Periodic Ground States Research Articles

GROUND STATES FOR THE SOS MODEL WITH COMPETING BINARY INTERACTIONS ON A CAYLEY TREE OF ORDER THREE

We consider a SOS (solid-on-solid) model with nearest-neighbor interaction prolonged next-nearest-neighbor interaction and one level next-nearest-neighbor interaction where the spin takes values in the set on a Cayley tree of order three. In the paper, we study translation-invariant and periodic ground states of the model SOS.

Periodic and Weakly Periodic Ground States Corresponding to the Subgroups of Index Three for the Ising Model on the Cayley Tree of Order Three

Periodic and Weakly Periodic Ground States Corresponding to the Subgroups of Index Three for the Ising Model on the Cayley Tree of Order Three

Periodic and weakly periodic ground states corresponding to the subgroups of index three for the Ising model on the Cayley tree of order three

UDC 517.9 We determine periodic and weakly periodic ground states with subgroups of index three for the Ising model on the Cayley tree of order three.

Periodic ground states for the mixed spin ising model with competing interactions on a Cayley tree

Periodic ground states for the mixed spin ising model with competing interactions on a Cayley tree

On the nonlinear Schrödinger equation with a toroidal-shaped trap in the strong confinement regime

We consider the 3D cubic nonlinear Schrödinger equation (NLS) with a strong toroidal-shaped trap. In the first part, we show that as the confinement is strengthened, a large class of global solutions to the time-dependent model can be described by 1D flows solving the 1D periodic NLS (theorem ). In the second part, we construct a steady state as a constrained energy minimizer, and prove its dimension reduction to the well-known 1D periodic ground state (theorems and ). Then, employing the dimension reduction limit, we establish the local uniqueness and the orbital stability of the 3D ring soliton (theorem ). These results justify the emergence of stable quasi-1D periodic dynamics for Bose–Einstein condensates on a ring in physics experiments.

Helfrich-Hurault elastic instabilities driven by geometrical frustration

The Helfrich-Hurault instability is a well-known mechanism behind the undulations that occur as a result of strain upon liquid crystal systems with periodic ground states. In this review, the Helfrich-Hurault elastic instability is examined with a focus on layered liquid crystals that are geometrically frustrated. The frustration is relieved by undulations in the layered structure to maintain the preferred layer spacing. Examples of cholesteric and smectic liquid crystals confined between two spherical fluid interfaces are described and the effects of topological constraints, anchoring conditions, and curvature on the instability are examined. Lastly, the Helfrich-Hurault instability is surveyed as a pattern formation mechanism across a range of materials, both biological and synthetic.

Optimality of the triangular lattice for Lennard–Jones type lattice energies: a computer-assisted method

It is well-known that any Lennard–Jones type potential energy must have a periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method that rigorously shows such global minimality result among 2-dimensional lattices once the exponents of the potential have been fixed. The method is applied to the widely used classical Lennard–Jones potential, which is the main result of this work. Furthermore, a new bound on the inverse density (i.e. the co-volume) for which the triangular lattice is minimal is derived, improving those found in (Bétermin and Zhang 2015 Commun. Contemp. Math. 17 1450049) and (Bétermin 2016 SIAM J. Math. Anal. 48 3236–3269). The same results are also shown to hold for other exponents as additional examples and a new conjecture implying the global optimality of a triangular lattice for any parameters is stated.

Solutions to a nonlinear Dirac–Maxwell system: from periodic waves to soliton-like waves

Using variational methods, we study the existence of stationary solutions to a periodic nonlinear Dirac–Maxwell system. Firstly, we prove the existence of n-periodic solutions to the system for all positive integers n. Then we use the concentration-compactness argument and the periodic approximation method for obtaining soliton-like solutions. We also investigate the existence of periodic ground states and soliton-like ground states, as well as their connection. Besides, results for the regularity of solutions and decay rate of the soliton-like solutions are given.

On Ground States for the SOS Model with Competing Interactions

We study periodic and weakly periodic ground states for the SOS model with competing interactions on the Cayley tree of order two and three. Further, we study non periodic ground states for the SOS model with competing interactions on the Cayley tree of order two

Periodic and weakly periodic ground states corresponding to subgroups of index three for the ising model on cayley tree

For the Ising model on a Cayley tree of order two, we describe a set of periodic and weakly periodic ground states that correspond to subgroups of index 3 of the group representation of the Cayley tree.

