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Dynamic Phase Transition, Universality, and Finite-size Scaling in the Two-dimensional Kine


Dynamic Phase Transition, Universality, and Finite-size Scaling in the Two-dimensional Kinetic Ising Model in an Oscillating Field
G. Korniss,1? C. J. White,1,2 P. A. Rikvold,1,2 and M. A. Novotny1
1

arXiv:cond-mat/0008155v1 [cond-mat.stat-mech] 9 Aug 2000

2

School of Computational Science and Information Technology, Florida State University, Tallahassee, Florida 32306-4120 Center for Materials Research and Technology and Department of Physics, Florida State University, Tallahassee, Florida 32306-4350 (February 1, 2008) We study the two-dimensional kinetic Ising model below its equilibrium critical temperature, subject to a square-wave oscillating external ?eld. We focus on the multi-droplet regime where the metastable phase decays through nucleation and growth of many droplets of the stable phase. At a critical frequency, the system undergoes a genuine non-equilibrium phase transition, in which the symmetry-broken phase corresponds to an asymmetric stationary limit cycle for the time-dependent magnetization. We investigate the universal aspects of this dynamic phase transition at various temperatures and ?eld amplitudes via large-scale Monte Carlo simulations, employing ?nite-size scaling techniques adopted from equilibrium critical phenomena. The critical exponents, the ?xed-point value of the fourth-order cumulant, and the critical order-parameter distribution all are consistent with the universality class of the two-dimensional equilibrium Ising model. We also study the cross-over from the multi-droplet to the strong-?eld regime, where the transition disappears. PACS numbers: 64.60.Ht, 75.10.Hk, 64.60.Qb, 05.40.-a

I. INTRODUCTION

Metastability and hysteresis are widespread phenomena in nature. Ferromagnets are common systems that exhibit these behaviors [1–5], but there are also numerous other examples ranging from ferroelectrics [6,7] to electrochemical adsorbate layers [8,9] to liquid crystals [10]. A simple model for many of these real systems is the kinetic Ising model (in either the spin or the lattice-gas representation). For example, it has been shown to be appropriate for describing magnetization dynamics in highly anisotropic single-domain nanoparticles and uniaxial thin ?lms [11–14].

(a)

(b)

(c)

FIG. 1. Metastable decay in the multi-droplet regime at T =0.8Tc for an L=128 square-lattice kinetic Ising system evolving under Glauber dynamics. The system is initialized with all spins si =1, and an applied ?eld, H=?0.3J, is set at t=0. Snapshots of the spin con?gurations are given at (a) t=30 Monte Carlo steps per spin (MCSS), (b) t=60 MCSS, (c) t=74 MCSS. The metastable lifetime (the average ?rst-passage time to zero magnetization) at this temperature and ?eld is τ =74.5 MCSS. Stable s=?1 (metastable s=+1) spins are represented by black (white).

Permanent address: Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, NY 12180-3590.

?

1

The system response to a single reversal of the “external ?eld” is fairly well understood [15]. In su?ciently large systems below the equilibrium critical temperature, Tc , the order parameter changes its value through the nucleation and growth of many droplets, inside which it has the equilibrium value consistent with the value of the applied ?eld, as shown in Fig. 1. This is the multi-droplet regime of phase transformation [15,16]. The well-known Avrami’s law [17] describes this process of homogeneous nucleation followed by growth quite accurately up to the time when the growing droplets coalesce and the stable phase becomes the majority phase [18]. The intrinsic time scale of the system is given by the metastable lifetime, τ , which is de?ned as the average ?rst-passage time to zero magnetization. It is a measure of the time it takes for the system to escape from the metastable region of the free-energy landscape. In this paper we will use the magnetic language in which the order parameter is the magnetization, m, and its conjugate ?eld is the external magnetic ?eld, H. Analogous interpretations, e.g., using the terms polarization and electric ?eld for ferroelectric systems [6,7], and coverage and chemical potential for adsorption problems [8,9], are straightforward. It is natural next to ask, “what is the response to an oscillating external ?eld?” The hysteretic behavior in ferromagnets has attracted signi?cant experimental interest, mainly focused on the characteristic behavior of the hysteresis loop and its area. Its dependence on the ?eld amplitude and frequency has been intensively studied and its scaling behavior (power law versus logarithmic) is still under investigation, both experimentally [11–13] and theoretically [19–27]. For the kinetic Ising ferromagnet in two dimensions it has been recently shown [25–27] that the true behavior is in fact a crossover, approaching a logarithmic frequency dependence only for extremely low frequencies. An important aspect of hysteresis in bistable systems, which is the focus of the present paper, is the dynamic competition between the two time scales in the system: the half-period of the external ?eld t1/2 (proportional to the inverse of the driving frequency) and the metastable lifetime τ . For low frequencies, a complete decay of the metastable phase almost always occurs in each half-period, just like it does after a single ?eld reversal. Consequently, the time-dependent magnetization reaches a limit cycle which is symmetric about zero [Fig. 2(a)]. For high frequencies, however, the system does not have enough time to switch during one half-period, and the symmetry of the hysteresis loop is broken. The magnetization then reaches an asymmetric limit cycle [Fig. 2(b)]. Avrami’s law [17,18] is a good approximation when the majority of the droplets do not overlap. Thus, it can be employed to estimate the timedependent magnetization and the dynamic order parameter (period-averaged magnetization) in the low-frequency (see the Appendix) and in the high-frequency [27] limits. However, it cannot describe the “critical regime” where t1/2 becomes comparable to τ , and which is dominated by coalescing droplets.

1.0

(a)

1.0

(b)

0.5

0.5

H(t)/H0 , m(t)

H(t)/H0 , m(t)

0.0

0.0

?0.5

?0.5

?1.0 0.0 400.0 800.0 1200.0 1600.0 2000.0

?1.0 0.0 40.0 80.0 120.0 160.0 200.0

t [MCSS]

t [MCSS]

FIG. 2. Monte Carlo magnetization time series (solid lines) in the presence of a square-wave external ?eld (dashed lines) for an L=128 system at T =0.8Tc and ?eld amplitude H0 =0.3J. (Tc is the two-dimensional equilibrium Ising critical temperature.) The metastable lifetime at this temperature and ?eld is τ =74.5 MCSS. (a) Dynamically disordered phase at dimensionless half-period Θ≡t1/2 / τ = 2.7. (b) Dynamically ordered phase at Θ=0.27.

