\magnification 1200
\vsize 23 truecm
\hsize 15truecm
\baselineskip 18truept
\centerline {\bf Off-diagonal Long Range Order}
\vskip 0.5cm
\centerline {\bf and Superconductive Electrodynamics}
\vskip 1cm
\centerline {{by Geoffrey L. Sewell}\footnote{$^{(a)}$}{Electronic mail:
G.L.Sewell@qmw.ac.uk}}
\vskip 1cm
\centerline {Department of Physics, Queen Mary and Westfield
College}
\vskip 0.5cm
\centerline {Mile End Road, London E1 4NS}
\vskip 1.5cm
\centerline {\bf Abstract}
\vskip 0.5cm\noindent
\vskip 1.5cm
We present a general, model independent, quantum statistical
derivation of superconductive electrodynamics from the
assumptions of off-diagonal long range order (ODLRO), local gauge
covariance and thermodynamic stability. On this basis, we obtain
the Meissner and Josephson effects, the quantisation of trapped
magnetic flux, and the metastability of supercurrents. A key to
these results is that the macroscopic wave function, specified
by the ODLRO condition, enjoys the rigidity property that
London$^{1,2}$ envisaged for the microstate of a superconductor.
\vskip 2cm
PACS Numbers 74.20.-z; 64.60.My; 05.30.-d
\vfill\eject
\centerline {\bf I.Introduction}
\vskip 0.2cm
The object of this article is to provide a model-independent
derivation of the electromagnetic properties of superconductors
from their order structure, the essential imput being the
assumptions of off-diagonal long range order, gauge covariance
and thermodynamic stability. Thus, our approach to the subject
is centred on very general principles, and is therefore
at the opposite pole from that provided by the computational
techniques of many-body theory.
\vskip 0.2cm
In order to explain the need for such an approach, let us first
recall that, in the case of metallic superconductivity, the
electron pairing hypothesis$^{3,4}$ has been amply substantiated,
at an empirical level, by the the measured values of the magnetic
flux quantisation$^5$ and of the Josephson tunnelling
frequency$^6$. Furthermore, the widely accepted microscopic
theory of superconductivity, devised by Bardeen, Cooper and
Schrieffer (BCS)$^7$ on the basis of this hypothesis, provides
an accurate picture of the thermodynamics$^{8-10}$ of
superconductors, especially of their second order phase
transitions. On the other hand, the electrodynamics of this
theory is seriously flawed in that, firstly, it violates the
principle of local gauge covariance$^{11,12},$ and consequently
does not even admit a precise definition of the local current
density; and, secondly, there is no indication that its ansatz
for current-carrying states have the metastability
properties$^{13,14}$ of supercurrents. In fact, the violation of
the
gauge principle by the BCS theory arises from its {\it
truncation} of a fully gauge invariant model
(Fr\"ohlich's electron-phonon system$^{15}$), that retains only
the interactions that give rise to the electron pairing.
Attempts$^{16,17}$ to remedy the situation by taking some account
of the residual interactions have led to derivations of the Meissner
effect that are only {\it approximately} gauge covariant. Since
exact gauge covariance is required for a consistent
electrodynamics, this is no solution of the problem.
As regards ceramic, i.e. high $T_{c},$ superconductors, the
microscopic theory is less developed, and has certainly not led
to an electrodynamics, even though various interesting ideas
concerning its underlying quantum mechanisms have been proposed,
e.g. in Refs. 18-23.
\vskip 0.2cm
Thus, there is a need for a quantum-based gauge covariant
electrodynamics of the superconducting phase. Since, at an
empirical level, this electrodynamics has such sharply defined
{\it qualitative} characteristics, common to the vast variety of
superconducting materials, a corresponding quantum theory should
surely isolate their origins in general, qualitative terms. We
remark here that the traditional techniques of many-body
theory$^{24,25}$ are unsuited to this purpose, since these are
designed for essentially approximative calculations rather than
precise classifications.
