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Added info on parameter space

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This commit is contained in:
Deepak Mallubhotla
2021-01-29 09:57:06 -06:00
parent dbcbc9722e
commit 14d7320bc5
2 changed files with 64 additions and 8 deletions
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@@ -45,7 +45,7 @@ $(USED_FIGS): $(FIG_DIR)/%.jpg: $(SCRIPT_DIR)/%.wls | $(FIG_DIR)
pdfs/os-free-energy-notes.pdf: main.tex bibliography.bib $(USED_FIGS)| $(PDF_DIR)
$(LATEXMK) $(<F)
cp $(@F) $@
cp main.pdf $@
# $(OUTPUTS): pdfs/%.pdf: tex/%.tex main.tex bibliography.bib | $(PDF_DIR)
# cd $(<D); $(LATEXMK) $(<F)

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main.tex
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@@ -97,13 +97,17 @@ I don't know if this assumption is correct, but I believe it is (and it handles
This assumes that we don't consider quasiparticles with energy above $\debye$ as participating in this interaction, and that $\debye \ll \epsF$, which are both reasonable assumptions for this problem.
This means that our term $A$ is then
\begin{align}
A &= \int_{0}^{\debye} 2 \dof \dd{\epsilon_k} \epsilon_k \left(1 + 2 f_k\frac{\epsilon_k}{E_k} - \frac{\epsilon_k}{E_k} \right) \\
&= 2 \dof \int_{0}^{\debye} \dd{\epsilon_k} \epsilon_k \left(1 + 2 f_k\frac{\epsilon_k}{E_k} - \frac{\epsilon_k}{E_k} \right) \\
&= 2 \dof \int_{0}^{\debye} \dd{\epsilon_k} \epsilon_k \left(1 + 2 \left( \frac{1}{1 + \exp\frac{E_k - \corr}{T}} \right)\frac{\epsilon_k}{E_k} - \frac{\epsilon_k}{E_k} \right) \\
&= \dof\debye^2 + 2 \dof \int_{0}^{\debye} \dd{\epsilon_k} 2 \epsilon_k \left( \frac{1}{1 + \exp\frac{E_k - \corr}{T}} \right)\frac{\epsilon_k}{E_k} - \frac{\epsilon_k^2}{E_k} \\
&= \dof\debye^2 + 2 \dof \int_{0}^{\debye} \dd{\epsilon_k} \frac{\epsilon_k^2}{E_k} \left( \frac{2}{1 + \exp\frac{E_k - \corr}{T}} - 1\right) \\;
&= \dof\debye^2 + 2 \dof \int_{0}^{\debye} \dd{\epsilon_k} \frac{\epsilon_k^2}{E_k} \frac{2}{1 + \exp\frac{E_k - \corr}{T}} - 2 \dof \int_0^{\debye} \dd{\epsilon_k} \frac{\epsilon_k^2}{\sqrt{k^2 + \Delta^2}} \\
&= \dof\debye^2 + 2 \dof \int_{0}^{\debye} \dd{\epsilon_k} \frac{\epsilon_k^2}{E_k} \frac{2}{1 + \exp\frac{E_k - \corr}{T}} - \dof \left( \debye \sqrt{\Delta^2 + \debye^2} + \Delta^2 \log \frac{\Delta}{\debye + \sqrt{\Delta^2 + \debye^2}} \right)
A &= \int_{-\debye}^{\debye} \dof \dd{\epsilon_k} \epsilon_k \left(1 + 2 f_k\frac{\epsilon_k}{E_k} - \frac{\epsilon_k}{E_k} \right) \\
&= \dof \int_{-\debye}^{\debye} \dd{\epsilon_k} \epsilon_k \left(1 + 2 f_k\frac{\epsilon_k}{E_k} - \frac{\epsilon_k}{E_k} \right) \\
&= \dof \int_{-\debye}^{\debye} \dd{\epsilon_k} \epsilon_k \left(1 + 2 \left( \frac{1}{1 + \exp\frac{E_k - \corr}{T}} \right)\frac{\epsilon_k}{E_k} - \frac{\epsilon_k}{E_k} \right)
\end{align}
We drop the first term, as it is odd.
The next two terms are even, as they only depend on $\epsilon_k^2$.
