We discuss the interplay between the degree of dynamical stochasticity, memory persistence and violation of the self-averaging property in the aging kinetics of quenched ferromagnets. We show that, in general, the longest possible memory effects, which correspond to the slowest possible temporal decay of the correlation function, are accompanied by the largest possible violation of self-averaging and a quasi-deterministic descent into the ergodic components. This phenomenon is observed in different systems, such as the Ising model with long-range interactions, including mean-field, and the short-range random field Ising model.
We consider the isotropic-to-nematic transition in liquid crystals confined to aerogel hosts, and assume that the aerogel acts as a random field. We generally find that self-averaging is violated. For a bulk transition that is weakly first-order, the violation of self-averaging is so severe, even the correlation length becomes non-self-averaging: no phase transition remains in this case. For a bulk transition that is more strongly first-order, the violation of self-averaging is milder, and a phase transition is observed.
Tracking tracer particles in heterogeneous environments plays an important role in unraveling the material properties. These heterogeneous structures are often static and depend on the sample realizations. Sample-to-sample fluctuations of such disorder realizations sometimes become considerably large. When we investigate the sample-to-sample fluctuations, fundamental averaging procedures are a thermal average for a single disorder realization and the disorder average for different disorder realizations. Here, we report on non-self-averaging phenomena in quenched trap models with finite system …
We investigate the non-self-averaging properties of the dynamics of Ising, 4-state Potts and 10-state Potts models in single-cluster Monte Carlo simulations on quenched ensembles of planar, trivalent Phi3 random graphs, which we use as an example of relevant quenched connectivity disorder. We employ a novel application of scaling techniques to the cumulative probability distribution of the autocorrelation times for both the energy and magnetisation in order to discern non-self-averaging. Although the specific results discussed here are for quenched random graphs, the method has quite general a…
We show that spatial quenched disorder affects polar active matter in ways more complex and far-reaching than believed heretofore. Using simulations of the 2D Vicsek model subjected to random couplings or a disordered scattering field, we find in particular that ergodicity is lost in the ordered phase, the nature of which we show to depend qualitatively on the type of quenched disorder: for random couplings, it remains long-range ordered, but qualitatively different from the pure (disorderless) case. For random scatterers, polar order varies with system size but we find strong non-self-averagi…
Local diffusion coefficients in disordered materials such as living cells are highly heterogeneous. Quenched disorder is utilized substantially to study such complex systems, whereas its analytical treatment is difficult to handle. We consider finite systems with quenched disorder in order to investigate the effects of sample disorder fluctuations and confinement on single-particle diffusivity. While the system is ergodic in a single disorder realization, the time-averaged mean squared displacement depends on the disorder, i.e., the system is ergodic but non-self-averaging. We find that the in…
The off-equilibrium probability distribution of the heat exchanged by a ferromagnet in a time interval after a quench below the critical point is calculated analytically in the large-N limit. The distribution is characterized by a singular threshold Qc < 0, below which a macroscopic fraction of heat is released by the k = 0 Fourier component of the order parameter. The mathematical structure producing this phenomenon is the same responsible of the order parameter condensation in the equilibrium low temperature phase. The heat exchanged by the individual Fourier modes follows a non trivial patt…
Critical behavior in short-time dynamics is investigated by a Monte Carlo study for the random-bond Potts ferromagnet with a trinary distribution of quenched disorders on two-dimensional triangular lattices. The dynamic scaling is verified and applied to estimate critical exponents $θ$, $z$ and $β/ν$ for several realizations of the trinary distribution. Our critical scaling analysis strongly indicates that the bond randomness influences the critical universality.
We report on large-scale Monte Carlo simulations of the three-dimensional 4-state Potts ferromagnet subject to quenched, random bond dilution. For small dilutions the rather strong first-order phase transition of the pure system is found to persist, whereas for larger dilutions the theoretically expected softening to a continuous transition is confirmed. The properties of the underlying disorder distributions of thermal observables are discussed and illustrated with a few selected examples.
