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Daniel L. Stein

Publications and source records attributed to Daniel L. Stein.

At least 19 recordsLinked to original sources
arXiv2025

Zero Temperature Dynamics of Ising Systems on Hypercubes

We study the zero-temperature Glauber dynamics of homogeneous Ising ferromagnets on hypercubes, as their dimension d varies. We investigate the asymptotic (d goes to infinity and time t goes to infinity) behavior of various quantities on hypercubes, such as the final magnetization, the probability for the system to enter a ground state, etc. Our numerical studies are carried out using a uniformly random initial state but with the constraint that the initial magnetization is zero. The final states can be divided into three categories: ground states, frozen states, and blinker states. We use the notion of a k-core to describe the geometry of the frozen states and give an exponential lower bound for the number of frozen states in terms of d. Blinker states -- which exist only in even d -- are final states containing at least one blinker (a permanently flipping spin). Blinker states can have rich local structures; we give explicit constructions for configurations that contain blinkers and prove that the lowest possible dimension for blinker configurations is d = 8. We also study the 'Nature vs. Nurture' problem on hypercubes, asking how much the final state depends on the information contained in the initial configuration, and how much depends on the realization of the dynamical evolution. Finally, we provide several conjectures and suggest some open problems based on the numerical results.

cond-mat.stat-mech↗
arXiv2020

Thermal Activation Barriers for Creation and Annihilation of Magnetic Droplet Solitons in the Presence of Spin Transfer Torque

We study noise-induced creation and annihilation of magnetic droplet solitons in experimental parameter regions in which they are linearly stable against drift. Exploiting the rotational symmetry of the problem, we transform to the reference frame rotating with the droplet soliton and introduce an effective potential energy that accounts for the work done against spin-transfer torque to rotate the magnetization between two different orientations. We use this function to compute the activation barrier in both directions between the uniform magnetization state and the droplet soliton state for a variety of nanocontact radii and currents. We investigate droplet soliton structures with both zero and nonzero spin-torque asymmetry parameter. Our approach can be applied to estimate activation barriers for dynamical systems where non-gradient terms can be absorbed by changes of reference frames, and suggests a technique applicable to extended systems that may not be uniformly magnetized.

cond-mat.mes-hall↗
arXiv2019

Local minima in disordered mean-field ferromagnets

We consider the complexity of random ferromagnetic landscapes on the hypercube $\{\pm 1\}^N$ given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to the Ising model on a dense random graph $\mathcal G(N,p)$. Previous results had shown that, with high probability as $N\to\infty$, the gradient search (energy-lowering) algorithm, initialized uniformly at random, converges to one of the homogeneous global minima (all-plus or all-minus). Here, we devise two modified algorithms tailored to explore the landscape at near-zero magnetizations (where the effect of the ferromagnetic drift is minimized). With these, we numerically verify the landscape complexity of random ferromagnets, finding a diverging number of (1-spin-flip-stable) local minima as $N\to\infty$. We then investigate some of the properties of these local minima (e.g., typical energy and magnetization) and compare to the situation where the edge-weights are drawn from a heavy-tailed distribution.

cond-mat.dis-nn↗
arXiv2019

Nature vs. Nurture: Dynamical Evolution in Disordered Ising Ferromagnets

We study the predictability of zero-temperature Glauber dynamics in various models of disordered ferromagnets. This is analyzed using two independent dynamical realizations with the same random initialization (called twins). We derive, theoretically and numerically, trajectories for the evolution of the normalized magnetization and twin overlap as the system size tends to infinity. The systems we treat include mean-field ferromagnets with light-tailed and heavy-tailed coupling distributions, as well as highly-disordered models with a variety of other geometries. In the mean-field setting with light-tailed couplings, the disorder averages out and the limiting trajectories of the magnetization and twin overlap match those of the homogenous Curie--Weiss model. On the other hand, when the coupling distribution has heavy tails, or the geometry changes, the effect of the disorder persists in the thermodynamic limit. Nonetheless, qualitatively all such random ferromagnets share a similar time evolution for their twin overlap, wherein the two twins initially decorrelate, before either partially or fully converging back together due to the ferromagnetic drift.

cond-mat.stat-mech↗
arXiv2018

Statistical Physics Models of Belief Dynamics: Theory and Empirical Tests

We build simple computational models of belief dynamics within the framework of discrete-spin statistical physics models, and explore how suitable they are for understanding and predicting real-world belief change on both the individual and group levels. We find that accurate modeling of real-world patterns requires attending to social interaction rules that people use, network structures in which they are embedded, distributions of initial beliefs and intrinsic preferences, and the relative importance of social information and intrinsic preferences. We demonstrate that these model parameters can be constrained by empirical measurement, and the resulting models can be used to investigate the mechanisms underlying belief dynamics in actual societies. We use data from two longitudinal studies of belief change, one on 80~individuals living in an MIT dorm during the 2008 presidential election season, and another on 94~participants recruited from Mechanical Turk during the 2016 presidential election primary season. We find that simple statistical physics-based models contain predictive value for real-world belief dynamics and enable empirical tests of different assumptions about the underlying network structure and the social interaction rules.

