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\begin{document}
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%****************************************************************************
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%******************* Please insert here the name(s) of the author(s). *******
\Author{Jorge Berger and Jacob Rubinstein}
\vspace*{0.4cm}
%********* Please insert here the title of the contribution. ****************
\Title{
On the zero set of the wave function in superconductivity}
\vspace*{0.4cm}
\today
\end{minipage}
\vspace*{3.5mm}
%********* Please insert now the abstract. ******************
\begin{Abstract}
\emph{Abstract.}
We consider the Ginzburg Landau functional in a multiply connected
planar domain with enclosed magnetic flux.
Particular attention is given to the zero set of the order parameter.
We show that there exist applied fields for which the zero set is of
codimension 1.
\end{Abstract}
\section {Introduction}
\setcounter{equation}{0}
The Ginzburg Landau (GL) model of
superconductivity concerns an order parameter $u$ which is a complex
valued function and the magnetic field vector
potential $A$. The problem is forced by an applied magnetic field
$H_e$. The local minimizers
describe the possible equilibrium configurations under the external
constraints. The square of the absolute
value of $u$ measures the density of
the superconducting electrons, while the phase of $u$ is related to the
supercurrent.
We shall consider here the GL functional in a two dimensional domain
$\Om$ that is
homotopic to an annulus. Of particular interest are the zeros of $u$.
It is often argued that the zero set, which we shall denote by $S$,
should consist of isolated points.
The heuristic reasoning is that since $u$ is complex, its zeros are the
intersection of the zero levels lines of its real and imaginary parts.
Morover, Elliott et al. have recently proved \cite{emq}
that under some smoothness
assumptions on the given data, $S$ is indeed a finite collection of
points.
We shall show that there exist fields $H_e$ for which $S$ consists
of a \it curve \rm connecting the inner boundary of the ring to its outer
boundary. When this occurs we say that $u$ is in the \it singly connected
state \rm \cite{beru1}.
The solution we discovered exists for some temperature range
below the critical temperature $T_c$ in which the transition to
superconductivity occurs. A by-product of our result is that the
Elliott - Matano - Qi \cite{emq}
theorem is valid only for simply connected domains.
When the ring is very thin, the GL functional can be approximated by
a one dimensional model on a curve \cite{rusc}.
We have studied this model in detail \cite{beru2}
and derived an intricate picture of phase transitions
when there exist nonuniformities in the ring's cross section.
Our results here give a rigorous justification to the phase transition
diagram of \cite{beru2}. In the thin ring limit the line of zeros collapses
to a point. The appearance of a singly connected state has
profound effects on the behavior of the supercurrent as
a function of $H_e$. Experimental aspects of the new phase
are discussed in \cite{beru3}.
\section {Formulation}
\setcounter{equation}{0}
We write the GL functional in nondimensional form
\be
G_\lm(u,A)=\int_\Om \left(\lm [-|u|^2 + |u|^4/2]
+|(\nabla-i A)u|^2 \right)
\, dx +
{\kappa^2 \over \lm }\int_{\Er^2} \bigl |\nabla \times A- H_e\bigr|^2\,dx.
\label{glnd}
\ee
Here $\lm$ is a parameter that vanishes at $T_c$ and
increases with $T_c-T$, where
$T\, (T_c)$ is the temperature (critical temperature), and $\ka$ is a
material parameter. We chose $R$, the inner radius
of the ring, as a lengthscale (rather than the
coherence length or the penetration length).
The external field is given in dimensional units by
$\Phi_0 H_e /2 \pi R^2$, and
the dimensional vector potential is $\Phi_0 A /2\pi R$, where $\Phi_0$
is the fundamental flux quantum.
We comment that the form (\ref{glnd}) is slightly
different than the form used in other mathematical texts
(\cite{dgp},\cite{rub}). In our scaling
the temperature appears explicitly in the functional, which is advantageous
for the purpose of studying phase transitions.
