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\begin{document}


\begin{slide}
\begin{center}
\Large Vortex sheets and solitons in superfluid $^3$He-A

\normalsize Superfluidity under Rotation 2003 \\
Lake Chuzenjiko, Japan 14.-17.5.2003

\vspace{1cm}
Erkki Thuneberg

Department of physical sciences, University of Oulu\\
and\\
Low temperature laboratory,
Helsinki university of technology\\
\end{center}
\end{slide} 

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

Introduction to superfluid $^3$He-A

Vortex sheet: analytic results

Solitons:  dissipation in NMR absorption 
 

\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 Simple model of the vortex sheet}

Minimize
\begin{eqnarray}
F=\int d^3r{\textstyle{1\over 2}}\rho_sv^2
+\sigma A.
\end{eqnarray}
with constraints \begin{equation}
\nabla\cdot{\bf v}=0,\  
\nabla\times{\bf v}= 2\mbox{\boldmath$\Omega$}.
\label{eq.nablav}\end{equation}
Here ${\bf v}\equiv{\bf v}_s-{\bf v}_n$.
$A$ is the
area of the sheet and
$\sigma$ its surface tension. ${\bf v}$ has tangential discontinuity
at the sheet.

Warning: this neglects the bending energy of the texture.

Exact corollary
\begin{equation}
\sigma K+\textstyle{1\over 2}\rho_s(v_1^2-v_2^2)=0.
\label{eq.curvat}
\end{equation}
where $K/2$ is the mean
curvature of the sheet ($K=R_a^{-1}+R_b^{-1}$, where
$R_a$ and
$R_b$ are the principal radii of curvature). $v_1$ and $v_2$ are the
(tangential) velocities on the two sides.
\end{slide} 

\begin{slide}
\scriptsize
{\bf Applications}

1) {\bf Planar sheets} with distance $b$:
\begin{equation}
{\bf v}=2\Omega x\hat{\bf y}
\end{equation}
\begin{eqnarray}
\frac{F}{V}=\frac{1}{b}\int_{-b/2}^{b/2}dx\frac{1}{2}\rho_sv^2
+\frac{\sigma}{ b}
=\frac{1}{6}\rho_s\Omega^2b^2+\frac{\sigma}{ b}
\end{eqnarray}
Minimization with respect to $b$ gives the equilibrium distance between
sheets
\begin{equation}
b=\left({3\sigma\over \rho_{\rm s}\Omega^2}\right)^{1/3}.
\label{e.distance}
\end{equation}

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

\onecolumn
\begin{slide}
\scriptsize
2) {\bf One cylindrical sheet} (cylindrical container, radius $R$)

Radius of equilibrium sheet $R_s$ for given circulation 
$\kappa=\oint {\rm d}{\bf
r}\cdot{\bf v}_s=2\pi
\Omega R_v^2$ obeys
\begin{eqnarray} {\sigma\over
R_s}+\rho_s\Omega^2R_v^2(1-{R_v^2\over 2R_s^2})=0,
\label{eq.curvat1}\end{eqnarray}
Stability against small deformations 
$r_s(\phi)=R_s+{\cal A}\cos n\phi$ is determined by
\begin{eqnarray} {(n^2-1)\sigma\over
R_s^2}+\rho_s\Omega^2R_s\left[1-n-(1-{R_v^2\over R_s^2})
[1+{R_v^2\over R_s^2}+n(1-{R_v^2\over R_s^2})
{R^{2n}+R_s^{2n}\over R^{2n}-R_s^{2n}}]\right]=0
\label{eq.curvat2}\end{eqnarray}