physics.soc-ph↗
arXiv2017

Zero-temperature dynamics in the dilute Curie-Weiss model

We consider the Ising model on a dense Erdős--Rényi random graph, $\mathcal G(N,p)$, with $p>0$ fixed---equivalently, a disordered Curie--Weiss Ising model with $\mbox{Ber}(p)$ couplings---at zero temperature. The disorder may induce local energy minima in addition to the two uniform ground states. In this paper we prove that, starting from a typical initial configuration, the zero-temperature dynamics avoids all such local minima and absorbs into a predetermined one of the two uniform ground states. We relate this to the local MINCUT problem on dense random graphs; namely with high probability, the greedy search for a local MINCUT of $\mathcal G(N,p)$ with $p>0$ fixed, started from a uniform random partition, fails to find a non-trivial cut. In contrast, in the disordered Curie--Weiss model with heavy-tailed couplings, we demonstrate that zero-temperature dynamics has positive probability of absorbing in a random local minimum different from the two homogenous ground states.

math.PR↗
arXiv2015

Large Fluctuations and Singular Behavior of Nonequilibrium Systems

We present a general geometrical approach to the problem of escape from a metastable state in the presence of noise. The accompanying analysis leads to a simple condition, based on the norm of the drift field, for determining whether caustic singularities alter the escape trajectories when detailed balance is absent. We apply our methods to systems lacking detailed balance, including a nanomagnet with a biaxial magnetic anisotropy and subject to a spin transfer torque. The approach described within allows determination of the regions of experimental parameter space that admit caustics.

math-ph↗
arXiv2014

Lifetimes of Metal Nanowires with Broken Axial Symmetry

We present a theoretical approach for understanding the stability of simple metal nanowires, in particular monovalent metals such as the alkalis and noble metals. Their cross sections are of order one nanometer so that small perturbations from external (usually thermal) noise can cause large geometrical deformations. The nanowire lifetime is defined as the time required for making a transition into a state with a different cross-sectional geometry. This can be a simple overall change in radius, or a change in the cross section shape, or both. We develop a stochastic field theoretical model to describe this noise-induced transition process, in which the initial and final states correspond to locally stable states on a potential surface derived by solving the Schrodinger equation for the electronic structure of the nanowire numerically. The numerical string method is implemented to determine the optimal transition path governing the lifetime. Using these results, we tabulate the lifetimes of sodium and gold nanowires for several different initial geometries.

cond-mat.mes-hall↗
arXiv2014

Spin Torque Oscillators with Thermal Noise: A Constant Energy Orbit Approach

We consider a biaxial macrospin with an easy and hard axis, and study its dynamical evolution under the combined effects of thermal noise and spin transfer torque. The spin-torque is associated with both a perpendicularly magnetized polarizer and an in-plane magnetized reference layer, leading to an effective tilt between the easy and spin polarization axes. Using techniques based on energy averaging over the relevant dynamical trajectories, we analyze the effects of tilt on the dynamics and derive the conditions for the occurrence of stable out-of-plane precessionary states. The presence of these states and their predicted stability boundaries can be tested in experiments on orthogonal spin-transfer devices, and may also serve as a test of the applicability of the macrospin model to real devices, which have internal magnetic degrees of freedom.

cond-mat.mes-hall↗
arXiv2014

Fluctuation Bounds For Interface Free Energies in Spin Glasses

We consider the free energy difference restricted to a finite volume for certain pairs of incongruent thermodynamic states (if they exist) in the Edwards-Anderson Ising spin glass at nonzero temperature. We prove that the variance of this quantity with respect to the couplings grows proportionally to the volume in any dimension greater than or equal to two. As an illustration of potential applications, we use this result to restrict the possible structure of Gibbs states in two dimensions.

math-ph↗
arXiv2013

Thermally-Assisted Spin-Transfer Torque Dynamics in Energy Space

We consider the general Landau-Lifshitz-Gilbert theory underlying the magnetization dynamics of a macrospin magnet subject to spin-torque effects and thermal fluctuations. Thermally activated dynamical properties are analyzed by averaging the full magnetization equations over constant- energy orbits. After averaging, all the relevant dynamical scenarios are a function of the ratio between hard and easy axis anisotropies. We derive analytically the range of currents for which limit cycles exist and discuss the regimes in which the constant energy orbit averaging technique is applicable.

cond-mat.mes-hall↗
arXiv2010

Stability of 2pi domain walls in ferromagnetic nanorings

The stability of 2pi domain walls in ferromagnetic nanorings is investigated via calculation of the minimum energy path that separates a 2pi domain wall from the vortex state of a ferromagnetic nanoring. Trapped domains are stable when they exist between certain types of transverse domain walls, i.e., walls in which the edge defects on the same side of the magnetic strip have equal sign and thus repel. Here the energy barriers between these configurations and vortex magnetization states are obtained using the string method. Due to the geometry of a ring, two types of 2pi walls must be distinguished that differ by their overall topological index and exchange energy. The minimum energy path corresponds to the expulsion of a vortex. The energy barrier for annihilation of a 2pi wall is compared to the activation energy for transitions between the two ring vortex states.