\vskip 0.2cm
In view of these considerations, we have adopted a different
approach$^{26-28}$ to superconductive electrodynamics, based on
the hypothesis of {\it off-diagonal long range order} (ODLRO),
which encapsulates the qualitative features of the electron
pairing in a gauge covariant way. This hypothesis was proposed
by Yang$^{29}$ as a characterisation of the structure of the
superconducting phase, following a similar proposal by
O.Penrose$^{30,31}$ in connection with the theory of superfluid
Helium. Formally, the ODLRO condition may be expressed in terms
of the relevant quantised matter field ${\psi}$ by the formulae
$$\lim_{{\vert}y{\vert}\to{\infty}}{\lbrack}{\langle}
{\psi}(x){\psi}^{\star}(x+y){\rangle}-{\Phi}(x)
{\Phi}^{\star}(x+y){\rbrack}=0 \ for \ bosons, \eqno(1.1a)$$
and
$$\lim_{{\vert}y{\vert}\to{\infty}}\bigl[{\langle}
{\psi}_{\uparrow}(x_{1})
{\psi}_{\downarrow}(x_{2})
{\psi}_{\downarrow}^{\star}(x_{2}^{\prime}+y)
{\psi}_{\uparrow}^{\star}(x_{1}^{\prime}+y){\rangle}
-{\Phi}(x_{1},x_{2}){\Phi}^{\star}(x_{1}+y,x_{2}+y)\bigr]$$
$$=0 \ for \ fermions, \eqno(1.1b)$$
where, in both cases, ${\Phi}$ is a complex-valued function,
often termed the {\it macroscopic wave function}, such that
${\Phi}(x+y),$ or ${\Phi}(x_{1}+y,x_{2}+y),$ does not tend to
zero as ${\vert}y{\vert}{\rightarrow}{\infty}.$ In the
bosonic case, ODLRO is simply a generalisation of Bose-
Einstein condensation$^{30,31}$ to interacting systems.
\vskip 0.2cm
We note here that the ODLRO condition is fulfilled not only by
the BCS ansatz, but also some of those for ceramic
superconductivity$^{19,21}$, as well as that of Feynman$^{32}$
for superfluid Helium (cf Ref. 31). It is also satisfied in the
low temperature phases of a number of tractable, gauge
covariant models, namely the ideal Bose gas$^{33}$, the hard
sphere Bose fluid on a lattice$^{34}$, and, at least at zero
temperature, the Hubbard model with {\it attractive} interaction
between electrons on the same site$^{35}$.
\vskip 0.2cm
Our project, then, is to derive the principal electrodynamic
properties of superconductors from precisely specified
assumptions of ODLRO, gauge covariance and thermodynamic
stability. In fact, progress towards this objective has already
been achieved in Refs. 26-28 and 36, which have provided
derivations of the Meissner effect$^{26},$ the quantisation of
trapped magnetic flux$^{27,36},$ and the Josephson effect$^{27},$
as well as a sketched approach to the theory of persistent
currents$^{28}$.
\vskip 0.2cm
The present article will be devoted to a review and further
development of this work. Our essential aim will be to provide
a coherent account of the way in which the electrodynamics of
superconductors, including the metastabilty of the supercurrents,
stems from their order structure. We shall keep the mathematics
quite simple, formulating the theory within the standard second
quantisation framework of condensed matter physics. We remark
here that one may put the whole theory onto a completely rigorous
basis by recasting it, as in Ref. 27, within the framework of
operator algebraic statistical mechanics.
\vskip 0.2cm
We shall organise the presentation of the theory as follows.
We start, in Sec. II, with a general formulation of our
gauge-covariant model of matter in interaction with a classical
magnetic field. This formulation covers the cases of both
continuous and lattice systems. A key result here is that the
dynamics of the model, in the presence of a uniform magnetic
induction $B,$ is covariant with respect to the {\it regauged
space translations}$^{37,26}$
$${\psi}(x){\rightarrow}{\psi}(x+a){\exp}\bigl({-ie(B{\times}a).x
\over 2{\hbar}c}\bigr),\eqno(1.2)$$
where $a$ is an arbitrary displacement and the sinusoidal factor
arises from the corresponding regauging of the magnetic vector
potential.