\begin{align}
A &= 2 \dof \int_{0}^{\debye} \dd{\epsilon_k} 2 \epsilon_k \left( \frac{1}{1 + \exp\frac{E_k - \corr}{T}} \right)\frac{\epsilon_k}{E_k} - \frac{\epsilon_k^2}{E_k} \\
&= 2 \dof \int_{0}^{\debye} \dd{\epsilon_k} \frac{\epsilon_k^2}{E_k} \left( \frac{2}{1 + \exp\frac{E_k - \corr}{T}} - 1\right) \\;
&= 2 \dof \int_{0}^{\debye} \dd{\epsilon_k} \frac{\epsilon_k^2}{E_k} \frac{2}{1 + \exp\frac{E_k - \corr}{T}} - 2 \dof \int_0^{\debye} \dd{\epsilon_k} \frac{\epsilon_k^2}{\sqrt{k^2 + \Delta^2}} \\
&= 2 \dof \int_{0}^{\debye} \dd{\epsilon_k} \frac{\epsilon_k^2}{E_k} \frac{2}{1 + \exp\frac{E_k - \corr}{T}} - \dof \left( \debye \sqrt{\Delta^2 + \debye^2} + \Delta^2 \log \frac{\Delta}{\debye + \sqrt{\Delta^2 + \debye^2}} \right)
\end{align}
Similarly, we can assume that $V_{kl} \rightarrow - V$ and $\Delta_k \rightarrow \Delta$, as part of our standard weak coupling assumption set, giving us
@@ -129,6 +133,58 @@ We can simplify this slightly further:
C &= 4 T \int_{0}^{\debye} \dof \dd{\epsilon_k} \log({\exp\frac{-E_k + \corr}{T}} + 1) + f_k \frac{E_k - \corr}{T}\\
\end{align}
\section{Owen-Scalapino Gap Calculation}
We are interested in being able to calculate the gap for different quasiparticle densities $n$, as shown in Owen-Scalapino's figure 1b in~\cite{OwenScalapino}.
To do this, we use the modified gap equation:
\begin{equation}
\left[ N(0) V \right]^{-1} = \int_{- \omega_D}^{\omega_D} \frac{\dd{\epsilon_k}}{\sqrt{\Delta^2 + \epsilon_k^2}} \tanh{\frac{\sqrt{\Delta^2 + \epsilon_k^2} - \corr}{2 T}}. \label{eq:gap}
\end{equation}
OS's equation 7 is the definition of a parameter $n$ measuring the excess quasiparticle density in units of $4 N(0) \Delta_0$:
\begin{equation}
n = \frac{1}{\Delta_0} \int_0^\infty \dd{\epsilon} \left( \frac{1}{1 + \exp(\frac{\sqrt{\Delta^2 + \epsilon_k^2} - \corr}{T})} - \frac{1}{1 + \exp(\frac{\sqrt{\Delta^2 + \epsilon_k^2}}{T})} \right). \label{eq:n}
\end{equation}
\begin{figure}[htp]
\centering
\includegraphics[width=14cm]{osGapVsTReproduction}
\caption{Reproduction of OS graph 1b, with parameters described in the text.} \label{fig:osRepro1b}
\end{figure}
First, to find $\Delta_0$, we solve the standard gap equation with $T=0$ and $\corr = 0$.
For \cref{fig:osRepro1b}, this is done for $N(0) V = 0.2$ and $\debye = 100$.
The values chosen have no special significance, but keep our energy scale at order 1, with $\Delta_0 = 1.34765$ and $T_c = 0.764083$.
Then, we numerically solve the pair of equations \cref{eq:gap} and \cref{eq:n} for the free parameters $\Delta$ and $\corr$ for our chosen value of $T$, keeping $N(0)$, $V$ and $\debye$ fixed.
Playing with the parameters slightly, I had success over the important part of the solution space starting at $(\Delta, \corr) = (\frac{\debye}{\sinh(\left(N(0) V\right)^{-1})}, n^2 \Delta_0)$.
A more sophisticated analysis could most likely be done to fix the starting space more accurately and avoid the messy behaviour in the region where the free energy is lower for the normal state.
However, as we discussed, for sufficiently small $n$ (n \lessapprox 0.1) this is unlikely to matter, as for small $T \rightarrow 0$ there is no return to the normal state, and for large $T$ the critical temperature is close to $T_c$ for $n = 0$ (OS find that the critical temperature only drops to around $0.9 T_{c, 0}$ for $n = 0.02$, judging by their plot).
The result of solving this pair of equations is the current correction to the chemical potential $\corr$ and the gap $\Delta$, which can be used in the Nam expression.
The numerical equations are not overly slow;
for the starting points given in the energetically favourable region the descent to the solution seems easy for Mathematica to handle.
\section{Nam usage}
Because the earlier result matches OS's curve, we are more confident that we can use this as a calculation for $\Delta$ in the Nam code.
However, the Nam code takes in the following parameters:
\begin{enumerate}
\item $\omega$ in \si{\per\s}
\item The Plasma frequency $\omega_p$, in \si{\per\s}
\item The impurity collision frequency $\tau$, in \si{\per\s}
\item $\vf$, in \si{\m\per\s}
\item A critical temperature $T_c$, in \si{\per\s}
\item The dipole moment $d$, in $\si{\coulomb \m}$
\end{enumerate}
The point here is the critical temperature $T_c$, which is essentially only used for calculating the gap $\Delta$ via $\Delta = 3.06 \sqrt{T_c \left( T_c - T\right)}$.
Because this is no longer the expression of interest, replacing this with the OS gap package should be enough to fulfill that role.
The OS gap package takes in its own parameters:
\begin{enumerate}
\item $T$, $\corr$ $\debye$ and the interaction $V$ in \si{\per\s}
\item $N(0)$ the density of states in \si{\s}.
\end{enumerate}
\printbibliography
\end{document}