-parameter field, which drives a classical ferromagnetic system across a continuous transition. For this purpose, we consider the paradigmatic two-dimensional (2D) Ising model in the presence of a periodic magnetic field h(t)=−A cos⁡(2πt/P)h(t)=-A\, \cos (2πt/P)h(t)=−Acos(2πt/P), evolving under a purely relaxational dynamics at the critical temperature. We show that the periodic driving gives rise to a peculiar dynamic scaling behavior in the thermodynamic limit, arising from a nontrivial interplay among the time ttt, the amplitude AAA and period PPP of h(t)h(t)h(t). The relevant scaling variables are τ=t/Pτ=t/Pτ=t/P and σ=APκσ=A P^κσ=APκ, with κ=yh/zκ= y_h/zκ=yh​/z, where yh=(d+2−η)/2y_h=(d+2-η)/2yh​=(d+2−η)/2 is the critical dimension of the magnetic field, and zzz is dynamic exponent for the critical relaxational dynamics (κ≈0.865κ\approx 0.865κ≈0.865 for the 2D Ising model). The dynamic scaling behaviors of the magnetization and bond-energy density show an oscillatory behavior around a smooth curve which approaches a large-τττ stationary behavior. We also briefly discuss the dynamic behavior of an Ising system driven across the critical point by a periodic time-varying temperature at zero magnetic field.