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\begin{document}
\twocolumn
\kalvo
\center{\Large VORTICES IN SUPERFLUID $^3$He}
\center{\normalsize Lectures for Kevo winter school
\\
20.-26.4.2002}

\vspace{1cm}
\center{\normalsize Erkki Thuneberg}
%\vspace{1cm}
\center{\normalsize Department of physical sciences
\\
University of Oulu}
\center{Thanks to the\\ Low temperature laboratory,
Helsinki university of technology\\
where most of this work was done}

\kalvoe 

\kalvo
\scriptsize
These lecture notes are available at\\
 
http://boojum.hut.fi/research/theory/

\normalsize
\subhead{Content}

Introduction to superfluid $^3$He\\
Vortices in superfluid B phase\\
Vortices in superfluid A phase\\
Conclusions



\kalvoe 
\onecolumn
\kalvo
\chaphead{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}
\kalvoe 

\kalvo
\chaphead{Length scales}
\includegraphics[width=0.9\linewidth]{pituusengl.eps}
\kalvoe 



\kalvo
\chaphead{Length scales in superfluid $^3$He}
\includegraphics[width=0.8\linewidth]{quasiclassical3he.eps}
\kalvoe 



\kalvo
\chaphead{Superfluidity}
Superfluids can flow without friction.

Superfluidity is a quantum phenomenon on a macroscopic scale.

Macroscopic number of particles is in the same quantum state.

\hspace{1cm}$\Rightarrow$ a macroscopic wave function (order parameter)

\subhead{Bosons}

A macroscopic number of bosons is in the same quantum state.

\hspace{1cm}$\Rightarrow$ superfluidity of liquid $^4$He and gases of
alkali atoms 

\subhead{Fermions}

Only one fermion can occupy one quantum state.

Fermions form pairs (Cooper pairs).

\hspace{1cm}$\Rightarrow$ superconductivity of metals, superfluidity of
liquid
$^3$He

\kalvoe 



\kalvo
\chaphead{Order parameter of superfluid $^3$He}

Cooper pairs form in p-wave state 

$\Rightarrow$ the spin state is triplet

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

$\textsf{A}_{ij}({\bf r})$ is the wave function for the center of mass
of a Cooper pair.
\kalvoe 

\kalvo
\chaphead{Ginzburg-Landau theory of $^3$He}

The order parameter   
\begin{eqnarray}
\textsf{A}=\left(\matrix{
A_{xx}&A_{x y}&A_{x z}\cr
A_{yx}&A_{yy}&A_{y z}\cr
A_{zx}&A_{z y}&A_{z
z}\cr  }\right) \nonumber
 \end{eqnarray}
is determined by a minimum of the free-energy functional 
\begin{eqnarray}
F&=&\int d^3r\Bigg[
\alpha A_{\mu i}^*A_{\mu i}\nonumber \\\mbox{}&&
+\beta_1\vert A_{\mu i}A_{\mu i}\vert^2
+\beta_2(A_{\mu i}A_{\mu i}^*)^2
+\beta_3 A_{\mu i}^*A_{\nu i}^*A_{\nu j} A_{\mu j}\nonumber \\\mbox{}&&
+\beta_4 A_{\mu i}^*A_{\nu i}A_{\nu j}^*A_{\mu j}
+\beta_5 A_{\mu i}^*A_{\nu i}A_{\nu j}A_{\mu j}^*
\nonumber \\\mbox{}&&+
K(\gamma -1)\partial_iA_{\mu i}^*\partial_jA_{\mu j}
+K\partial_iA_{\mu j}^*\partial_iA_{\mu j}\Bigg].
\label{e.gl}
\end{eqnarray} 
where $\alpha=\alpha'({T\over T_{\rm
c}}-1)$.                                                                   
Input parameters $\alpha'$, $\beta_1$, $\beta_2$, $\beta_3$, $\beta_4$,
$\beta_5$, $K$ and $\gamma$.

