Is electromagnetic gauge invariance spontaneously violated in superconductors?

We aim to give a pedagogical introduction to those elementary aspects of superconductivity which are not treated in the classic textbooks. In particular, we emphasize that global U (1) phase rotation symmetry, and not gauge symmetry, is spontaneously violated, and show that the BCS wave function is, contrary to claims in the literature, fully gauge invariant. We discuss the nature of the order parameter, the physical origin of the many degenerate states, and the relation between formulations of superconductivity with fixed particle numbers vs. well-defined phases. We motivate and to some extend derive the effective field theory at low temperatures, explore symmetries and conservation laws, and justify the classical nature of the theory. Most importantly, we show that the entire phenomenology of superconductivity essentially follows from the single assumption of a charged order parameter field. This phenomenology includes Anderson’s characteristic equations of superfluidity, electric and magnetic screening, the Bernoulli Hall effect, the balance of the Lorentz force, as well as the quantum effects, in which Planck’s constant manifests itself through the compactness of the U (1) phase field. The latter effects include flux quantization, phase slippage, and the Josephson effect.     

Color Superconductivity in Supersymmetric Gauge Theories

The studies of superconductivity, dual superconductivity and color superconductivity have been undertaken through the breaking of supersymmetric gauge theories which automatically incorporate the condensation of monopoles and dyons leading to confining and superconducting phases. Constructing the total effective Lagrangian of N=2 SU(2) gauge theory with N _f =2 quark multiplets and quark chemical potential at classical and quantum levels, it has been demonstrated that baryon number symmetry is spontaneously broken as a consequence of the SU(2) strong gauge dynamics and the color superconductivity dynamically takes space at the non-SUSY vacuum.     

Superconductivity in Supersymmetric Theories

The study of superconductivity has been undertaken through the breaking of supersymmetric gauge theories which automatically incorporate the condensation of monopoles and dyons leading to confining and superconducting phases. Constructing the effective Lagrangian near a singularity in moduli space for N=2 supersymmetric theory with SU(2) gauge group, it has been shown that when a mass term is added to this Lagrangian, the N=2 Supersymmetry is reduced to N=1 supersymmetry yielding the dyonic condensation which leads to confinement and superconductivity as the consequence of generalized Meissner effect. In the Coulomb phase of N=2 SU(3) Yang–Mills theory the gauge symmetry has been broken down to SU(2)×U(l) and it has been shown that on perturbing it by suitable tree-level superpotential this supersymmetry theory breaks to N=1 SU(2) Yang-Mills theory described by Higgs field in confining phase incorporating superconductivity.     

Superconformal supergravity and internal symmetry

The Weyl-like gauging of Wess-Zumino superconformal symmetry and of its internal symmetric generalizations SU(2,2|N) are treated. The prolification of gauge fields is much more modest than in super-Riemannian theories. Unlike supergravity, a field theory of this type is still a unified theory that provides its own sources.     

Critical behavior at finite-temperature confinement transitions

Confinement in a pure gauge theory at non-zero temperature may be discussed in terms of an order parameter which transforms under a global symmetry group, the center of the gauge group. Integrating out all degrees of freedom except this order parameter generates an effective scalar field theory for the order parameter, globally invariant under the center symmetry. We argue that the effective theory possesses only short-range couplings, and hence that the finite-temperature confinement phase transition (when continuous) is accompanied by long-range fluctuations only in the order parameter. Universality ideas then lead to predictions for the critical properties of U(1), Z(N), and SU(N) gauge theories for all dimensionalities of space-time. An explicit renormalization-group calculation is presented for the U(1) gauge theory in (2 + 1) dimensions, the results of which fit the general picture.     

Gauged N = 8 supergravity in five dimensions

We construct gauged N = 8 supergravity theories in five dimensions. Instead of the twenty-seven vector fields of the ungauged theory, the gauged theories contain fifteen vector fields and twelve second-rank antisymmetric tensor fields satisfying self-dual field equations. The fifteen vector fields can be used to gauge any of the fifteen-dimensional semisimple subgroups of SL(6,R), specially SO(p, 6?p) for p = 0, 1, 2, 3. The gauged theories also have a physical global SU(1,1) symmetry which survives from the E₆₍₆₎ symmetry of the ungauged theory. This SU(1,1) for the SO(6) gauging is presumably related to that of the chiral N = 2 theory in ten dimensions. In our formalism we maintain a composite local USp(8) symmetry analogous to SU(8) in four dimensions.     

