Kaon nucleon scattering in lattice QCD

In this paper I discuss why one would want to study meson baryon interactions within the framework of lattice QCD, and what I have contributed to this area of research. I begin with a little background on QCD, and Chiral theories. I analyze Kaon Nucleon scattering, and calculate the Kaon Nucleon scattering length from data obtained using a lattice QCD simulation. Therefore, I show that it is possible to use Lattice QCD simulation to extract observables. This is important in areas where experiments are challenging or impossible with current technologies

Emergent particle-hole symmetry in spinful bosonic quantum Hall systems

When a fermionic quantum Hall system is projected into the lowest Landau level, there is an exact particle-hole symmetry between filling fractions $\nu$ and $1-\nu$. We investigate whether a similar symmetry can emerge in bosonic quantum Hall states, where it would connect states at filling fractions $\nu$ and $2-\nu$. We begin by showing that the particle-hole conjugate to a composite fermion `Jain state' is another Jain state, obtained by reverse flux attachment. We show how information such as the shift and the edge theory can be obtained for states which are particle-hole conjugates. Using the techniques of exact diagonalization and infinite density matrix renormalization group, we study a system of two-component (i.e., spinful) bosons, interacting via a $\delta$-function potential. We first obtain real-space entanglement spectra for the bosonic integer quantum Hall effect at $\nu=2$, which plays the role of a filled Landau level for the bosonic system. We then show that at $\nu=4/3$ the system is described by a Jain state which is the particle-hole conjugate of the Halperin (221) state at $\nu=2/3$. We show a similar relationship between non-singlet states at $\nu=1/2$ and $\nu=3/2$. We also study the case of $\nu=1$, providing unambiguous evidence that the ground state is a composite Fermi liquid. Taken together our results demonstrate that there is indeed an emergent particle-hole symmetry in bosonic quantum Hall systems.Comment: 10 pages, 8 figures, 4 appendice

CosmoLattice

This is the user manual for CosmoLattice, a modern package for lattice simulations of the dynamics of interacting scalar and gauge fields in an expanding universe. CosmoLattice incorporates a series of features that makes it very versatile and powerful: $i)$ it is written in C++ fully exploiting the object oriented programming paradigm, with a modular structure and a clear separation between the physics and the technical details, $ii)$ it is MPI-based and uses a discrete Fourier transform parallelized in multiple spatial dimensions, which makes it specially appropriate for probing scenarios with well-separated scales, running very high resolution simulations, or simply very long ones, $iii)$ it introduces its own symbolic language, defining field variables and operations over them, so that one can introduce differential equations and operators in a manner as close as possible to the continuum, $iv)$ it includes a library of numerical algorithms, ranging from $O(\delta t^2)$ to $O(\delta t^{10})$ methods, suitable for simulating global and gauge theories in an expanding grid, including the case of `self-consistent' expansion sourced by the fields themselves. Relevant observables are provided for each algorithm (e.g.~energy densities, field spectra, lattice snapshots) and we note that remarkably all our algorithms for gauge theories always respect the Gauss constraint to machine precision. In this manual we explain how to obtain and run CosmoLattice in a computer (let it be your laptop, desktop or a cluster). We introduce the general structure of the code and describe in detail the basic files that any user needs to handle. We explain how to implement any model characterized by a scalar potential and a set of scalar fields, either singlets or interacting with $U(1)$ and/or $SU(2)$ gauge fields. CosmoLattice is publicly available at www.cosmolattice.net.Comment: 111 pages, 3 figures and O(100) code file

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

The iterated Carmichael \lambda-function and the number of cycles of the power generator

Iteration of the modular l-th power function f(x) = x^l (mod n) provides a common pseudorandom number generator (known as the Blum-Blum-Shub generator when l=2). The period of this pseudorandom number generator is closely related to \lambda(\lambda(n)), where \lambda(n) denotes Carmichael's function, namely the maximal multiplicative order of any integer modulo n. In this paper, we show that for almost all n, the size of \lambda(\lambda(n)) is n/exp((1+o(1))(log log n)^2 log log log n). We conjecture an analogous formula for the k-th iterate of \lambda. We deduce that for almost all n, the psuedorandom number generator described above has at least exp((1+o(1))(log log n)^2 log log log n) disjoint cycles. In addition, we show that this expression is accurate for almost all n under the assumption of the Generalized Riemann Hypothesis for Kummerian fields. We also consider the number of iterations of \lambda it takes to reduce an integer n to 1, proving that this number is less than (1+o(1))(log log n)/log 2 infinitely often and speculating that log log n is the true order of magnitude almost always.Comment: 28 page