Weakly Periodic Ground States for the λ-Model

For the λ-model on a Cayley tree of order k ≥ 2, we describe the set of periodic and weakly periodic ground states corresponding to normal divisors of index 2 of the group representation of Cayley tree.

Ground states for the SOS model with an external field on the Cayley tree

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values $0,1,2,...,m, m\geq2$ and non zero external field, on a Cayley tree of order $k$. In the case $k=2, m=2$, we describe translation-invariant ground states for the SOS model with a translation-invariant external field. Some periodic ground states for the SOS model with periodic external field are described.

Слабо периодические основные состояния для λ-модели

УДК 517.98 + 530.1 Для λ -моделі на дереві Келі порядку k ≥ 2 описано множину періодичних і слабко періодичних основних станів, що відповідають нормальним дільникам індексу 2 групового зображення дерева Келі.

Ground state criteria in one-dimensional antiferromagnetic Ising model with long range interaction

One-dimensional long-range Ising models with antiferromagnetic, convex pair interactions are investigated. A new criterion characterizing ground states is given. The criterion and a new transformation yield short proofs identifying and characterizing the ground states. The uniqueness of periodic ground states up to shifts is shown.

A note on inhomogeneous ground states at large global charge

In this note we search for the ground state of the D = 3 Wilson-Fisher conformal O(4) model, at nonzero values of the two independent charge densities ρ1,2, on the torus spatial slice. Using an effective theory valid on scales longer than the scale defined by the charge density, we show that the ground-state configuration is inhomogeneous for generic ratios ρ1/ρ2. This result confirms, within the context of a well-defined effective theory, a recent no-go result of [1]. We also show that any spatially periodic ground state solutions have an energetic preference towards longer periods, within some range of ρ1/ρ2 containing a neighborhood of zero. This suggests that the scale of variation of the ground state solution in finite volume will be the infrared scale, and that the use of the effective theory at large charge in finite volume is self-consistent. Note added: the statements in this paper are true for arbitrary ratio of ρ1/ρ2, which we proved after we uploaded this paper. See [2].

On Orbital Stability of Ground States for Finite Crystals in Fermionic Schrödinger--Poisson Model

We consider the Schrödinger--Poisson--Newton equations for finite crystals under periodic boundary conditions with one ion per cell of a lattice. The electron field is described by the $N$-particle Schrödinger equation with an antisymmetric wave function. Our main results are (i) the global dynamics with moving ions, and (ii) the orbital stability of periodic ground states under the novel Jellium and Wiener-type conditions on the ion charge density. Under the Jellium condition both ionic and electronic charge densities of the ground state are uniform.

Periodic and Weakly Periodic Ground States for the λ-Model on Cayley Tree

In this paper we consider the λ-model on the Cayley tree of order two. We describe the periodic and weakly periodic ground states for the considered model.

Periodic ground states of the electron gas in two and three dimensions

In this paper, we review the possibility of periodic ground states of the two‐ and three‐dimensional electron gas at high and intermediate densities. At the Hartree–Fock level, we obtain explicit solutions at all densities having lower energies than the homogeneous Fermi gas. In the high‐density region, the electrons only partially fill up the band approaching spin density waves as predicted by Overhauser. Preliminary variational Monte Carlo calculations on the polarized two‐dimensional electron gas confirm the existence of these new types of ground states.

On stability of ground states for finite crystals in the Schrödinger–Poisson model

We consider the Schrödinger–Poisson–Newton equations for finite crystals under periodic boundary conditions with one ion per cell of a lattice. The electrons are described by one-particle Schrödinger equation. Our main results are (i) the global dynamics with moving ions and (ii) the orbital stability of periodic ground state under a novel Jellium and Wiener-type conditions on the ion charge density. Under the Jellium condition, both ionic and electronic charge densities for the ground state are uniform.

Periodic and weakly periodic ground states for the Potts model with competing interactions on the Cayley tree

We consider the Potts model with competing interactions on the Cayley tree of order k with k ≥ 2. We describe the sets of periodic and weakly periodic ground states corresponding to normal subgroups of the group representation of the Cayley tree of index 4.