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We present an optimization principle for the Harper-Hofstadter model that naturally yields the critical value 𝜆=2 for the Harper parameter. We provide proofs for this principle and its corollaries. We demonstrate that it can be applied to a continuum model, where it can be used to find the physical parameters for criticality. 

We examine the spectrum and quantum states of small lattices with cylindrical and toroidal topology subject to a scalar gauge potential that introduces a position dependent phase in the inter-site coupling. Equivalency of gauges assumed in infinite lattices is generally lost due to the periodic boundary conditions, and conditions that restore it are identified. We trace the impact of various system parameters including gauge choice, boundary conditions and inter-site coupling strengths, and an additional axial field. We find gauge dependent appearance of avoided crossings and persistent degeneracies, and we show their impact on the associated eigenstates. Smaller lattices develop prominent gaps in spectral lines associated with edge states, which are suppressed in the thermodynamic limit. Toroidal lattices have counterparts of most of the features observed in cylindrical lattices, but notably they display a transition from localization to delocalization determined by the relation between the field parameter and the number of lattice sites. 

We examine the spectrum for lattices with cylindrical and toroidal topology subject to Abelian gauge potentials. Gauges that are equivalent in planar lattices with trivial topology develop differences due to the periodic boundary conditions. But some residual gauge equivalency, evident in the spectrum, is found to remain under specific conditions. This is associated with the gauge structure being commensurate with the lattice periodicity, and the behavior of the associated field along the symmetry axes. The interplay of the gauge choice and the periodic boundary conditions leads to a class of persistent degeneracies that are found to be robust against vast changes in system parameters and even under change of topology from cylinder to torus. 

We study the spectrum and stationary states in a ring-shaped lattice potential in the context of ultracold atoms with attractive interatomic interactions. We determine analytical solutions in the absence of a lattice by mapping them to those for repulsive interactions, and then we numerically follow the transformation of those solutions as the lattice is introduced and strengthened. Several features emerge that are specific to negative nonlinearity, which include soliton branches detaching to create new ground states, gaps opening up at the bottom of the primary spectral branch, and multiple splitting and rejoining of some branches. We correlate the spectral features with the behavior of the density and phase of the corresponding eigenstates, and track them along branches and as various system parameters change. We find that the phase is sensitive to how a specific point in the spectrum is approached, particularly relevant at certain persistent gaps in the spectrum. The symmetry and stability properties are generally found to be opposite of that found for repulsive interactions. 

Two species of mutually interacting ultracold bosonic atoms are studied in a ring-shaped trap with a species-selective azimuthal lattice which may rotate. We examine the spectrum and the states in a collective spin formalism. The system can be modeled as a pair of coupled Lipkin-Meshkov-Glick Hamiltonians, and can be used to generate a high degree of entanglement. The Hamiltonian has two components: a linear part that can be controlled by manipulating the azimuthal lattice, and an interaction-dependent quadratic part. Exact solutions are found for the quadratic part for equal strengths of intraspecies and interspecies interactions. In different regimes the Hamiltonian can emulate a beam splitter or a two-mode squeezer of quantum optical systems. We study entanglement properties of the ground state of the Hamiltonian in dependence on various parameters with the prospect of possible quantum information and metrology applications. 

Two species of mutually interacting ultracold bosonic atoms are studied in a ring-shaped trap with a species-selective azimuthal lattice which may rotate. We examine the spectrum and the states in a collective spin formalism. The system can be modeled as a pair of coupled Lipkin-Meshkov-Glick Hamiltonians, and can be used to generate a high degree of entanglement. The Hamiltonian has two components: a linear part that can be controlled by manipulating the azimuthal lattice, and an interaction-dependent quadratic part. Exact solutions are found for the quadratic part for equal strengths of intraspecies and interspecies interactions. In different regimes the Hamiltonian can emulate a beam splitter or a two-mode squeezer of quantum optical systems. We study entanglement properties of the ground state of the Hamiltonian in dependence on various parameters with the prospect of possible quantum information and metrology applications. 

We present a comprehensive study of stationary states in a coherent medium with a quadratic or Kerr nonlinearity in the presence of localized potentials in one dimension for both positive and negative signs of the nonlinear term as well as for barriers and wells. The description is in terms of the nonlinear Schrödinger equation and hence applicable to a variety of systems, including interacting ultracold atoms in the mean field regime and light propagation in optical fibers. We determine the full landscape of solutions in terms of a potential step and build solutions for rectangular barrier and well potentials. It is shown that all the solutions can be expressed in terms of a Jacobi elliptic function with the inclusion of a complex-valued phase shift. Our solution method relies on the roots of a cubic polynomial associated with a hydrodynamic picture, which provides a simple classification of all the solutions, both bounded and unbounded, while the boundary conditions are intuitively visualized as intersections of phase space curves. We compare solutions for open boundary conditions with those for a barrier potential on a ring, and also show that numerically computed solutions for smooth barriers agree qualitatively with analytical solutions for rectangular barriers. A stability analysis of solutions based on the Bogoliubov equations for fluctuations shows that persistent instabilities are localized at sharp boundaries and are predicated by the relation of the mean density change across the boundary to the value of the derivative of the density at the edge. We examine the scattering of a wave packet by a barrier potential and show that at any instant the scattered states are well described by the stationary solutions we obtain, indicating applications of our results and methods to nonlinear scattering problems.