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\begin{document}


\begin{slide}
\begin{center}
\Large RECENT TOPICS IN THE THEORY OF SUPERFLUID $^3$He

\normalsize Lecture at ISSP Tokyo 13.5.2003

\vspace{1cm}
Erkki Thuneberg

Department of physical sciences, University of Oulu

\vspace{1cm}
Risto H\"anninen and Janne Viljas 

Low temperature laboratory,
Helsinki university of technology\\
\end{center}
\end{slide} 

\begin{slide}
{\Large\bf Content}

Introduction to superfluid $^3$He

I. Josephson effect

II. Vortex sheet in $^3$He-A

III. $^3$He in aerogel

\vspace{1cm}
These lecture notes will be placed at\\
\href{http://boojum.hut.fi/research/theory/}
{http://boojum.hut.fi/research/theory/}

\end{slide}


\begin{slide}
{\Large \bf Phase diagram of superfluid $^3$He}

$^3$He is the less common isotope of helium. It is a fermion.

\includegraphics[width=0.6\linewidth]{He3PD.eps}
\end{slide} 

\begin{slide}
{\Large \bf Length scales}

\includegraphics[width=0.9\linewidth]{pituusengl.eps}
\end{slide} 



\begin{slide}
{\Large \bf Length scales in superfluid $^3$He}\par
\includegraphics[width=0.9\linewidth]{quasiclassical3he.eps}
\end{slide}


\begin{slide}
{\Large \bf  Cooper pairs in superfluid $^3$He}

\includegraphics[width=0.8\linewidth]{OP3He.eps}

\end{slide} 

\begin{slide}
{\Large\bf  Part I. JOSEPHSON EFFECT}\par
\includegraphics[width=0.60\linewidth]{JosephsonBackground2.eps}\par
Experiments:\\
- superconductors: 1960's\\
- $^3$He: Avenel and Varoquaux 1988 
\end{slide} 

\begin{slide}
{\Large\bf New experiments in $^3$He-B}\par\scriptsize
 - S. Backhaus, S. Pereverzev, R. Simmonds, A. Loshak, J. Davis,
	R. Packard, A. Marchenkov (1998, 1999)\\
- O. Avenel, Yu. Mukharsky, E. Varoquaux (1999)\par
\includegraphics[width=0.60\linewidth]{JosephsonBackground2b.eps}
\end{slide} 

\begin{slide}
{\Large \bf Superfluid B phase}\par
\includegraphics[width=1.0\linewidth]{OP3He2.eps}
\end{slide} 

\begin{slide}
{\Large \bf Pinhole}\par
\includegraphics[width=0.90\linewidth]{PinHole3.eps}
\end{slide} 

\begin{slide}
{\Large\bf Pinhole current-phase relation}\par
\includegraphics[width=1.0\linewidth]{PinholeCP.eps}
\par $\pi$ state occurs only at very low temperatures\\
$\Rightarrow$ not consistent with experiments \par
\end{slide} 

\begin{slide}
{\Large \bf Aperture array}\par
\includegraphics[width=0.80\linewidth]{JosIdeaPuolikas.eps}\par
``Anisotextural'' $\pi$ state 
\end{slide} 

\begin{slide}
{\Large \bf Results}\par
- asymmetry of the junction\\
- one fitting parameter $\eta$\par
\includegraphics[width=0.95\linewidth]{JosAsymm.eps}\par
Theoretically strong dependence on magnetic field and surroundings\\
- to be tested experimentally\end{slide} 

\begin{slide}
{\Large\bf Large aperture}\par
\includegraphics[width=0.25\linewidth]{hole3d.eps}
\includegraphics[width=0.60\linewidth]{JosCalc2.eps}\par
\includegraphics[width=0.35\linewidth]{CPGL.eps}
\scriptsize\href{http://boojum.hut.fi/research/theory/jos.html}
{http://boojum.hut.fi/research/theory/jos.html}

\end{slide} 

\begin{slide}
\baselineskip=0.7truecm
{\Large\bf Part II. ROTATING SUPERFLUID}\par
An uncharged superfluid cannot rotate homogeneously\par
Rotation takes place via vortex lines\par
\includegraphics[width=0.4\linewidth]{VMacroOlli.eps}
\end{slide} 