Expansion 1) in the amplitude of the order parameter and 2) in gradients.
\kalvoe 


\twocolumn
\begin{slide}
\scriptsize\baselineskip=0.7truecm
\subhead{Exercises}
B-phase
\begin{eqnarray}
\textsf{A}=\Delta\left(\matrix{
1&0&0\cr
0&1&0\cr
0&0&1\cr  }\right). \nonumber
 \end{eqnarray}
Substitute in the free energy (\ref{e.gl}). Minimize with respect to
$\Delta$ to obtain
\begin{equation}
\Delta^2={\vert\alpha\vert\over 6\beta_{12}+2\beta_{345}},
\end{equation}
where $\beta_{ij...}=\beta_i+\beta_j+...$. This gives the free energy
density
\begin{equation}
f={F\over\rm volume}=- {3\alpha^2\over 12\beta_{12}+4\beta_{345}}.
\end{equation}

A-phase
\begin{eqnarray}
\textsf{A}=\Delta\left(\matrix{
1&i&0\cr
0&0&0\cr
0&0&0\cr  }\right). \nonumber
 \end{eqnarray}
Substitution and minimization give
\begin{equation}
f=- {\alpha^2\over 4\beta_{245}}, \quad
\Delta^2={\vert\alpha\vert\over 4\beta_{245}}.
\end{equation}

The same energies can be obtained by rotating the spin and orbital
coordinates into arbitrary orientation. Also an arbitrary phase factor
$\exp(i\chi)$ can be included. Thus for the
B-phase
\begin{eqnarray}
\textsf{A}=\Delta\exp(i\chi)\left(\matrix{
R_{xx}&R_{x y}&R_{x z}\cr
R_{yx}&R_{yy}&R_{y z}\cr
R_{zx}&R_{z y}&R_{z
z}\cr  }\right),
\label{e.Bop} \end{eqnarray}
where $\textsf{R}$ is a rotation matrix.
The simplest representation for the A phase 
uses an orthonormal triad 
$\hat{\bf m}$, $\hat{\bf n}$, $\hat{\bf l}$  and a
unit vector $\hat{\bf d}$: 
\begin{equation}
A_{\mu j} = \Delta \hat{d}_{\mu} (\hat{m}_{j} + i\hat{n}_{j}).
\end{equation}

\includegraphics[width=0.8\linewidth]{3HePhaseDiagram2D.eps}
\kalvoe 

\kalvo

\subhead{Problem of a rotating superfluid}
An uncharged superfluid cannot rotate homogeneously:
\begin{eqnarray}
&&\textsf{A}({\bf r})=\textsf{A}_0\exp[ i\chi({\bf r})]\nonumber\\&&
\Rightarrow {\bf v}={{\bf p}\over 2m}={-i\hbar\mbox{\boldmath
$\nabla$}\over 2m}={\textstyle{\hbar\over 2m}}\mbox{\boldmath
$\nabla$}\chi\nonumber\\&&
\Rightarrow\mbox{\boldmath $\nabla$}\times{\bf v}=0.\nonumber
\end{eqnarray}

Rotation takes place via vortex lines

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



\kalvo
\subhead{Simple model for a vortex}
\includegraphics[width=0.4\linewidth]{VortexProb.eps}\\
\begin{eqnarray}
\textsf{A}(r,\phi)=\textsf{A}_0e^{{\rm i}\phi}f(r)\nonumber
\end{eqnarray}
\includegraphics[width=0.5\linewidth]{TanhFunc.eps}

It follows that
\begin{eqnarray}
{\bf v}={\textstyle{\hbar\over 2m}}\mbox{\boldmath
$\nabla$}\phi={\textstyle{\hbar\over
2m}}\,{\hat{\mbox{\boldmath$\phi$}}\over r}.\nonumber
\end{eqnarray}
Circulation around a vortex line
\begin{eqnarray}
\kappa=\oint d{\bf l}\cdot{\bf v}={\textstyle{\hbar\over
2m}}\oint d{\bf l}\cdot\mbox{\boldmath
$\nabla$}\phi={h\over 2m}.\nonumber
\end{eqnarray}

\kalvoe 

\kalvo
\subhead{Density of vortex lines $n$}

Angular velocity $\Omega$.

