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\centerline {\bf MACROSCOPIC QUANTUM THEORETIC APPROACH}
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\centerline {{\bf TO SUPERCONDUCTIVE
ELECTRODYNAMICS}\footnote*{Based on talk given at the
Amalfi Conference of October 14-16, 1993, on "Superconductivity
and Strongly Correlated Electronic Systems"}}
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\centerline {{by Geoffrey L. Sewell}\footnote{**}{Partially
supported by European Capital and Mobility Contract No. CHRX-
CT92-0007}}
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\centerline {Department of Physics, Queen Mary and Westfield
College,}
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\centerline {Mile End Road, London E1 4NS}
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{\bf Dedication.} It is a pleasure to contribute to this volume,
dedicated to Maria Marinaro. I shall use this opportunity to
present a rather personal approach to superconductivity theory,
which I hope will appeal to Maria's eclectic tastes.
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{\bf Abstract.} I present a general, quantum statistical
derivation of superconductive electrodynamics from the
assumptions of off-diagonal long range order (ODLRO) and gauge
covariance of the second kind, without reference to the
microscopic mechanism responsible for the ordering. On this
basis, I prove that the macroscopic wave function, specified by
the ODLRO condition, enjoys the London rigidity property [Lo];
and, from this result, I derive the Meissner and Josephson
effects, and the quantisation of trapped magnetic flux. I also
outline a framework for the treatment of the open problem of
the metastability of supercurrents.
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{\bf 1. Introduction.} The object of this article is to present
an approach to superconductive electrodynamics, leading to a
derivation of the electromagnetic properties of superconductors
from their order structure. This approach is based on a quantum
treatment of macroscopic variables, and so is at the opposite
pole from the standard microscopic, many-body theory.
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In order to explain the need for such an approach, let me first
recall that, even in the case of metallic superconductivity, the
widely accepted Bardeen-Cooper-Schrieffer [BCS] theory does not
provide a satisfactory electrodynamics, because it fails to meet
the basic requirement of gauge covariance of the {\it second kind}
[Sc1, Fr1]. Here, the essential point is that, although one starts
with a fully gauge invariant model, given by Fr\"ohlich's
electron-phonon system [Fr2], the BCS ansatz is based on a
truncated, gauge-dependent version of this model, that retains
only those interactions that give rise to Cooper pairing.
Attempts [An, Ri] to overcome this difficulty by taking account
of the remaining interactions have led to derivations of the
Meissner effect that are only {\it approximately} gauge
invariant. Since exact gauge invariance is required for the very
definition of local electric currents, this is no solution to the
problem. As regards ceramic, i.e. high $T_{c},$ superconductors,
the microscopic theory is still less developed and has not led
to an electrodynamics.
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Thus, there is a need for a need for a quantum-based gauge
invariant electrodynamics of superconductors. Since, at the
observational level, this electrodynamics has such sharply
defined {\it qualitative} characteristics, the task of a
corresponding quantum theory is surely to exhibit them in a
precise form. It is clear that the traditional techniques of
many-body theory [Pi, Th] are unsuited to this purpose, since
they are designed for essentially approximative calculations
rather than precise classifications. Moreover, the shortcomings
of these techniques are quite radical, since, except in the very
special case of exactly solvable models, they are based on
approximations, which are renderd uncontrollable by the extreme
microscopic instability at the root of statistical mechanics
(cf. [VK].)
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I shall now present a different approach to this problem [Se1,
2], that is based on the characterisation, proposed by Yang [Ya],
of the order structure of superconductors. This is completely
gauge covariant and circumvents the radical problems of many-body
theory. To explain what is involved here, let us first note that
the BCS characterisation of the metallic superconductive phase
by electron pairing, first proposed by Schafroth [Sc2], has been
amply substantiated by experiments on the Josephson effect [Jo]
and the quantisation of trapped magnetic flux in
multiply-connected superconductors [DF]. Further, it was pointed
out by Yang [Ya] that this characterisation is captured by the
hypothesis of {\it off-diagonal long range order} (ODLRO), first
introduced by O. Penrose [Pe, PO] for the theory of superfluid
Helium. Moreover, the ODLRO hypothesis is fulfilled not only by
the BCS ansatz, but by the bipolaron model [AR, AM] of high
$T_{c}$ superconductors, as well as Feynman's [Fe] theory of
superfluid Helium [PO]. These observations suggest an approach
to superconductive electrodynamics based on the assumption of
ODLRO.
