Spin-flip Dynamics Research Articles

Large Deviations for Finite State Markov Jump Processes with Mean-Field Interaction Via the Comparison Principle for an Associated Hamilton–Jacobi Equation

We prove the large deviation principle for the trajectory of a broad class of mean field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie-Weiss spin flip dynamics with singular jump rates. The main step in the proof of the large deviation principle, which is of independent interest, is the proof of the comparison principle for an associated collection of Hamilton-Jacobi equations. Additionally, we show that the large deviation principle provides a general method to identify a Lyapunov function for the associated McKean-Vlasov equation.

Ab initio two-component Ehrenfest dynamics.

We present an ab initio two-component Ehrenfest-based mixed quantum/classical molecular dynamics method to describe the effect of nuclear motion on the electron spin dynamics (and vice versa) in molecular systems. The two-component time-dependent non-collinear density functional theory is used for the propagation of spin-polarized electrons while the nuclei are treated classically. We use a three-time-step algorithm for the numerical integration of the coupled equations of motion, namely, the velocity Verlet for nuclear motion, the nuclear-position-dependent midpoint Fock update, and the modified midpoint and unitary transformation method for electronic propagation. As a test case, the method is applied to the dissociation of H2 and O2. In contrast to conventional Ehrenfest dynamics, this two-component approach provides a first principles description of the dynamics of non-collinear (e.g., spin-frustrated) magnetic materials, as well as the proper description of spin-state crossover, spin-rotation, and spin-flip dynamics by relaxing the constraint on spin configuration. This method also holds potential for applications to spin transport in molecular or even nanoscale magnetic devices.

A mathematical perspective on metastable wetting

In this paper we investigate the dynamical behavior of an interface or polymer, in interaction with a distant attractive substrate. The interface is modeled by the graph of a nearest neighbor path with non-negative integer coordinates, and the equilibrium measure associates to each path \eta a probability proportional to \lambda^{H(\eta)} where \lambda is non-negative and H(\eta) is the number of contacts between \eta and the substrate at zero. The dynamics is the natural spin flip dynamics associated to this equilibrium measure. We let the distance to the substrate at both polymer ends be equal to aN, where 0 2/(1-2a) the mixing time is exponential in N and the system relaxes in an exponential manner which is typical of metastability.

Melting of an Ising quadrant

We consider an Ising ferromagnet endowed with zero-temperature spin-flip dynamics and examine the evolution of the Ising quadrant, namely the spin configuration when the minority phase initially occupies one quadrant while the majority phase occupies the three remaining quadrants. The two phases are then always separated by a single interface, which generically recedes into the minority phase in a self-similar diffusive manner. The area of the invaded region grows (on average) linearly with time and exhibits non-trivial fluctuations. We map the interface separating the two phases onto the one-dimensional symmetric simple exclusion process and utilize this isomorphism to compute basic cumulants of the area. First, we determine the variance via an exact microscopic analysis (the Bethe ansatz). Then we turn to a continuum treatment by recasting the underlying exclusion process into the framework of the macroscopic fluctuation theory. This provides a systematic way of analyzing the statistics of the invaded area and allows us to determine the asymptotic behaviors of the first four cumulants of the area.

Variational Description of Gibbs-Non-Gibbs Dynamical Transitions for Spin-Flip Systems with a Kac-Type Interaction

We continue our study of Gibbs-non-Gibbs dynamical transitions. In the present paper we consider a system of Ising spins on a large discrete torus with a Kac-type interaction subject to an independent spin-flip dynamics (infinite-temperature Glauber dynamics). We show that, in accordance with the program outlined in \cite{vEFedHoRe10}, in the thermodynamic limit Gibbs-non-Gibbs dynamical transitions are \emph{equivalent} to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the empirical density \emph{conditional} on their endpoint. More precisely, the time-evolved measure is non-Gibbs if and only if this set is not a singleton for \emph{some} value of the endpoint. A partial description of the possible scenarios of bifurcation is given, leading to a characterization of passages from Gibbs to non-Gibbs and vice versa, with sharp transition times. Our analysis provides a conceptual step-up from our earlier work on Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss model, where the mean-field interaction allowed us to focus on trajectories of the empirical magnetization rather than the empirical density.

