Dynamic phase transition in the periodically driven one-dimensional Ising chain: existence, the dimensional mechanism, and the stationarity scaling

In kinetic Ising ferromagnets driven by a zero-mean periodic field, a dynamic phase transition (DPT) can occur: below a dynamic phase boundary the hysteresis loop spontaneously breaks its half-cycle symmetry and the period-averaged magnetization $Q$ becomes nonzero. This phenomenon is well established by Monte Carlo simulations in two and three dimensions, but in the one-dimensional Glauber–Ising chain under cyclic driving the Monte Carlo study of Petukhov (2015) finds $Q$ fluctuating around zero for every driving period, with only a qualitative attribution of this absence to the simpler topology of one-dimensional phase boundaries. The open problem is to turn this empirical absence into a quantitative theory: (i) establish whether the infinite one-dimensional chain under symmetric periodic driving at $T>0$ can ever support $Q\neq 0$, or prove $Q=0$ identically; (ii) give a quantitative account of how the geometry of domain walls (interfaces) in $d\geq 2$ enables the dynamic symmetry breaking that one-dimensional kink proliferation destroys; and (iii) derive the scaling law $N_c\sim t_c^{-n}$ for the number of cycles needed to reach the stationary switching regime, and account for the empirically reported exponent $n\approx 4.6\pm 0.4$.

Background

The kinetic Ising model — Ising spins $s_i=\pm 1$ with ferromagnetic nearest-neighbor coupling, updated by Glauber single-spin-flip stochastic dynamics toward local equilibrium — is the standard model for nonequilibrium magnetization dynamics. When the external field $h(t)$ varies periodically (sinusoidally, or as a square wave alternating between $+F_0$ and $-F_0$ with half-period $t_c$), the magnetization lags the field and traces a hysteresis loop. Tomé and de Oliveira (1990), and independently Lo and Pelcovits (1990) and Acharyya and Chakrabarti (1995), discovered that such a driven kinetic Ising ferromagnet has its own nonequilibrium phase transition, the dynamic phase transition (DPT): in a region of the (field amplitude, temperature) plane bounded by a dynamic phase boundary at each driving period, the loop spontaneously loses its symmetry between the two half-cycles, so that the dynamic order parameter $Q$ — the magnetization averaged over one full field period — takes a nonzero value of either sign, although the field has zero time average. Subsequent Monte Carlo work established the DPT in two and three dimensions (Sides, Rikvold and Novotny 1998, 1999; Korniss et al. 2000; Buendía and Rikvold 2008; Park and Pleimling 2013): it lies in the equilibrium Ising universality class, its transition line $T_D(h_0,\omega)$ sits below the equilibrium critical temperature $T_c$, and its half-cycle switching is described by Avrami (Kolmogorov–Johnson–Mehl–Avrami) nucleation-and-growth theory, in which domains of the field-favored phase nucleate stochastically and grow until they impinge and coalesce.

In one dimension this picture fails for a structural reason. The equilibrium nearest-neighbor Ising chain has $T_c=0$, and its phase boundaries are kinks — point-like domain walls separating domains of opposite magnetization — which are thermally nucleated in pairs at a rate $\Gamma\propto\exp(-2E_k/kT)$ (with $E_k$ the kink formation energy) and drift along the chain under a driving force $F$. Any magnetized state of the chain is therefore dismantled stochastically by kink proliferation rather than swept coherently, which is also the physical basis of Petukhov's kink model of state switching in quasi-one-dimensional nanosystems (nanowires, polymer chains, and single-chain magnets, i.e., one-dimensional arrays of anisotropic magnetic molecules): applying Kolmogorov's statistical theory of phase transformations to the nucleation, drift, and coalescence of domains in a cyclically driven Glauber chain, checked against Monte Carlo simulations of a $10^4$-site chain with periodic boundary conditions. The simulations confirm the kink-model switching curves, but they also show that the cycle-averaged magnetization fluctuates around zero for every driving period studied: no symmetry-breaking DPT appears in one dimension, in contrast to the two-dimensional simulations. Petukhov could only attribute this absence qualitatively to the 'simpler topology' of one-dimensional phase boundaries, and reported in addition that the number of field cycles $N_c$ after which the switching kinetics become stationary — that is, reproducible from cycle to cycle — scales with the half-period as $N_c\sim t_c^{-n}$, with $n\approx 4.6\pm 0.4$ extracted empirically from the finite-chain Monte Carlo simulations and without theoretical derivation. The 1999 review of Chakrabarti and Acharyya already reflects the same gap: the dynamic hysteresis (loop-area scaling) of the Ising chain was simulated in one to four dimensions, but the symmetry-breaking DPT itself was characterized only in $d\geq 2$, and no theory delimits the lowest dimension in which dynamic order can survive. This matters beyond the minimal model because an analytic criterion would apply to periodically driven quasi-one-dimensional magnets, where field-cycled state switching controls functional behavior, and because it would isolate precisely which geometric property of higher-dimensional interfaces — their line/area tension, their nucleation barriers, or the shape of growing domains — is responsible for nonequilibrium dynamic order.

Problem Statement

The problem is to convert the empirical absence of a dynamic phase transition in the periodically driven one-dimensional Ising chain into a quantitative theory of dynamic ordering versus dimensionality, with three linked deliverables serving one common objective — explaining, rather than merely observing, why dynamic order dies in one dimension, how higher-dimensional domain-wall geometry restores it, and how a repeatedly cycled chain relaxes to its stationary regime. Deliverable (1) is a one-dimensional existence determination in its own right, distinct from the general criterion of (2).

(1) Existence/absence in 1D. Determine whether the stationary state of the infinite nearest-neighbor Glauber–Ising chain driven by a zero-mean periodic field (square wave alternating between $+F_0$ and $-F_0$ with half-period $t_c$, or sinusoidal) at temperature $T>0$ can spontaneously break the half-cycle symmetry — that is, whether any point of the $(F_0,T,t_c)$ parameter space supports a nonzero cycle-averaged magnetization $Q\neq 0$ — or whether $Q=0$ identically in the thermodynamic limit, as existing Monte Carlo data suggest. The $T\to 0$ limit is included in the question.

(2) Dimensional mechanism. Provide a quantitative account of how domain-wall geometry enables the symmetry breaking in $d\geq 2$. Two requirements are mandatory. First, a criterion that separates systems sustaining a dynamically ordered phase under periodic driving from those that do not, and that distinguishes one-dimensional kink proliferation from two- and three-dimensional interface motion. Second, a reproduction of the known dynamic phase boundaries of the two- and three-dimensional models, with the predicted dynamic transition temperature at each sampled field amplitude and period agreeing with the published Monte Carlo result to within approximately ten percent.

(3) Stationarity scaling. Derive the scaling $N_c\sim t_c^{-n}$ for the number of field cycles $N_c$ after which the switching kinetics become stationary (reproducible from cycle to cycle), as reported by Petukhov (2015), and account for the empirically reported exponent $n\approx 4.6\pm 0.4$. That value was extracted from Monte Carlo simulations of the finite $10^4$-site chain, not from an asymptotic solution of the analytic model.

References

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