We study in detail the dynamic scaling of the three-dimensional Ising model under cooling on finite-sized lattices subject to an externally applied field whose magnitude fixes a scaled variable pertinent to it. Three different protocols, protocols A, B, and C, in which the field is applied either only below or only above the critical point besides during the whole process, respectively, are investigated. Anomalous finite-time scaling (FTS) are found in protocols B and C on a large lattice only and in protocol A when extrinsic self-similarity is broken. However, these anomalous scalings in cooling can be rectified unexpectedly by the recently found breaking-of-extrinsic-self-similarity exponent of FTS in zero-field heating, except in the case of protocol B in which a reduced zero-field cooling exponent is additionally required. This provides a distinctive source to the breaking-of-extrinsic-self-similarity exponents and thus confirms their validity. The different scaling behaviors of the three different protocols also shows that the so-called phases fluctuations at and above the critical point are more important than those below it. The previously found qualitative difference between zero-field heating and zero-field cooling is essentially the symmetry of the system states, namely whether they are ordered or disordered. We also confirm that there exists a revised FTS regime-the regime in which both the lattice sizes and driving rates are indispensable-in between the two end regimes of FTS and finite-size scaling even in the presence of the field. Its crossover to the end FTS regime is responsible for the anomalous scalings of the large lattice size in protocols B and C. However, our results show that the revised FTS is never needed for the usual order parameter in which no absolute value is taken. In addition, the end FTS regime in cooling is found to exhibit a special feature different from that in heating. Our results demonstrate that new exponents are needed for scaling in the whole driven process even in the case in which an external field is applied.
We study in detail the dynamic scaling of the three-dimensional Ising model under cooling on finite-sized lattices subject to an externally applied field whose magnitude fixes a scaled variable pertinent to it. Three different protocols, protocols A, B, and C, in which the field is applied either only below or only above the critical point besides during the whole process, respectively, are investigated. Anomalous finite-time scaling (FTS) are found in protocols B and C on a large lattice only and in protocol A when extrinsic self-similarity is broken. However, these anomalous scalings in cooling can be rectified unexpectedly by the recently found breaking-of-extrinsic-self-similarity exponent of FTS in zero-field heating, except in the case of protocol B in which a reduced zero-field cooling exponent is additionally required. This provides a distinctive source to the breaking-of-extrinsic-self-similarity exponents and thus confirms their validity. The different scaling behaviors of the three different protocols also shows that the so-called phases fluctuations at and above the critical point are more important than those below it. The previously found qualitative difference between zero-field heating and zero-field cooling is essentially the symmetry of the system states, namely whether they are ordered or disordered. We also confirm that there exists a revised FTS regime-the regime in which both the lattice sizes and driving rates are indispensable-in between the two end regimes of FTS and finite-size scaling even in the presence of the field. Its crossover to the end FTS regime is responsible for the anomalous scalings of the large lattice size in protocols B and C. However, our results show that the revised FTS is never needed for the usual order parameter in which no absolute value is taken. In addition, the end FTS regime in cooling is found to exhibit a special feature different from that in heating. Our results demonstrate that new exponents are needed for scaling in the whole driven process even in the case in which an external field is applied.
Abstract We study in detail the dynamic scaling of the three-dimensional (3D) Ising model driven through its critical point on finite-sized lattices. We show explicitly that while finite-size scaling (FSS) at fixed driving rates and finite-time scaling (FTS) on fixed lattice sizes are satisfied for one set of all four observables we measure in their respective scaling regimes, they can be violated for the other set of the observables even in the same regimes. The different behaviors of the two sets of the observables indicate that the usual critical fluctuations can be divided into the so-called phases fluctuations and magnitude fluctuations. The self-similarity of criticality can also be divided into intrinsic and extrinsic self-similarities. The numerical results show that the phases fluctuations lead to the different behaviors while breaking the extrinsic self-similarity gives rise to the violations of the scalings. The set of the observables that violate the scalings is further divided into a primary observable and a secondary observable. Their different leading behaviors enable us to identify four breaking-of-extrinsic-self-similarity exponents for rectifying the violations of either FSS or FTS in either heating or cooling. Crossovers from the extrinsic-self-similarity-breaking-controlled regimes to the usual FSS or FTS regimes are also discussed. In addition, qualitatively different behaviors of the magnitude fluctuations in cooling and in heating and their origin are revealed. Moreover, both FTS and FSS are good down to quite low temperatures in cooling with the extrinsic self-similarity. This indicates that phase ordering can only have an effect at even lower temperatures. Besides, from the quality of curve collapses, we find that the 3D dynamic critical exponent z in heating and in cooling appears identical but the two-dimensional ones different. Our results demonstrate that new exponents are generally required for scaling in the whole driven process near criticality once the lattice size is taken into account. This opens a new door in critical phenomena and suggest that much is yet to be explored in driven nonequilibrium critical phenomena.
