Competing quantum paramagnetic ground states of the Heisenberg antiferromagnet on the star lattice

We investigate various competing paramagnetic ground states of the Heisenberg antiferromagnet on the two dimensional star lattice which exhibits geometric frustration. Using slave particle mean field theory combined with a projective symmetry group analysis, we examine a variety of candidate spin liquid states on this lattice, including chiral spin liquids, spin liquids with Fermi surfaces of spinons, and nematic spin liquids which break lattice rotational symmetry. Motivated by connection to large-N SU(N) theory as well as numerical exact diagonalization studies, we also examine various valence bond solid (VBS) states on this lattice. Based on a study of energetics using Gutzwiller projected states, we find that a fully gapped spin liquid state is the lowest energy spin liquid candidate for this model. We also find, from a study of energetics using Gutzwiller projected wave functions and bond operator approaches, that this spin liquid is unstable towards two different VBS states -- a VBS state which respects all the Hamitonian symmetries and a VBS state which exhibits $\sqrt{3}\times\sqrt{3}$ order -- depending on the ratio of the Heisenberg exchange couplings on the two inequivalent bonds of the lattice. We compute the triplon dispersion in both VBS states within the bond operator approach and discuss possible implications of our work for future experiments on candidate materials.Comment: 23 pages, 21 figures, 2 table

A formula for the central value of certain Hecke L-functions

Let N = 1 mod 4 be the negative of a prime, K=Q(sqrt{N}) and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters \psi_{\D} of K with conductor (\D) and infinite type $(1,0)$. Their L-series L(\psi_\D,s) are associated to a CM elliptic curve E(N,\D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(\psi_\D,s) of the form L(\psi_\D,1) = \Omega \sum_{[\A],I} r(\D,[\A],I) m_{[\A],I}([\D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [\A] are class group representatives of K$. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level$7|D|$over$K$.Comment: 43 page

FlexibleSUSY -- A spectrum generator generator for supersymmetric models

We introduce FlexibleSUSY, a Mathematica and C++ package, which generates a fast, precise C++ spectrum generator for any SUSY model specified by the user. The generated code is designed with both speed and modularity in mind, making it easy to adapt and extend with new features. The model is specified by supplying the superpotential, gauge structure and particle content in a SARAH model file; specific boundary conditions e.g. at the GUT, weak or intermediate scales are defined in a separate FlexibleSUSY model file. From these model files, FlexibleSUSY generates C++ code for self-energies, tadpole corrections, renormalization group equations (RGEs) and electroweak symmetry breaking (EWSB) conditions and combines them with numerical routines for solving the RGEs and EWSB conditions simultaneously. The resulting spectrum generator is then able to solve for the spectrum of the model, including loop-corrected pole masses, consistent with user specified boundary conditions. The modular structure of the generated code allows for individual components to be replaced with an alternative if available. FlexibleSUSY has been carefully designed to grow as alternative solvers and calculators are added. Predefined models include the MSSM, NMSSM, E$_6$SSM, USSM, R-symmetric models and models with right-handed neutrinos.Comment: 56 pages, 3 figures, 3 tables; v3: correcting typos, matches version accepted for publication by CP

Universal optimality of the E8E_8 and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers $n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory.Comment: 95 pages, 6 figure

The monomial representations of the Clifford group

We show that the Clifford group - the normaliser of the Weyl-Heisenberg group - can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions for SIC vectors, and has other applications to SICs and to Mutually Unbiased Bases. Exact solutions for SICs in dimension 16 are presented for the first time.Comment: Additional author and exact solutions to the SIC problem in dimension 16 adde

