On minima of sum of theta functions and Mueller-Ho Conjecture

Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $\theta (s;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-s \frac{\pi }{y }|mz+n|^2}$ be the theta function associated with the lattice $\Lambda ={\mathbb Z}\oplus z{\mathbb Z}$. In this paper we consider the following pair of minimization problems $$\min_{ \mathbb{H} } \theta (2;\frac{z+1}{2})+\rho\theta (1;z),\;\;\rho\in[0,\infty),$$ $$\min_{ \mathbb{H} } \theta (1; \frac{z+1}{2})+\rho\theta (2; z),\;\;\rho\in[0,\infty),$$ where the parameter $\rho\in[0,\infty)$ represents the competition of two intertwining lattices. We find that as $\rho$ varies the optimal lattices admit a novel pattern: they move from rectangular (the ratio of long and short side changes from $\sqrt3$ to 1), square, rhombus (the angle changes from $\pi/2$ to $\pi/3$) to hexagonal; furthermore, there exists a closed interval of $\rho$ such that the optimal lattices is always square lattice. This is in sharp contrast to optimal lattice shapes for single theta function ($\rho=\infty$ case), for which the hexagonal lattice prevails. As a consequence, we give a partial answer to optimal lattice arrangements of vortices in competing systems of Bose-Einstein condensates as conjectured (and numerically and experimentally verified) by Mueller-Ho \cite{Mue2002}.Comment: 42 pages; comments welcom

Novel Ground-State Crystals with Controlled Vacancy Concentrations: From Kagom\'{e} to Honeycomb to Stripes

We introduce a one-parameter family, $0 \leq H \leq 1$, of pair potential functions with a single relative energy minimum that stabilize a range of vacancy-riddled crystals as ground states. The "quintic potential" is a short-ranged, nonnegative pair potential with a single local minimum of height $H$ at unit distance and vanishes cubically at a distance of \rt. We have developed this potential to produce ground states with the symmetry of the triangular lattice while favoring the presence of vacancies. After an exhaustive search using various optimization and simulation methods, we believe that we have determined the ground states for all pressures, densities, and $0 \leq H \leq 1$. For specific areas below 3\rt/2, the ground states of the "quintic potential" include high-density and low-density triangular lattices, kagom\'{e} and honeycomb crystals, and stripes. We find that these ground states are mechanically stable but are difficult to self-assemble in computer simulations without defects. For specific areas above 3\rt/2, these systems have a ground-state phase diagram that corresponds to hard disks with radius \rt. For the special case of H=0, a broad range of ground states is available. Analysis of this case suggests that among many ground states, a high-density triangular lattice, low-density triangular lattice, and striped phases have the highest entropy for certain densities. The simplicity of this potential makes it an attractive candidate for experimental realization with application to the development of novel colloidal crystals or photonic materials.Comment: 25 pages, 11 figure

Ground-state properties of the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice: A variational study based on entangled-plaquette states

We study, on the basis of the general entangled-plaquette variational ansatz, the ground-state properties of the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice. Our numerical estimates are in good agreement with available exact results and comparable, for large system sizes, to those computed via the best alternative numerical approaches, or by means of variational schemes based on specific (i.e., incorporating problem dependent terms) trial wave functions. The extrapolation to the thermodynamic limit of our results for lattices comprising up to N=324 spins yields an upper bound of the ground-state energy per site (in units of the exchange coupling) of $-0.5458(2)$ [$-0.4074(1)$ for the XX model], while the estimated infinite-lattice order parameter is $0.3178(5)$ (i.e., approximately 64% of the classical value).Comment: 8 pages, 3 tables, 2 figure

Fractal space frames and metamaterials for high mechanical efficiency

A solid slender beam of length $L$, made from a material of Young's modulus $Y$ and subject to a gentle compressive force $F$, requires a volume of material proportional to $L^{3}f^{1/2}$ [where $f\equiv F/(YL^{2})\ll 1$] in order to be stable against Euler buckling. By constructing a hierarchical space frame, we are able to systematically change the scaling of required material with $f$ so that it is proportional to $L^{3}f^{(G+1)/(G+2)}$, through changing the number of hierarchical levels $G$ present in the structure. Based on simple choices for the geometry of the space frames, we provide expressions specifying in detail the optimal structures (in this class) for different values of the loading parameter $f$. These structures may then be used to create effective materials which are elastically isotropic and have the combination of low density and high crush strength. Such a material could be used to make light-weight components of arbitrary shape.Comment: 6 pages, 4 figure

Superfluid-Insulator Transitions on the Triangular Lattice

We report on a phenomenological study of superfluid to Mott insulator transitions of bosons on the triangular lattice, focusing primarily on the interplay between Mott localization and geometrical charge frustration at 1/2-filling. A general dual vortex field theory is developed for arbitrary rational filling factors f, based on the appropriate projective symmetry group. At the simple non-frustrated density f=1/3, we uncover an example of a deconfined quantum critical point very similar to that found on the half-filled square lattice. Turning to f=1/2, the behavior is quite different. Here, we find that the low-energy action describing the Mott transition has an emergent nonabelian SU(2)\times U(1) symmetry, not present at the microscopic level. This large nonabelian symmetry is directly related to the frustration-induced quasi-degeneracy between many charge-ordered states not related by microscopic symmetries. Through this ``pseudospin'' SU(2)symmetry, the charged excitations in the insulator close to the Mott transition develop a skyrmion-like character. This leads to an understanding of the recently discovered supersolid phase of the triangular lattice XXZ model (cond-mat/0505258, cond-mat/0505257, cond-mat/0505298) as a ``partially melted'' Mott insulator. The latter picture naturally explains a number of puzzling numerical observations of the properties of this supersolid. Moreover, we predict that the nearby quantum phase transition from this supersolid to the Mott insulator is in the recently-discovered non-compact CP^1 critical universality class (PRB 70, 075104 (2004)). A description of a broad range of other Mott and supersolid states, and a diverse set of quantum critical points between them, is also provided.Comment: 24 pages, 14 figure