Symmetry breaking plays a pivotal role in modern physics. Although self-similarity is also a symmetry, and appears ubiquitously in nature, a fundamental question arises as to whether self-similarity breaking makes sense or not. Here, by identifying an important type of critical fluctuation, dubbed ‘phases fluctuations’, and comparing the numerical results for those with self-similarity and those lacking self-similarity with respect to phases fluctuations, we show that self-similarity can indeed be broken, with significant consequences, at least in nonequilibrium situations. We find that the breaking of self-similarity results in new critical exponents, giving rise to a violation of the well-known finite-size scaling, or the less well-known finite-time scaling, and different leading exponents in either the ordered or the disordered phases of the paradigmatic Ising model on two- or three-dimensional finite lattices, when subject to the simplest nonequilibrium driving of linear heating or cooling through its critical point. This is in stark contrast to identical exponents and different amplitudes in usual critical phenomena. Our results demonstrate how surprising driven nonequilibrium critical phenomena can be. The application of this theory to other classical and quantum phase transitions is also anticipated.
The Kibble-Zurek (KZ) mechanism has been applied to a variety of systems ranging from low-temperature Bose-Einstein condensations to grand unification scales in particle physics and cosmology and from classical phase transitions to quantum phase transitions. Here, we show that finite-time scaling (FTS) provides a detailed improved understanding of the mechanism. In particular, the finite time scale, which is introduced by the external driving (or quenching) and results in FTS, is the origin of the division of the adiabatic regimes from the impulse regime in the KZ mechanism. The origin of the KZ scaling for the defect density, generated during the driving through a critical point, is not that the correlation length ceases growing in the nonadiabatic impulse regime, but rather, is that it is taken over by the effective finite length scale corresponding to the finite time scale. We also show that FTS accounts well for and improves the scaling ansatz proposed recently by Liu, Polkovnikov, and Sandvik, [Phys. Rev. B 89, 054307 (2014)]. Further, we show that their universal power-law scaling form applies only to some observables in cooling but not to heating. Even in cooling, it is invalid either when an appropriate external field is present. However, this finite-time-finite-size scaling calls for caution in application of FTS. Detailed scaling behaviors of the FTS and finite-size scaling, along with their crossover, are explicitly demonstrated, with the dynamic critical exponent z being estimated for two- and three-dimensional Ising models under the usual Metropolis dynamics. These values of z are found to give rise to better data collapses than the extant values do in most cases but take on different values in heating and cooling in both two- and three-dimensional spaces.
Critical points with emergent symmetry exhibit intriguing scaling properties induced by two divergent length scales, attracting extensive investigations recently. We study the driven critical dynamics in a three-dimensional q-state clock model, in which the ordered phase breaks the Zq discrete symmetry, while an emergent U(1) symmetry appears at the critical point. By increasing the temperature at a finite velocity v to traverse the critical point from the ordered phase, we uncover rich dynamic scaling properties beyond the celebrated Kibble-Zurek mechanism. Our findings reveal the existence of two finite-time scaling (FTS) regions, characterized by two driving-induced time scales ζd∝v−z/r and ζd′∝v−z/r′, respectively. Here z is the dynamic exponent, r is the usual critical exponent of v, and r′ represents an additional critical exponent of v associated with the dangerously irrelevant scaling variable. While the square of the order parameter M2 obeys the usual FTS form, the angular order parameter ϕq shows remarkably distinct scaling behaviors controlled by both FTS regions. For small v, ϕq is dominated by the time scale ζd, whereas for large v, ϕq is governed by the second time scale ζd′. We verify the universality of these scaling properties in models with both isotropic and anisotropic couplings. Our theoretical insights provide a promising foundation for further experimental investigations in the hexagonal RMnO3 (R=rare earth) materials.
A variety of phenomena is observed experimentally in random-field (RF) systems realized by the application of an external field to diluted antiferromagnets. At low temperatures, infinitely long hysteretic effects are manifested by the history dependence of the final states: long-range order is observed if the field is applied after cooling, while domain states are reached when field cooled. While no indications for critical fluctuations are detected in 2-D systems, scaling behavior, for both the correlation length and the specific heat, is observed in 3-D systems over an intermediate range of temperatures. The related critical properties seem to be well described by the corresponding ones in the 2-D pure Ising model. The renormalization-group approach, which yields for the equilibrium critical exponents their values of the pure model in d−2 dimensions, is reviewed. A generalization of the dimensional-reduction approach, which accounts self-consistently for renormalized responses of the RF system, is presented. The dynamical effects are implicitly incorporated through the variation in the critical response between the local and the global regimes, associated with short- and long-time scales, respectively. In both regimes the lower critical dimension is found to be d=2 in accordance with stability arguments. The short-time critical behavior indicates a dimensional reduction by one for the 3-D thermal exponents, in agreement with the experimental results.
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.