\begin{slide}
{\Large \bf Vortex cores in $^3$He-B}\par
Numerical minimization of Ginzburg-Landau functional in 2 D
\begin{eqnarray}
\partial_j\partial_jA_{\mu i}+&(\gamma -1)\partial_i\partial_jA_{\mu j}         
=[-\textsf{A}
+\beta_1\textsf{A}^*Tr(\textsf{A}\textsf{A}^T)+\beta_2\textsf{A}
Tr(\textsf{A}\textsf{A}^{T*})\cr                  
&+\beta_3\textsf{A}\textsf{A}^T\textsf{A}^*+\beta_4\textsf{A}\textsf{A}^{T*}
\textsf{A}+\beta_5\textsf{A}^*\textsf{A}^T\textsf{A}]_{\mu         
i}.
\end{eqnarray}
$\Rightarrow$ broken symmetry in vortex cores\par
\includegraphics[width=0.38\linewidth]{Bcores.eps}
\end{slide} 

\begin{slide}
{\Large\bf Virtual half-quantum vortices}\par
\includegraphics[width=0.45\linewidth]{hole3dQuantum.eps}
\end{slide} 

\begin{slide}
{\Large \bf The A phase}

The order parameter $A_{\mu j} = \Delta \hat{d}_{\mu} (\hat{m}_{j} +
i\hat{n}_{j})$ 

\includegraphics[width=0.50\linewidth]{AphasOP.eps}

A phase factor $e^{i\chi}$ corresponds to rotation of $\hat{\bf m}$
and $\hat{\bf n}$ around $\hat{\bf l}$:
\begin{eqnarray}
e^{i\chi}(\hat{\bf m}+i\hat{\bf n})&=&(\cos\chi+i\sin\chi)(\hat{\bf
m}+i\hat{\bf n})\nonumber\\
&=&(\hat{\bf m}\cos\chi-\hat{\bf n}\sin\chi)
+i(\hat{\bf m}\sin\chi+\hat{\bf n}\cos\chi).\nonumber
\end{eqnarray}

Superfluid velocity 
\begin{equation} 
{\bf v}_{\rm s}={\hbar\over 2m}\mbox{\boldmath$\nabla$}\chi= 
{\hbar\over 2m}\sum_j\hat m_j\mbox{\boldmath$\nabla$}\hat n_j. 
\label{supvel}
\end{equation}
\end{slide} 



\begin{slide}
{\Large \bf Vortices in the A phase}

Consider the structure

\includegraphics[width=0.70\linewidth]{CUV0.eps}

Here $\hat{\bf l}$ sweeps once trough all orientations (once a unit
sphere).

$\Rightarrow$ $\hat{\bf m}$
and $\hat{\bf n}$ circle twice around $\hat{\bf l}$ when one goes
around this object. 

$\Rightarrow$ This is a two-quantum vortex. It is called {\em
continuous}, because
$\Delta$ (the amplitude of the order parameter) vanishes nowhere. 
\end{slide} 

\begin{slide}\baselineskip=0.9truecm
{\Large \bf Hydrostatic theory of $^3$He-A}

Assume the order parameter ($\hat{\bf m}$, $\hat{\bf n}$, $\hat{\bf l}$,
$\hat{\bf d}$)
changes slowly in space. Then we can make gradient expansion of the
free energy
\begin{eqnarray}
F&=& \int d^3r\Bigg[-{\textstyle{1\over 2}}\lambda_{\rm D}(\hat{\bf
d}\cdot\hat{\bf l})^2 +{\textstyle{1\over 2}}\lambda_{\rm H}(\hat{\bf
d}\cdot{\bf H})^2
\nonumber \\
&&
+{\textstyle{1\over 2}}\rho_\perp{\bf v}^2
+{\textstyle{1\over 2}}(\rho_\parallel-\rho_\perp)(\hat{\bf l}\cdot{\bf v})^2
+C{\bf v}\cdot\nabla\times\hat{\bf l}
-C_0(\hat{\bf l}\cdot{\bf v})
(\hat{\bf l}\cdot\nabla\times\hat{\bf l})\nonumber \\  &&
 +{\textstyle{1\over 2}}K_{\rm s}(\nabla\cdot\hat{\bf l})^2
+{\textstyle{1\over 2}}K_{\rm t}\vert\hat{\bf l}\cdot\nabla\times\hat{\bf l}\vert^2
+{\textstyle{1\over 2}}K_{\rm b}\vert\hat{\bf
l}\times(\nabla\times\hat{\bf l})\vert^2
\nonumber \\
&&+{\textstyle{1\over 2}}K_5 \vert(\hat{\bf l}\cdot\nabla)\hat{\bf d}\vert^2+
{\textstyle{1\over 2}}K_6[(\hat{\bf l}\times\nabla)_i\hat{\bf d}_j)]^2\ 
\Bigg].\label{e.grad}
\end{eqnarray}
\end{slide} 

\begin{slide}
{\Large \bf Vortex phase diagram in $^3$He-A}

\includegraphics[width=0.80\linewidth]{AphaseVortices&DiagramRev.eps}
\end{slide} 

\twocolumn
\begin{slide}
%\scriptsize
\baselineskip=0.9truecm
{\Large \bf Vortex sheet}

Vortex sheets are possible in $^3$He-A

\includegraphics[width=0.7\linewidth]{vsmakro.eps}

Sheets were first suggested to exist in $^4$He, but they were found to
be unstable.