Frequency evolutions of hysteresis loop area and hysteresis tools such as remanence and coercivity of a kinetic Ising model in the presence of quenched bond dilution are investigated in detail. The kinetic equation describing the time dependence of the magnetization is derived by means of effective-field theory with single-site correlations. It is found that the frequency dispersions of hysteresis loop area, remanence and coercivity strongly depend on the quenched bond randomness, as well as applied field amplitude and oscillation frequency. In addition, the shape of the hysteresis curves for …
Effects of the averaging over disorder realizations (samples) on the phase behavior are analyzed in terms of the mean field approximation for the random field Ising model with infinite range interactions. It is found that the averaging is equivalent to a drastic modification in the statistics of the quenched variables. In its turn, this lowers the critical temperature of a second-order phase transition or, depending on the sampling, even suppresses the ordered phase. Possible first order transitions are shown to be softened by the sample averaging. Common issues and differences in the interpre…
The two-dimensional random-bond Q-state Potts model is studied for Q near 2 via the perturbative renormalisation group to one loop. It is shown that weak disorder induces cross-correlations between the quenched-averages of moments of the two-point spin/spin and energy/energy correlation functions, which should be observable numerically in specific linear combinations of various quenched correlation functions. The random-bond Ising model in (2+epsilon) dimensions is similarly treated. As a byproduct, a simple method for deriving the scaling dimensions of all moments of the local energy operator…
We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size $L$ like $\sim L \ln\ln(L)$. In contrast, the specific heat is shown to be self-ave…
The study of quenched random systems is facilitated by the idea that the ensemble averages describe the thermal averages for any specific realization of the couplings, provided the system is large enough. Careful examination suggests that this idea might have a follow, when the correlation length becomes of the order of the size of the system. We find certain bound quantities are not self-averaging when the correlation length becomes of the order of the size of the system. This suggests that the strength of self-averaging, expressed in terms of properly chosen signal to noise ratios, may serve…
We investigate by means of Monte Carlo simulations the dynamic phase transition of the two-dimensional kinetic Blume-Capel model under a periodically oscillating magnetic field in the presence of a quenched random crystal-field coupling. We analyze the universality principles of this dynamic transition for various values of the crystal-field coupling at the originally second-order regime of the corresponding equilibrium phase diagram of the model. A detailed finite-size scaling analysis indicates that the observed nonequilibrium phase transition belongs to the universality class of the equilib…
We study the partition functions of multiple replicas (copies) of D-brane configurations in the type IIB (IKKT) matrix model. We consider the quenched regime, where small fluctuations of the matrices are superimposed onto the slow (quenched) dynamics of the background, so the partition function is an ensemble average over the background. Interacting D-branes always factorize in a simple way. On the other hand, the non-interacting BPS configurations may or may not factorize depending on the number of replicas, and their factorization mechanism is more involved as the corresponding saddle-point …
When a two-dimensional Ising ferromagnet is quenched from above the critical temperature to zero temperature, the system eventually converges to either a ground state (all spins aligned) or an infinitely long-lived metastable stripe state. By applying results from percolation theory, we analytically determine the probability to reach the stripe state as a function of the aspect ratio and the form of the boundary conditions. These predictions agree with simulation results. Our approach generally applies to coarsening dynamics of non-conserved scalar fields in two dimensions.
We discuss the non-self-averaging phenomena in the critical point of weakly disordered Ising ferromagnet. In terms of the renormalized replica Ginzburg-Landau Hamiltonian in dimensions D <4, we derive an explicit expression for the probability distribution function (PDF) of the critical free energy fluctuations. In particular, using known fixed-point values for the renormalized coupling parameters, we obtain the universal curve for such PDF in the dimension D=3. It is demonstrated that this function is strongly asymmetric: its left tail is much slower than the right one.
Recent renormalization group results predict non self averaging behaviour at criticality for relevant disorder. However, we find strong self averaging(SA) behaviour in the critical region of a quenched Ising model on an ensemble of small-world networks, despite the relevance of the random bonds at the pure critical point.
