\magnification 1200 \vsize 23truecm \hsize 15truecm \baselineskip 18truept \centerline {\bf MACROSCOPIC QUANTUM THEORETIC APPROACH} \vskip 0.5cm \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"}} \vskip 1cm \centerline {{by Geoffrey L. Sewell}\footnote{**}{Partially supported by European Capital and Mobility Contract No. CHRX- CT92-0007}} \vskip 0.5cm \centerline {Department of Physics, Queen Mary and Westfield College,} \vskip 0.5cm \centerline {Mile End Road, London E1 4NS} \vskip 0.5cm {\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. \vskip 1cm {\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. \vskip 1cm {\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. \vskip 0.2cm 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. \vskip 0.2cm 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].) \vskip 0.2cm 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. \vskip 0.2cm 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]. \vskip 0.2cm 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. \vskip 0.2cm 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.} \vskip 0.2cm 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. \vskip 0.2cm 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. \vskip 0.2cm In ${\S}6,$ I shall briefly summarise the conclusions to be drawn from this work. \vskip 0.5cm {\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. \vskip 0.2cm 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 $t.$ \vskip 0.2cm 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. \vskip 0.2cm 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.} \vskip 0.2cm ${\bf Note}$ that, although ${\Psi}$ is not an observable, the quantity in angular brackets in (2.8) is one. \vskip 0.3cm ${\bf Lemma \ 2.1.}$ {\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.} \vskip 0.3cm {\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 ${\exp}(i{\eta}).$ \vskip 0.5cm {\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. \vskip 0.3cm ${\bf Proposition \ 3.1.}$ {\it The system cannot support uniform (non-zero) magnetic induction in translationally invariant states possessing the property of ODLRO.} \vskip 0.3cm The following Corollary follows immediately from this Propostion and elementary thermodynamics. \vskip 0.3cm ${\bf Corollary \ 3.2.}$ {\it Assuming that there is no translational symmetry breakdown in the equilibrium state, then either \vskip 0.2cm (1) ODLRO prevails and $B=0,$ or \vskip 0.2cm (2) the system is normally diamagnetic and does not possess ODLRO. \vskip 0.2cm 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}