\includegraphics[width=0.25\linewidth]{SheetCyl.eps}
\hspace{1cm}
\includegraphics[width=0.45\linewidth]{RingPhaseD.eps}
\hspace{1cm}
$r_0=\rho_s\kappa^2/(8\pi^2\sigma)$
\end{slide} 

\begin{slide}
\scriptsize
3) {\bf Rectangular container} 

\includegraphics[width=0.4\linewidth]{SheetRec.eps}\\
Kinetic energy for vortexfree rectangle (area $d_1d_2$) (Fetter 1974)
\begin{eqnarray}
f_{\rm k}&=&{F_{\rm k}\over d_1d_2d_3}=\rho_s\Omega^2d_1d_2g({d_1\over
d_2}),
\label{e.kin}\\
g(x)&=&g({1\over x})={x\over 6}-{x^2\over\pi^5}\sum_{j=1}^\infty
{1\over(j-{1\over 2})^5}\tanh{\pi(j-{1\over 2})\over x}.
\label{e.g}
\end{eqnarray}
Applying to $n$ sheets in configurations (a) and (b) gives
\begin{eqnarray}
f_{{\rm a},n}&=&{f_{\rm s}\over b}\left[{nb\over a}+{\alpha\over
n+1}{a\over b} g({a\over(n+1)b})\right],
\label{e.an}\\
f_{{\rm b},n}&=&{f_{\rm s}\over b}\left[n+{\alpha\over
n+1}{a\over b} g({b\over(n+1)a})\right],
\label{e.bn}
\end{eqnarray}
where $\alpha=\rho_{\rm s}\Omega^2b^3/f_{\rm s}$ and $f_{\rm
s}\approx\sigma$.

Sequence for increasing $\Omega$ ($b=0.9a$): 0, 1a, 1b, 2a, 2b, \ldots,
5a, 5b, 6b, 7b,
\ldots
\end{slide} 

\begin{slide}
\scriptsize
4) {\bf Bending of sheets at A-B interface} (H\"anninen et al 2003)

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

\begin{equation}
{\bf v}=2\Omega x\hat{\bf y}
\end{equation}
\begin{eqnarray}
\frac{F}{L_xL_y}=\int _0^\infty
dz\left(\frac{1}{b}\int_{-b/2+\zeta}^{b/2+\zeta}dx\frac{1}{2}\rho_sv^2
+\frac{\sigma}{
b}\sqrt{1+\left(\frac{d\zeta}{dz}\right)^2}\right)\nonumber\\ =\int
_0^\infty dz\left[\frac{1}{6}\rho_s\Omega^2(b^2+12\zeta^2)
+\frac{\sigma}{b}\sqrt{1+\left(\frac{d\zeta}{dz}\right)^2}\right]
\end{eqnarray}
\begin{equation}
\Rightarrow \frac{z}{a}=1-\sqrt{2-(\zeta/a)^2}-
\frac{1}{\sqrt{2}}\ln\frac{\sqrt{2}-\sqrt{2-(\zeta/a)^2}}{(\sqrt{2}-1)\zeta/a}
\end{equation}
where $a=b/\sqrt{6}$ ($b$ is the equilibrium distance).

Surprisingly the Bekarevich-Khalatnikov model gives exactly the same
form for vortex lines.

\vspace{1cm}
{\bf Summary}

Analytical calculations for sheets are simpler than for vortex lines.

Textural bending energy? 
\end{slide} 


\begin{slide}
{\Large \bf Solitons}

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=0.35\linewidth]{domainwall.eps}

Structure of splay soliton (${\bf B}=B\hat{\bf z}$)


