\begin{document}

\title{Nonuniversal Effects in the Homogeneous Bose Gas}

\author{Shawn Hermans}

\address{Saint John's University, Collegeville, MN 56321}

%Professor Eric Braaten\thanks{{\tt The Ohio State University}} %Professor Thomas Kirkman\thanks{{\tt Saint John's University}} \author{Advisor: Professor Eric Braaten}

\address{The Ohio State University, Columbus, OH 43210}

\maketitle

\begin{abstract}

In 1924 Albert Einstein predicted the existence of a special type of matter now known as Bose-Einstein condensation. However, it was not until 1995 that simple BEC (Bose-Einstein condensation) was observed in a low-density Bosonic gas. This recent experimental breakthrough has led to renewed theoretical interest in BEC. The focus of my research is to more accurately determine basic properties of homogeneous Bose gases. In particular nonuniversal effects of the energy density and condensate fraction will be explored. The validity of the theoretical predictions obtained is verified by comparison to numerical data from the paper \begin{it}Ground State of a Homogeneous Bose Gas: A Diffusion Monte Carlo Calculation \end{it} by Giorgini, Boronat, and Casulleras. \end{abstract}

%\dedicate{To my parents for their supporting me through college, %to God for all the mysteries of physics, and to Jammie for her %unconditional love.}

%\newpage

%\tableofcontents

\newpage

\section{Introduction}

The Bose-Einstein condensation of trapped atoms allows the experimental study of Bose gases with high precision. It is well known that the dominant effects of interactions between the atoms can be characterized by a single number $a$ called the S-wave scattering length. This property is known as \begin{it}universality\end{it}. Increasingly accurate measurements will show deviations from universality. These effects are due to sensitivity to aspects of the interatomic interactions other than the scattering length. These effects are known as \begin{it}nonuniversal\end{it} effects. Intensive theoretical investigations into the homogeneous Bose gas revealed that properties could be calculated using a low-density expansion in powers of $\sqrt{na^3}$, where $n$ is the number density. For example the energy density has the expansion \begin{equation}

\frac{E}{N} = \frac{2 \pi na {\hbar}^2}{m} \Bigg( 1 + \frac{128}{15\sqrt{\pi}}\sqrt{na^3} + \frac{8(4\pi-3\sqrt{3})}{3}na^3 (\ln(na^3)+c) + ... \Bigg) \label{en}

\end{equation}

The first term in this expansion is the mean-field approximation and was calculated by Bogoliubov \cite{Bog}. The corrections to the mean-field approximation can be calculated using perturbation theory. The coefficient of the $(na^3)^{3/2}$ term was calculated by Lee, Huang, and Yang \cite{LHY} and the last term was first calculated by Wu \cite{wu}. Hugenholtz and Pines \cite{hp} have shown that the constant $c_1$ and the higher-order terms in the expansion are all nonuniversal. Giorgini, Boronat, and Casulleras \cite{GBC} have studied the ground state of a homogeneous Bose gas by exactly solving the N-bodied Schr"odinger (to within statistical error) using a diffusion Monte Carlo method.

In section II of this paper, theoretical background relevant to this problem is presented. Section III is a brief summary of the numerical data from Giorgini, Boronat, and Casulleras. Section IV briefly presents the higher-order approximations and Section V analyzes the higher order approximations by fitting to data from Ref. \cite{GBC}. Section VI presents conclusions to date and prospects for future analysis. \section{Theoretical Background}

This section discusses the background relevant to understanding the calculations and theory discussed later in the paper.

\subsection{Qualitative Properties of Bose-Einstein Condensation}

All elementary particles fall into two exclusive classes, bosons and fermions. The...