Spin-dependent lifetimes and nondegenerate spin-orbit driven hybridization points in Heusler compounds

We present an ab initio calculation of the $\stackrel{P\vec}{k}$- and spin-resolved electronic lifetimes in the half-metallic Heusler compounds ${\mathrm{Co}}_{2}\mathrm{Mn}\mathrm{Si}$ and ${\mathrm{Co}}_{2}\mathrm{Fe}\mathrm{Si}$. We determine the spin-flip and spin-conserving contributions to the lifetimes and study in detail the behavior of the lifetimes around states that are strongly spin mixed by spin-orbit coupling. We find that for nondegenerate bands the spin mixing alone does not determine the energy dependence of the (spin-flip) lifetimes. Qualitatively, the lifetimes reflect the lineup of electron and hole bands. We predict that different excitation conditions lead to drastically different spin-flip dynamics of excited electrons and may even give rise to an enhancement of the nonequilibrium spin polarization.

Charge Ordering in a Pure Spin Model: Dipolar Spin Ice

We study the dipolar spin-ice model at fixed density of single excitations, ρ, using a Monte Carlo algorithm where processes of creation and annihilation of such excitations are banned. In the limit of ρ going to zero, this model coincides with the usual dipolar spin-ice model at low temperatures, with the additional advantage that a negligible number of monopoles allows for equilibration even at the lowest temperatures. Thus, the transition to the ordered fundamental state found by Melko, den Hertog, and Gingras in 2001 is reached using simple local spin flip dynamics. As the density is increased, the monopolar nature of the excitations becomes apparent: the system shows a rich ρ vs T phase diagram with "charge" ordering transitions analogous to that observed for Coulomb charges in lattices. A further layer of complexity is revealed by the existence of order both within the charges and their associated vacuum, which can only be described in terms of spins--the true microscopic degrees of freedom of the system.

Zero-temperature random-field Ising model on a bilayered Bethe lattice

The zero-temperature random-field Ising model is solved analytically for magnetization versus external field for a bilayered Bethe lattice. The mechanisms of infinite avalanches which are observed for small values of disorder are established. The effects of variable interlayer interaction strengths on infinite avalanches are investigated. The spin-field correlation length is calculated and its critical behavior is discussed. Direct Monte Carlo simulations of spin-flip dynamics are shown to support the analytical findings. We find, paradoxically, that a reduction of the interlayer bond strength can cause a phase transition from a regime with continuous magnetization reversal to a regime where magnetization exhibits a discontinuity associated with an infinite avalanche. This effect is understood in terms of the proposed mechanisms for the infinite avalanche.

Limiting shapes in two-dimensional Ising ferromagnets

We consider an Ising model on a square lattice with ferromagnetic spin-spin interactions spanning beyond nearest neighbors. Starting from initial states with a single unbounded interface separating ordered phases, we investigate the evolution of the interface subject to zero-temperature spin-flip dynamics. We consider an interface which is initially (i) the boundary of the quadrant or (ii) the boundary of a semi-infinite bar. In the former case the interface recedes from its original location in a self-similar diffusive manner. After a rescaling by √[t], the shape of the interface becomes more and more deterministic; we determine this limiting shape analytically and verify our predictions numerically. The semi-infinite bar acquires a stationary shape resembling a finger, and this finger translates along its axis. We compute the limiting shape and the velocity of the Ising finger.