We study the early time dynamics of the 2d ferromagnetic Ising model instantaneously quenched from the disordered to the ordered, low temperature, phase. We evolve the system with kinetic Monte Carlo rules that do not conserve the order parameter. We confirm the rapid approach to random critical percolation in a time-scale that diverges with the system size but is much shorter than the equilibration time. We study the scaling properties of the evolution towards critical percolation and we identify an associated growing length, different from the curvature driven one. By working with the model …
Current theories of heteropolymers are inherently macrpscopic, but are applied to folding proteins which are only mesoscopic. In these theories, one computes the averaged free energy over sequences, always assuming that it is self-averaging -- a property well-established only if a system with quenched disorder is macroscopic. By enumerating the states and energies of compact 18, 27, and 36mers on a simplified lattice model with an ensemble of random sequences, we test the validity of the self-averaging approximation. We find that fluctuations in the free energy between sequences are weak, and …
It is shown that the variance of a perturbation Hamiltonian density vanishes in the infinite-volume limit of the perturbed spin systems with quenched disorder. This is proven in a simpler way and under less assumptions than before. A corollary of this theorem indicates the impossibility of non-spontaneous replica symmetry-breaking in disordered spin systems. The commutativity between the infinite-volume limit and the switched-off limit of a replica symmetry-breaking perturbation implies that the variance of the spin overlap vanishes in the replica symmetric Gibbs state.
We investigate and contrast, via the Wang-Landau (WL) algorithm, the effects of quenched bond randomness on the self-averaging properties of two Ising spin models in 2d. The random bond version of the superantiferromagnetic (SAF) square model with nearest- and next-nearest-neighbor competing interactions and the corresponding version of the simple ferromagnetic Ising model are studied. We find that, the random bond SAF model shows a strong violation of self-averaging, much stronger than that observed in the case of the random bond Ising model. Our analysis of the asymptotic scaling behavior of…
We report some results on the quenched disordered Spherical multi-$p$-Spin Model in presence of ferromagnetic couplings. In particular, we present the phase diagrams of some representative cases that schematically describe, in the mean-field approximation, the behavior of most known transitions in glassy materials, including dynamic arrest in super-cooled liquids, amorphous-amorphous transitions and spin-glass transitions. A simplified notation is introduced in order to compute systems properties in terms of an effective, self-induced, field encoding the whole ferromagnetic information.
Correlation inequalities have played an essential role in the analysis of ferromagnetic models but have not been established in spin glass models. In this study, we obtain some correlation inequalities for the Ising models with quenched randomness, where the distribution of the interactions is symmetric. The acquired inequalities can be regarded as an extension of the previous results, which were limited to the local energy for a spin set, to the local energy for a pair of spin sets. Besides, we also obtain some correlation inequalities for asymmetric distribution.
In a previous paper we found that in the random field Ising model at zero temperature in three dimensions the correlation length is not self-averaging near the critical point and that the violation of self-averaging is maximal. This is due to the formation of bound states in the underlying field theory. We present a similar study for the case of disordered Potts and Ising ferromagnets in two dimensions near the critical temperature. In the random Potts model the correlation length is not self-averaging near the critical temperature but the violation of self-averaging is weaker than in the rand…
We present a theory to describe domain formation observed very recently in a quenched Rb-87 gas, a typical ferromagnetic spinor Bose system. An overlap factor is introduced to characterize the symmetry breaking of M_F=\pm 1 components for the F=1 ferromagnetic condensate. We demonstrate that the domain formation is a co-effect of the quantum coherence and the thermal relaxation. A thermally enhanced quantum-oscillation is observed during the dynamical process of the domain formation. And the spatial separation of domains leads to significant decay of the M_F=0 component fraction in an initial …
We perform a detailed study of the critical behavior of the mean field diluted Ising ferromagnet by analytical and numerical tools. We obtain self-averaging for the magnetization and write down an expansion for the free energy close to the critical line. The scaling of the magnetization is also rigorously obtained and compared with extensive Monte Carlo simulations. We explain the transition from an ergodic region to a non trivial phase by commutativity breaking of the infinite volume limit and a suitable vanishing field. We find full agreement among theory, simulations and previous results.
