Merges

main.tex

@@ -197,7 +197,7 @@ We can keep the same expressions as earlier with the same reduced dimensionless
 \end{align}
 And then our expression, with the changes becomes:
 \begin{equation}
-	\sigma(\kappa, \xi) = -i \frac{3 \sigma_0}{4} \frac{1}{\xi}\left[\int_{1 - \xi}^{1}\dd{\xi} \tanh(\frac{\xi + \xi' - \mu}{2 t}) I_1 + \int_{1}^{\infty} \dd{\xi'} \left( \tanh(\frac{\xi + \xi' - \mu}{2t}) I_1  - \tanh(\frac{\xi' - \mu}{2t})I_2 \right) \right]
+	\sigma(\kappa, \xi) = -i \frac{3 \sigma_0}{4} \frac{1}{\xi}\left[\int_{1 - \xi}^{1}\dd{\xi} \tanh(\frac{\xi + \xi' - \mu}{2 t}) I_1 + \int_{1}^{\infty} \dd{\xi'} \left( \tanh(\frac{\xi + \xi' - \mu}{2t}) I_1  - \tanh(\frac{\xi' - \mu}{2t})I_2 \right) \right] \label{eq:nam}
 \end{equation}
 with
 \begin{align}
@@ -210,22 +210,38 @@ with
 	g  &= \frac{\xi' \left( \xi + \xi'\right) + 1}{\sqrt{\xi'^2 - 1}\sqrt{(\xi + \xi')^2 - 1}}
 \end{align}
 
-\section{Noise calculation}
+\section{Mathematica implementation}
 
-For our noise calculation, we assume a material parameterised by a Debye frequency $\debye$ and the interaction parameter $N(0) V$.
+We want to dimensionally reduce these when we implement the code, by writing $\corr$, $\Delta$, $\omega_D$ and $T$ in units of $\Delta_0$:
-Because these aren't necessarily experimentally accessible, I tried to keep their values such that they lead to a physically reasonable $T_c$.
+\begin{align}
-In order to keep $N(0) V$ small and $\debye$ bigger than the other parameters, I chose $N(0)V = 0.25$ and $\debye = \SI{1e13}{\per\s}$.
+	o_D &= \frac{\omega_D}{\Delta_0} \\
-This leads to $T_{c0} = T_c(\mu = 0) = \SI{1.44e11}{\per\s}$, similar to the values used for the equilibrium Nam case.
+	m &= \frac{\corr}{\Delta_0} \\
+	d &= \frac{\Delta}{\Delta_0} \\
+	t &= \frac{T}{\Delta_0}
+\end{align}
+giving us
+\begin{equation}
+	\left[ N(0) V \right]^{-1} = \int_{- o_D}^{o_D} \frac{\dd{\varepsilon}}{\sqrt{d^2 + \varepsilon^2}} \tanh{\frac{\sqrt{d^2 + \varepsilon^2} - m}{2 t}}
+\end{equation}
+\begin{equation}
+	n = \int_0^\infty \dd{\epsilon} \left( \frac{1}{1 + \exp(\frac{\sqrt{d + \varepsilon^2} - m}{t})} - \frac{1}{1 + \exp(\frac{\sqrt{d^2 + \varepsilon^2}}{t})} \right)
+\end{equation}
 
-\begin{enumerate}
+We can look at the Nam conductivity now:
-	\item Use the Owen Scalapino coupled integral equations \cref{eq:gap,eq:n}, find $\mu$ and $\Delta$ for fixed $n$.
+\begin{figure}[htp]
-	\item Find the expected gap from the approximation in OS, $T_c(n) \approx (1 - 4n) T_{c0}$.
+	\centering
-	If $T > T_c(n)$, then the calculation is skipped (a more complete handling would use either the Lindhard form or use a Nam expression that's been extended to $\Delta = 0$).
+	\includegraphics[width=14cm]{realpartofconductivity}
-	This is necessary because the coupled integral equations are very hard to solve
+	\caption{$\frac{\Re[\sigma_{SC}]}{\sigma_N}$} \label{fig:real}
-	\item Using the modified Nam equations, calculate the dielectric function and create the approximated interpolation form, similar to the equilibrium case.
+\end{figure}
-	\item Calculate the noise as usual.
+\begin{figure}[htp]
-\end{enumerate}
+	\centering
+	\includegraphics[width=14cm]{imagpartofconductivity}
+	\caption{$\frac{\Im[\sigma_{SC}]}{\sigma_N}$} \label{fig:imag}
+\end{figure}
 
+As there is as yet no filtering based on the free energy, this is messy, so the comparison to \cref{fig:osRepro1b} is necessary to see which combinations of $T$ and $n$ are valid.
+However, this is still quite numerically unstable, even with the dimensional reduction procedure described above for $n$ and $\corr$.
+The information that higher $n$ leads to lower $\sigma$ is expected, and does suggest that the implementation is correct, although still very noisy.
 \printbibliography
 
 \end{document}