\vskip 0.2cm
In Sec. III, we employ this result to derive the Meissner
effect, as in Ref. 26. The essential point is that the formula
(1.2) for regauged space translations, together with the
assumption of ODLRO, leads to the conclusion that {\it either}
${\Phi}$ {\it or} $B$ must vanish. The requirement of
thermodynamic stability of the ordered phase then implies that
the latter condition prevails, for sufficiently weak applied
fields. Thus, we have a Meissner effect, since normally
diamagnetic systems can accommodate non-zero uniform induction.
The key to this result is that the macroscopic wave function
exhibits a rigidity, in the face of the applied field, of the
kind envisaged by London$^1$ at a more microscopic level. It is
worth remarking here that the principle of local gauge
covariance, which was an obstacle to the previous theories, is
an essential ingredient of our argument leading to the Meissner
effect!
\vskip 0.2cm
In Sec. IV, we extend that argument, along the lines of Ref. 36,
to derive the quantisation of trapped magnetic flux in
multiply-connected systems.
\vskip 0.2cm
In Sec. V, we formulate the theory of persistent currents in a
multiply-connected body, such as a ring. Here, the supercurrents
are none other than the currents that implement the Meissner
effect by screening the trapped magnetic flux from the interior
of the body$^2,$ and the essential problem is that of the
stabilisation of both the trapped flux and its screening current.
In fact, this is a problem of {\it metastability}, since the
current-carrying state has higher free energy than that of true
thermal equilibrium at the same temperature. Thus, adopting our
earlier characterisation$^{14}$ of metastable states by
thermodynamic stability against strictly local, rather than
global, disturbances, we formulate the condition for the
persistence of currents in terms of a variational principle, that
is subject to the constraint of the flux quantisation. In this
way, we characterise the phenomenon of superconductivity itself,
i.e. the persistence of currents, by a relatively simple
thermodynamic assumption, together with that of ODLRO.
Furthermore, we show that this phenomenon is intimately related
to a superselection rule, that forbids locally induced
transitions between states with different flux quantum numbers.
\vskip 0.2cm
In Sec. VI, we formulate the Josephson effect and derive its
tunnelling frequency form the properties of the macroscopic
wave function.
\vskip 0.2cm
We conclude, in Sec. VII, with a brief discussion of open
problems of the theory.
\vskip 0.5cm
\centerline {\bf II. The General Model}
\vskip 0.2cm
Our model, ${\Sigma},$ is an infinitely extended system,
consisting of electrons and possibly another species of
particles, e.g. phonons or ions, in a space, $X,$ which may be
either a Euclidean continuum or a lattice. Points in $X$ will
usually be denoted by $x,$ sometimes by $y,a$ or $b.$ We shall
denote the electronic component of ${\Sigma}$ by ${\Sigma}_{el},$
and the other component, if any, by ${\Sigma}^{\prime}.$ We shall
assume that the dynamics of the model is covariant w.r.t. gauge
transformations of both first and second kind, space translations
and time reversals. These assumptions represent general demands
of quantum mechanics and electromagnetism, and are fulfilled by
particular models, such as those of Fr\"ohlich$^{15}$ and
Hubbard$^{38}$, on which theories of metallic and high $T_{c}$
superconductivity, respectively, are based.
\vskip 0.2cm
For simplicity, we shall generally employ a notation appropriate
to the case where $X$ is a continuum. This may easily be
translated into the corresponding one for lattice case by
standard procedures employed in gauge theories$^{39}$.