CONSERVATION LAWS OF A NONLINEAR INCOMPRESSIBLE TWO-FLUID MODEL

We study the conservation laws of the Choi-Camassa two-fluid model (1999) which is developed by approximating the two-dimensional (2D) Euler equations for incompressible motion of two non-mixing fluids in a channel. As preliminary work of this thesis, we compute the basic local conservation laws and the point symmetries of the 2D Euler equations for the incompressible fluid, and those of the vorticity system of the 2D Euler equations. To serve the main purpose of this thesis, we derive local conservation laws of the Choi-Camassa equations with an explicit expression for each locally conserved density and corresponding spatial flux. Using the direct conservation law construction method, we have constructed seven conservation laws including the conservation of mass, total horizontal momentum, energy, and irrotationality. The conserved quantities of the Choi-Camassa equations are compared with those of the full 2D Euler equations of incompressible fluid. We review periodic solutions, solitary wave solutions and kink solutions of the Choi-Camassa equations. As a result of the presence of Galilean symmetry for the Choi-Camassa model, the solitary wave solutions, the kink and the anti-kink solutions travel with arbitrary constant wave speed. We plot the local conserved densities of the Choi-Camassa model on the solitary wave and on the kink wave. For the solitary waves, all the densities are finite and decay exponentially, while for the kink wave, all the densities except one are finite and decay exponentially

Dynamics of early planetary gear trains

A method to analyze the static and dynamic loads in a planetary gear train was developed. A variable-variable mesh stiffness (VVMS) model was used to simulate the external and internal spur gear mesh behavior, and an equivalent conventional gear train concept was adapted for the dynamic studies. The analysis can be applied either involute or noninvolute spur gearing. By utilizing the equivalent gear train concept, the developed method may be extended for use for all types of epicyclic gearing. The method is incorporated into a computer program so that the static and dynamic behavior of individual components can be examined. Items considered in the analysis are: (1) static and dynamic load sharing among the planets; (2) floating or fixed Sun gear; (3) actual tooth geometry, including errors and modifications; (4) positioning errors of the planet gears; (5) torque variations due to noninvolute gear action. A mathematical model comprised of power source, load, and planetary transmission is used to determine the instantaneous loads to which the components are subjected. It considers fluctuating output torque, elastic behavior in the system, and loss of contact between gear teeth. The dynamic model has nine degrees of freedom resulting in a set of simultaneous second order differential equations with time varying coefficients, which are solved numerically. The computer program was used to determine the effect of manufacturing errors, damping and component stiffness, and transmitted load on dynamic behavior. It is indicated that this methodology offers the designer/analyst a comprehensive tool with which planetary drives may be quickly and effectively evaluated

Modelling the Flow and Transport Properties of Two-Dimensional Fracture Networks, Including the Effect of Stress

This thesis focuses on the effective hydraulic transmissivity of two-dimensional fracture networks in rocks. The main simulation tool used in this work is the discrete fracture network code NAPSAC. There are four main topics in this thesis: (1) estimating permeability from network properties, (2) comparing discrete fracture network with effective continuum models, (3) using DFN for hydro-mechanical coupled modelling, and (4) solute transport simulations. For fracture networks with uniform aperture, the permeability can be estimated using segment density, fracture density, and fracture lengths of the fracture network. For fracture networks with apertures directly proportional to their lengths, the individual conductance of each of the fracture segments was used to calculate an effective conductance for the whole network. The arithmetic mean of the segment conductance gives a good approximation for the effective conductance of the whole network. A series of effective continuum models of a fracture network were created using different element sizes, and their flow behaviours were compared against results obtained from discrete fracture network model. The permeability tensors of each of the elements in the effective continuum meshes were calculated using discrete fracture network methods. It was found that the flow through effective continuum model with any element size gave good agreement with the discrete fracture network results. Hydro-mechanical coupled simulations were carried out using NAPSAC, where the applied far field stresses are applied to each fractures independently. Simulations were then carried out using the distinct element code UDEC to justify the simplified physics used in NAPSAC. It was shown that for random 2D fracture networks under a range of loadings, NAPSAC and UDEC seem to predict similar overall flows. Different ways for modelling the effects of rock matrix diffusion were explored. The significance of rock matrix diffusion, as well as the diffusion distance, was linked to the magnitude of the pressure gradient across the fracture network. A semi-analytical method for estimating the diffusion distance was proposed: using the perimeter and the area of each of the matrix blocks, it is possible to estimate the diffusion distance using the 'shape factor' concept