## Dissertations, Theses, and Capstone Projects

## Date of Degree

9-2015

## Document Type

Dissertation

## Degree Name

Ph.D.

## Program

Physics

## Advisor

Eugene M. Chudnovsky

## Subject Categories

Physics

## Keywords

Nanomagnatism; Quenched Randomness; Random Magnets; Topological Defects

## Abstract

We explore multiple different examples of quenched randomness in systems with a continuous order parameter. In all these systems, it is shown that understanding the effects of topology is critical to the understanding of the effects of quenched randomness.

We consider *n*-component fixed-length order parameter interacting with a weak random field in *d *= 1,2,3 dimensions. Relaxation from the initially ordered state and spin-spin correlation functions have been studied on lattices containing hundreds of millions sites. At *n *- 1 < *d* presence of topological structures leads to metastability, with the final state depending on the initial condition. At *n *- 1 > *d*, when topological objects are absent, the final, lowest-energy, state is independent of the initial condition. It is characterized by the exponential decay of correlations that agrees quantitatively with the theory based upon the Imry-Ma argument. In the borderline case of *n* - 1 = *d*, when topological structures are non-singular, the system possesses a weak metastability with the Imry-Ma state likely to be the global energy minimum.

We study random-field *xy* spin model at *T* = 0 numerically on lattices of up to 1000 x 1000 x 1000 spins with the accent on the weak random field. Our numerical method is physically equivalent to slow cooling in which the system is gradually losing the energy and relaxing to an energy minimum. The system shows glass properties, the resulting spin states depending strongly on the initial conditions. Random initial condition for the spins leads to the vortex glass (VG) state with short-range spin-spin correlations defined by the average distance between vortex lines. Collinear and some other vortex-free initial conditions result in the vortex-free ferromagnetic (F) states that have a lower energy. The energy difference between the F and VG states correlates with vorticity of the VG state. Correlation functions in the F states agree with the Larkin-Imry-Ma theory at short distances. Hysteresis curves for weak random field are dominated by topologically stable spin walls raptured by vortex loops. We find no relaxation paths from the F, VG, or any other states to the hypothetical vortex-free state with zero magnetization.

XY and Heisenberg spins, subjected to strong random fields acting at few points in space with concentration *c _{r}* << 1, are studied numerically on 3d lattices containing over four million sites. Glassy behavior with strong dependence on initial conditions is found. Beginning with a random initial orientation of spins, the system evolves into ferromagnetic domains inversely proportional to

*c*in size. The area of the hysteresis loop,

_{r}*m(H)*, scales as

*c*

_{r}^{2}. These findings are explained by mapping the effect of strong dilute random field onto the effect of weak continuous random field. Our theory applies directly to ferromagnets with magnetic impurities, and is conceptually relevant to strongly pinned vortex lattices in superconductors and pinned charge density waves.

The random-anisotropy Heisenberg model is numerically studied on lattices containing over ten million spins. The study is focused on hysteresis and metastability due to topological defects, and is relevant to magnetic properties of amorphous and sintered magnets. We are interested in the limit when ferromagnetic correlations extend beyond the size of the grain inside which the magnetic anisotropy axes are correlated. In that limit the coercive field computed numerically roughly scales as the fourth power of the random anisotropy strength and as the sixth power of the grain size. Theoretical arguments are presented that provide an explanation of numerical results. Our findings should be helpful for designing amorphous and nanosintered materials with desired magnetic properties.

## Recommended Citation

Proctor, Thomas Chapman, "Ordering and topological defects in solids with quenched randomness" (2015). *CUNY Academic Works.*

https://academicworks.cuny.edu/gc_etds/1100