\vskip 0.2cm
We shall describe the electronic subsystem ${\Sigma}_{el}$ in
terms of a quantised field,
$${\psi}=\pmatrix{{\psi}_{1}\cr {\psi}_{-1}\cr}{\equiv}
\pmatrix{{\psi}_{\uparrow}\cr {\psi}_{\downarrow}\cr},
\eqno(2.1)$$
which satisfies the canonical anticommutation relations
$$[{\psi}_{s}(x),{\psi}_{s^{\prime}}^{\star}(x^{\prime})]_{+}=
{\delta}_{ss^{\prime}}{\delta}(x-x^{\prime}); \
[{\psi}_{s}(x),{\psi}_{s^{\prime}}(x^{\prime})]_{+}=0\eqno(2.2)$$
We shall sometimes use the symbol ${\psi}^{\#}$ to denote
either ${\psi}$ or ${\psi}^{\star}.$
\vskip 0.2cm
The observables$^{40}$ of ${\Sigma}_{el}$ are formed from
the polynomials in ${\psi}$ and ${\psi}^{\star}$ that are
invariant under gauge transformations of the first
kind, ${\psi}{\rightarrow}{\psi}e^{i{\alpha}},$ with ${\alpha}$
constant. Thus, they are generated algebraically by operators
of the form
$${\psi}_{s_{1}}^{\star}(x_{1}).. \
.{\psi}_{s_{n}}^{\star}(x_{n}){\psi}_{s_{n+1}}(x_{n+1}).. \
.{\psi}_{s_{2n}}(x_{2n}).$$
\vskip 0.2cm
We shall be concerned with the properties of the the system in
the presence of a classical electromagnetic field $(E,B),$
represented by a scalar potential ${\phi}$ and a vector
potential, $A:$ thus, $E=-{\nabla}{\phi}-c^{-
1}{\partial}A/{\partial}t$ and $B=curlA.$ We assume that the
dynamics of the model is covariant w.r.t. gauge transformations
of the second kind, as given by the formula
$$A{\rightarrow}A+{\nabla}{\chi}, \ {\phi}{\rightarrow}
{\phi}-c^{-1}{{\partial}{\chi}\over {\partial}t}, \
{\psi}{\rightarrow}{\psi}{\exp}({ie{\chi}\over {\hbar}c})
\eqno(2.3),$$
where ${\chi}$ is an arbitrary function of position and time. We
assume that the observables representing the position-dependent
densities of electronic charge, current and magnetic polarisation
in the presence of this field are given by the standard formulae
$${\rho}=-e{\psi}^{\star}{\psi}; \ j(x)=
{ie\over 2}({\psi}^{\star}{\nabla}{\psi}-
({\nabla}{\psi}^{\star}){\psi})+{e\over c}A{\psi}^{\star}{\psi};
\ and \ m={e{\hbar}\over mc}{\psi}^{\star}{\sigma}{\psi},
\eqno(2.4)$$
where $-e$ is the electronic charge and ${\sigma}$ is the spin
vector, whose components
$({\sigma}_{1},{\sigma}_{2},{\sigma}_{3})$
are the Pauli matrices. We note here that ${\rho}, \ j$ and $m$
are invariant w.r.t. all gauge transformations.
\vskip 0.2cm
We assume that the formal Hamiltonian of the model is of the form
$$H={\int}({\hbar}{\nabla}{\psi}^{\star}+
{ie\over c}A{\psi}^{\star}).({\hbar}{\nabla}{\psi}-{ie\over
c}A)dx-
{\int}(B.m+{\rho}{\phi})dx+V_{el}+H_{int}+H^{\prime},\eqno(2.5)$$
where $V_{el}$ is the potential energy of the interelectronic
interactions, $H^{\prime}$ is the Hamiltonian for
${\Sigma}^{\prime}$ and $H_{int}$ is the energy of interaction
between ${\Sigma}_{el}$ and ${\Sigma}^{\prime}.$ We assume that
these three contributions to $H$ are all independent of the
potentials ${\phi}$ and $A,$ and thus that $H$ is gauge
invariant. Further, we stipulate that the magnetic interactions
between the electrons, that stem from the sources $j(x)$ and
$m(x)$ according to Maxwell's equations, are not incorporated
into either $V_{el}$ or $H_{int},$ but are represented by the
dependence of $B$ on these sources (cf. Comment at the end of
this Section). In the theory that follows, we shall confine our
considerations to situations where $B$ is static, $E=0,$ and,
except in Sec. 6, we shall take it that ${\phi}=0$ and ${\chi}$
is time-independent.