First and second order quantum phase transitions in multi-band superconductors

In multi-band and inter-metallic materials superconductivity can be destroyed by applying external pressure in these systems. In many cases the critical temperature is driven continuously to zero, the superconducting to normal transition being associated with a superconducting quantum critical point (SQCP). In this paper we propose a model for this type of SQCP based on the increase of hybridization as pressure is applied in the material. We study a two-band superconductor with hybridization V between these bands. We use a BCS approximation and include both inter- and intra-band attractive interactions. We show that for negligible inter-band interactions, as hybridization increases there is a second order phase transition from a superconductor to a normal state at zero temperature at a critical value of the hybridization V_c. This SQCP can be reached by pressure, since this external parameter controls hybridization in the system. We also find discontinuous transitions at zero temperature and the appearance of a gapless superconducting (GS) phase in a certain range of hybridization in the case of inter-band interactions being dominant.     

Heavy-particle-induced effective lagrangian in spontaneously broken theories (II). The abelian Higgs-Kibble model

In the context of the abelian Higgs-Kibble model with a charged fermion, we study in detail low-energy effective field theories of light particles when the heavy mass scales in the theory are generated by the Higgs-Kibble mechanism. Our analysis is based on the systematic use of factorization methods, and is valid to all orders in renormalized perturbation theory. Emphasis is given to finding the vestiges of the original (spontaneously broken) local gauge symmetry left in low-energy effective field theories, and general techniques are developed for that purpose. When only Fermi fields or / and physical Higgs fields correspond to light particles, low-energy effective field theories do not exhibit such signs. On the other hand, when physical gauge fields (together with other unphysical fields) correspond to light particles, the original local gauge symmetry restricts the resulting low-energy effective local action to a non-trivial form.     

CRAVITATION AND SPINOR GAUGE FIELD

Basing on the Lorentz covariance and SO (4, 2) symmetry of Dirac theory, anobvious covariant theory of spinor gauge field is obtained by expanding the Lorentztransformation to general coordinate tranformation and making the SO (4, 2) to belocalized. We have proved that, by the gauge independence, the symmetrygroup is reduced to the localized rotation of Lorentz group in Riemann space automa-tically. So our theory is the natural generalization of Dirac theory in curved space.We have also proved that, the spinor gauge field can not appear in flat space, thenthe existence of spinor gauge field is closely related to the curvature. The differencesbetween our theory and Utiyama and Kibble theories are also discussed, and it is poin-ted out that the so-called scalar property of Dirac wave function in general relativity isa misunderstanding caused by the unobvious covariance of those theories, even inthose theories We can not distinguish what is the genuine gauge. field and what is theeffect of the structure of space. In obvious covariant theory this paradox disappears.     

Remarks on Wilson lines,modular invariance and possible string relics in Calabi-Yau compactifications

When one mods out a (2,2) conformal field theory by the action of a discrete group, it is possible to include Wilson lines to break the gauge symmetry. We simplify and generalize an earlier analysis by Witten of the constraints that modular invariance places on the allowed symmetry breaking patterns. The analysis does not depend on the details of the original conformal field theory. We then consider the fractionally charged states in such theories, first discussed by Wen and Witten. We note that these are rather generic, and consider the possibilities for their detection. We also note that, while in general they are expected to be massive (∼M_Planck), in models based on free fields, such as orbifold compactifications, there are likely to be massless (very light) fractionally charged states.     

Some aspects of the geometry of first-quantized theories

The ECSK and Yang-Mills theories are constructed with emphasis on their fiber bundle structure. In particular, the momentum tensor is derived as the Noether current of translational symmetry. The structure of the ECSK theory as a gauge theory of the Poincaré group is discussed. A theory of a Dirac field exhibiting internal affine symmetry, i.e., full internal Poincaré symmetry, is described. Aspects of the topological-geometric foundations of these theories are discussed, and some intuitive interpretations are presented.     

Topology and lattice gauge fields

A general discussion of the topology of continuum gauge fields and the problems involved in defining and computing the topology of a lattice gauge field configuration is given. Two definitions of the topological charge for 4-dimensional SU(n) lattice gauge theory are presented. The first of these is geometrically the most straightforward, the second the most useful for efficient numerical calculations.     

Weak equivalence principle from a spontaneously broken gauge theory of gravity

Further consequences of a finite topological field theory for gravity based on the SL(5,R) gauge group are reported. After symmetry breaking, it induces four-dimensional Einstein spaces with a cosmological constant related to the tiny scale of the symmetry breaking. It is shown that not only a ‘background’ metric emerges from a Higgs-like mechanism, but also consistently the geodesic equation central to Einstein?s equivalence principle. In next order of the symmetry breaking scale, the induced torsion could even provoke a tiny Lorentz violation.     