This symmetry breaking between the symmetric and asymmetric limit cycles has been the subject of intensive research over the last decade. It was ?rst observed during numerical integration of a mean-?eld equation of motion for the magnetization of a ferromagnet in an oscillating ?eld [21,22]. Since then, it has been observed and studied in numerous Monte Carlo (MC) simulations of kinetic Ising systems [24,27–33], as well as in further mean-?eld studies [23,29,31,32,34]. It may also have been experimentally observed in ultrathin ?lms of Co on Cu(001) [12]. The results of these studies suggest that this symmetry breaking corresponds to a genuine continuous non-equilibrium phase transition. For recent reviews see Refs. [35,36]. Associated with the transition is a divergent time scale (critical slowing down) [31] and, for spatially extended systems, a divergent correlation length [27,28]. Estimates for the critical 2

exponents and the universality class of the transition have recently become available [27,28,37] after the successful application of ?nite-size scaling techniques borrowed from equilibrium critical phenomena [38–41]. The purpose of the present paper is to extend preliminary results [37] and to provide more accurate estimates of the exponents for two-dimensional kinetic Ising systems in a square-wave oscillating ?eld. The use of the squarewave ?eld tests the universality of the dynamic phase transition (DPT) [27,28], and it also signi?cantly increases computational speed, compared to the more commonly used sinusoidal ?eld. We further explore the universal aspects of the transition by varying the temperature and ?eld amplitude within the multi-droplet regime, and we study the cross-over to the strong-?eld regime where the transition disappears. In obtaining our results, we rely on dynamic MC simulations. Computational methods are always helpful, especially when theoretical ideas are largely missing. There are cases, however, when even the use of standard equilibrium techniques, such as ?nite-size scaling requires some insight and building analogies between equilibrium and non-equilibrium systems [27,28]. This is the case for our present study. No e?ective “Hamiltonian” was known before the completion of this work for the dynamic orderparameter (in the coarse-grained sense), from which the long-distance behavior of the model could be derived. This is a typical di?culty when dealing with systems far from equilibrium [42,43]. Recently, however, a coarse-grained Hamiltonian has been derived [44] for the dynamic order-parameter, supporting our results for the DPT. Similar to the previous work for sinusoidly oscillating ?elds [27,28], we perform large-scale simulations and ?nite-size scaling to investigate the universal properties of the DPT. The remainder of the paper is organized as follows. In Sec. II we de?ne the model and the observables of interest. Section III contains the Monte Carlo results and analyses. Conclusions and outlook are given in Sec. IV.
II. MODEL AND RELEVANT OBSERVABLES

To model spatially extended bistable systems in two dimensions, we study a nearest-neighbor kinetic Ising ferromagnet on a L×L square lattice with periodic boundary conditions. The model is de?ned by the Hamiltonian H = ?J si sj ? H(t) si ,
i

(1)

ij

where si =±1 is the state of the ith spin, J > 0 is the ferromagnetic interaction, ij runs over all nearest-neighbor pairs, i runs over all L2 lattice sites, and H(t) is an oscillating, spatially uniform applied ?eld. The magnetization per site, 1 m(t) = 2 L
L2

si (t) ,
i=1

(2)

is the density conjugate to H(t). The temperature T is ?xed below its zero-?eld critical value Tc (J/kB Tc =ln(1 + √ 2)/2 [45], where kB is Boltzmann’s constant), so that the magnetization for H=0 has two degenerate spontaneous equilibrium values, ±msp (T ). For nonzero ?elds the equilibrium magnetization has the same sign as H, while for H not too strong, the opposite magnetization direction is metastable and decays slowly towards equilibrium with time, as described in Sec. I. The dynamic used in this study, as well as in Refs. [27,28], is the Glauber single-spin-?ip MC algorithm with updates at randomly chosen sites. Note that the random sequential update scheme corresponds to independent Poisson arrivals for the update attempts (discrete events) at each site. Thus, the arrival pattern is strongly asynchronous. The time unit is one MC step per spin (MCSS). Each attempted spin ?ip from si to ?si is accepted with probability W (si → ?si ) = exp(?β?Ei ) . 1 + exp(?β?Ei ) (3)

Here ?Ei is the energy change resulting from acceptance, and β = 1/kB T . For the largest system studied (L=512) we used a scalable massively parallel implementation of the algorithm for this asynchronous dynamics [46–49]. The parallel discrete-event scheme ensures that the underlying dynamic is not changed (that is, the update attempts are identical, independent Poisson arrivals at each site), while a substantial amount of parallelism is exploited. The parallel implementation [47] was carried out on a Cray T3E, employing up to 256 processing elements. The dynamic order parameter is the period-averaged magnetization [21], Q= 1 2t1/2 3 m(t)dt , (4)

where t1/2 is the half-period of the oscillating ?eld, and the beginning of the period is chosen at a time when H(t) changes sign. Although the phase of the ?eld does not in?uence the results reported in this paper, the choice made here is convenient in studies of the hysteresis loop-area distributions and consistent with Refs. [25–28]. Analogously we also de?ne the local order parameter Qi = 1 2t1/2 si (t)dt , (5)

which is the period-averaged spin at site i. For slowly varying ?elds the probability distribution of Q is sharply peaked at zero [27,28]. We shall refer to this as the dynamically disordered phase. It is illustrated by the evolution of the magnetization in Fig. 2(a) and by the Q≈0 time series in Fig. 3. For rapidly oscillating ?elds the distribution of Q becomes bimodal with two sharp peaks near ±msp (T ), corresponding to the broken symmetry of the hysteresis loops [27,28]. We shall refer to this as the dynamically ordered phase. It is illustrated in Fig. 2(b) and by the Q≈msp ?O(1) time series in Fig. 3. Near the DPT we use ?nite-size scaling analysis of MC data to estimate the critical exponents that characterize the transition. We also keep track of the normalized period-averaged internal energy (in units of J) [31], E=? 1 2t1/2 1 L2 si (t)sj (t)dt ,
ij

(6)

since it also exhibits important characteristics of the DPT. Previous studies of the DPT have used an applied ?eld which varies sinusoidally in time. While sinusoidal or linear saw-tooth ?elds are the most common in experiments and are necessary to obtain a vanishing loop area in the low-frequency limit [25–27], the wave form of the ?eld should not a?ect universal aspects of the DPT. This should be so because the transition essentially depends on the competition between two time scales: the half-period t1/2 of the applied ?eld, and the average time it takes the system to leave the metastable region near one of its two degenerate zero-?eld equilibria when a ?eld of magnitude H0 and sign opposite to the magnetization is applied. This metastable lifetime, τ (T, H0 ) , is estimated as the average ?rst-passage time to zero magnetization. In the present paper we use a square-wave ?eld of amplitude H0 . This has signi?cant computational advantages over the sinusoidal ?eld variation since we can use two look-up tables to determine the acceptance probabilities: one for H = +H0 and one for H = ?H0 . In terms of the dimensionless half-period, Θ = t1/2 / τ (T, H0 ) , (7)

the DPT should occur at a critical value Θc of order unity. Although Θ can be changed by varying either t1/2 , H0 , or T , in a ?rst approximation we expect Θc to depend only weakly on H0 and T . This expectation will be con?rmed in Sec. IV by simulations carried out at several values of H0 and T for di?erent system sizes. In many studies of the DPT the transition has been approached by changing H0 or T [24,29,30,32]. While this is correct in principle, τ (T, H0 ) depends strongly and nonlinearly on its arguments [15]. We therefore prefer changing t1/2 at constant H0 and T [27,28], as this gives more precise control over the distance from the transition. We focus on systems which are not only larger than the critical droplet, but also signi?cantly larger than the typical droplet separation [15]. In this regime many supercritical droplets form and contribute to the decay of the metastable phase (the KJMA or Avrami theory for homogeneous nucleation [17,18]), as seen in Fig. 1. This is the only regime where the DPT is expected to exist. For small systems one observes subtle ?nite-size e?ects, not related to the DPT but rather to the stochastic single-droplet decay mode [15,50]. In the single-droplet regime, subject to a periodic applied ?eld, the system exhibits stochastic resonance [26].
III. SIMULATION RESULTS A. Signs of the dynamic phase transition