\vskip 0.2cm
We shall assume that the dynamics is covariant w.r.t. space
translations and time reversals. The transformations of the
fields ${\psi}, \ A,$ corresponding to space translations
$a({\in}X),$ are given by
$$A(x){\rightarrow}(x+a), \
{\psi}(x){\rightarrow}{\psi}(x+a).\eqno(2.6)$$
The operation of time reversal serves to transform ${\psi}_{s},
\ {\psi}_{s}^{\star}$ and $A$ to ${\psi}_{-s}^{\star}, \
{\psi}_{-s}$ and $-A,$ respectively, and to invert the order of
the terms in the operator products. Thus, its effective
action on the electronic observables and vector potential is
given by
$${\psi}_{s_{1}}^{\#}(x_{1}). \
.{\psi}_{s_{n}}^{\#}(x_{n})
{\rightarrow}{\psi}_{-s_{n}}^{{\#}{\star}}(x_{n}). \ .
{\psi}_{-s_{1}}^{{\#}{\star}}(x_{1}); \ A{\rightarrow}
-A.\eqno(2.7)$$
\vskip 0.2cm
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 ${\chi}(x)=-
(B{\times}x).a,$ we have the relation
$A(x)+{\nabla}{\chi}(x)={\chi}(x-a).$ Hence, by (2.3) and (2.6),
the dynamics of ${\Sigma}$ is covariant w.r.t. the
transformation$^{37,26}$
$${\psi}(x){\rightarrow}{\psi}_{a}(x){\equiv}
{\psi}(x+a){\exp}({-ie(B{\times}x).a)\over e{\hbar}c}), \
A(x){\rightarrow}A(x),\eqno(2.8)$$
together with the transformation of the ${\Sigma}^{\prime}-$
observables corresponding to space translations. Since (2.6)
consists of a space translation, compensated by a gauge
transformation in such a way as to leave $A$ unchanged, we term
it a {\it regauged space translation}.
\vskip 0.2cm
We denote by $Q_{t}$ the time-translate, in Heisenberg
representation, of an arbitrary observable, $Q,$ of ${\Sigma}.$
A dynamical characterisation of thermal equilibrium states of
the system at inverse temperature ${\beta}$ is given by the Kubo-
Martin-Schwinger (KMS) condition$^{41,42}$, i.e.,
$${\langle}Q_{t}Q^{\prime}{\rangle}_{A}=
{\langle}Q^{\prime}Q_{t+i{\hbar}{\beta}}{\rangle}_{A},
\eqno(2.9)$$
for arbitrary observables $Q$ and $Q^{\prime},$ where the angular
brackets denote expectation value and the suffix $A$ indicates
its dependence on $A$. Most importantly, this condition is valid
even for infinite systems$^{43}$, where the traditional Gibbs
canonical formulation is not directly applicable. In the case
where $B$ is uniform, the covariance of the dynamics w.r.t regauged
space translations ensures that the model supports
equilibrium states that are invariant under these translations.
\vskip 0.2cm
Turning now to the thermodynamics of the model, we define
$f(B,T)$ to be its global free energy density at temperature $T,$
under the action of the uniform induction, $B.$ This function can
be formulated by standard statistical mechanical methods$^{44}$
in terms of the microscopic description of ${\Sigma},$ with $B$
taken to be a given control variable.