\includegraphics[width=0.65\linewidth]{ldvect3.eps}
\end{slide} 


\begin{slide}\scriptsize
{\Large \bf NMR resonance frequencies}
\begin{eqnarray}
(\mathcal{D}+U_\parallel) d_\theta &=&
\alpha_\parallel d_\theta \label{e.schrod1}\\ 
(\mathcal{D}+U_\perp) d_z &=&
\alpha_\perp d_z .  
\label{e.schrod2}
\end{eqnarray} 
\begin{equation}
\mathcal{D}f=-\frac{K_6}{\lambda_{\rm d}}\nabla^2f
%-\frac{K_5-K_6}{\lambda_{\rm d}}\mbox{\boldmath$\nabla$}\cdot\lbrack\hat{\bf l} 
%(\hat{\bf l}\cdot\mbox{\boldmath$\nabla$}) f\rbrack.
-\frac{K_5-K_6}{\lambda_{\rm d}}
\mbox{\boldmath$\nabla$}\cdot\left\lbrack\hat{\bf l}  (\hat{\bf l}\cdot
\mbox{\boldmath$\nabla$}) f\right\rbrack.
\label{e.ddef}\end{equation} 
 \begin{eqnarray}
U_\parallel &=& 1-l_z^2-2(\hat{\bf l}\times\hat{\bf d}_0)_z^2 
 \\
U_\perp &=&1-2\hat{l}_z^2-(\hat{\bf l}\times\hat{\bf d}_0)_z^2
-\frac{K_6}{\lambda_{\rm d}}
(\mbox{\boldmath$\nabla$}\theta)^2-\frac{K_5-K_6}{\lambda_{\rm
d}}(\hat{\bf l}\cdot\mbox{\boldmath$\nabla$}\theta)^2. 
\label{e.pot}
\end{eqnarray} 
\includegraphics[width=0.45\linewidth]{potwell_splay.eps}
\end{slide}
 
\begin{slide}
{\Large \bf Results (no dissipation)}

Effect of $F_1^a=0$, -1 and strong coupling

\includegraphics[width=0.65\linewidth]{lambdaexp.eps}


\end{slide} 


\begin{slide}
{\Large \bf Dissipation}

Normal-superfluid conversion (Leggett-Takagi) and spin diffusion
(simple model)
\begin{eqnarray}
\dot{\bf S}_q &=& \gamma{\bf S}_q \times \left( {\bf B} -
\mu_0\gamma\frac{F_0^a}{\chi_{0}} {\bf S}_p \right) +
\frac{1}{\tau}\left[ (1-\lambda) {\bf S}_p - \lambda {\bf S}_q \right]
+ \kappa \nabla^2 {\bf S}_q\label{e.ltequ2a}
 \\ 
\dot{\bf S}_p &=& \gamma{\bf S}_p \times \left( {\bf B} -
\mu_0\gamma \frac{F_0^a}{\chi_{0}} {\bf S}_q \right) -
\frac{1}{\tau}\left[ (1-\lambda) {\bf S}_p - \lambda {\bf S}_q \right]
-\hat{\bf d}
\times \frac{\delta f}{\delta\hat{\bf d}}  \\ 
\dot{\hat{\bf d}} &=& \gamma\hat{\bf d}\times \left[ {\bf B} -
\mu_0\gamma\frac{F_0^a}{\chi_{0}}{\bf S}_q
-\mu_0\gamma
\left(\frac{F_0^a}{\chi_{0}}+\frac{1}{\lambda\chi_0}\right)
{\bf S}_p
\right] .  
\end{eqnarray}

\end{slide} 


\begin{slide}
{\Large \bf Results}

\includegraphics[width=0.45\linewidth]{parallelsplayabs.eps}
\includegraphics[width=0.45\linewidth]{pabsL50t75H28tratwist.eps}

\includegraphics[width=0.65\linewidth]{alphawith50and75.eps}

R. H\"anninen and E. T. \href
{http://arXiv.org/abs/cond-mat/0103052}{cond-mat/0103052}

\end{slide} 

\begin{slide}
{\Large\bf Conclusion}

Vortex sheet: analytic results\\
\hspace{1cm} - maybe can be tested at high rotation speed

Solitons:  including dissipation\\
\hspace{1cm} - a problem in transverse splay resonance\\
\hspace{1cm} - measurement of longitudinal resonance of splay soliton?\\
 
\vspace{2cm}
\scriptsize
These lecture notes will be available at\\
\href{http://boojum.hut.fi/research/theory/}
{http://boojum.hut.fi/research/theory/}

\end{slide}







\end{document}