# Formation of Stripes and Slabs Near the Ferromagnetic Transition

@article{Giuliani2013FormationOS, title={Formation of Stripes and Slabs Near the Ferromagnetic Transition}, author={Alessandro Giuliani and Elliott H. Lieb and Robert Seiringer}, journal={Communications in Mathematical Physics}, year={2013}, volume={331}, pages={333-350} }

We consider Ising models in d = 2 and d = 3 dimensions with nearest neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as (distance)−p, p > 2d, at large distances. If the strength J of the ferromagnetic interaction is larger than a critical value Jc, then the ground state is homogeneous. It has been conjectured that when J is smaller than but close to Jc, the ground state is periodic and striped, with stripes of constant width h = h(J), and h → ∞ as $${J…

## 15 Citations

### Periodic Striped Ground States in Ising Models with Competing Interactions

- Physics
- 2015

We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of…

### Periodic striped states in Ising models with dipolar interactions

- Mathematics
- 2022

. We review the problem of determining the ground states of 2D Ising models with nearest neighbor ferromagnetic and dipolar interactions, and prove a new result supporting the conjecture that, if the…

### Crystalline Motion of Interfaces Between Patterns

- Materials Science
- 2016

We consider the dynamical problem of an antiferromagnetic spin system on a two-dimensional square lattice $$\varepsilon \mathbb {Z}^2$$εZ2 with nearest-neighbour and next-to-nearest neighbour…

### One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension

- MathematicsJournal of Functional Analysis
- 2022

### One-dimensional minimizers for a diffuse interface generalized antiferromagnetic model in general dimension

- Mathematics
- 2019

In this paper we consider the diffuse interface version of the generalized antiferromagnetic model studied in its sharp interface limit as ε → 0 in [7, 8, 9] in the discrete and in [13] and [4] in…

### Exact Periodic Stripes for Minimizers of a Local/Nonlocal Interaction Functional in General Dimension

- MathematicsArchive for Rational Mechanics and Analysis
- 2018

We study the functional considered in Giuliani et al. (Phys Rev B 84:064205, 2011, Commun Math Phys 331(1):333–350, 2014) and Giuliani and Seiringer (Commun Math Phys 347:983–1007, 2016) and a…

### Exact Periodic Stripes for Minimizers of a Local/Nonlocal Interaction Functional in General Dimension

- MathematicsArchive for Rational Mechanics and Analysis
- 2018

We study the functional considered in Giuliani et al. (Phys Rev B 84:064205, 2011, Commun Math Phys 331(1):333–350, 2014) and Giuliani and Seiringer (Commun Math Phys 347:983–1007, 2016) and a…

### The antiferromagnetic XY model on the triangular lattice: chirality transitions at the surface scaling

- PhysicsCalculus of Variations and Partial Differential Equations
- 2021

We study the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice. The system is fully frustrated and displays two families of ground…

### On the optimality of stripes in a variational model with non-local interactions

- MathematicsCalculus of Variations and Partial Differential Equations
- 2019

We study pattern formation for a variational model displaying competition between a local term penalizing interfaces and a non-local term favoring oscillations. By means of a $$\Gamma $$Γ-convergence…

### Crystallinity of the Homogenized Energy Density of Periodic Lattice Systems

- MathematicsMultiscale Modeling & Simulation
- 2023

We study the homogenized energy densities of periodic ferromagnetic Ising systems. We prove that, for finite range interactions, the homogenized energy density, identifying the effective limit, is…

## References

SHOWING 1-10 OF 46 REFERENCES

### Checkerboards, stripes, and corner energies in spin models with competing interactions

- Physics
- 2011

We study the zero temperature phase diagram of Ising spin systems in two dimensions in the presence of competing interactions, long range antiferromagnetic and nearest neighbor ferromagnetic of…

### One-dimensional Ising ferromagnet frustrated by long-range interactions at finite temperatures

- Physics
- 2009

We consider a one-dimensional lattice of Ising-type variables where the ferromagnetic exchange interaction J between neighboring sites is frustrated by a long-ranged anti-ferromagnetic interaction of…

### Striped phases in two-dimensional dipole systems

- Physics
- 2007

We prove that a system of discrete 2D in-plane dipoles with four possible orientations, interacting via a 3D dipole-dipole interaction plus a nearest neighbor ferromagnetic term, has periodic striped…

### Ising nematic phase in ultrathin magnetic films : A Monte Carlo study

- Physics
- 2006

We study the critical properties of a two--dimensional Ising model with competing ferromagnetic exchange and dipolar interactions, which models an ultra-thin magnetic film with high out--of--plane…

### Ising models with long-range antiferromagnetic and short-range ferromagnetic interactions

- Materials Science, Mathematics
- 2006

We study the ground state of a $d$-dimensional Ising model with both long-range (dipole-like) and nearest-neighbor ferromagnetic (FM) interactions. The long-range interaction is equal to…

### Phase diagram of an Ising model for ultrathin magnetic films: Comparing mean field and Monte Carlo predictions

- Physics
- 2007

We study the critical properties of a two-dimensional Ising model with competing ferromagnetic exchange and dipolar interactions, which models an ultrathin magnetic film with high out-of-plane…

### On the Absence of Ferromagnetism in Typical 2D Ferromagnets

- Physics
- 2007

We consider the Ising systems in d dimensions with nearest-neighbor ferromagnetic interactions and long-range repulsive (antiferromagnetic) interactions that decay with power s of the distance. The…

### Stripe width and nonlocal domain walls in the two-dimensional dipolar frustrated Ising ferromagnet

- Physics
- 2008

We describe a novel type of magnetic domain wall which, in contrast to Bloch or Neel walls, is non-localized and, in a certain temperature range, non-monotonic. The wall appears as the mean-field…

### Modulated phases of a one-dimensional sharp interface model in a magnetic field

- Physics
- 2009

We investigate the ground states of 1D continuum models having short-range ferromagnetic type interactions and a wide class of competing longer-range antiferromagnetic type interactions. The model is…

### Striped Periodic Minimizers of a Two-Dimensional Model for Martensitic Phase Transitions

- Computer Science
- 2012

Improved Conti’s results are improved, by computing exactly the minimal energy and by proving that minimizers are periodic one-dimensional sawtooth functions.