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This is precisely the approach I pursue. My essential objective
is to relate the electromagnetic properties to the order
structure of these systems in purely macroscopic quantum terms.
Here, I shall formulate the theory within the standard framework
of condensed matter physics, by contrast with the mathematically
more abstract treatment of [Se2].
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I shall organise the treatment as follows. In ${\S}2,$ I shall
formulate the generic quantum model of a system of interacting
particles of one or more species, satisfying the requirement of
gauge covariance of the second kind. Here, I shall specify the
condition of ODLRO.
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In ${\S}3,$ I shall derive the Meissner effect from the
assumptions of ODLRO, gauge covariance and translational
invariance. The key to this is the incompatibility of ODLRO with
a non-zero uniform magnetic induction (Prop. 3.1). This
corresponds to a London rigidity [Lo], at the {\it macroscopic
quantum level.}
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In ${\S}4,$ I shall extend the above treatment to derive both the
quantisation of trapped magnetic flux, in multiply-connected
superconductors, and the Josephson effect, from the assumptions
of ODLRO and gauge covariance.
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In ${\S}5,$ I shall formulate, in outline, a framework for a
treatment of the metastability of persistent currents, which,
remarkably, remains an open problem.
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In ${\S}6,$ I shall briefly summarise the conclusions to be drawn
from this work.
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{\bf 2. The Model.} We take the quantum model, ${\Sigma},$ to be
an infinitely extended system of particles in a Euclidean space
$X:$ lattice systems may be formulated analogously. It will be
assumed that ${\Sigma}$ consists of a system, ${\Sigma}_{el},$
of electrons, and possibly of another component, ${\Sigma}_{i},$
consisting of ions or phonons. Points in $X$ will generally be
denoted by $x,$ but sometimes by $y,a$ or $b$. It will be assumed
that the model enjoys the properties of gauge covariance of the
second kind, and that its interactions are translationally
invariant. These assumptions are satisfied by Fr\"ohlich's [Fr2]
electron-phonon model and Hubbard's [Hu] strong repulsion model,
on which the theories of metallic and ceramic superconductivity,
respectively, are usually based. Further, at a more fundamental
level, they are also satisfied by the electron-ion model with
Coulomb interactions.
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The electronic part, ${\Sigma}_{el},$ of ${\Sigma}$ is formulated
in terms of a quantised field
${\psi}=({\psi}_{\uparrow},{\psi}_{\downarrow}),$
satisfying the canonical anticommutation relations
$${\lbrack}{\psi}_{\alpha}(x),{\psi}_{\beta}(y)^{\star}
{\rbrack}_{+}= {\delta}_{{\alpha},{\beta}}{\delta}(x-y); \
{\lbrack}{\psi}_{\alpha}(x),{\psi}_{\beta}(y){\rbrack}_{+}
=0\eqno(2.1)$$
The observables of ${\Sigma}_{el}$ are generated by the
polynomials in ${\psi}$ and ${\psi}^{\star}$ that are invariant
under gauge transformations of the first kind, i.e.,
${\psi}{\rightarrow}{\psi}e^{i{\alpha}},$ with ${\alpha}$
constant. Thus, they are generated algebraically by the monomials
$${\psi}^{\star}(x_{1}).. \
.{\psi}^{\star}(x_{n}){\psi}(x_{n+1}).. \ .{\psi}(x_{2n}).$$
A dynamical characterisation of equilibrium states at inverse
temperature ${\beta}$ is then given by the Kubo-Martin-Schwinger
(KMS) condition, which constitutes a generalisation to infinite
systems of the standard Gibbsian one [HHW; Se3,4], and is given
by
$${\langle}Q_{1}(t)Q_{2}{\rangle}=
{\langle}Q_{2}Q_{1}(t+i{\hbar}{\beta}){\rangle}\eqno(2.2)$$
for all observables $Q_{1}, \ Q_{2},$ where $Q(t)$ is the
Heisenberg operator representing the evolute of $Q$ at time $t.$
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We shall be concerned with the properties of ${\Sigma}$ in the
presence of a classical magnetic induction $B=curlA,$ and, as
stated above, we assume that its dynamics is covariant w.r.t.
