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\section*{Introduction}

\begin{figure}[hbt]
  \label{eg-phase-diagram}
  \includegraphics[width=\columnwidth]{Phase-diag2.png}
  \caption{An example phase diagram of water showing different phase transitions and phases separating them. The critical point a the end of the liquid-gaseous phase is clearly shown, beyond this point a supercritical fluid is observed and the thermodynamic variables can be varied without a corresponding discontinuity in a state function. Source \parencite{wiki:phase-digram-eg}}
\end{figure}

In statistical mechanics phase transitions occur when there is a discontinuity singularity in the free energy or one of its derivatives, such as specific heat capacity, density, or other thermodynamic observables. Examples include the liquid-gas transition, the appearance super conductors, or ferromagnetic materials at the Curie Temperature $T_C$. Phase diagrams such as the one shown in Figure \ref{eg-phase-diagram} are used to represent these transitions, with the lines representing where discontinuities are present.

As the magnitude of the discontinuity varies across the extent of the phase transition, it will often reach a point such that it becomes $0$ and thus the phase transition line abruptly ends. Beyond this point one can continuously transition between different phases without crossing the now terminated phase transition line.

Of these transitions one of the easiest to model is that of ferromagnets at their Curie Temperature. This involves heating a ferromagnetic material up, possibly in the presence of an external magnetic field $H$, and observing the magnetisation $M$ of the material itself, the order parameter of the system. All phase transitions occur when $H = 0$ due to the symmetry of the system to sign reversal of the external magnetic field, making this a useful model for exploring critical points themselves.

This has birthed a number of models for ferromagnetic materials around their critical point including: the Ising Model which models the magnetic as a lattice of interacting magnetic spin sites\fnmark{ising-model-generalisations}; Mean Field Theory which models interactions of spins with a hypothetical mean field of all other spins in the material; and Renormalisation Group Theory which is a more general mathematical framework for understanding systems which can exhibit scale invariance.

\fntext{ising-model-generalisations}{
There are a number of generalisations of the Ising Model including the Heisenberg Model, XY model, Potts Model, among others which increase the possible states of spins. These however are not discussed in this paper.
}

% https://en.wikipedia.org/wiki/Classical_Heisenberg_model
% https://en.wikipedia.org/wiki/Classical_XY_model
% https://en.wikipedia.org/wiki/Potts_model

%7. **Renormalization Group Theory**: The Renormalization Group (RG) theory is a powerful mathematical framework for studying the critical behavior of ferromagnetic materials. It involves a systematic procedure for studying the behavior of systems at different length scales, enabling the determination of critical exponents, fixed points, and the universality of phase transitions. The RG theory has significantly advanced our understanding of critical phenomena in ferromagnetic systems and beyond.

Of these we will focus most on the Ising Model, which we will use to model ferromagnetic materials under various conditions, comparing it to Mean Field Theory. (\TKK will we mention renormalisation), with the aim of extracting quantitative data (\TKK we are not mentioning critical) about the behaviour of systems at their critical points (\TKK Do we want to mention universality classes? Is that a thing which we can?).

In the discussion we will explore the theory behind the models discussed, as well as phase transitions and critical points themselves. This will be followed by a brief discussion of the Monte-Carlo (\TKK do I need to mention this before) method employed in the simulation and relevant statistical methods. Finally we will present data from these simulations, comparing it to the result from theory and concluding on the effectiveness of our model in desired exploration.




%All transitions occur at zero magnetic field, H = 0, because of the symmetry of a ferromagnet to reversals in the field. The additional symmetry means that it is often easier to work in magnetic language and we shall do so throughout most of this book.

%Of these te 

%These phase transitions are occur at phase transition lines on phase diagrams, which termi

%A phase transition occurs when there is a singularity in the free energy or one of its derivatives. What is often visible is a sharp change in the properties of a substance. The transitions from liquid to gas, from a normal conductor to a superconductor, or from paramagnet to ferromagnet are common examples. The phase diagram of a typical fluid is shown in Fig. 1.1. 