We performed extensive simulations and ?nite-size scaling analysis of the data on square lattices with L between 64 and 512 at T =0.8Tc and H0 =0.3J. We also investigated the universality of the DPT within the multi-droplet regime for various ?elds and temperatures below the equilibrium critical temperature, using smaller systems with L from 64 to 128. Typical runs near the DPT consist of 2×105 full periods. For example, at T =0.8Tc and H0 =0.3J, where the critical half-period of the ?eld is about 70 MCSS, this corresponds to 2.8×107 MCSS. Away from the transition point, an order of magnitude shorter runs were su?cient to obtain high-quality statistics. 4

The system was initialized with all spins up and the square-wave external ?eld started with the half-period in which H=?H0 . After some relaxation the system magnetization would reach a limit cycle [Fig. 2] (except for thermal ?uctuations). In other words, Q [Eq. (4)] (together with other period-averaged quantities) becomes a stationary stochastic process [Fig. 3]. We discarded the ?rst 1000 periods of the time series to exclude transients from the stationary-state averages. For large half-periods (Θ ? Θc ) the magnetization switches every half-period [Fig. 2(a)] and Q≈0, while for small half-periods (Θ ? Θc ) the magnetization does not have time to switch during a single half-period [Fig. 2(b)], resulting in |Q|≈msp , as can be seen from the time series in Fig. 3. The transition between the high- and low-frequency regimes is characterized by large ?uctuations in Q near Θc [Fig. 3].

1.0
Θ=0.27 Θ=0.98 Θ=2.7

0.5

Q(j)
0.0 ?0.5 ?1.0 500

1000

1500

2000

j [periods]
FIG. 3. Time series of the order parameter Q at T = 0.8Tc and H0 = 0.3J for L = 128. Horizontal trace near Q = +1: Θ = 0.27 < Θc [dynamically ordered phase, corresponding to Fig. 2(b)]. Strongly ?uctuating trace: Θ = 0.98 ≈ Θc (near the DPT). Horizontal trace near Q = 0: Θ = 2.7 > Θc [dynamically disordered phase, corresponding to Fig. 2(a)].

(a)

(b)

(c)

FIG. 4. Con?gurations of the local order parameter {Qi } at T =0.8Tc and H0 =0.3J for L=128. (a) Θ = 0.27 < Θc (dynamically ordered phase). (b) Θ = 0.98 ≈ Θc (near the DPT). (c) Θ = 2.7 > Θc (dynamically disordered phase). On the gray-scale black (white) corresponds to ?1 (+1).

To illustrate the spatial aspects of the transition we also show con?gurations of the local order parameter {Qi } in Fig. 4. Below Θc [Fig. 4(a)] the majority of spins spend most of their time in the +1 state, i.e., in the metastable phase during the ?rst half-period, and in the stable equilibrium phase during the second half-period (except for equilibrium

5

?uctuations). Thus, most of the Qi ≈+1. Droplets of si =?1 that nucleate during the negative half-period and then decay back to +1 during the positive half-period show up as roughly circular gray spots in the ?gure. Since the spins near the center of such a droplet become negative ?rst and revert to positive last, these spots appear darkest in the middle. Also, for not too large lattices one occasionally observes the full reversal of an ordered con?guration {Qi }→{?Qi }, typical of ?nite, spatially extended systems undergoing symmetry breaking. Above Θc [Fig. 4(c)] the system follows the ?eld in every half-period (with some phase lag) and Qi ≈0 at all sites i. Near Θc [Fig. 4(b)] there are large clusters of both Qi ≈+1 and Qi ≈?1 separated by “interfaces” where Qi ≈0. These large-scale structures remain reasonably stationary over several periods. For ?nite systems in the dynamically ordered phase the probability density of Q becomes bimodal. Thus, to capture symmetry breaking, one has to measure the average norm of Q as the order parameter, i.e., |Q| [39]. Figure 5(a) shows that this order parameter is of order unity for Θ < Θc and vanishes for Θ > Θc , except for ?nite-size e?ects. To characterize and quantify this transition in terms of critical exponents we employ the well-known technique of ?nite-size scaling [38–41]. The quantity analogous to the susceptibility is the scaled variance of the dynamic order parameter [27,28],
Q XL = L 2

Q2

L

? |Q|

2 L

.

(8)

Note that for our system the ?eld conjugate to Q and a corresponding ?uctuation-dissipation theorem are not known, hence we cannot measure the susceptibility directly. For ?nite systems XL has a characteristic peak near Θc [see Fig. 5(b)] which increases in height with increasing L, while no ?nite-size e?ects can be observed for Θ ? Θc and Θ ? Θc . This implies the existence of a divergent length scale, possibly the correlation length which governs the Q long-distance behavior of the local order-parameter correlations Qi Qj . The location of the maximum in XL also shifts with L, which gives further important information about the critical exponents.

1.0

10

4

(a)
0.8

<|Q|>L

0.6

XL

0.4

XL

L=64 L=90 L=128 L=180 L=256 L=512

(b)
6000

(c) L=64 L=90 L=128 L=180 L=256 L=512
10
3

4000
Q

10
Q

2

L=64 L=90 L=128 L=180 L=256 L=512

2000
0.2

10

1

10
0.0 0.50 0.75 1.00 1.25 1.50

0

Θ

0 0.70

0.80

0.90

Θ

1.00

1.10

1.20

0.00

0.50

1.00

1.50

Θ

2.00

2.50

3.00

FIG. 5. Finite-size behavior of the order parameter at T =0.8Tc and H0 =0.3J for various system sizes. (a) The order Q parameter |Q| L . (b) The scaled variance of the order parameter, XL , as de?ned in Eq. (8). (c) Same as (b) on lin-log scale to provide an enhanced view of the peaks for smaller systems.

The normalized stationary time-displaced autocorrelation function of the order parameter,
Q CL (n) =

Q(j)Q(j + n) ? Q(j) Q2 (j) ? Q(j) 2

2

,

(9)

provides further insights into the DPT as the system exhibits critical slowing down [Fig. 6]. This can be seen as increasing correlation times with increasing system sizes. In Sec. III.D we provide a quantitative analysis of the correlation times. We also measured the period-averaged internal energy [Eq. (6)] and its ?uctuations [31]
E XL = L 2

E2

L

? E

2 L

,

(10)

as can be seen in Fig. 7. The peaks of these ?uctuations exhibit a slow increase with the system size (compared to the order-parameter ?uctuations), as one may anticipate by analogy with the equilibrium heat capacity.

6

0.75

CL(n)

L=64 L=90 L=128 L=180 L=256

Q

0.25

?0.25

0

2000

n [periods]

4000

6000

8000

10000

FIG. 6. Critical slowing down for the order parameter at T =0.8Tc and H0 =0.3J at Θ=Θc , as shown by the normalized Q autocorrelation function CL (n).