The Euler Lagrange equations associated with (\ref{glnd}) are
\begin{eqnarray}
&&-(\nabla-iA)^2 u + \lm u(|u|^2-1)=0,
\quad x\in \Om \label{minE1}\\
&& \nabla \times (\nabla \times A- H_e) =
\lm \kappa^{-2}\Im(\bar u (\nabla - i A)u) 1_{\Om},
\quad x\in \Er^2\label{minE2}\\
&&(\nabla - i A)u \cdot \nu =0, \quad x\in \partial \Om,
\label{minE3}
\end{eqnarray}
where $\nu$ is the unit normal to $\partial \Om$.
We can remove the gauge invariance of (\ref{glnd}) \cite{dgp} by
imposing the constraint $\nabla \cdot A=0$.
We assume in sections 3 and 4 that $H_e$ is a smooth vector field in the
direction orthogonal to the plane, and that \it its support does not
intersect $\Om$. \rm It is easy to
verify that for sufficiently small $\lm$ the global
minimizer is $(u,A)=(0,A_e)$, where $A_e$ is a solution of $\nabla
\times A_e=H_e$. Increasing $\lm$ (i.e. decreasing the temperature), we
reach a critical (bifurcation) value $\lm_p$ where a nonzero solution
emerges. The bifurcation equation is
\begin{eqnarray}
&&-(\nabla -iA_e)^2 u - \lm_p u=0,
\quad x\in \Om \label{bifur1}\\
&&(\nabla - i A_e)u \cdot \nu =0, \quad x\in \partial \Om.
\label{bifur2}
\end{eqnarray}
Since $\lm_p$ is the first $\lm$ for which there is a nontrivial
solution for (\ref{bifur1}), there is a variational characterization
for it:
\be
\lm_p={\rm inf}\,\,\, GL(u)={\rm inf}\,\,\,
\int_\Om |(\nabla- iA_e)u|^2 \, dx,
\label{varcar}
\ee
where the minimization is taken under the constraint $\int_\Om |u|^2 dx=1$.
It has been recognized since the classical work of Little and Parks
\cite{lipa} that the flux of $H_e$ through the hole bounded by the ring,
i.e. the integral of $H_e$ over this area,
is an important physical quantity. The flux times $2 \pi$ will be
denoted by $\Phi$. It will be shown that a necessary condition
for the singly
connected state to appear is that $\Phi$ is as
far as possible from an integer.
The following geometrical - spectral characterization will
be useful in the sequel:
\theo{Definition }{Q}
{Let $\Om$ be a ring-like domain, and let $\Gm$
be a curve connecting the inner boundary of the ring to its outer
boundary. We denote the class of such curves by $\call$.
We introduce the scalar eigenvalue problem
\be
\mu_\Om(\Gm)={\rm inf}_{y \in \cala} \int_\Om |\nabla y|^2\,dx,
\label{eigen}
\ee
where $\cala =\{y \in H^1(\Om),\,\int_\Om y^2\,dx=1,\,\,y|_\Gm =0\}$.
Let
\be
\mu_\Om={\rm inf}\,\mu_\Om(\Gm).
\label{inf}
\ee
A domain $\Om$ is said to be of type Q if the infimum in (\ref{inf})
is achieved at a finite discrete set of isolated curves $\{\Gm_1, \Gm_2,
..., \Gm_q\}$.}
\section {The linear problem}
\setcounter{equation}{0}
We first establish our result for the linear bifurcation problem
(\ref{bifur1})-(\ref{bifur2}).
\theo{\Theorem}{1}{Assume $\Om$ is of type Q.
Let
\be
2\Phi=2l+1
\label{condphi}
\ee
for some integer $l$. Then every solution of (\ref{bifur1})-(\ref{bifur2})
vanishes along a curve $\Gm$ of class
$\call$. Moreover, the polar form of the solution is
$u(x)=Y(x)e^{i \phi(x)}$, where $\phi(x)$ satisfies
\be
\nabla \phi =A_e.
\label{phase}
\ee
The curve $\Gm$, the amplitude $Y$ and the
eigenvalue $\lm_p$ are the solution to the
minimization problem (\ref{eigen})-(\ref{inf}).