%As the temperature and pressure are varied water can exist as a solid, a liquid, or a gas. Well-defined phase boundaries separate the regions in which each state is stable. Crossing the phase boundaries there is a jump in the density and a latent heat, signatures of a first-order transition. Consider moving along the line of liquid-gas coexistence. As the temperature increases the difference in density between the liquid and the gas decreases continuously to zero as shown in Fig. 1.2. It becomes zero at the critical point beyond which it is possible to move continuously from a liquid-like to a gas-like fluid. The difference in densities, which becomes non-zero below the critical temperature, is called the order parameter of the liquid- gas transition. Seen on the phase diagram of water the critical point looks insignificant. However, there are clues that this might not be the case. Fig. 1.3 shows the specific heat of argon measured along the critical isochore, p = p-. There is a striking signature of criticality: the specific heat diverges and is infinite at the critical temperature itself. Analogous behaviour is seen in magnetic phase transitions. The phase diagram of a simple ferromagnet is shown in Fig. 1.4. Just as in the case of liquid—gas coexistence there is a line of first-order transitions ending in a critical point. All transitions occur at zero magnetic field, H = 0, because of the symmetry of a ferromagnet to reversals in the field. The additional symmetry means that it is often easier to work in magnetic language and we shall do so throughout most of this book.

%In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. One example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas cannot be liquefied by pressure alone. At the critical point, defined by a critical temperature Tc and a critical pressure pc, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures, and the ferromagnet–paramagnet transition (Curie temperature) in the absence of an external magnetic field.[2]
%
%Ferromagnetic systems are a useful and widely used model for phase transitions
%
%Phase diagrams, critical points
%
%There are a number of models for ferromagnetic materials critical,

\section*{Discussion}

Here we will discuss the theory behind the different models mentioned as well as phase transitions themselves.

\subsection*{Thermodynamics}

\begin{figure}[hbt]
  \label{thermodynamic-variables}
  \includegraphics[width=\columnwidth]{thermodynamic-variables.png}
  \caption{The relationship between the partition function and various thermodynamic observables. From \parencite{yeomansStatisticalMechanicsPhase1992}}
\end{figure}

Whichever model we apply standard statistical mechanics to give us relations between the partition function, itself derived from the energy of the system, and other thermodynamic observables. These relationships can be seen in Figure \ref{thermodynamic-variables}, from \cite[p. 17]{yeomansStatisticalMechanicsPhase1992}.

We obtain the magnetisation and energy from our chosen model, deriving the specific heat capacity, and magnetic susceptibility (labeled as \enquote{Isothermal susceptibility} in the diagram) from these base quantities by the equations,

\begin{align}
c    &= \frac{\p}{\p T} \left\langle\frac{E}{N}\right\rangle \\
\chi &= \frac{\p}{\p h} \left\langle\frac{M}{N}\right\rangle.
\end{align}

where $E$ is the total energy of the system, $M$ is total magnetisation of the system and then $N$ is the number of cells in the Ising Model. For convenience we write $m = \frac MN$.

We can then apply the Fluctuation-dissipation theorem \cite{???} to transform the derivatives in these equations into expressions about the thermodynamic variance of the property at equilibrium. These are much more useful for the purposes of simulation and are given by,

\begin{align}
c    &= \frac{1}{N k_B T^2} \Var(E) \\
\chi &= \frac{N}{k_B T} \Var(m).
\end{align}

Which in terms of our dimensionless energy gives us \TKK.

\subsection*{The Ising Model}

%The Ising model was invented by the physicist Wilhelm Lenz (1920), who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by Ising (1925) alone in his 1924 thesis;[2] it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by Lars Onsager (1944). It is usually solved by a transfer-matrix method, although there exist different approaches, more related to quantum field theory.

The Ising Model will be the main model used to explore the behaviour of our system. It was developed by \TKK in \TKK for \TKK.

It is comprised of a grid of spin-sites which can take a value of either $+1$ or $-1$, representing up or down spins respectively. These spins interact in a local manner with their nearest neighbours in the grid. We write the interaction energy as,

\begin{equation}
E = -J\sum_{\ip{ij}} s_i s_j
\end{equation}

where $J > 0$ is the strength of the interaction, $\sum_{\ip{ij}}$ represents a sum over all pairs of nearest neighbours, with the neighbours being written as sites $i, j$\fnmark{sites}. We can see trivially that this energy expression preferences same spin neighbours having $(\pm 1)^2 = 1 \implies E_i < 0$.

\fntext{sites}{When performing a calculation a \emph{site} is mapped to two integers $(x, y) \in \N^2$ representing the location of this site in the overall grid, however in mathematics we abstract this to a single index for conceptual ease.}

In the fully general model this adds to energy from the the externally imposed magnetic field $H$ giving a total energy of,

\begin{equation*}
  E = -J\sum_{\ip{ij}} s_i s_j - H \sum_{i} s_i.
\end{equation*}

From these expressions we can obtain the partition function, in the standard manner as,

\begin{equation}
  Z(T) = \sum_{\omega \in \Omega} \exp\left(\frac{E(\omega)}{k_B T}\right)
\end{equation}

where $\omega \in \Omega$ is a particular micro-state, corresponding to a particular set of choices for $s_i \in \set{\pm 1}$. This has a total phase space of $\Omega = \set{1, -1}^N$ where $N$ is the number of cells in the grid.