Variational Description of Gibbs-non-Gibbs Dynamical Transitions for the Curie-Weiss Model

We perform a detailed study of Gibbs-non-Gibbs transitions for the Curie-Weiss model subject to independent spin-flip dynamics (“infinite-temperature” dynamics). We show that, in this setup, the program outlined in van Enter et al. (Moscow Math J 10:687–711, 2010) can be fully completed, namely, Gibbs-non-Gibbs transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the magnetization conditioned on their endpoint. As a consequence, we show that the time-evolved model is non-Gibbs if and only if this set is not a singleton for some value of the final magnetization. A detailed description of the possible scenarios of bifurcation is given, leading to a full characterization of passages from Gibbs to non-Gibbs and vice versa with sharp transition times (under the dynamics, Gibbsianness can be lost and can be recovered). Our analysis expands the work of Ermolaev and Külske (J Stat Phys 141:727–756, 2010), who considered zero magnetic field and finite-temperature spin-flip dynamics. We consider both zero and non-zero magnetic field, but restricted to infinite-temperature spin-flip dynamics. Our results reveal an interesting dependence on the interaction parameters, including the presence of forbidden regions for the optimal trajectories and the possible occurrence of overshoots and undershoots in time of the initial magnetization of the optimal trajectories. The numerical plots provided are obtained with the help of MATHEMATICA.

Extraordinary variability and sharp transitions in a maximally frustrated dynamic network

Using Monte Carlo and analytic techniques, we study a minimal dynamic network involving two populations of nodes, characterized by different preferred degrees. Reminiscent of introverts and extroverts in a population, one set of nodes, labeled introverts (I), prefers fewer contacts (a lower degree) than the other, labeled extroverts (E). As a starting point, we consider an extreme case, in which an I simply cuts one of its links at random when chosen for updating, while an E adds a link to a random unconnected individual (node). The model has only two control parameters, namely, the number of nodes in each group, NI and NE. In the steady state, only the number of crosslinks between the two groups fluctuates, with remarkable properties: Its average (X) remains very close to 0 for all NI > NE or near its maximum () if NI < NE. At the transition (NI = NE), the fraction wanders across a substantial part of [0,1], much like a pure random walk. Mapping this system to an Ising model with spin-flip dynamics and unusual long-range interactions, we note that such fluctuations are far greater than those displayed in either first- or second-order transitions of the latter. Thus, we refer to the case here as an “extraordinary transition”. Thanks to the restoration of detailed balance and the existence of a “Hamiltonian”, several qualitative aspects of these remarkable phenomena can be understood analytically.

Collective non-equilibrium dynamics at surfaces and the spatio-temporal edge

Symmetries represent a fundamental constraint for physical systems and relevant new phenomena often emerge as a consequence of their breaking. An important example is provided by space- and time-translational invariance in statistical systems, which hold at a coarse-grained scale in equilibrium and are broken by spatial and temporal boundaries, the former being implemented by surfaces —unavoidable in real samples— the latter by some initial condition for the dynamics which causes a non-equilibrium evolution. While the separate effects of these two boundaries are well understood, we demonstrate here that additional, unexpected features arise upon approaching the effective edge formed by their intersection. For this purpose, we focus on the classical semi-infinite Ising model with spin-flip dynamics evolving out of equilibrium at its critical point. Considering both subcritical and critical values of the coupling among surface spins, we present numerical evidence of a scaling regime with universal features which emerges upon approaching the spatio-temporal edge and we rationalise these findings within a field-theoretical approach.

Transition path sampling for discrete master equations with absorbing states

Transition path sampling (TPS) algorithms have been implemented with deterministic dynamics, with thermostatted dynamics, with Brownian dynamics, and with simple spin flip dynamics. Missing from the TPS repertoire is an implementation with kinetic Monte Carlo (kMC), i.e., with the underlying dynamics coming from a discrete master equation. We present a new hybrid kMC-TPS algorithm and prove that it satisfies detailed balance in the transition path ensemble. The new algorithm is illustrated for a simplified Markov State Model of trp-cage folding. The transition path ensemble from kMC-TPS is consistent with that obtained from brute force kMC simulations. The committor probabilities and local fluxes for the simple model are consistent with those obtained from exact methods for simple master equations. The new kMC-TPS method should be useful for analysis of rare transitions in complex master equations where the individual states cannot be enumerated and therefore where exact solutions cannot be obtained.

