Nature of Long-Range Order in Stripe-Forming Systems with Long-Range Repulsive Interactions

Abstract

We study two dimensional stripe forming systems with competing repulsive interactions decaying as r(-α). We derive an effective Hamiltonian with a short-range part and a generalized dipolar interaction which depends on the exponent α. An approximate map of this model to a known XY model with dipolar interactions allows us to conclude that, for α<2 long-range orientational order of stripes can exist in two dimensions, and establish the universality class of the models. When α≥2 no long-range order is possible, but a phase transition in the Kosterlitz-Thouless universality class is still present. These two different critical scenarios should be observed in experimentally relevant two dimensional systems like electronic liquids (α=1) and dipolar magnetic films (α=3). Results from Langevin simulations of Coulomb and dipolar systems give support to the theoretical results.

Periodic Striped Ground States in Ising Models with Competing Interactions

“…A particularly interesting and poorly understood phenomenon is that of periodic stripe formation [29,30,37,39]. In a series of papers, this phenomenon was studied in Ising and related models with short range attractive and long range repulsive interactions.…”

Critical exponents and scaling invariance in the absence of a critical point

“…All the literature we are aware of supports the absence of long-range positional order of domains at finite temperature171821, thus suggesting that not even a kind of staggered magnetization associated with the striped pattern is expected to display the 2D-Ising critical behaviour (nor the MF critical behaviour foreseen by Wasilevsky1112). In the prevailing understanding of critical phenomena the fulfilment of equation (2) and other scaling relations is associated directly with spontaneous breaking of a specific symmetry of the Hamiltonian in b =0.…”

“…Concrete realizations of this model involve either Coulomb repulsion ( α =1) or dipole–dipole antiferromagnetic interaction ( α =3) that competes with a ferromagnetic short-ranged interaction22. This competition leads to the formation of a striped ground state whose elementary excitations are described by an elastic-like Hamiltonian associated with the displacement of domain walls—as a result of the subtle interplay between the two interactions1721. The experimental results presented in the next subsections refer to a ferromagnetic model system representative of the Hamiltonian (1) with α =3 (refs 18, 19, 27, 33, 34, 35, 36).…”

“…It is not clear yet how to call the phase with mobile stripes. Possibly the most appropriate definition is that of stripe liquid20 or floating solid53, which highlight the lack of positional order in this phase21. With certainty, the temperature corresponding to the red line cannot be identified with a Curie temperature.…”

“…In particular, the spatial period of the modulated phase provides, alongside the correlation length, a second long length scale that transforms the critical point into a so-called avoided critical point. The more general theoretical framework of avoided criticality was developed in the 1990s (refs 13, 14, 15) and foresees that even a small amount of frustration changes the critical point of an unfrustrated model of collective order into a completely different object—generally not a second-order transition point—depending on details like the spatial range and the strength of the competing interactions involved, the dimensionality of the system and the number of components of the order parameter16171819202122. A non-exhaustive list of pattern-forming systems for which the phenomenology of avoided criticality has been proposed comprises magnetic films, ferrofluids, diblock copolymers, amphiphilic solutions, systems undergoing Turing-like phase separating chemical reactions and charged stripes in cuprate high- T c superconductors2223242526.…”