?1.60

(a)
?1.65

6.00

(b)

XL

?1.70

?1.75

L=64 L=90 L=128 L=180 L=256 L=512
1.00 1.50

4.00
E

<E>L

L=64 L=90 L=128 L=180 L=256 L=512

2.00

?1.80 0.50

Θ

0.00 0.50

1.00

Θ

1.50

FIG. 7. Finite-size behavior of the period-averaged internal energy at T =0.8Tc and H0 =0.3J for various system sizes. (a) E The period-averaged internal energy. (b) The scaled energy variance, XL as de?ned in Eq. (10).

B. Finite-size scaling

Scaling laws and ?nite-size scaling for equilibrium systems with an a-priori known Hamiltonian can be systematically derived using the concepts of the free energy and the renormalization group [51]. The kinetic Ising model with the explicitly time-dependent Hamiltonian, Eq. (1), is driven far from equilibrium. Although the order-parameter distribution P (Q) is stationary, the e?ective Hamiltonian controlling its ?xed-point behavior has not been known until recently [44] (after the completion of this study). Motivated by the similarity of the ?nite-size e?ects shown in Figs. 5-7 to those characteristic of a typical continuous phase transition, we borrow the corresponding scaling assumptions from equilibrium ?nite-size scaling. For our model the quantity analogous to the reduced temperature in equilibrium systems (i.e., the distance from the in?nite-system critical point) is θ= |Θ ? Θc | . Θc (11)

Finite-size scaling theory provides simple scaling relations for the observables for ?nite systems in the critical regime [39,40]: |Q|
L Q XL

=L

= L?β/ν F± (θL1/ν )
γ/ν

(12) (13) , (14)

E XL = c1 ln LJ± (θL1/ν )

G± (θL

1/ν

)

7

where F± , G± , and J± are scaling functions and the + (?) index refers to Θ > Θc (Θ < Θc ). The logarithmic E scaling in XL is motivated by the very slow divergence of the scaled period-averaged energy variance [Fig. 7(b)]. The above formulation of scaling is explicitly based on the in?nite-system critical point Θc , which can be estimated with far greater accuracy than the location of the maximum of the order-parameter ?uctuations for the individual ?nite system sizes. We use the fourth-order cumulant intersection method [39,40] to estimate the value of Θc at which the transition occurs in an in?nite system. In order to do this, we plot UL = 1 ? Q4 L 3 Q2 2 L (15)

as a function of Θ for several system sizes as shown in Fig. 8. For the largest system (L=512) the statistical uncertainty in UL was too large to use it to obtain estimates for the crossing. Our estimate for the dimensionless critical halfperiod, based on the remaining ?ve system sizes, is Θc =0.918±0.005 with a ?xed-point value U ? =0.611±0.003 for the cumulant [Fig 8(b)].

0.70

0.67

(a)
0.65 0.50 0.63

(b)

0.30

0.10

L=64 L=90 L=128 L=180 L=256 L=512
0.40 0.60 0.80 1.00 1.20 1.40

UL

UL

0.61

0.59

0.57

L=64 L=90 L=128 L=180 L=256
0.87 0.89

?0.10 0.20

Θ

0.55 0.85

Θ

0.91

0.93

0.95

FIG. 8. (a) The fourth-order cumulant as de?ned in Eq. (15) at T =0.8Tc and H0 =0.3J for various system sizes. (b) The region around the cumulant crossing in (a) enlarged. The horizontal and vertical dashed lines indicate the ?xed-point value U ? =0.611 and the scaled critical half-period Θc =0.918, respectively.

Then at Θc the scaling forms Eqs. (12-14) yield |Q|
L Q XL E XL

∝L

∝ L?β/ν
γ/ν

(16) (17) (18)

∝ c2 + c1 ln(L) ,

which enable us to estimate the exponent ratios β/ν and γ/ν, and to directly check the postulated logarithmic Q divergence in the period-averaged energy ?uctuations. Plotting |Q| L and XL at Θc and utilizing a weighted linear least-squares ?t to the logarithmic data yields β/ν=0.126 ± 0.005 [Fig. 9(a)] and γ/ν=1.74 ± 0.05 [Fig. 9(b)]. Note that these values are extremely close (within statistical errors) to the corresponding ratios for the equilibrium twodimensional Ising universality class, β/ν=1/8=0.125 and γ/ν=7/4=1.75. Further, the straight line in Fig. 9(c) E indicates the slow logarithmic divergence of XL at the critical point. In addition to the scaling at Θc , we also checked Q E the divergences of the peaks of the ?uctuations, (XL )peak and (XL )peak , since they asymptotically should follow the Q same scaling laws, Eqs. (17) and (18), respectively. The measured exponent γ/ν=1.78 ± 0.05 for (XL )peak and the E logarithmic divergence for (XL )peak agree to within the statistical errors with the results obtained at Θc , as can be seen in Fig. 9(b) and (c), respectively. From the ?nite-system shifting of the transition one can estimate the correlation-length exponent ν by tracking the Q shift in the location of the maximum in XL : |Θc (L) ? Θc | ∝ L?1/ν , (19)

where Θc (L) is the location of the peak for ?nite systems. However, the precision of this method for our data is very poor, due to limited resolution in ?nding the locations of the maxima and consequently the large relative errors in 8

|Θc (L) ? Θc |. Excluding the smallest (due to strong corrections to scaling) and the largest systems (due to very poor resolution and extremely large statistical error), we obtain ν=0.87 ± 0.4, but the large error estimate obviously implies rather poor accuracy [Fig. 10].
0.7
10
4

6

(a)
0.6

XL at Θc Q (XL)peak (slope) γ/ν=1.74±0.05 (slope) γ/ν=1.78±0.05
Q

(b)

(c)

<|Q|>L

XL

Q

5

10

3

XL

E

0.5

<|Q|>L at Θc (|slope|) β/ν=0.126±0.005
10
2

4

XL at Θc E (XL)peak
E

0.4

50

100

500

50

100

500

3

50

100

500

L

L

L

FIG. 9. Critical exponent estimates at T =0.8Tc and H0 =0.3J. Straight lines are the weighted least-square ?ts. (a) Determining β/ν through the ?nite-size e?ects of the order parameter, based on Eq. (16) (log-log plot). (b) Determining γ/ν through the ?nite-size e?ects of the order-parameter ?uctuations, based on Eq. (17) (log-log plot). (c) Showing the logarithmic divergence of the period-averaged energy ?uctuations, based on Eq. (18) (log-lin plot).

0.10

(|slope|)=1/ν=1.15±0.6 (|slope|)=1/ν=1.05 (|slope|)=1/ν=1.0

|Θc(L)?Θc|
0.01 50

100

L

500

FIG. 10. Exponent estimate for ν at T =0.8Tc and H0 =0.3J, based on Eq. (19) (log-log plot). The dot-dashed line is a weighted least-squares ?t (excluding the smallest and the largest system) yielding the slope 1/ν=1.15 ± 0.6 (ν=0.87 ± 0.4). The dashed line represents the “optimal” value for this exponent, using the best quality data collapse for the scaling function Eq. (12) as discussed in the text, yielding 1/ν=1.05 (ν=0.95). The solid line represents the two-dimensional equilibrium Ising exponent ν=1.0.