cond-mat.mes-hall↗
arXiv2009

Annular Spin-Transfer Memory Element

An annular magnetic memory that uses a spin-polarized current to switch the magnetization direction or helicity of a magnetic region is proposed. The device has magnetic materials in the shape of a ring (1 to 5 nm in thickness, 20 to 250 nm in mean radius and 8 to 100 nm in width), comprising a reference magnetic layer with a fixed magnetic helicity and a free magnetic layer with a changeable magnetic helicity. These are separated by a thin non-magnetic layer. Information is written using a current flowing perpendicular to the layers, inducing a spin-transfer torque that alters the magnetic state of the free layer. The resistance, which depends on the magnetic state of the device, is used to read out the stored information. This device offers several important advantages compared to conventional spin-transfer magnetic random access memory (MRAM) devices. First, the ring geometry offers stable magnetization states, which are, nonetheless, easily altered with short current pulses. Second, the ring geometry naturally solves a major challenge of spin-transfer devices: writing requires relatively high currents and a low impedance circuit, whereas readout demands a larger impedance and magnetoresistance. The annular device accommodates these conflicting requirements by performing reading and writing operations at separate read and write contacts placed at different locations on the ring.

cond-mat.mes-hall↗
arXiv2008

A Micromagnetic Study of Magnetization Reversal in Ferromagnetic Nanorings

We present results of micromagnetic simulations of thin ferromagnetic rings undergoing magnetization reversal. This geometry is one of few examples in micromagnetics in which the transition states have been found analytically in a 1D model. According to this model, at low fields and large ring sizes, the energetically preferred transition state is a localized magnetization fluctuation (instanton saddle). At high fields and small ring size, the preferred saddle state is a uniformly rotated magnetization (constant saddle). In the first part of this paper, we use numerical micromagnetic simulations to test these predictions of the 1D analytical model for more realistic situations, including a variety of ring radii, annular widths and magnetic fields. The predicted activation energies for magnetization reversal are found to be in close agreement with numerical results, even for rings with a large annular width where the 1D approximation would be expected to break down. We find that this approximation breaks down only when the ring's annular width exceeds its radius. In the second part, we present new metastable states found in the large radius limit and discuss how they provide a more complete understanding of the energy landscape of magnetic nanorings.

cond-mat.mes-hall↗
arXiv1996

Oscillatory Behavior of the Rate of Escape through an Unstable Limit Cycle

Suppose a two-dimensional dynamical system has a stable attractor that is surrounded by an unstable limit cycle. If the system is additively perturbed by white noise, the rate of escape through the limit cycle will fall off exponentially as the noise strength tends to zero. By analysing the associated Fokker-Planck equation we show that in general, the weak-noise escape rate is non-Arrhenius: it includes a factor that is periodic in the logarithm of the noise strength. The presence of this slowly oscillating factor is due to the nonequilibrium potential of the system being nondifferentiable at the limit cycle. We point out the implications for the weak-noise limit of stochastic resonance models.

cond-mat↗
arXiv1995

A Scaling Theory of Bifurcations in the Symmetric Weak-Noise Escape Problem

We consider the overdamped limit of two-dimensional double well systems perturbed by weak noise. In the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double well system are varied, a unique MPEP may bifurcate into two equally likely MPEP's. At the bifurcation point in parameter space, the activation kinetics of the system become non-Arrhenius. In this paper we quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our analysis relies on the development of a new scaling theory, which yields `critical exponents' describing weak-noise behavior near the saddle, at the bifurcation point.

cond-mat↗
arXiv1994

Asymptotic Exit Location Distributions in the Stochastic Exit Problem

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point $S$. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength $ε$, the system state will eventually leave the domain of attraction $Ω$ of $S$. We analyse the case when, as $ε\to0$, the exit location on the boundary $\partialΩ$ is increasingly concentrated near a saddle point $H$ of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on $\partialΩ$ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter $μ$, equal to the ratio $|λ_s(H)|/λ_u(H)$ of the stable and unstable eigenvalues of the linearized deterministic flow at $H$. If $μ<1$ then the exit location distribution is generically asymptotic as $ε\to0$ to a Weibull distribution with shape parameter $2/μ$, on the $O(ε^{μ/2})$ length scale near $H$. If $μ>1$ it is generically asymptotic to a distribution on the $O(ε^{1/2})$ length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden' region: a wedge-shaped subset of $Ω$ with $H$ as vertex, which is reached from $S$, in the $ε\to0$ limit, only via `bent' (non-smooth) fluctuational paths that first pass through the vicinity of $H$. We deduce from the presence of this forbidden region that the classical Eyring formula for the small-$ε$ exponential asymptotics of the mean first exit time is generically inapplicable.

adap-org↗