Spontaneous symmetry breaking in local gauge quantum field theory; the Higgs mechanism

Spontaneous symmetry breakings in indefinite metric quantum field theories are analyzed and a generalization of the Goldstone theorem is proved. The case of local gauge quantum field theories is discussed in detail and a characterization is given of the occurrence of the Higgs mechanism versus the Goldstone mechanism. The Higgs phenomenon is explained on general grounds without the introduction of the so-called Higgs fields. The basic property is the relation between the local internal symmetry group and the local group of gauge transformations of the second kind. Spontaneous symmetry breaking ofc-number gauge transformations of the second kind is shown to always occur if there are charged local fields. The implications about the absence of mass gap in the Wightman functions and the occurrence of massless particles associated with the unbroken generators in the Higgs phenomenon are discussed.     

Maximal symmetry and mass generation of Dirac fermions and gravitational gauge field theory in six-dimensional spacetime

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [1] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincar′e symmetry P(1,5)= SO(1,5) P~(1,5) as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.     

The Three Avtars of the Chern-Simons Term in 3 Space-Time Dimensions

I discuss in detail the three “avtars” of the Chern-Simons (C-S) term in 2 + 1 dimensions i.e. (i) C-S term as gauge field mass term (ii) C-S term as a purely kinematic term (iii) gauge theories with purely C-S action. In the first case we find that because of the C-S term one has massive gauge quanta and still the theory is gauge invariant. Such a C-S term can be generated either by spontaneous symmetry breaking or by radiative corrections. The dramatic effect of this term is that the vortices of the abelian (or nonabelian) Higgs model now have finite, quantized charge and angular momentum. In the second case the C-S term is not really independent but can be expressed in terms of the basic quanta of the 0(3) nonlinear σ-model or CP¹ model. In either case one finds that due to this term the soliton of the model has fractional spin and statistic interpolating between fermions and bosons. The relevance of this in the context of high-T_c, superconductivity is discussed in some detail. Finally the third avtar has to do with some recent work of Witten where he has shown that the Hilbert spaces of the quantum Yang-Mills theory with pure C-S action can be interpreted as the spaces of the conformal blocks in 1 + 1 dimensional conformal theories.     

Local symmetries in gauge theories in a finite-dimensional setting

It is shown that the correct mathematical implementation of symmetry in the geometric formulation of classical field theory leads naturally beyond the concept of Lie groups and their actions on manifolds, out into the realm of Lie group bundles and, more generally, of Lie groupoids and their actions on fiber bundles. This applies not only to local symmetries, which lie at the heart of gauge theories, but is already true even for global symmetries when one allows for fields that are sections of bundles with (possibly) non-trivial topology or, even when these are topologically trivial, in the absence of a preferred trivialization.     

Dielectric properties of multiband electron systems II. Collective modes

Starting from the tight-binding dielectric matrix in the random phase approximation we examine the collective modes and electron-hole excitations in a two-band electronic system. For long wavelengths (q → 0), for which most of the analysis is carried out, the properties of the collective modes are closely related to the symmetry of the atomic orbitals involved in the tight-binding states. In insulators there are only inter-band charge oscillations. If atomic dipolar transitions are allowed, the corresponding collectivemodes reduce in the asymptotic limit of vanishing bandwidths to Frenkel excitons for an atomic insulator with weak on-site interactions. The finite bandwidths renormalize the dispersion of these modes and introduce a continuum of incoherent inter-band electron-hole excitations. The possible Landau damping of collective modes due to the presence of this continuum is discussed in detail. In conductors the intra-band charge fluctuations give rise to plasmons. If the atomic dipolar transitions are forbidden, the coupling of inter-band collective modes and plasmons tends to zero as q → 0. On the contrary, in dipolar conductors this coupling is strong and nonperturbative, due to the long range monopole-dipole interactions between intra-band and inter-band charge fluctuations. The resulting collective modes are hybrids of intra-band plasmons and inter-band dipolar oscillations. It is shown that the frequency of the lower hybridized longitudinal mode is proportional to the frequency of the transverse dipolar mode when the latter is small. The dielectric instability in a multi-band conductor is therefore characterized by the simultaneous softening of a transverse and a longitudinal mode, which is an important, directly measurable consequence of the present theory.