\subsection*{Mean Field Theory}

Whereas the Ising Model restrict interactions of spins to nearest-neighbours, Mean Field Theory, also known as Curie-Weiss theory (\TKK consistent capitalisation of terms), instead chooses to average the effect that each other site on the cell has on a chosen site. In this sense a site interacts with a \enquote{mean field}, representing all other sites in the lattice. 

Taking this spin to be $s_0 = \pm 1$ we can determine its energy as

\begin{equation}
  E(s_0) = -s_0\left(J\sum_{\ip{0 j}} s_j + H \right)
\end{equation}

where this time one site, $0$, is fixed in our nearest neighbour sum. On a square 2D grid this sum has a constant number of terms, $q = 4$ which allows us to re-write this expression as,

\begin{equation}
  E(s_0) = -s_0(qJm + H) - Js_0 \sum_{\ip{0 j}}(s_j - m)
\end{equation}

where $s_j - m$ is the effect of the variation of site $s_j$ from the overall mean on the energy of $s_0$. If we take this variation to be $0$ for all $s_j$ then we obtain,

\begin{equation}
  E(s_0) = -s_0(qJm + H)
\end{equation}

which is an expression containing no cross terms and hence allows for treatment as a non-interacting thermodynamic system with

\begin{equation}
\begin{split}
  Z_1(s_0) &= \sum_{s \in \set{\pm 1}} \exp(-\beta E(s)) \\
         &= 2\cosh({(\beta(qJm + H))}) \\ 
         &= 2\cosh({\beta_0(qm + H)})
\end{split}
\end{equation}

where $\beta_0$ is the dimensionless $\beta$ discussed in \TKK.

\begin{equation}
  Z = (Z_1(s_0))^N
\end{equation}

where $N$ is the number of sites in the system. Applying the identities seen in Figure \ref{thermodynamic-variables} we obtain an expression for the magnetisation as,

\begin{equation}
  m = \tanh(\beta_0(qJm + H))
\end{equation}

which must be solved to find a self consistent value of $m$ for a given $H, \beta_0$.

\subsection*{Phase transitions and Critical Points}

%
%In dimensions greater than four, the phase transition of the Ising model is described by mean-field theory. The Ising model for greater dimensions was also explored with respect to various tree topologies in the late 1970’s, culminating in an exact solution of the zero-field, time-independent Barth (1981) model for closed Cayley trees of arbitrary branching ratio, and thereby, arbitrarily large dimensionality within tree branches. The solution to this model exhibited a new, unusual phase transition behavior, along with non-vanishing long-range and nearest-neighbor spin-spin correlations, deemed relevant to large neural networks as one of its possible applications.
%
%The Ising problem without an external field can be equivalently formulated as a graph maximum cut (Max-Cut) problem that can be solved via combinatorial optimization.

%A phase transition is defined as a defined by a singularity in thermodynamic potential or its derivatives. 
%
%When crossing a critical point we use an order parameter, in this case $m$ the mean magnetisation, to characterise which phase we in. For the ordered phase this parameter has a non-zero value, in the ordered phase it has a zero value, up to thermodynamic variance.
%
%A phase transition occurs when there is a singularity in the free energy or one of its derivatives. What is often visible is a sharp change in the properties of a substance. The transitions from liquid to gas, from a normal conductor to a superconductor, or from paramagnet to ferromagnet are common examples. The phase diagram of a typical fluid
%
%
%
%In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point.
%
%
%A wide variety of physical systems undergo rearrangements of their internal constituents in response to the thermodynamic conditions to which they are subject. Two classic examples of systems displaying such phase transitions are the ferromagnet and fluid systems. As the temperature of a ferromagnet is increased, its magnetic moment is observed to decrease smoothly, until at a certain temperature known as the critical temperature, it vanishes altogether.


\section*{Method}

\begin{enumerate}
	\item Monte-Carlo
	\item relevant statistical methods
	\item Anything about how we fit shit I suppose.
	\item Breaking symmetry with the initial $s_i = -1$ state.
\end{enumerate}