}
\begin{Proof}
Since the solutions of (\ref{bifur1})-(\ref{bifur2}) are critical points of
(\ref{varcar}), the real and imaginary parts of $u$ are analytic
functions \cite{mor}.
We recall Theorem 2.6 of \cite{emq} that states that $S$ consists
only of isolated points or curves which end on the boundary of $\Om$.
Thus the phase is well defined almost everywhere.
Moreover, using the same arguments as in Corollary 2.7 of \cite{emq},
$S$ can at most contain curves of type $\call$.
Let $\phi_c$ be the family of solutions of (\ref{phase}) that differ
from each other by a constant.
Consider first solutions of
(\ref{bifur1}) of the form $u_c=Y e^{i \phi_c}$ for some $c$.
Let $\calc$ be a curve
circulating once around the ring. Integrating (\ref{phase}) along $\calc$,
using (\ref{condphi}), and applying Stokes theorem,
we find that $u_c$ is not single valued unless it
has a zero somewhere on $\calc$. It follows that $Y$ must vanish along
some curve $\Gm_c$ of type $\call$.
Also, normalizing $Y$ to have a unit $L^2$ norm, we see by
substituting (\ref{phase}) into (\ref{varcar}) that
\be
GL(u_c)=\mu_\Om(\Gm_c).
\ee
Assume now in contradiction that there exists a
solution $u_1=Y_1 e^{i \phi_1}$ of (\ref{bifur1})-(\ref{bifur2})
whose phase does not satisfy (\ref{phase}).
Using (\ref{condphi}), we see that
$u_2=Y_1 e^{i (2\phi_c-\phi_1)}$ is also a solution.
But since (\ref{bifur1})-(\ref{bifur2})
is linear, any linear combination of $u_1$ and $u_2$ is a solution too.
Consider for example
\be
u^*_c=Y^*_c e^{i\phi^*_c}=u_1-u_2=
Y_1 e^{i\phi_1}(1-e^{2i(\phi_c-\phi_1)})
\label{lincomb}
\ee
for some $c$.
Clearly we can find a value $c$ such that
the equation $\phi_c(x)-\phi_1(x)=0$ defines a curve $\gm_c$.
Since $\gm_c$ consists of zeros of $u^*_c$, it must be of class $\call$.
Equation (\ref{bifur1}) implies that
the supercurrent $J=Y_c^{*2}(\nabla \phi^*_c -A_e)$ is a divergence free
vector field. Therefore the streamlines satisfy the dynamical system
\begin{eqnarray}
&& \dot{x_1} \doteq J_1=-\psi_{x_2} \label{stream1} \\
&& \dot{x_2} \doteq J_2= \psi_{x_1} \label{stream2}
\end{eqnarray}
for some stream function $\psi(x)$. Consider now the domain
$\Om \setminus \gm_c$.
The boundary conditions (\ref{bifur2}) imply that $\psi$ is
constant on its boundary.
We shall first prove that the dynamical system
(\ref{stream1})-(\ref{stream2})
has no periodic solutions. Assume in contradiction that
there exists some periodic orbit.
Dividing $J$ by $Y_c^{*2}$, integrating along the periodic orbit
and denoting the tangential direction on the orbit by $\tau$, we get
\be
\int {J \over Y_c^{*2}} \cdot d\tau = \int (\nabla \phi^*_c -A_e)d \tau =0,
\label{curin}
\ee
where we used in the last equality the single valuedness of $u^*_c$ and
the fact that $H_e=0$ in $\Om$. But since the integrand in
(\ref{curin}) is nonnegative, we conclude that $J \equiv 0$,
and therefore all the points on the orbits are in fact critical.
>From the Poincare-Bendixon theorem we now infer that all the orbits are
either critical points, or converging to critical points.
But since $\psi$ is constant on the orbit, it follows that all the values
taken by $\psi$ are critical. If $\psi$ takes more than one value, it must
then take values in some interval, which is impossible according to Sard's
theorem. Hence $\psi$ is constant in $\Om \setminus \gm_c$,
which implies $J \equiv 0$ and therefore
$\nabla \phi^*_c=A_e$ there.