Limiting shapes of Ising droplets, Ising fingers, and Ising solitons

We examine the evolution of an Ising ferromagnet endowed with zero-temperature single spin-flip dynamics. A large droplet of one phase in the sea of the opposite phase eventually disappears. An interesting behavior occurs in the intermediate regime when the droplet is still very large compared to the lattice spacing, but already very small compared to the initial size. In this regime the shape of the droplet is essentially deterministic (fluctuations are negligible in comparison with characteristic size). In two dimensions the shape is also universal, that is, independent of the initial shape. We analytically determine the limiting shape of the Ising droplet on the square lattice. When the initial state is a semi-infinite stripe of one phase in the sea of the opposite phase, it evolves into a finger which translates along its axis. We determine the limiting shape and the velocity of the Ising finger on the square lattice. An analog of the Ising finger on the cubic lattice is the translating Ising soliton. We show that far away from the tip, the cross-section of the Ising soliton coincides with the limiting shape of the two-dimensional Ising droplet and we determine a relation between the cross-section area, the distance from the tip, and the velocity of the soliton.

Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach

We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in Z^3 with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O(L^2polylog(L)) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean curvature motion such that each point of the surface feels a drift which tends to minimize the local surface tension. Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to D.B. Wilson in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (logarithmic for monotone surfaces and L^{1/2} for the SOS model). Finally, combining techniques from kinetically constrained spin systems together with the above mixing time result, we prove an almost diffusive lower bound of order 1/L^2 up to logarithmic corrections for the spectral gap of the SOS model with horizontal size L and unbounded heights.

An Extraordinary Transition in a Minimal Adaptive Network of Introverts and Extroverts

Abstract We study a minimal adaptive network involving two populations, modeling the behavior of extreme introverts (I) and extroverts (E). When chosen to update, an I simply cuts one of its links at random while an E adds a link to any other yet-to-be-connected individual (node). In the steady state, the active links in the system are obviously only the cross-links between the I's and the E's. With no free parameters other than the numbers of each population (NI, NE), this minimal model displays remarkable properties: Through simulations using O(10)-O(1000) nodes, we find that the typical number of cross-links (X) fluctuates surprisingly close to the minimum or the maximum allowed values, depending on whether NI > NE or otherwise. At the transition point (i.e., NI = NE), the fraction X/(NINE) wanders across a substantial part of the unit interval, much like a pure random walk confined between two walls. Since this system can be mapped to a NINE Ising model with spin flip dynamics, we note that such fluctuations are far greater than those in the standard Ising model (at either first or second order transitions). Thus, we refer to the case here as an “extraordinary transition.” Thanks to the restoration of detailed balance and the existence of a “Hamiltonian,” several qualitative aspects of these remarkable phenomena can be understood analytically

Magnetic friction: From Stokes to Coulomb behavior

We demonstrate that in a ferromagnetic substrate, which is continuously driven out of equilibrium by a field moving with constant velocity $v$, at least two types of friction may occur when $v$ goes to zero: The substrate may feel a friction force proportional to $v$ (Stokes friction), if the field changes on a time scale which is longer than the intrinsic relaxation time. On the other hand, the friction force may become independent of $v$ in the opposite case (Coulomb friction). These observations are analogous to e.g. solid friction. The effect is demonstrated in both, the Ising (one spin dimension) and the Heisenberg model (three spin dimensions), irrespective which kind of dynamics (Metropolis spin-flip dynamics or Landau-Lifshitz-Gilbert precessional dynamics) is used. For both models the limiting case of Coulomb friction can be treated analytically. Furthermore we present an empiric expression reflecting the correct Stokes behavior and therefore yielding the correct cross-over velocity and dissipation.