To obtain a more complete picture of how well the scaling relations in Eqs. (12) and (13) hold, we plot |Q| L Lβ/ν Q [Fig. 11(a)] and XL L?γ/ν [Fig. 11(b)] vs θL1/ν [41]. For the exponent ratios we used β/ν=1/8=0.125, and γ/ν=7/4=1.75, since our estimate for those (within small statistical errors) implied that they take on the equilibrium two-dimensional Ising universal values. Most importantly, we used various values of ν between 0.5 and 1.2 to ?nd the best data collapse as observed visually, since our estimate for this exponent was far from reliable. The “optimal” value obtained this way (by showing scaling plots to group members who did not know the particular values of ν used), and used in Fig. 11(a) and (b), is ν=0.95 ± 0.15. Full scaling plots using the exact Ising exponents are also shown in Fig. 11(c) and (d), and they result in similarly good data collapse.

9

10

0

(a)
10
β/ν
0

Θ<Θc slope=β

(b)
10
?1

Θ>Θc slope=?γ Θ<Θc slope=?γ

<|Q|>LL

10

?2

10

?1

L=64 L=90 L=128 L=180 L=256 L=512

Θ>Θc slope=β?ν

?γ/ν

10

?3

10

?4

L=64 L=90 L=128 L=180 L=256 L=512

XLL

10

?1

10

0

θL

1/ν

10

1

10

2

10

3

10

?5

10

?1

10

0

θL

1/ν

10

1

10

2

(c)
10
1/8
0

10

0

Θ<Θc slope=1/8

(d)
10
?1

Θ>Θc slope=?7/4 Θ<Θc slope=?7/4
L=64 L=90 L=128 L=180 L=256 L=512

<|Q|>LL

10

?2

10

?1

L=64 L=90 L=128 L=180 L=256 L=512

Θ>Θc slope=?7/8

XLL
2 3

?7/4

10

?3

10

?4

10

?1

10

0

θL

10
1

1

10

10

10

?5

10

?1

10

0

θL

1

10

1

10

2

FIG. 11. Finite-size scaling (full data collapse) at T =0.8Tc and H0 =0.3J using β/ν=1/8, γ/ν=7/4 (two-dimensional equilibrium Ising values), and ν=0.95 (which yields the best quality data collapse). (a) For the order parameter |Q| L (log-log Q plot). (b) For the scaled order-parameter variance XL (log-log plot). Straight lines in both graphs represent the asymptotic large-argument behaviors of the scaling functions F± and G± given in Eqs. (12) and (13), respectively. Figures (c) and (d) are the same as (a) and (b), except that the exact Ising exponent ν=1.0 is used.

C. Order-parameter histograms at criticality

We devote this subsection to analyzing the universal characteristics of the full order-parameter distribution, P (Q), at the critical point. This distribution is bimodal for ?nite systems if observed for su?ciently long times (Fig. 12) [52]. It is more convenient to focus on the distribution of |Q|, avoiding the e?ect of the insu?cient number of switching events between the two symmetry-broken phases for large systems, which causes the skewness in Fig. 12(b). Figure 13(a) shows the order-parameter distributions PL (|Q|) at the critical point for various system sizes. Finite-size scaling arguments [39,40] suggest that at Θc PL (|Q|) = Lβ/ν P(Lβ/ν |Q|) . (20)

Thus, the scaled distributions, L?β/ν PL (|Q|) vs x=|Q|Lβ/ν , should fall on the same curve P(x) for di?erent system sizes. Again, we used β/ν = 1/8. The quality of the data collapse is quite impressive [Fig. 13(b)], with deviations mainly observed for the smallest L and the largest values of |Q| [Fig. 13(c)], possibly as a result of corrections to scaling.

10

3
1

(a)
0.5

(b)
2

Q(j)

0

PL(Q)
1 0 ?1

?0.5

?1 1.5e+05

j [periods]

1.75e+05

2e+05

?0.5

Q

0

0.5

1

FIG. 12. (a) Short segment of the order-parameter time series at T =0.8Tc and H0 =0.3J for an L=128 system at Θc [52]. (b) Order-parameter histogram for the same parameters.

What we ?nd somewhat surprising, is that the distribution appears to be identical (except for stronger corrections to scaling at the DPT) to that of the equilibrium two-dimensional Ising model on a square lattice with periodic boundary conditions at criticality, without a need for any additional scaling parameters. We checked this by performing standard equilibrium two-dimensional Ising simulations with Glauber dynamics and system sizes ranging from L=64 to L=128, and also by comparing our scaled DPT order-parameter histograms to the high-precision two-dimensional equilibrium Ising MC data of Ref. [53] (Fig. 13). We had expected the shapes of the distributions to be identical for the DPT and equilibrium Ising model, as a consequence of the identical values for the cumulant ?xed-point value U ? . However, it is not obvious to us why the microscopic length scales in the DPT and the equilibrium Ising model also appear to be identical, as evidenced by the absence of the need for an additional scaling, L→L/a (a being the microscopic length scale in the DPT).

6
3

4

PL(|Q|)

PL(|Q|)

PL(|Q|)

L=64 L=90 L=128 L=180 L=256 L=512

(a)

2

L=64 L=90 L=128 L=180 L=256 L=512

(b)

10

1

(c)
10
0

10

?1

?β/ν

L

2

L

1

10

?2

10

?3

L=64 L=90 L=128 L=180 L=256 L=512

0

?β/ν

0

0.2

0.4

0.6

0.8

1

0

0

0.25

0.5

0.75

|Q|

|Q|L

β/ν 1

1.25

1.5

1.75

10

?4

0

0.25

0.5

0.75

|Q|L

β/ν 1

1.25

1.5

1.75

FIG. 13. Order-parameter histograms PL (|Q|) at T =0.8Tc and H0 =0.3J for various system sizes at Θc . The thin solid lines represent two-dimensional equilibrium Ising order-parameter histograms at the critical point for L=64, 90, and 128 on all three graphs. (a) Order-parameter distributions. (b) Scaled order-parameter distributions, according to Eq. (20) with β/ν=1/8. The bold solid line is the corresponding (Monte Carlo) two-dimensional equilibrium Ising distribution without any additional scaling parameters [53]. (c) Same as (b) on lin-log scales to enhance the view of the corrections to scaling for small systems at large |Q|.