gauge transformations of the second kind, i.e.,
$$A(x){\rightarrow}A(x)+{\nabla}{\phi}(x); \ {\psi}(x)
{\rightarrow}{\psi}(x){\exp}(ie{\phi}(x)/{\hbar}c)\eqno(2.3)$$
where ${\phi}$ is an arbitrary function of position and $-e$ is
the electronic charge. Further, the assumption of translationally
invariant interactions implies the covariance of the dynamics
w.r.t. space translations
$$A(x){\rightarrow}A(x+a); \ {\psi}(x){\rightarrow}{\psi}(x+a),
\eqno(2.4)$$
with $a$ an arbitrary spatial displacement, together with a
corresponding transformation for ${\Sigma}_{i}.$
Specialising now to the case where the magnetic induction $B$ is
uniform, and so may be represented by the vector potential
$A(x)={1\over 2}(B{\times}x),$ and choosing ${\phi}(x)=-{1\over
2}(B{\times}x).a,$ we have the relation
$A(x)+{\nabla}{\phi}(x){\equiv}A(x-a).$ Hence, by (2.3) and
(2.4), the dynamics is covariant w.r.t
$${\psi}(x){\rightarrow}{\psi}(x+a)
{\exp}({-ie(B{\times}x).a\over 2{\hbar}c});
\ A(x){\rightarrow}A(x)\eqno(2.5)$$
together with the corresponding transformation for
${\Sigma}_{i}.$ Thus, the space translation for the
electronic part of ${\Sigma},$ in the presence of a uniform
magnetic induction are given by (2.5). We term these the {\it
regauged space translations.} It will be seen that the sinusoidal
factor plays a crucial role in our derivation of the Meissner
effect in ${\S}3,$ and, subsequently, of other electromagnetic
properties of superconductors.
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We define the {\it pair field}
$${\Psi}(x_{1},x_{2})={\psi}_{\uparrow}(x_{1})
{\psi}_{\downarrow}(x_{2})\eqno(2.6)$$
The property of ODLRO may then be expressed in terms of this
field
by the condition that
$${\lim}_{{\vert}y{\vert}\to\infty}[{\omega}({\Psi}(x_{1},x_{2})
{\Psi}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)
-{\Phi}(x_{1},x_{2}){\Phi}^{\star}
(x_{1}^{\prime}+y,x_{2}^{\prime}+y)]=0\eqno(2.8)$$
for all $x_{1}, \ x_{2}, \ x_{1}^{\prime}$ and $x_{2}^{\prime}$
in $X,$ where ${\Phi}$ is a classical field, that does not tend
to zero at infinity, i.e., for some $x_{1},x_{2}, \
{\Phi}(x_{1}+y,x_{2}+y)$ does not tend to zero as
${\vert}y{\vert}{\rightarrow}{\infty}. \ {\Phi}$ is then termed
the {\it macroscopic wave function.}
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${\bf Note}$ that, although ${\Psi}$ is not an observable, the
quantity in angular brackets in (2.8) is one.