Molecular orbital concept on spin-flip transport in molecular junctions

The spin-dependent electron transport correlated with spin-flip dynamics in a molecular junction was investigated in the wave-packet and Green’s function approaches. The molecular junction adopted in this work is described by a simple one-dimensional tight-binding chain including a localized spin. The spin exchange coupling J between the localized and conduction electron spins was taken into account through the s-d Hamiltonian. The wave-packet simulations showed that the transmission probabilities in both the spin-flip and no-flip processes show large peaks at the eigenvalues of the spin singlet (−3J/4) and triplet (J/4) states, and that, different transmission properties appear at the mid-gap of the two eigenvalues: the spin-flip process shows a moderate decrease, whereas the no-flip process an abrupt drop. Dividing the s-d Hamiltonian into two submatrices and referring to the molecular orbital concept for the coherent electron transport, we found that the moderate decrease in the spin-flip process at the mid-gap is the result of a coherent-and-cooperative contribution from the singlet and triplet states of the conduction and localized electron spins, and that, the abrupt drop in the no-flip process at the mid-gap is mainly caused by the coherent cancellation from the singlet and triplet states. The molecular orbital concept available for the electron transport including spin-flip scattering processes is described in Green’s function method, in analogy to the one derived for the spinless electron transport.

Low-Temperature Dynamics of the Curie-Weiss Model: Periodic Orbits, Multiple Histories, and Loss of Gibbsianness

We consider the Curie-Weiss model at initial temperature 0<β −1≤∞ in vanishing external field evolving under a Glauber spin-flip dynamics with temperature 0<β′−1≤∞. We study the limiting conditional probabilities and their continuity properties and discuss their set of points of discontinuity (bad points). We provide a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending earlier work for the case of independent spin-flip dynamics.For initial temperature β −1>1 we prove that the time-evolved measure stays Gibbs forever, for any (possibly low) temperature of the dynamics.In the regime of heating to low-temperatures from even lower temperatures, 0<β −1<min {β′−1,1} we prove that the time-evolved measure is Gibbs initially and becomes non-Gibbs after a sharp transition time. We find this regime is further divided into a region where only symmetric bad configurations exist, and a region where this symmetry is broken.In the regime of further cooling from low-temperatures, β′−1<β −1<1 there is always symmetry-breaking in the set of bad configurations. These bad configurations are created by a new mechanism which is related to the occurrence of periodic orbits for the vector field which describes the dynamics of Euler-Lagrange equations for the path large deviation functional for the order parameter.To our knowledge this is the first example of the rigorous study of non-Gibbsian phenomena related to cooling, albeit in a mean-field setup.

A kinetic Ising model study of dynamical correlations in confined fluids: Emergence of both fast and slow time scales

Experiments and computer simulation studies have revealed existence of rich dynamics in the orientational relaxation of molecules in confined systems such as water in reverse micelles, cyclodextrin cavities, and nanotubes. Here we introduce a novel finite length one dimensional Ising model to investigate the propagation and the annihilation of dynamical correlations in finite systems and to understand the intriguing shortening of the orientational relaxation time that has been reported for small sized reverse micelles. In our finite sized model, the two spins at the two end cells are oriented in the opposite directions to mimic the effects of surface that in real system fixes water orientation in the opposite directions. This produces opposite polarizations to propagate inside from the surface and to produce bulklike condition at the center. This model can be solved analytically for short chains. For long chains, we solve the model numerically with Glauber spin flip dynamics (and also with Metropolis single-spin flip Monte Carlo algorithm). We show that model nicely reproduces many of the features observed in experiments. Due to the destructive interference among correlations that propagate from the surface to the core, one of the rotational relaxation time components decays faster than the bulk. In general, the relaxation of spins is nonexponential due to the interplay between various interactions. In the limit of strong coupling between the spins or in the limit of low temperature, the nature of relaxation of the spins undergoes a qualitative change with the emergence of a homogeneous dynamics where decay is predominantly exponential, again in agreement with experiments.