D. Critical slowing down

Computing the stationary autocorrelation function given by Eq. (9), we already pointed out that at Θc the correlation time increases fast with system size [Fig. 6]. Correlation times are typically extracted from an exponential decay as
Q CL (n) ∝ e?n/τL ,
Q

(21)

11

Q and they are expected to be ?nite for ?nite systems. The correlation time τL is also well de?ned in the L→∞ limit away from the transition. However, it diverges with L at the transition point as Q τL ∝ Lz ,

(22)

where z is the dynamical critical exponent. For not too late times we had reasonable statistics including the larger Q systems (up to L=256) to ?t the usual exponential decay [Fig. 14(a)]. Then plotting the correlation times τL vs L yields the dynamic exponent z = 1.91 ± 0.15, as shown in Fig. 14(b). This value is within two standard deviations of most estimates for the dynamic exponent of the two-dimensional equilibrium Ising model with local dynamics [54].
1.0
4

10

(b)
0.8

τL at Θc (slope) z=1.91±0.15
Q

(a)
0.6

0.4

L=64 L=90 L=128 L=180 L=256
20 40 60 80

τL [periods]
Q
100 120 140 160 180 200

CL(n)

Q

10

3

10

2

n [periods]

50

100

L

500

FIG. 14. Critical slowing down for the order parameter at T =0.8Tc and H0 =0.3J at Θ=Θc . (a) The normalized autocorrelation function on lin-log scale for early times. The straight lines are ?ts to exponential decays according to Eq. (21). (b) Determining the dynamic exponent, z, using a power-law ?t to Eq. (22) (log-log plot).

E. Universality for various temperatures and ?elds and cross-over to the strong-?eld regime

The underlying ingredient for the spatially extended bistable systems exhibiting a DPT is the local metastability (and the corresponding characteristic time spent in the metastable “free-energy well”) in the presence of an external ?eld. This, in turn, provides a competition between time scales if the system is driven by a periodic ?eld. Based on this, we expect that su?ciently large systems (in which many droplets contribute to the decay of the metastable phase) exhibit the DPT at a half-period t1/2 comparable to the metastable lifetime τ (T, H0 ) . In other words, we expect the critical dimensionless half-period to be of order one, Θc ? O(1). To test this expectation, we performed simulations at T =0.8Tc for ?eld amplitudes ranging from 0.3J to in?nity with system sizes L=64, 90, and 128. [H0 =∞ corresponds to the Glauber spin-?ip probabilities Eq. (3) being equal to 0 (1) depending on whether the spin is parallel (anti parallel) to the external ?eld, with no in?uence from the con?guration of the neighboring spins.] We further performed runs at T =0.9Tc, T =0.6Tc, and T =0.5Tc for various ?eld amplitudes and system sizes L = 64 and 90. The typical run length was 2×104 periods. The purpose of these runs was to explore the universal nature of the DPT in the multi-droplet regime, and the crossover to the strong-?eld regime where the DPT should disappear. In the strong-?eld regime the droplet picture breaks down since the individual spins are decoupled. Thus, the metastable phase no longer exists, and the decay of the phase having opposite sign to the external ?eld approaches a simple exponential form (which becomes exact in the H0 →∞ limit). In the Appendix we show that under these conditions the system magnetization always relaxes to a symmetric limit cycle with Q=0 for all frequencies, thus, no DPT can exist. Figures 15(a), (b), and (c) show the order parameter vs the dimensionless half-period for L = 64 and a range of ?eld amplitudes at T =0.8Tc, T =0.6Tc, and T =0.5Tc, respectively. The typical order-parameter pro?le where the system exhibits the DPT prevails up to some temperature dependent cross-over ?eld amplitude H× (T ) [?lled symbols in Fig. 15]. For H0 > H× (T ) the underlying decay mechanism belongs to the strong-?eld regime, and correspondingly the DPT disappears as expected.

12

1.00

1.00

1.00

(a)
0.80

0.60

0.40

H0=0.3J, <τ>=74.8 H0=0.4J, <τ>=41.9 H0=0.50J, <τ>=27.6 H0=0.75J, <τ>=13.6 H0=1J, <τ>=8.41 H0=1.125J, <τ>=6.92 H0=1.25J, <τ>=5.82 H0=1.5J, <τ>=4.30 H0=2J, <τ>=2.67 H0=2.5J, <τ>=1.86 H0=3J, <τ>=1.41 H0=∞, <τ>=0.69

(b)
0.80

(c)
0.80

0.60

0.40

H0=0.6J, <τ>=64.64 H0=0.8J, <τ>=29.12 H0=1.0, <τ>=16.59 H0=1.2J, <τ>=10.69 H0=1.4J, <τ>=7.48 H0=1.6J, <τ>=5.51 H0=1.8J, <τ>=4.23 H0=2.0J, <τ>=3.36 H0=∞, <τ>=0.69

0.60

0.40

H0=0.79J, <τ>=62.88 H0=1.0J, <τ>=29.78 H0=1.2J, <τ>=17.05 H0=1.4J, <τ>=10.87 H0=1.6J, <τ>=7.42 H0=1.75J, <τ>=5.81 H0=1.8J, <τ>=5.40 H0=2.0J, <τ>=4.08 H0=2.2J, <τ>=3.20 H0=∞, <τ>=0.69

<|Q|>

<|Q|>

0.20

0.20

<|Q|>
0.20 3.00 0.00 0.00

0.00 0.00

0.50

1.00

1.50

Θ

2.00

2.50

3.00

0.00 0.00

1.00

Θ

2.00

1.00

Θ

2.00

3.00

FIG. 15. DPT in the multi-droplet regime and strong-?eld cross-over for various temperatures and ?eld amplitudes with L=64. Order-parameter pro?les with ?lled symbols exhibit a DPT. Corresponding lifetimes in units of MCSS are also indicated in the ?gures. (a) T =0.8Tc . (b) T =0.6Tc . (c) T =0.5Tc .

One can trace this crossover from the multi-droplet to the strong-?eld regime by plotting Θc vs H0 /J [Fig. 16(a)] and vs τ [Fig. 16(b)] for ?xed temperatures. Here again we employed the fourth-order cumulant intersection method using L=64 and L=90 to identify the in?nite-system transition point, Θc . The crossover to the strong-?eld regime is indicated by the drop in Θc for large ?elds (small lifetimes).

1.2

1.2

(a)
1
1.0

(b)

3

(c)

SF

0.8

0.8

2

Θc

0.6

0.4

T=0.5Tc T=0.6Tc T=0.8Tc T=0.9Tc

Θc

0.6

H/J

0.4

0.2

0.2

T=0.5Tc T=0.6Tc T=0.8Tc T=0.9Tc

MD
1

SD
0
80

0

0

0.5

1

1.5

2

0.0

0

20

40

60

0

0.2

0.4

0.6

0.8

1

1.2

H0/J

<τ>

T/Tc

FIG. 16. Scaled critical half-period in the multi-droplet/DPT regime: (a) as a function of the external ?eld amplitude H0 , (b) as a function of the lifetime τ . In (a) the corresponding empty symbols for each temperature on the horizontal axis represent the points which are the MC upper bounds for the cross-over ?eld amplitude, H× (T ), beyond which no DPT can be found for any non-zero Θ. In (b) the corresponding empty symbols are the lower bound for the lifetime below which no DPT is observed. (c) Metastable phase diagram for the kinetic Ising model. The stars connected by a dashed line give a numerical estimate for the dynamic spinodal for our smallest system, L=64, that separates the multi-droplet (MD) from the single-droplet (SD) regime [15]. The solid line is an analytic estimate for the “mean-?eld spinodal” which separates the MD from the strong-?eld (SF) regime [15]. The circles represent the temperature and ?eld amplitude values at which we ran our simulations. The ?lled circles indicate points where the system exhibits a DPT, and empty circles represent points where it does not.