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${\bf Lemma \ 2.1.}$ {\it The ODLRO conditions define the
macroscopic wave function up to a constant phase factor, i.e.,
if ${\Phi}_{1},{\Phi}_{2}$ both satisfy these conditions, for the
same state of ${\Sigma},$ then
${\Phi}_{2}={\Phi}_{1}{\exp}(i{\eta}),$ where ${\eta}$ is a
real-valued constant.}
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{\bf Proof.} Assuming that ${\Phi}_{1},{\Phi}_{2}$ both
satisfy the ODLRO conditions with respect to the same state, it
follows from (2.8) that
$${\lim}_{{\vert}y{\vert}\to\infty}[{\Phi}_{1}(x_{1},x_{2})
{\Phi}_{1}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)-
{\Phi}_{2}(x_{1},x_{2})
{\Phi}_{2}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)]=0
\eqno(2.9)$$
Since this is valid for all
$x_{1},x_{2},x_{1}^{\prime},x_{2}^{\prime}{\in}X,$ we may
replace $x_{1},x_{2}$ here by arbitrary points
$x_{1}^{{\prime}{\prime}},x_{2}^{{\prime}{\prime}},$
thereby obtaining
$${\lim}_{{\vert}y{\vert}\to\infty}[{\Phi}_{1}
(x_{1}^{{\prime}{\prime}},x_{2}^{{\prime}{\prime}})
{\Phi}_{1}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y))-
{\Phi}_{2}(x_{1}^{{\prime}{\prime}},x_{2}^{{\prime}{\prime}})
{\Phi}_{2}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)]=0
\eqno(2.10)$$
On multiplying (2.9) by
${\Phi}_{2}(x_{1}^{{\prime}{\prime}},x_{2}^{{\prime}{\prime}})$
and (2.10) by ${\Phi}_{2}(x_{1},x_{2})),$ and then taking the
difference, we see that
$${\lim}_{{\vert}y{\vert}\to\infty}
{\Phi}_{1}^{\star}(x_{1}^{\prime}+y,x_{2}+y)
[{\Phi}_{1}(x_{1},x_{2}){\Phi}_{2}(x_{1}^{{\prime}{\prime}},
x_{2}^{{\prime}{\prime}})-
{\Phi}_{1}(x_{1}^{{\prime}{\prime}},x_{2}^{{\prime}{\prime}})
{\Phi}_{2}(x_{1},x_{2})]=0$$
Since, by the above definition of ODLRO, ${\Phi}$ does not tend
to zero at infinity, it follows that the quantity in the square
brackets of this last equation vanishes. Therefore, since
${\Phi}_{1,2}$ are non-zero, by the same stipulation,
$${\Phi}_{2}(x_{1},x_{2})=c{\Phi}_{1}(x_{1},x_{2}) \
{\forall}x_{1},x_{2}{\in}X$$
where $c$ is a complex-valued constant. Consequently, as
${\Phi}_{1},{\Phi}_{2}$ both satisfy (2.8), it follows
immediately that $c$ is just a constant phase factor
${\exp}(i{\eta}).$
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{\bf 3. The Meissner Effect.} The essential distinction
between normal diamagnetism and the Meissner effect is that the
former can support a uniform, static, non-zero magnetic induction
and the latter cannot. Thus, we base our derivation of the
Meissner effect on considerations of the response of a state
possessing ODLRO to the action of a uniform magnetic field.
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${\bf Proposition \ 3.1.}$ {\it The system cannot support uniform
(non-zero) magnetic induction in translationally invariant states
possessing the property of ODLRO.}
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The following Corollary follows immediately from this Propostion
and elementary thermodynamics.
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${\bf Corollary \ 3.2.}$ {\it Assuming that there is no
translational symmetry breakdown in the equilibrium state, then
either
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(1) ODLRO prevails and $B=0,$ or
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(2) the system is normally diamagnetic and does not possess
ODLRO.
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Further, assuming that, in the absence of a magnetic field,
the free energy density of the ODLRO phase is lower, by
${\Delta},$ than that of the normal one, the former phase will
prevail, and thus the system will exhibit the Meissner effect,
provided that the applied field, $H,$ satisfies the
condition that} ${\vert}H{\vert}