Returning to (\ref{varcar}), normalizing $Y^*_c$ to have a unit $L^2$
norm, and using (\ref{phase}), we obtain the following identity for $u^*_c$:
\be
\lm_p=GL(u^*_c)=\mu_\Om(\gm_c).
\label{enermu}
\ee
But varying $c$ we obtain a smooth family of curves $\gm_c$ satisfying
(\ref{enermu}) in contradiction to our assumption that $\Om$ is of
type Q. Moreover, property Q implies that the zero set of $u$ consists of
any single member $\Gm_j$ of the set $\{\Gm_1, \Gm_2,..., \Gm_q\}$.
For each one of them there corresponds a unique amplitude $Y_j$.
\end{Proof}
\section {The nonlinear equations}
\setcounter{equation}{0}
The existence of minimizers of (\ref{glnd}) in the singly connected
state is established in the following theorem
\theo{\Theorem}{2}{Assume $\Om$ is of type Q, and (\ref{condphi})
holds. Then there exists an interval $I_\delta=(\lm_p,\lm_p+\delta),\,\,
\delta>0$, such that
for every $\lm \in I_\delta$, the global minimizer of (\ref{glnd})
vanishes along a curve of class $\call$.
The phase of the global minimizer satisfies (\ref{phase}) and
the current vanishes everywhere.}
\begin{Proof}
Fix $\lm > \lm_p$, and let $(u,A)=(Ye^{i\phi},A_e+A_i)$ be a
global minimizer of $G_\lm$.
If $\phi$ satisfies (\ref{phase}), then $A_i=0$, and the result follows
by the same arguments as in Theorem 1.
If, on the other hand, $\phi$
does not satisfy (\ref{phase}), then from equation (\ref{minE2}) $A_i$
cannot vanish identically. Consider now
\be (v,B)=(Ye^{i(2\phi_c-\phi)},A_e-A_i),
\label{new}
\ee
where $\phi_c$ is a solution of (\ref{phase}). Condition (\ref{condphi})
implies that $(v,B)$ is smooth, and it is easy to check that
$G_\lm(v,B)=G_\lm(u,A)$. We remark that $u$ and $v$ are of
different homotopy types: Let the homotopy type of $u$ be $d_1$. This
means $\int_{\calc} \nabla \phi =d_1$, where $\calc$ is any curve
circulating once about the ring. But then the homotopy type
of $v$ is $d_2=2l+1-d_1$, and since $d_1$ and $l$ are
integers, $d_1 \neq d_2$.
We have shown (Theorem 1) that a unique solution $u_p$ bifurcates
at $\lm=\lm_p,\,2\Phi=2l+1$. Applying the Crandall - Rabinowitz
abstract bifurcation theory (\cite{crra}, Theorem 2.4),
one can show that there is a unique branch
originating at $u_p$. The details are similar to the examples in
\cite{crra} and to Theorem 8..6 of
\cite{bpq}, so we do not spell them out. It follows that at least
at a small $\lm$ interval to the right of $\lm_p$, the phase $\phi$ must
satisfy (\ref{phase})
Substituting (\ref{phase}) and $A_i=0$ in (\ref{minE1})-(\ref{minE3})
we find that the amplitude $Y$ solves
\be
-\Delta Y +\lm Y(Y^2-1)=0,\;\;x \in \Om
\label{sol1}
\ee
\be
\nu \cdot \nabla Y=0,\,\,x\in \partial \Om
\label{sol2}
\ee
Multiplying (\ref{sol1}) by $Y$, integrating by parts and using
(\ref{phase}) again, we get
\be
G=-{\lm \over 2} \int_{\Om} Y^4\,dx \,<\,0.
\ee
Thus the pair $(u,A_e)$, where $u$ is in the singly connected state,
is the global minimizer of $G$.
\end{Proof}
\section {The one dimensional model}
\setcounter{equation}{0}
When the ring is very thin, the GL model can be approximated by
a functional on a curve $M$ forming the ring's skeleton. The one
dimensional model was used by several authors (see e.g. \cite{pan}).