Why stable in $^3$He-A?
\end{slide} 


\begin{slide}\baselineskip=0.9truecm
Dipole-dipole interaction (\ref{e.grad}) 
\begin{eqnarray}
f_D=-{\textstyle{1\over 2}}\lambda_{\rm D}(\hat{\bf
d}\cdot\hat{\bf l})^2\nonumber
\end{eqnarray}
$\Rightarrow$\\
\includegraphics[width=0.95\linewidth]{domainwall.eps}

Vortex sheet = soliton wall to which the vortices are bound. 

\includegraphics[width=1.1\linewidth]{vssketch.eps}
\end{slide} 


\begin{slide}
\scriptsize
{\Large \bf Shape of the vortex sheet}

The equilibrium configuration of the sheet is
determined by the
minimum of 
\begin{eqnarray}
F=\int d^3r{\textstyle{1\over 2}}\rho_s({\bf v}_n-{\bf v}_s)^2
+\sigma A.\nonumber
\end{eqnarray}
Here $A$ is the area of the sheet and $\sigma$ its surface tension.

The equilibrium distance $b$ between two sheets is 
\begin{equation}
b=\left({3\sigma\over \rho_{\rm s}\Omega^2}\right)^{1/3}.
\label{e.distance}
\end{equation}
This gives 0.36 mm at $\Omega = 1$ rad/s.

The area of the sheet
\begin{equation}
A\propto{1\over b}\propto \Omega^{2/3},\nonumber
\end{equation}
as compared to $N_{\rm vortex\ line}\propto \Omega$.

\includegraphics[width=1\linewidth]{sheetdist.eps}

Connection lines with the side wall allow the vortex sheet to grow and
shrink when angular velocity $\Omega$ changes.
 
\end{slide} 

\onecolumn
\begin{slide}
{\Large \bf NMR spectra of vortices}

\includegraphics[width=0.50\linewidth]{ASpectraShrodinger.eps}
\scriptsize\href{http://boojum.hut.fi/research/theory/sheet.html}
{http://boojum.hut.fi/research/theory/sheet.html}
\end{slide} 

\begin{slide}
{\Large \bf Part III. IMPURE SUPERFLUID $^3$He}

$^3$He is a naturally pure substance

Impurity can be introduced by porous aerogel\par
\includegraphics[width=0.7\linewidth]{aerogel_1.eps}\\
- strands of SiO$_2$\\
- typically 98\% empty\\
- small angle x-ray scattering $\Rightarrow$
homogeneous on scale above $\approx 100$ nm
\end{slide} 

\begin{slide}
{\Large \bf
Experimental phase diagram}\par
\includegraphics[width=0.50\linewidth]{AerogelPhaseDiag.eps}\par
\scriptsize
Porto et al (1995),
Sprague et al (1995),
Matsumoto et al (1997)
\end{slide} 

\twocolumn
\begin{slide}
{\Large\bf Models}\par
Main effect of aerogel is to scatter quasiparticles

Simplest model: homogeneous scattering model

\includegraphics[width=0.3\linewidth]{HSMillustration.eps}

$\Rightarrow$ only qualitative agreement with experiments
\end{slide} 

\begin{slide}
Better models:

\includegraphics[width=1.0\linewidth]{IISModel.eps}
\end{slide}
 
\onecolumn
\begin{slide}
{\large\bf Isotropic inhomogeneous scattering model (IISM)}\par

Isotropic on large scale\\
$\Rightarrow$ Hydrodynamics the same as in bulk but renormalized
parameter values

Better fit with experiment than in HSM\\
$\Rightarrow$ realistic parameter values

\includegraphics[width=0.5\linewidth]{TcComp.eps}

\scriptsize\href{http://boojum.hut.fi/research/theory/aerogel.html}
{http://boojum.hut.fi/research/theory/aerogel.html}
\end{slide} 


\begin{slide}
\baselineskip=0.9truecm
{\Large \bf Conclusion}

Superfluid $^3$He forms a very rich system because of
the several different length scales.

\includegraphics[width=0.7\linewidth]{quasiclassical3heEND.eps}
\end{slide} 



\end{document}
 