\includegraphics[width=0.8\linewidth]{vortexdens.eps}\\
On the average, the superfluid rotates as a normal fluid.
The circulation around the container equals
\begin{eqnarray}
2\pi R \Omega R=n\kappa \pi R^2\nonumber
\end{eqnarray}
\begin{eqnarray}
\Rightarrow n={2\Omega\over \kappa}.
\label{e.nvortex}\end{eqnarray}
\kalvoe 


\kalvo
\subhead{The vortex core}
\includegraphics[width=0.4\linewidth]{VortexProb.eps}

Boundary condition (B phase)
\begin{eqnarray}
\textsf{A}(r\rightarrow\infty,\phi)=e^{{\rm i}\phi}\left(\matrix{
1&0&0\cr 0&1&0\cr
0&0&1\cr  }\right)
\label{e.bvorbc}\end{eqnarray}
Simple ansatz:
\begin{eqnarray}
\textsf{A}(r,\phi)=f(r)e^{{\rm i}\phi}\left(\matrix{
1&0&0\cr 0&1&0\cr
0&0&1\cr  }\right)\nonumber
\end{eqnarray}
\includegraphics[width=0.5\linewidth]{TanhFunc.eps}
\kalvoe 

\onecolumn
\kalvo\footnotesize
\chaphead{Symmetry classification}

Axially symmetric vortices (Salomaa \& Volovik 1983)
\begin{eqnarray}
\begin{array}{c|c}{\rm Symmetry}&\ \textsf{A}(r,\phi)=\left(\matrix{
A_{rr}&A_{r \phi}&A_{r z}\cr
A_{\phi r}&A_{\phi\phi}&A_{\phi z}\cr
A_{zr}&A_{z \phi}&A_{z z}\cr  }\right)\\\hline
{\infty\over m}{2'\over m'}&e^{{\rm
i}\phi}\left(\matrix{ R_1(r)&I_1(r)&0\cr
I_2(r)&R_2(r)&0\cr
0&0&R_3(r)\cr  }\right)\\
 {\infty\over m}&e^{{\rm i}\phi}\left(\matrix{
C_1(r)&C_2(r)&0\cr
C_3(r)&C_4(r)&0\cr
0&0&C_5(r)\cr  }\right)\\
 \infty m'&e^{{\rm i}\phi}\left(\matrix{
R_1(r)&I_1(r)&R_2(r)\cr
I_2(r)&R_3(r)&I_3(r)\cr
R_4(r)&I_4(r)&R_5(r)\cr  }\right)\\
 \infty 2'&e^{{\rm i}\phi}\left(\matrix{
R_1(r)&I_1(r)&I_2(r)\cr
I_3(r)&R_2(r)&R_3(r)\cr
I_4(r)&R_4(r)&R_5(r)\cr  }\right)\\
 \infty&e^{{\rm i}\phi}\left(\matrix{
C_1(r)&C_2(r)&C_3(r)\cr
C_4(r)&C_5(r)&C_6(r)\cr
C_7(r)&C_8(r)&C_9(r)\cr  }\right)
\end{array}\nonumber\end{eqnarray}
\kalvoe 


\kalvo
\chaphead{Vortex cores in $^3$He-B}

Numerical minimization of Ginzburg-Landau functional (\ref{e.gl}) with
boundary condition (\ref{e.bvorbc})  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}
\includegraphics[width=0.41\linewidth]{Bcores.eps}
\kalvoe 


\twocolumn
\kalvo\footnotesize
\chaphead{Josephson effect}
\includegraphics[width=0.9\linewidth]{jwall.eps}

Numerical calculation with Ginzburg-Landau theory (Janne Viljas)

$\Rightarrow$ $\pi$-state
\includegraphics[width=0.5\linewidth]{pistate.eps}

\kalvoe 

\kalvo\footnotesize
The order parameter in the orifice for a phase difference $\pi$\\
\includegraphics[width=1.0\linewidth]{JosOP.eps}

This is very similar as the order parameter in the double-core vortex
on an axis that passes between the two cores.

The two cores can be interpreted as two {\em half-quantum} vortices.