It was rigorously derived in \cite{rusc}:
\be
G^1_\lm(u)=\int_M \left(\lm [-|u|^2 + |u|^4/2]
+|(\partial_\theta-i A_e)u|^2 \right)
\, D(\theta)\,d\theta.
\label{gl1d}
\ee
Here $\theta$ is a variable along $M$ and $D(\theta)$ is a strictly
positive function that measures the
thickness of the ring at $\theta$.
The Euler Lagrange equation associated with (\ref{gl1d}) is
\be
(\partial_\theta - A_e)D(\partial_\theta-A_e)u+\lm Du(1-|u|^2)=0,
\label{el1}
\ee
with periodic boundary conditions.
The Q property is now replaced by
\theo{Definition }{Q1}
{Consider the scalar eigenvalue problem
\be
\mu_M(\al)={\rm inf}_{y \in \calb} \int_M |\partial_\theta
y(\theta)|^2\,D(\theta+\al)
\,d\theta
\label{eigen1}
\ee
where $\calb =\{y \in H^1(M),\,\int_M y^2=1,\,\,y(0) =0\}$.
Let
\be
\mu_M={\rm inf}\,\mu_M(\al).
\label{inf1}
\ee
A pair $(M,D)$ is said to be of type Q1 if the infimum in (\ref{inf1})
is achieved at a discrete set of points $\al$.}
The phase diagram for the model (\ref{gl1d}) was systematically
studied in \cite{beru2}. The following theorem provides theoretical
support to the overall structure of the diagram:
\theo{\Theorem}{3}{Assume $(M,D)$ is of type Q1, and $\Phi$ satisfies
(\ref{condphi}). Then
\begin{enumerate}
\item
There exists a critical value $\lm^1_p$
where the trivial solution to (\ref{el1}) bifurcates into a
singly connected state, and an
interval $I^1_\delta=(\lm_p^1,\lm_p^1+\delta),\,\,\delta >0$,
where the global minimizer for
(\ref{gl1d}) is singly connected.
\item
There is another value $\lm^1_q$, such that for all $\lm \in
(\lm_q^1,\infty)$ the global minimizer is not singly connected
\end{enumerate}}
\begin{Proof}
The first part of the theorem follows by the same arguments as in
Theorem 2. Notice, though, that in the one dimensional setup, the
equation $\partial_\theta \phi =A_e$, replacing (\ref{phase}), is always
solvable.
To prove the second part, we recall \cite{beru2}
that in the one dimensional case
the Euler Lagrange equation for the phase of the order parameter
can be integrated explictly in terms of the applied magnetic field.
The energy functional for doubly connected functions (i.e. functions
that do not vanish on $M$ \cite{beru1}) takes the form
\be
G_\lm^1(Y)=\int_M \left(\lm [-Y^2 + Y^4/2]
+(\partial_\theta Y)^2 \right) \, D(\theta)\,d\theta+
{(2\pi k)^2 \over \Lm}.
\label{gl1da}
\ee
Here $Y$ is the amplitude of $u$,
$k=N -\Phi$, where $N$ is the circulation of the phase,
and $\Lm=\int_M {d\theta \over DY^2}$. Clearly, to minimize the energy,
$N$ is chosen to minimize $k^2$. Our assumption on $\Phi$ implies
$k^2=1/4$.
The energy functional for singly connected functions is similar except
that the last term is ommitted.
The Euler Lagrange equation for the amplitude of the
singly connected state is
\be
\partial_\theta(D \partial_\theta Y)+\lm D(Y-Y^3)=0,\,\,\,Y(0)=Y(2\pi)=0,
\label{scel}
\ee
where, without loss of generality, we chose the angle at which
$Y$ vanishes to be $\theta=0$.
It is easy to verify that the solution to (\ref{scel}) satisfies
$0 \leq Y \leq 1$.
Set $e=D(\partial_\theta Y)^2+\lm D (Y^2-Y^4/2)$. Differentiating $e$
with respect to $\theta$ and using (\ref{scel}) we get
\be
\partial_\theta e+D^{-1}\partial_\theta De=
2\lm D^{-1}\partial_\theta D(Y^2-Y^4/2).
