%% LyX 2.0.3 created this file.  For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[11pt,fleqn]{article}
\usepackage[latin9]{inputenc}
\setlength{\parskip}{0cm}
\setlength{\parindent}{0pt}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{esint}

\makeatletter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% template that does not use Revtex4
%%% but allows special fonts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% Please use this template.
%%% Edit it using e.g. Notepad
%%% Ignore the header (do not change it)
%%% Process the file in the Latex site 
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% Page setup
\topmargin -1.5cm      
\oddsidemargin -0.04cm   
\evensidemargin -0.04cm  
\textwidth 16.59cm
\textheight 24cm 
 
 


% Fonts
\usepackage{latexsym}\usepackage{bm}


% Math symbols I
\newcommand{\sinc}{\mbox{sinc}}
\newcommand{\const}{\mbox{const}}
\newcommand{\trc}{\mbox{trace}}
\newcommand{\intt}{\int\!\!\!\!\int }
\newcommand{\ointt}{\int\!\!\!\!\int\!\!\!\!\!\circ\ }
\newcommand{\ar}{\mathsf r}
\newcommand{\im}{\mbox{Im}}
\newcommand{\re}{\mbox{Re}}

% Math symbols II
\newcommand{\eexp}{\mbox{e}^}
\newcommand{\bra}{\left\langle}
\newcommand{\ket}{\right\rangle}

% Mass symbol
\newcommand{\mass}{\mathsf{m}} 
\newcommand{\Mass}{\mathsf{M}} 

% More math commands
\newcommand{\tbox}[1]{\mbox{\tiny #1}}
\newcommand{\bmsf}[1]{\bm{\mathsf{#1}}} 
\newcommand{\amatrix}[1]{\begin{matrix} #1 \end{matrix}} 
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}

% Other commands
\newcommand{\hide}[1]{}
\newcommand{\drawline}{\begin{picture}(500,1)\line(1,0){500}\end{picture}}
\newcommand{\bitem}{$\bullet$ \ \ \ }
\newcommand{\Cn}[1]{\begin{center} #1 \end{center}}
\newcommand{\mpg}[2][1.0\hsize]{\begin{minipage}[b]{#1}{#2}\end{minipage}}
\newcommand{\Dn}{\vspace*{3mm}}

% Figures
\newcommand{\putgraph}[2][0.30\hsize]{\includegraphics[width=#1]{#2}}

% heading
\newcommand{\heading}[1]{\begin{center} \Large {#1} \end{center}}
\newcommand{\auname}[1]{\begin{center} \bf Submitted by: #1 \end{center}}




\makeatother

\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\heading{E5966: Long Range Interaction Ising Model 1D, Domain Walls}
\auname{Yaakov Kleeorin}

\textbf{The problem:}

\Dn

Consider the one dimensional Ising model with the hamiltonian

\[
H=-\sum_{n,n'}J(n-n')\sigma(n)\sigma(n')
\]
with $\sigma(n)=\pm1$ at each site $n$ and long range interaction
$J(n)=b/n^{\gamma}$ with $b>0$. Find the energy of a domain wall
at $n=0$, i.e. all the $n<0$ spins are down and the others are up.
Show that the standard argument for the absence of spontaneous magnetization
at finite temperature fails if $\gamma<2$.

\Dn\Dn

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{The solution:}

\Dn

The standard argument for the absence of spontaneous magnetization
is that in $1D$, it is always favourable (for any finite Temperature)
to create a domain wall at some location (lets call it $n=0)$ thus
gaining entropy at the expense of a small energy increase.

~

$F=E-TS$ is the free energy we need to minimize. Since the energy
gain (in comparison to the ground state) in creating a domain wall
is $2J$ (in the standard argument) and the entropy becomes $\ln(N)$
where $N$ is the number of spins and also the number of positions
the wall can assume. the change in free energy is $\Delta F=2J-T\ln(N)$
and since $N$ is huge, this is negative for any finite temperature,
making this transition favorable.

~

%%%%%%%%%%%%%%%
\Dn

In our long range interaction model, the ground state energy is the
summing of all interactions in any pair of spins, all being the same
sign. We will count over all $N$ spins, making sure we dont double
count by demanding $n'>n$ in our sum:

\begin{equation}
E_{0}=-\sum_{n=-\infty}^{\infty}\sum_{n'=n+1}^{\infty}\frac{b}{(n'-n)^{\gamma}}
\end{equation}


We will not solve this sum.

%%%%%%%%%%%%%%
\Dn

When creating a domain wall at $n=0$ we flip the sign of the interaction
between every pair on opposite sides of the wall, thus paying a double
price (-1 goes to 1, the difference is 2). We sum over all $N$ spins
(we assume $N$ spins on every side, but obviously it doesnt have
to be and it doesnt matter):

\begin{equation}
E_{1}=E_{0}+2\sum_{n=0}^{N}\sum_{n'=1}^{N}\frac{b}{(n'+n)^{\gamma}}
\end{equation}


%%%%%%%%%%%%%
\Dn

In order to go to an integral form, we make $n$ continuous by going
to new variables representing spin location. We introduce spin separation
$a$ and define $x=na$ and $y=n'a$. $dx=dy=a$. 

\begin{equation}
E_{1}=E_{0}+2ba^{\gamma-2}\int_{0}^{Na}dx\int_{a}^{Na}dy\frac{1}{(x+y)^{\gamma}}
\end{equation}


This integral is solved differently for different $\gamma$.

%%%%%%%%%%%%%
\Dn

For $\gamma>2$:

\begin{equation}
\Delta E=2\frac{b}{(\gamma-1)(\gamma-2)}
\end{equation}


%%%%%%%%%%%%%
\Dn

For $\gamma=2$:

\begin{equation}
\Delta E=2b\ln(N)
\end{equation}


%%%%%%%%%%%%%
\Dn

Under the value of $\gamma=2$ the divergence in $N$ is even stronger.
For example the $1<\gamma<2$ case gives:

\begin{equation}
\Delta E=2N^{2-\gamma}\frac{b}{(\gamma-1)(\gamma-2)}
\end{equation}


%%%%%%%%%%%%%
\Dn

for the cases of $\gamma\leq2$ we see that it is not favorable, in
terms of free energy, to create a domain wall:

\begin{equation}
\Delta F=-T\ln(N)+2b\ln(N)
\end{equation}


\begin{equation}
\Delta F=-T\ln(N)+2bN^{2-\gamma}/(\gamma-1)(\gamma-2)
\end{equation}


in the first case it depends on the value of $T$. In the second case
it is never (as $N$ is huge) favorable to create domain walls, thus
breaking the argument.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