\vskip 0.2cm
On the other hand, this induction is {\it not} given, a priori,
in the physical situation where ${\Sigma}$ is subjected to a
magnetic field, $H_{ex},$ due to a fixed source of current
density $J_{ex}.$ For, in this situation, the induction, $B,$ and
the equilibrium state of ${\Sigma}$ co-determine one another.
\vskip 0.2cm
To formulate the thermodynamics of ${\Sigma}$ under these
conditions, we have to incorporate the energy of the field, $B,$
and of its interaction with the source, $J_{ex},$ into the
picture. Thus, we have to consider the system under consideration
to comprise the composite, ${\tilde {\Sigma}},$ of ${\Sigma}$ and
the field $B.$ The contribution to the internal energy of
${\tilde {\Sigma}}$ due to the induction, $B,$ and its coupling
to the source is then
$${\int}({1\over 2}B^{2}-c^{-1}A.J_{ex})dx,$$
and, in view of the Maxwell equation $curlH_{ex}=c^{-1}J_{ex},$
this reduces to
$${\int}({1\over 2}B^{2}-H_{ex}.B)dx.\eqno(2.10)$$
Thus, in the case where $B$ and $H_{ex}$ are uniform, the free
energy density of ${\tilde {\Sigma}}$ is
$${\phi}(B,T,H_{ex})=f(B,T)+{1\over
2}B^{2}-B.H_{ex}.\eqno(2.11)$$
The equilibrium value of $B,$ corresponding to given $H_{ex}$ and
$T,$ is then obtained by minimising ${\phi},$ and the resultant
Gibbs free energy density is
$${\tilde f}(H_{ex},T)=min_{B}{\phi}(B,T,H_{ex}).\eqno(2.12)$$
\vskip 0.2cm
{\bf Comment.} This formulation of the thermodynamics of ${\tilde
{\Sigma}}$ is, of course, semi-classical, since the field $B$ is
treated as classical. In this picture, the field energy (2.10)
stems
from the magnetic interactions of the currents and spins in
${\Sigma}$
both with one another and with the external source. For a
discussion
of the problem of a fully quantum formulation, see Ref. 27,
Sec.4.
\vfill\eject
\centerline {\bf III. ODLRO and the Meissner Effect.}
\vskip 0.2cm
In order to formulate ODLRO, we introduce the {\it pair field}
$${\Psi}(x_{1},x_{2})={\psi}_{\uparrow}(x_{1})
{\psi}_{\downarrow}(x_{2}).\eqno(3.1)$$
We then say that a state of ${\Sigma}$ possesses the property of
ODLRO if there is a {\it classical} two-point field
${\Phi}(x_{1},x_{2})$ such that$^{29}$
$$\lim_{{\vert}y{\vert}\to{\infty}}\bigl[{\langle}
{\Psi}(x_{1},x_{2})
{\Psi}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y){\rangle}_{A}-
{\Phi}(x_{1},x_{2})
{\Phi}^{\star}(x_{1}^{\prime}+y,x_{2}^{\prime}+y)\bigr]=0,\eqno(3.2)$$
and, further, ${\Phi}(x_{1}+y,x_{2}+y),$ does not tend to zero
as ${\vert}y{\vert}{\rightarrow}{\infty}.$ In this case, ${\Phi}$
is termed the {\it macroscopic wave function} of the state.
\vskip 0.2cm
{\bf Note.} Although ${\Psi}$ is not an an observable, in the
sense specified in Sec. II, the quantity in angular brackets
in the ODLRO condition (3.2) is.
\vskip 0.2cm
In order to relate ODLRO to superconductive electrodynamics, we
note that 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.
\vskip 0.2cm
The following two Propositions, which we shall prove at the end
of this Section, provide us with the key to the relationship
between ODLRO and the electromagnetic properties of our model.