The DPT in the multidroplet regime (H0 <H× (T )) appears to be universal, as can be seen from the scaled orderparameter plot in Fig. 17. Note that both graphs contain 28 DPT data sets: three di?erent temperatures, with four di?erent ?eld amplitudes for each, and at least two di?erent system sizes (L = 64, 90, and 128 at T =0.8Tc, and L = 64 and 90 at T =0.6Tc and T =0.5Tc) for all these parameters! While the slopes in the asymptotic scaling regime appear to be the same, the small parallel shift may be the result of the non-universal critical amplitudes at di?erent temperatures and ?elds, or simply our inaccuracy in determining Θc (T, H0 ), due to the relatively short runs.

13

(a)
10
0

Θ<Θc slope=β
<|Q|>L

(b)
10
0

Θ<Θc slope=1/8 Θ>Θc slope=?7/8

<|Q|>L

Θ>Θc slope=β?ν
10
?1

β/ν

1/8

10

?1

10

?2

10

?1

10

0

θL

10 1/ν

1

10

2

10

3

10

?2

10

?1

10

0

θL

10 1

1

10

2

10

3

FIG. 17. Universality of the DPT in the multi-droplet regime. (a) Finite-size scaling (full data collapse) for the order parameter |Q| L (log-log plot). The ?gure contains 28 DPT data sets: three di?erent temperatures, with four di?erent ?eld amplitudes for each, and at least two di?erent system sizes (L = 64, 90, and 128 at T =0.8Tc , and L = 64 and 90 at T =0.6Tc and T =0.5Tc ) for all these parameters. The temperature and ?eld values were chosen such that the system is in the multi-droplet (MD) regime [?lled circles in Fig. 16(c)]. We used β/ν=1/8, γ/ν=7/4 (two-dimensional equilibrium Ising values), and ν=0.95, the optimal value, as described in the Section III.B. Straight lines represent the asymptotic large-argument behaviors of the scaling functions F± given by Eq. (12). Figure (b) is the same as (a), except that the exact Ising exponent ν=1.0 is used.

On the other hand, as expected, the system shows no singularity and |Q| ?→ 0 for any non-zero frequency in the strong-?eld regime, H0 >H× (T ) (see the Appendix). Here, the ?nite-size e?ects simply re?ect the central-limit theorem, i.e., Q2 ?O(1/L2 ), or |Q| ?O(1/L). Figures 18(a) and (b) illustrate this at T =0.8Tc, H0 =1.5J and at T =0.8Tc, H0 =∞, respectively. One also expects that a similar behavior prevails at T >Tc for any ?eld amplitude, i.e., H× (T ) vanishes for T >Tc. We checked this for T =1.1Tc and H0 =0.05J, and the results con?rm the |Q| ?O(1/L) scaling of the order parameter for all frequencies, showing none of the characteristic ?nite-size e?ects of a DPT [Fig. 18(c)]. This result is in distinct disagreement with older work such as Refs. [29,35], which report observations of the DPT at temperatures considerably above Tc . For smaller system sizes, however, one may observe nontrivial resonance [55].

L→∞

0.14

(a)
L=64 L=90 L=128

6.0

0.18

<|Q|>LL
4.0

0.014

0.8

7.00

(b)
0.6

0.12

<|Q|>LL
0.13

(c)
L=64 L=90 L=128
6.00

<|Q|>LL

0.012

0.10

2.0

0.010

L=64 L=90 L=128

0.4 0.2

5.00

0.08 0.0 0.0 0.06 1.0 2.0

<|Q|>L

0.008 0.0 0.0 0.006 5.0

<|Q|>L

<|Q|>L

Θ

3.0

4.0

Θ

10.0

15.0

4.00 0.00 0.08

0.50

Θ

1.00

1.50

0.04

0.004

0.02

0.002

0.00 0.0

1.0

2.0

Θ

3.0

4.0

0.000 0.0

5.0

Θ

10.0

15.0

0.03 0.00

0.50

Θ

1.00

1.50

FIG. 18. Order parameter |Q| L behavior outside the multi-droplet regime. (a) Strong-?eld behavior at T =0.8Tc , H0 =1.5J ( τ =4.3 MCSS). (b) Strong-?eld behavior at T =0.8Tc , H0 =∞ ( τ =ln 2 MCSS). (c) High-temperature behavior at T =1.1Tc , H0 =0.05 ( τ =81.8 MCSS). The insets in all three graphs show data collapse for the scaled order parameter |Q| L L.

Due to the expected complications of a divergent equilibrium correlation length, we did not perform simulations at Tc . However, we conjecture that H× (T ) vanishes at Tc . We expect this conjecture to be extremely di?cult to prove or disprove numerically, due to very large and possibly complicated ?nite-size e?ects.

14

IV. CONCLUSIONS AND OUTLOOK

In this paper we have studied the hysteretic response of a spatially extended bistable system exhibiting a DPT. Our model system is the two-dimensional kinetic Ising ferromagnet below its equilibrium critical temperature subject to a periodic square-wave applied ?eld. The results indicate that for ?eld amplitudes and temperatures such that the metastable phase decays via the multi-droplet mechanism, the system undergoes a continuous dynamic phase transition when the half-period of the ?eld, t1/2 , is comparable to the metastable lifetime, τ (T, H0 ) . Thus, the critical value Θc of the dimensionless half-period de?ned in Eq. (7) is of order unity. As Θ is increased beyond Θc , the order parameter |Q| (the expectation value of the norm of the period-averaged magnetization) vanishes [Fig. 5(a)], displaying singular behavior at the critical point, as shown in Fig. 5(b) and (c). The characteristic ?nite-size e?ects in the order parameter and its ?uctuations indicate that there is a divergent correlation length associated with the transition. We used standard ?nite-size scaling techniques adopted from the theory of equilibrium phase transitions. We estimated Θc and the critical exponents β, γ, and ν from relatively high precision data for system sizes between L = 64 and 512 at T =0.8Tc and H0 =0.3J. Our best estimates are β/ν=0.126 ± 0.005, γ/ν=1.74 ± 0.05, and ν=0.95 ± 0.15. These values agree within statistical errors with those previously obtained with a sinusoidally oscillating ?eld [27,28], providing strong evidence that the shape of the ?eld oscillation does not a?ect the universal aspects of the DPT. Observing the stationary autocorrelation function we also saw that at the transition point the system exhibits critical slowing down governed by the dynamic exponent z=1.91±0.15. This is also very close to the corresponding exponent z=2.12(5), measured in standard two-dimensional Ising simulations with local dynamics [54]. Our best values for the exponent ratios β/ν and γ/ν are given with relatively high con?dence, while for ν it is rather poor. In this sense tracking down the exponent ν and obtaining an accurate estimate for it remains elusive. Note, however, that we could only rely on the standard (single spin-?ip) MC algorithm, since we had to preserve the underlying dynamics. Using more sophisticated algorithms to avoid critical slowing down as seen in Figs. 6 and 14, would require an underlying “Hamiltonian” for the corresponding local order parameter {Qi }, which is not yet known. While in a coarse-grained/universal sense a φ4 Hamiltonian is supported by our data, it does not point to any one particular microscopic Hamiltonian for the microscopic order parameter {Qi }. However, a φ4 coarse-grained Hamiltonian for {Qi } has been recently derived starting from the time-dependent Ginzburg-Landau equation for the magnetization [44]. Of the known universality classes, our exponent estimates for the DPT are closest (and within the statistical errors) to those of the the two-dimensional equilibrium Ising model: β/ν=1/8=0.125, γ/ν=7/4=1.75, and ν=1. Consequently, our measured exponent ratios satisfy the hyperscaling relation 2(β/ν) + γ/ν = 1.99 ± 0.05 ≈ d , (23)

