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\begin{document}

\begin{center}
$\ $
\vspace{-4\baselineskip}
\\
\textbf{\Large Advanced statistical mechanics II}\\
{\large (SS 16, FU Berlin)}\\ [0.5em]
\textbf{Problem sheet 8}\\
Due date: May 21, 2016 
\end{center}


\subsection*{Problems}
\begin{enumerate}
\setcounter{enumi}{21}

%%%=====================================================================
%%%========================== 1111111111111 =============================
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\item {\bf Variations on Gaussian states (1+1.5+1.5+1+1+1+2+2+1*+1*=11+2*)}\\
In this exercise we will get acquainted with thermal states of free systems.
To start with, consider a single fermion described by a state in $\mathcal H_k=\text{span}\{\ket{0_k},\ket{1_k}\}$ with $c_k, c_k^\dagger, N_k=c_k^\dagger c_k$ being the annihilation, creation and numbering operators respectively.
\begin{enumerate}
\item Use the definition of the exponential of an operator to show that 
\begin{equation}
\quad N_k(a)=e^{\ln(1+a)N_k}=\id+aN_k\;.
\end{equation}
\item By taking  appropriate limits show that 
\begin{equation}
\hspace{-2cm}\lim_{a\rightarrow -1^+}\frac{N_k(a)}{\tr[N_k(a)]}=\id-N_k \quad\text{and}\quad \lim_{a\rightarrow \infty }\frac{N_k(a)}{\tr[N_k(a)]}=N_k\quad\text{and}\quad \lim_{a\rightarrow 0}\frac{N_k(a)}{\tr[N_k(a)]}=\id/2\;.
\end{equation}
\item Argue that $N_k=\ketbra{0_k}{0_k}$ and $\id-N_k=\ketbra{1_k}{1_k}$. What is $N_k^2$ and $(\id-N_k)^2$? How would you summarize these observations?
\end{enumerate}
Let us now consider a system of $L$ fermions with the notation as in previous exercises. For $N=\sum_{k=1}^L N_k$ define
 $N(a)=e^{\ln(1+a)N}$.
 It can be shown (cf. [1]) that any free fermionic thermal state is of this form and such states are also called Gaussian states.
 \begin{enumerate}[resume]
\item Argue that you may use the Baker-Hausdorff relation to show that $N(a)=\prod_{k=1}^L N_k(a)$. Calculate $\tr[N(a)]$. What physical quantity did you just calculate?
\item Use the above result to show that 
\begin{align}\sandwich{n_1,\ldots,n_L}{N(a)}{n_1,\ldots,n_L}&=e^{\ln(1+a)\sum_{k=1}^L \sandwich{n_1,\ldots,n_L}{N_k}{n_1,\ldots,n_L}}\label{eq}\\
&=e^{\ln(1+a)\sum_{k=1}^L n_k}\label{eq2}\;.
\end{align}
When does $\sandwich{\psi}{\exp(-\beta H)}{\psi}=\exp(-\beta \sandwich{\psi}{H}{\psi})$ hold? 
\item  Show \eqref{eq} and \eqref{eq2} without using the product factorization, but rather by explicitly using the series of the exponential.
\item Prove that $ \rho_G(a)=N(a)/\tr[N(a)]$ represents a quantum state for $a\in [-1,\infty]$.
\item Calculate the partition function of a Hamiltonian $H=-\sum_k N_k$ at a temperature $T$ by evaluating the trace in the energy eigenbasis. Compare your results with the method used above.
\item \bonus $N(-2)$
\item \bonus Show that $N(a)^2=N(2a+a^2)$. Find all fixed points of this equation i.e. values of $a$ s.t. $N(a)^2=N(a)$.
\end{enumerate}

[1] A. Kitaev, "Anyons in an exactly solved model and beyond", \href{https://arxiv.org/abs/cond-mat/0506438}{Arxiv: cond-mat/0506438} 


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\item {\bf Fermi energy and neutron stars (.5+.5+1+3+1+1+1+5*=8+5*)}\\
The concept of Fermi energy is of paramount importance to solid-state physics, but it is essentially a corollary to the exclusion principle and correct use of quantum mechanics and therefore it can be applied in different settings e.g. neutron stars.

\begin{enumerate}
\item Show that
%a degenerate gas of $N$ fermions with mass $m$ occupying a 3 dimensional volume $V$ has the Fermi energy given by
\begin{align}
  n_k=\frac{1}{e^{\beta(E_k-\mu)}+1}\xrightarrow{T\rightarrow 0} \theta(\mu-E_k)=\begin{cases}0& E_k>\mu\\
  1&E_k<\mu
  \end{cases}
\end{align}
\end{enumerate}
The Fermi energy $E_F$ for $N$ fermions $@ T=0$ is defined implicitly by
\begin{align}
  N=\sum_{k:\,E_k<E_F} 1\;.
\end{align}
\begin{enumerate}[resume]
\item Use the standing wave relation $\Delta k=\frac{(2\pi)^3}{V}$ and the fact that for $N,V\rightarrow \infty$ \& $n_f=N/V=const$ we have $\sum_{k:\,E_k<E_F} 1 \rightarrow \frac{(2s+1)}{\Delta k} \int_{\{k: E_k< E_F\}} d^3 k$ to show that 
\begin{align}
  k_F=\left(\frac{6\pi^2}{2s+1}n_f\right)^{1/3}
\end{align}
\end{enumerate}
We now consider a charge neutral stellar object of radius $R=10$km and $M=1.4 M_S$ where $M_S$ is the mass of the star closest to our planet.
  \begin{enumerate}[resume]
\item Estimate $n_f=N/V$ in the case that our object consists of helium atoms and in the case of neutrons.
\item Calculate the non-relativistic Fermi energy $E_F=\frac{\hbar^2k_F^2}{2m}$ for an ideal gas of 
\begin{enumerate}[i]
\item electrons
\item protons
\item neutrons.
\end{enumerate}
Do the same calculation for the ultra-relativistic dispersion relation $E_F=\hbar k_F c$.
Collect your results in the table below.\\
\vspace{0.1cm}

\begin{tabular}{ l | c | c | c | c }
 & electrons & protons & neutrons & non-degeneracy temperature \\ 
 \hline
mass $m$ & & &\\[0.2cm] 
$E_f=\frac{\hbar^2k_F^2}{2m}$ &&& \\[0.2cm]
 $E_F=\hbar k_F c$ &&& 
 \label{tab}
\end{tabular}
\item A Fermi gas is degenerate if $E_f\gg k_B T$. In what temperatures are the above cases non-degenerate? Of what order is $E_F$ compared to temperatures that are typically found in stars?
\item We need to use the relativistic dispersion relation if the non-relativistic Fermi energy would give $E_F \ll mc^2$. Is this condition fulfilled for the object we are considering if it is made up of an electron-proton gas? What if it consist of a neutron gas?
\item What is the energy output of the reaction $n\rightarrow e^- + p^+ +\overline \nu_e$? Do you expect our object to consists of an electron-proton gas or rather neutrons?
\item \bonus Sum up this exercise into a coherent statement of what state of matter you expect for an object of radius $R$ and mass $M$. Try to cover the loopholes that we didn't consider like interactions or mean distance of the constituents etc.
\end{document}