\vskip 0.2cm
{\bf Proposition 3.1.} {\it (1) The ODLRO condition (3.2) defines
the macroscopic wave function ${\Phi},$ up to a phase factor,
i.e., if ${\Phi}_{1}$ and ${\Phi}_{2}$ are two such functions
which satisfy this condition for the same state, then
${\Phi}_{1}=e^{i{\alpha}}{\Phi}_{2},$ where ${\alpha}$ is
some real constant.
\vskip 0.2cm
(2) In the case where $A=0$ and where the state is invariant
with respect to time reversals, the function ${\Phi}$ is
real-valued, up to a constant phase factor.}
\vskip 0.2cm
{\bf Proposition 3.2.} {\it If the system is in a translationally
invariant state possessing the property of ODLRO, then
\vskip 0.2cm
(1) in the case where $X$ is a continuum, the magnetic induction
$B$ vanishes, and
\vskip 0.2cm
(2) in the case where $X$ is a lattice, $B$ is restricted to the
discrete set of values ${\lbrace}B^{(n)}{\rbrace},$ given by the
condition that $B^{(n)}.(a{\times}b)$ is an integral multiple of
${\pi}e/{\hbar}c,$ for any $a, \ b$ in $X.$ Thus, the vectors
$B^{(n)}$ are the sites of an associated lattice, ${\cal B},$
defined by this condition.}
\vskip 0.2cm
{\bf Comments.} (1) Prop. 3.2 demonstrates that, in the
continuum case, translationally invariant (including equilibrium)
states with ODLRO do not admit uniform magnetic fields, i.e.,
they exhibit the Meissner effect.
\vskip 0.2cm
(2) In the case where $X$ is a lattice, this conclusion must be
modified by the possibility that ODLRO might be compatible with
a non-zero quantised induction $B^{(n)}.$
\vskip 0.2cm
(3) The question of whether ODLRO, with or without the quantised
magnetic induction $B^{(n)},$ prevails is a thermodynamic one.
\vskip 0.2cm
(4) The removal of ODLRO by even infinitessimal changes in $B$
from the value $0$ or, in the lattice case, $B^{(n)}$ suggests
the
advent of a phase transition.
\vskip 0.2cm
(5) In the case of lattice systems, the non-zero $B^{(n)}$'s are
of the order of ${\hbar}c/el^{2},$ where $l$ is the spacing of
the lattice $X.$ Thus, for typical values of $l,$ e.g.
$10^{-8}$cm., they are of the order of $10^{9}$G, which is not
only many orders of magnitude larger than any known critical
fields for superconductors, but also much larger than the
internal fields in ferromagnets.
\vskip 0.2cm
We base our treatment of the thermodynamics and phase structure
of
the system, which is evidently needed in view of Comment (3),
on the following assumptions.
\vskip 0.2cm
{\bf (III.1)} {\it (a) The equilibrium state of ${\Sigma}$
corresponding to any given $(A,T)$ is unique and therefore
translationally invariant.
\vskip 0.2cm
(b) Similarly, the equilibrium state of ${\tilde {\Sigma}}$
corresponding to given $(H_{ex},T)$ is unique and translationally
invariant.}
\vskip 0.2cm
{\bf Note.} This assumption precludes the applicability of our
treatment to the mixed phase of type II superconductors, since
this breaks the space translational symmetry of the system.
\vskip 0.2cm
The next assumption, prompted by the above Comment (5), excludes
ODLRO states with non-zero quantised induction $B^{(n)}$ from the
theory, and thus puts the results of Prop. 3.2 for continuous and
lattice systems onto the same footing.
\vskip 0.2cm
{\bf (III.2)} {\it Even in the case of lattice systems, the
equilibrium states of ${\tilde {\Sigma}}$
that possess the property of ODLRO carry no magnetic induction.}
\vskip 0.2cm
The next assumption follows the line suggested by the above
Comment (4).
\vskip 0.2cm
{\bf (III.3)} {\it (a) For $B=0,$ the system ${\Sigma}$ undergoes
a phase transition at a temperature $T_{c},$ such that, for
$T