where d=2 is the spatial dimension [57]. Further, the ?xed-point value of the fourth-order cumulant, U ? =0.611±0.003, is also extremely close to that of the Ising model, U ? =0.6106901(5) [56]. These ?ndings provide conclusive evidence that the DPT indeed corresponds to a non-trivial ?xed point. We tested the full data collapse for the scaled order parameter and its variance, as shown in Fig. 11, and it con?rmed the existence of the universal scaling functions given by Eqs. (12) and (13). Also, at the critical frequency, the order-parameter distributions follow ?nite-size scaling predictions, Eq. (20), as shown in Fig. 13. More surprisingly, the critical DPT order-parameter distributions fall on that of the two-dimensional equilibrium Ising model at the critical temperature (except for stronger corrections to scaling), without any additional ?tting of the underlying microscopic length scale. While our ?nite-size scaling data clearly indicate the existence of a divergent length scale, we did not measure the correlation length for the local order parameter directly. Future studies may include extracting the correlation length, ξQ , from the Qi Qj correlations (or from the corresponding structure factor). This approach would also provide another way to measure the exponent ν by plotting ξQ vs θ in the critical regime for large systems, and assuming ξQ ?θ?ν . We also studied the universal aspects of the DPT at other temperatures and ?elds. Shorter runs at T =0.8Tc, T =0.6Tc, and T =0.5Tc for ?eld amplitudes H0 <H× (T ) also con?rmed scaling and the universality of the DPT [Fig. 17]. The condition for the ?eld amplitude implies that the system only exhibits a DPT in the multi-droplet regime. For H0 >H× (T ) strong-?eld behavior governs the decay of the magnetization, and the DPT disappears, as indicated by Fig. 16 and Fig. 18(a),(b). We also found that the high-temperature phase is qualitatively similar to the strong-?eld regime in that there is no sign of a DPT for T >Tc [Fig. 18(c)]. One may ask how general the phenomenon of a DPT is in spatially extended bistable systems, subject to a periodic applied “?eld” which drives the system between its metastable and stable “wells.” It is possible that having up-down symmetry of the period-averaged “magnetization” is su?cient for possessing a Hamiltonian at the coarse-grained level, even if the system is driven far away from equilibrium and is microscopically irreversible [58]. Future research can address this question by studying other systems (not necessarily ferromagnets) that exhibit hysteresis. 15

ACKNOWLEDGMENTS

We thank S. W. Sides, S. J. Mitchell, and G. Brown for stimulating discussions. We would like to thank W. Janke for providing us with data from Ref. [53] for comparison of the critical order-parameter distributions. We acknowledge support by the US Department of Energy through the former Supercomputer Computations Research Institute, by the Center for Materials Research and Technology at Florida State University, and by the US National Science Foundation through Grants No. DMR-9634873, DMR-9871455, and DMR-9981815. This research also used resources of the National Energy Research Scienti?c Computing Center, which is supported by the O?ce of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.
APPENDIX: LOW-FREQUENCY AND STRONG-FIELD MEAN-FIELD APPROXIMATION

For simplicity, in the following we assume that the magnetization decays from m=+1 to m=?1 after a single ?eld reversal (H0 →?H0 ). This is a good approximation below the equilibrium critical temperature (msp ≈1), and for any temperature when H0 →∞. Further, we assume that the volume fraction of meta- or unstable spins follows a simple monotonic decay, ?(t). ? In terms of the volume fraction of positive spins, φ(t), the magnetization can be written as m(t) = 2φ(t) ? 1 . Subject to a square-wave ?eld, H(t) = ?H0 0 ≤ t < t1/2 , +H0 t1/2 ≤ t < 2t1/2 (A2) (A1)

in the ?rst (second) half-period the volume fraction of the positive (negative) spins decays according to ?(t). Thus, ? in each period (measuring time t from the beginning of the period) φ(t) ≈ φ(0)?(t) ? 0 ≤ t ≤ t1/2 . 1 ? 1 ? φ(t1/2 ) ?(t ? t1/2 ) t1/2 ≤ t ≤ 2t1/2 ? (A3)

Using this approximation, one directly obtains a linear mapping φn+1 = 1 ? [1 ? φn ?(t1/2 )]?(t1/2 ) , ? ? (A4)

where φn ≡φ(2nt1/2 ) is the volume fraction of the positive spins at the beginning of the nth period, n = 0, 1, 2, . . .. The stationary value of this quantity is φ? = lim φn =
n→∞

1 . 1 + ?(t1/2 ) ?

(A5)

Consequently, the magnetization reaches a stationary limit cycle m(t) ≈
2 ? 1+?(t1/2 ) ?(t) ? 1 ? 2 1 ? 1+?(t1/2 ) ?(t ? t1/2 ) ? ?

0 ≤ t ≤ t1/2 . t1/2 ≤ t ≤ 2t1/2

(A6)

In this limit cycle the magnetization oscillates about zero, m(0) = ?m(t1/2 ) = 1 ? ?(t1/2 ) ? 1 + ?(t1/2 ) ? (A7)

and the symmetry of the magnetization, m(t ± t1/2 )=?m(t) implies Q= 1 2t1/2 m(t)dt = 0 . (A8)

This corresponds to the symmetric (dynamically disordered) phase. Note that this symmetric phase is always reached when ? decreases monotonically from unity at t=0 to zero as t→∞ ? 16

In the multi-droplet regime, the volume fraction of the metastable phase decays according to Avrami’s law [17,18]. In the low-frequency limit, t1/2 ? τ (Θ?1), each half-period almost always contains a complete metastable decay [Fig. 2(a)]. Avrami’s law for the metastable volume fraction in each half-period can then be directly applied using 3 3 ?(t)=?ms (t)≈e?(ln 2)t / τ [17,18]. This functional form for ?ms (t) breaks down when t1/2 becomes comparable to ? τ (Θ≈1), thus this simple mean-?eld approximation cannot predict any instability related to the DPT. In the H0 →∞ limit the individual spins become decoupled. Then one can obtain ?(t)=e?(ln 2)t/ τ which is exact ? for all frequencies. Further, this exponential decay is a good approximation everywhere in the strong-?eld regime, thus, no DPT can exist there.

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