\kalvoe 

\onecolumn
\kalvo
\chaphead{Other line defects in $^3$He-B}
In the order parameter
$$
\textsf{A}=\Delta\exp(i\chi)\textsf{R}(\hat{\bf
n},\theta)\eqno(\ref{e.Bop})
$$
the rotation matrix $\textsf{R}$ can be parametrized with an axis of
rotation $\hat{\bf n}$ and an angle $\theta$.

Symmetry $U(1)\times SO_3$. Homotopy groups $\pi(U(1))=Z$,
$\pi(SO_3)=Z_2$.

Variation of $\textsf{R}$ results in spin currents.

\includegraphics[width=0.65\linewidth]{SMV3.eps}

Ginzburg-Landau calculation\\
$\Rightarrow$
The spin-mass vortex is stable against dissociation: 
$F_{\rm sm}<F_{\rm s}+F_{\rm m}$.

Spin current is not conserved on a large scale.

\kalvoe 
\kalvo
\chaphead{Additional terms in Ginzburg-Landau functional}

\begin{eqnarray}
F_{\rm basic}&=&\int d^3r\Bigg[
\alpha A_{\mu i}^*A_{\mu i}\nonumber \\\mbox{}&&
+\beta_1\vert A_{\mu i}A_{\mu i}\vert^2
+\beta_2(A_{\mu i}A_{\mu i}^*)^2
+\beta_3 A_{\mu i}^*A_{\nu i}^*A_{\nu j} A_{\mu j}\nonumber \\\mbox{}&&
+\beta_4 A_{\mu i}^*A_{\nu i}A_{\nu j}^*A_{\mu j}
+\beta_5 A_{\mu i}^*A_{\nu i}A_{\nu j}A_{\mu j}^*
\nonumber \\\mbox{}&&+
(\gamma -1)\partial_iA_{\mu i}^*\partial_jA_{\mu j}
+\partial_iA_{\mu j}^*\partial_iA_{\mu j}\Bigg].
\nonumber
\end{eqnarray} 
Dipole-dipole interaction of $^3$He nuclear moments ($g_D\sim
10^{-6}\alpha'$)
\begin{equation}
F_{\rm D}=g_{\rm D}\int d^3r(A_{i i}^*A_{j j}+A_{i j}^*A_{j
i}-{\textstyle{2\over 3}}A_{\mu i}^*A_{\mu i}).
\label{e.GLdipole}
\end{equation}
External magnetic field
\begin{equation}
F_{\rm H}=\int d^3r\left(-{\rm i}g_{\rm H1}
\epsilon_{\kappa\mu\nu}H_\kappa A_{\mu i}^*A_{\nu i}+g_{\rm H}H_\mu
A_{\mu i}^*A_{\nu i}H_\nu
+g_{\rm H}'H^2A_{\mu i}^*A_{\mu i}\right).
\label{e.GLfield}
\end{equation}
Rotation of the liquid
\begin{eqnarray}
F_{\rm G}&=&-{\rm
i}(2\eta-1)K\epsilon_{kij}\int d^3r(\mbox{\boldmath$\nabla$}
\times{\bf v}_{\rm n})_kA_{\mu i}^*A_{\mu j}. \label{e.GLgradient2}
\end{eqnarray}
\kalvoe 

\kalvo
\chaphead{Large-scale structure of a spin-mass vortex}

\includegraphics[width=0.50\linewidth]{SMVplanar.eps}
\hfill\includegraphics[width=0.35\linewidth]{smvcont.eps}

A few spin-mass vortices are created in fast A$\rightarrow$B
transitions. 
\kalvoe 



\kalvo
\chaphead{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}
\kalvoe 



\kalvo
\chaphead{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).

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

Exercise: verify this by drawing $\hat{\bf m}$
and $\hat{\bf n}$ on the figure. 

Conclusion: this is a two-quantum vortex. It is called {\em continuous},
because
$\Delta$ (the amplitude of the order parameter) vanishes nowhere. 


\kalvoe 


\kalvo\baselineskip=0.9truecm
\chaphead{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}\nonumber
\end{eqnarray}
Needs as input $\lambda_{\rm D}=4g_D\Delta^2$, $\lambda_{\rm h}$,
$\rho_\perp$,
$\rho_\parallel$, $C$, $C_0$, $K_{\rm t}$,
$K_{\rm s}$,
$K_{\rm b}$,
$K_5$, and $K_6$.