The Ising and BEG models critical behavior is analyzed in 2D and 3D by means of a renormalization group scheme on small clusters made of a few lattice cells. Different kinds of cells are proposed for both ordered and disordered model cases. In particular, cells preserving a possible antiferromagnetic ordering under decimation allow for the determination of the Néel critical point and its scaling indices. These also provide more reliable estimates of the Curie fixed point than those obtained using cells preserving only the ferromagnetic ordering. In all studied dimensions, the present procedure…
We study the free energy distribution function of weakly disordered Ising ferromagnet in terms of the D-dimensional random temperature Ginzburg-Landau Hamiltonian. It is shown that besides the usual Gaussian "body" this distribution function exhibits non-Gaussian tails both in the paramagnetic and in the ferromagnetic phases. Explicit asymptotic expressions for these tails are derived. It is demonstrated that the tails are strongly asymmetric: the left tail (for large negative values of the free energy) is much more slow than the right one (for large positive values of the free energy). It is …
The dynamic evolution at zero temperature of a uniform Ising ferromagnet on a square lattice is followed by Monte Carlo computer simulations. The system always eventually reaches a final, absorbing state, which sometimes coincides with a ground state (all spins parallel), and sometimes does not (parallel stripes of spins up and down). We initiate here the numerical study of ``Chaotic Time Dependence'' (CTD) by seeing how much information about the final state is predictable from the randomly generated quenched initial state. CTD was originally proposed to explain how nonequilibrium spin glasse…
Using high-precision Monte-Carlo simulations based on a parallel version of the Wang-Landau algorithm and finite-size scaling techniques we study the effect of quenched disorder in the crystal-field coupling of the Blume-Capel model on the square lattice. We mainly focus on the part of the phase diagram where the pure model undergoes a continuous transition, known to fall into the universality class of the pure Ising ferromagnet. A dedicated scaling analysis reveals concrete evidence in favor of the strong universality hypothesis with the presence of additional logarithmic corrections in the s…
A comprehensive theory for electronic transport in itinerant ferromagnets is developed. We first show that the Q-field theory used previously to describe a disordered Fermi liquid also has a saddle-point solution that describes a ferromagnet in a disordered Stoner approximation. We calculate transport coefficients and thermodynamic susceptibilities by expanding about the saddle point to Gaussian order. At this level, the theory generalizes previous RPA-type theories by including quenched disorder. We then study soft-mode effects in the ferromagnetic state in a one-loop approximation. In three-…
An effective field theory is derived that describes the quantum critical behavior of itinerant ferromagnets in the presence of quenched disorder. In contrast to previous approaches, all soft modes are kept explicitly. The resulting effective theory is local and allows for an explicit perturbative treatment. It is shown that previous suggestions for the critical fixed point and the critical behavior are recovered under certain assumptions. The validity of these assumptions is discussed in the light of the existence of two different time scales. It is shown that, in contrast to previous suggesti…
The effects of quenched disorder on the critical properties of itinerant quantum magnets are considered. Particular attention is paid to locally ordered rare regions that are formed in the presence of quenched disorder even when the bulk system is still in the nonmagnetic phase. It is shown that these local moments or instantons destroy the previously found critical fixed point in the case of antiferromagnets. In the case of itinerant ferromagnets, the critical behavior is unaffected by the rare regions due to an effective long-range interaction between the order parameter fluctuations.
We study the dynamics of the q-state random bond Potts ferromagnet on the square lattice at its critical point by Monte Carlo simulations with single spin-flip dynamics. We concentrate on q=3 and q=24 and find, in both cases, conventional, rather than activated, dynamics. We also look at the distribution of relaxation times among different samples, finding different results for the two q values. For q=3 the relative variance of the relaxation time tau at the critical point is finite. However, for q=24 this appears to diverge in the thermodynamic limit and it is ln(tau) which has a finite relat…
We investigate the effects of quenched bond randomness on the critical properties of the two-dimensional ferromagnetic Ising model embedded in a triangular lattice. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method. In the first part of our study we present the finite-size scaling behavior of the pure model, for which we calculate the critical amplitude of the specific heat's logarithmic expansion. For the disordered system, the numerical data and the relevant detailed finite-size scaling analysis along the lines of the two well-k…
For quicker formation of ice, before inserting inside a refrigerator, heating up of a body of water can be beneficial. We report first observation of a counterpart of this intriguing fact, referred to as the Mpemba effect (ME), during ordering in ferromagnets. By performing Monte Carlo simulations of a generic model, we have obtained results on relaxation of systems that are quenched to sub-critical state points from various temperatures above the critical point. For a fixed final temperature, a system with higher starting temperature equilibrates faster than the one prepared at a lower temper…