\label{gron}
\ee
If the singly connected
state is a global minimizer, its energy
should be negative. Therefore there must be at least
one point where $|\partial_\theta Y| \leq C \lm^{1/2}$ for some constant
$C$. Applying Gronwall's lemma to (\ref{gron}), we conclude that
$|\partial_\theta Y| \leq C \lm^{1/2}$ for all $\theta$.
Multiplying (\ref{scel}) by $Y$ and integrating by parts, we get
for the singly connected state
\cite{beru2}
\be
G_\lm^1(Y) =-{\lm \over 2}\int_M DY^4\,d\theta.
\label{energid}
\ee
We also compute
\be
G_\lm^1(Y=1) =-{\lm \over 2}\int_M D +{\pi^2 \over \int_M D^{-1}}.
\label{it1}
\ee
The singly connected state vanishes at some point, and
the derivative estimate we have obtained implies that, for large $\lm$,
$Y$ is small at some interval of size $\lm^{-p}$ about this
point for every $p\in (1/2,1)$. Thus
\be
{\lm \over 2}\int_M D-{\lm \over 2}\int_M DY^4\,d\theta \geq
O(\lm^{1-p}),\,\,\,p \in (1/2,1).
\label{estim}
\ee
Therefore for $\lm$ sufficiently large the constant solution $Y \equiv 1$
has a lower energy than any singly connected state.
\end{Proof}
\section {Discussion}
\setcounter{equation}{0}
We have shown that the zero set of the order parameter can be of
codimension 1. Our construction is based on the solvability of the
phase equation (\ref{phase}), which requires the applied magnetic
field to vanish in $\Om$, while still allowing for nonzero flux.
Therefore the topology of the ring is crucial.
Elliot, Matano and Qi proved in \cite{emq} that the zero set of
the minimizers to the Ginzburg Landau functional in two dimensional
domains consists of isolated points. We have given here an example of
domains for which the zero set consists of a curve. The reason for
the disagreement is that the construction in Lemma 2.7 of \cite{emq} is
valid only for singly connected domains.
The bifurcation equation (\ref{bifur1})-(\ref{bifur2}) can also
be interpreted as the Schr\"odinger equation for a charged particle
in a ring under an external magnetic field and
with zero probability current through the boundary.
Thus Theorem 1 implies that whenever the magnetic flux, measured
in fundamental flux units, is an integer plus half, then
there exists a `forbidden curve' for the ground state.
In \cite{beru1} and \cite{beru2} we have used asymptotic and numerical
calculations to study a weighted GL energy on a circle.
We predicted the existence of a temperature range in which
a new superconducting phase will emerge if the weight is nontrivial.
The new phase is characterized by a smooth transition between
states with different angular quantum numbers. Theorem 3 provides a
theoretical support to the phase transition picture we computed.
On the other hand, the standard GL energy in thin annular domains can be
shown \cite{rusc} to converge to the weighted GL energy on a circle, where
the weight is determined by the local thickness of the annulus.
Thus we have rigourosly established the existence of the new phase for
asymmetric thin rings.
The geometrical-spectral characterization of $\Om$ we introduced
in Definition Q poses a novel challenging problem in calculus of
variations. Essentially condition Q (or Q1) means that the domain is not
symmetric. It does not hold for the
symmetric annulus confined between two concentric circles.
We conjecture that Q is a generic property.
\vspace{3mm}%
\hspace*{18mm}%
{\bf Acknowledgments}\\[0.3cm]%
\begin{minipage}[t]{167mm}%
\small \em
This work was supported by the US-Israel Binational Science
Foundation. We thank G. Wolansky for helpful conversations.
\end{minipage}
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\end{thebibliography}
\begin{Address}
J.B.: Department of Physics, Technion,
32000 Haifa, Israel;
{\tt e-mail:phr76jb@vmsa.technion.ac.il}
J.R.: Department of Mathematics, Technion,
32000 Haifa, Israel;
{\tt e-mail:koby@leeor.technion.ac.il}
\end{Address}
\end{document}