
\sect{Fermions in magnetic field, quantum phase transition}

%Baruch's  B29

A $d$ dimensional container ($d=1,2,3$)  contains fermions of density $n$, temperature $T=0$, mass $m$ and spin $\frac{1}{2}$, having a magnetic moment ${\bar m}$. The container is placed in a magnetic field $H/{\bar m}$ so that the fermion spectra is $\epsilon_{{\bf p}}=\frac{{\bf p}^2}{2m}\pm H$ where ${\bf p}$ is the momentum. (Note that orbital effects are neglected, possible e.g. at d=2 with the field parallel to the layer).
\begin{itemize}
\item[(a)]  Evaluate the chemical  potential $\mu(H)$, for small $H$:
Consider first an expansion to lowest order in $H$ and then evaluate $d\mu/dH$ to note the change at finite $H$.
\item[(b)] Beyond which $H_c$ does  the consideration in (a) fail? Find $\mu(H)$ at $H>H_c$ and plot qualitatively $\mu(H)/\mu_0$ as function of $H/\mu_0$ (where $\mu_0=\mu(H=0)$) for $d=1,2,3$, indicating the values of $\mu(H)/\mu_0$ at $H_c$.
\item[(c)] Of what order is the  phase transition at $H_c$, at either $d=1,2,3$? Does the phase transition survive at finite $T$? (no need for finite $T$ calculations -- just note analytic properties of thermodynamic functions).
\item[(d)] The container above, called A, with $H\ne 0$ is now attached to an identical container B (same fermions at density $n$, $T=0$), but with $H=0$. In which direction will the fermions flow initially? Specify your answer for $d=1,2,3$ at relevant ranges of $H$.

\end{itemize}
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