# XY model with weak random anisotropy in a symmetry-breaking magnetic field.

TL;DR: A numerical study of the two-dimensional classical XY model with weak random anisotropy at zero temperature with an important aftereffect is observed, which should give smaller remanence after much longer computer time.

Abstract: We present a numerical study of the two-dimensional classical XY model with weak random anisotropy at zero temperature. Zero-field configurations obtained by ultrafast cooling, first-magnetization curves, and hysteresis loops have been calculated for different random-anisotropy-to-exchange ratios. In zero field, a pinning of vortices by the random-anisotropy field occurs. It prevents the binding then collapsing of pairs of opposite charges and thus leads to a nonferromagnetic ground state. Applying a magnetic field causes a progressive depinning of vortices that disappear in pairs until saturation. Starting from saturation and decreasing the applied field leads, in zero-field, to a magnetic state of large remanent magnetization. However, an important aftereffect is observed. It should give smaller remanence after much longer computer time. Then, the reversal of magnetization in negative fields occurs through a peculiar process that involves the formation and collapse of new kinds of topological defects (infinite strings). These linear defects are in fact the ultimate stage in the shrinking of domains oriented in the initial direction of saturation. Their collapse occurs abruptly through the creation and propagation in opposite directions, along the defect, of an unbound vortex-antivortex pair.

## Summary (2 min read)

### INTRODUCTION

- Both the classical isotropic XYmodel and the magnetic properties of random-anisotropy systems have attracted great interest during the last decade.
- Magnetic properties of random-field systems (randomanisotropy fields of continuous symmetry, p-fold symmetry-breaking fields, random-exchange fields, etc.) have been studied theoretically and experimentally in 3D and sometimes 2D systems.
- Decreasing the applied field from saturation leads to a phase of large remanent magnetization in which the magnetization varies continuously in a semicircle oriented along the saturation field.
- Above a certain critical field the string abruptly collapses through the creation and propagation along the defect of a vortex-antivortex pair.
- This paper is divided into four sections.

### jjj%$

- All the calculations are carried out at zero temperature on a square lattice of 100X100 spins with cyclic boundary conditions.
- A cycle in the calculation consists of exploring all of the 10000 spins in a random order and setting each explored spin in its local minimum of energy between its anisotropy axis and local-field direction.
- Net magnetization and contain regions with local ferromagnetic order (Imry and Ma domains' ) the size of which decreases with D/J.
- These domains are not specific of two dimensions; they have already been studied theoretically and experimentally in 3D randomanisotropy systems.
- At the thermal equilibrium the number of vortices is determined by a counterbalance between vortex-antivortex annihilations and vortex-antivortex pair creations by thermal ac-tivation.

### ZERO-FIELD CONFIGURATIONS

- The number of vortices of each sign is equal (periodic boundary conditions) and increases rapidly with D/J, especially around D /J-0.
- Concerning the effect of D/J on the number of vortices, the authors have already mentioned that in zero field, a nearly linear relationship (naD/J) breaks down for weak random anisotropy due to important slowing down of the dynamics when D/J~O (Fig. 2 ).
- In the simulation, the authors observe that the string begins breaking at the point where it is the most symmetrical with respect to the field with a symmetry point located on a bound between two neighboring spins.
- On the contrary, the field H"which characterizes the disappearing of the last vortice pair in the first magnetization process is associated with a depinning phenomenon and thus depends on the strength of the random anisotropy D. This constitutes an important difference between these two characteristic fields.
- In the presence of random anisotropy, these defects appear naturally in a hysteresis cycle due to nucleation of domains rotating in opposite directions.

### CONCLUSION

- The authors focus on the magnetization behavior in the first magnetization process and hysteresis cycle.
- Then applying an increasing magnetic field leads to a progressive depinning and annihilating of vortices, together with progressive rotations of the magnetization inside Imry and Ma domains towards the direction of the field.
- The width of the strings decreases down to a few interatomic distances when the strength of the field increases.
- To their knowledge, such linear defects have never been characterized in a real sample.
- Then, by applying a magnetic field along one of the two magnetization directions, one kind of domain will shrink by propagation of the Neel walls.

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