Low field ($H< 1$ mT): $\hat{\bf d}\parallel\hat{\bf l}$ $\Rightarrow$
vortex cores fills all space

High field ($H> 1$ mT): $\hat{\bf d}\perp{\bf H}$ $\Rightarrow$
vortex cores limited by $g_D$ to size $\xi_D\approx 10\ \mu$m. 
\kalvoe 

\kalvo
\chaphead{Continuous vortex lattices}

Projection of $\hat{\bf l}$ field on $x$-$y$ plane  

\begin{tabular}{cc}
low field vortex & high field vortex\\
Locked vortex 1 (LV1) & Continuous unlocked vortex (CUV)\\
\includegraphics[width=0.40\linewidth]{lv1.eps}&
\includegraphics[width=0.40\linewidth]{cuv.eps}\\
square lattice & centered rectangular lattice\\
$P{4\over n}b'm'$ & $C2'$
\end{tabular}
\kalvoe 

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

Also 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?
\kalvoe 


\kalvo\baselineskip=0.9truecm
Dipole-dipole interaction (\ref{e.GLdipole}) 
\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}
\kalvoe 


\kalvo
\scriptsize
\chaphead{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 $N_{\rm vortex\ line}\propto \Omega$ (\ref{e.nvortex}).

\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. 
\kalvoe 

\onecolumn
\kalvo
\chaphead{Vortex phase diagram in $^3$He-A}
\includegraphics[width=0.80\linewidth]{AphaseVortices&DiagramRev.eps}
\kalvoe 


\kalvo
\chaphead{NMR spectra of vortices}
\includegraphics[width=0.55\linewidth]{ASpectraShrodinger.eps}
\kalvoe 

\kalvo\baselineskip=0.9truecm
\chaphead{Nucleation of vortices}

The length scale of vorticity in $^3$He-A is 10 $\mu$m.  \\
Surface roughness can be controlled on this level.\\
$\Rightarrow$ possibility to study intrinsic vortex formation
 
\includegraphics[width=0.35\linewidth]{VortexNuclA.eps}
\kalvoe 


\kalvo
\chaphead{Summary of periodic vortex structures}
\tiny

\begin{tabular}{cccccccc}%{c|cclccrc}
\hline\vspace{0.3cm}
material&vortex name&Bravais lattice&space group&$N$&$\nu_{\rm
d}$&theory&experiment\\ \hline\hline\vspace{0.3cm}
$^4$He-II&vortex line&hexagonal&$P6/mm'm'$&1&-&1949-55&1950-79\\
\hline\vspace{0.3cm}
s.c. metals&flux line&several&several&1&-&1957&1936-66\\
\hline\vspace{0.3cm}
$^3$He-B&A-phase-core v.&hexagonal&$P6m'm'$&1&-&1983&1982\\
\vspace{0.3cm}&double-core v.&cent. rectang.&$Cm'm'2$&1&-&1986&1982\\
\hline\vspace{0.3cm}
$^3$He-A&locked vortex 1 (LV1)&square&$P4/nb'm'$&4&2&1978&1990\\
\vspace{0.3cm}
&cont. unlocked v. (CUV)&cent. rectang.&$C2'$&2&0&1983&1982\\
\vspace{0.3cm}&singular vortex (SV)&cent.
rectang.&$Cm'$&1&-&1983&1987-95\\
\vspace{0.3cm} &locked vortex 2
(LV2)&cent. rectang.&$C2'$&2&1&1985&-\\
\vspace{0.3cm} &vortex sheet (VS)&prim.
rectang.&$Pb'a'n$&4&0&1994&1985-94\\
\hline
\end{tabular}

$N$: number of circulation quanta per unit cell

$\nu_d$: topological quantum number for $\hat{\bf d}$.
\kalvoe 




\kalvo\baselineskip=0.9truecm
\chaphead{Conclusion}

Vorticity in superfluid $^3$He forms a very rich system because of
the several length scales that are relevant in this system.

\includegraphics[width=0.7\linewidth]{quasiclassical3he.eps}
\kalvoe 



\end{document}
 