Showing papers on "Symmetry (physics) published in 1998"
TL;DR: In this paper, the spontaneous appearance of an internal magnetic field below the transition temperature of the superconductor Sr2RuO4 was observed, which indicated that the superconducting state in this material is characterized by the breaking of time-reversal symmetry.
Abstract: Although the properties of most superconducting materials are well described by the theory1 of Bardeen, Cooper and Schrieffer (BCS), considerable effort has been devoted to the search for exotic superconducting systems in which BCS theory does not apply. The transition to the superconducting state in conventional BCS superconductors involves the breaking of gauge symmetry only, whereby the wavefunction describing the Cooper pairs—the paired electron states responsible for superconductivity—adopt a definite phase. In contrast, a signature of an unconventional superconducting state is the breaking of additional symmetries2, which can lead to anisotropic pairing (such as the ‘d-wave’ symmetry observed in the copper oxide superconductors) and the presence of multiple superconducting phases (as seen in UPt3 and analogous behaviour in superfluid 3He; 3–5). Here we report muon spin-relaxation measurements on the superconductor Sr2RuO4 that reveal the spontaneous appearance of an internal magnetic field below the transition temperature: the appearance of such a field indicates that the superconducting state in this material is characterized by the breaking of time-reversal symmetry. These results, combined with other symmetry considerations, suggest that superconductivity in Sr2RuO4 is of ‘p-wave’ (odd-parity) type, analogous to superfluid 3He.
813 citations
TL;DR: In this paper, the pseudospin symmetry and its energy dependence in real nuclei is discussed. But the authors focus on the competition between the CB and PSOP, which is mainly decided by the derivative of the difference between the scalar and vector potentials.
Abstract: Relating the pseudospin symmetry back to the Dirac equation through the framework of relativistic Hartree-Bogoliubov (RHB) theory, the pseudospin approximation in real nuclei is discussed. From the Dirac equation, the mechanism behind the pseudospin symmetry was studied and the pseudospin symmetry is shown to be connected with the competition between the centrifugal barrier (CB) and the pseudospin orbital potential (PSOP), which is mainly decided by the derivative of the difference between the scalar and vector potentials. With the scalar and vector potentials derived from a self-consistent relativistic Hartree-Bogoliubov calculation, the pseudospin symmetry and its energy dependence in real nuclei is discussed.
316 citations
01 Nov 1998
TL;DR: In this article, the search for the universality class of confining strings is discussed, and some new tests for the equivalence between gauge fields and strings are proposed, based on a talk given at the "Strings '97" conference.
Abstract: This article is based on a talk given at the “Strings '97” conference. It discusses the search for the universality class of confining strings. The key ingredients include the loop equations, the zigzag symmetry, the non-linear renormalization group. Some new tests for the equivalence between gauge fields and strings are proposed.
297 citations
TL;DR: In this article, the present stage of the density functional theory for noncollinear magnetic states and its application to particular physical problems is reviewed and a generalized approach based on the notion of spin space groups is presented which allows a consequent treatment of the symmetry properties of both nonrelativistic and relativistic problems.
Abstract: The article attempts to review the present stage of the density functional theory for noncollinear magnetic states and its application to particular physical problems. The discussion starts with basic theorems of the theory and derivation of the Kohn-Sham equation for a noncollinear magnet. Special features of solving this equation are illustrated using the augmented-spherical-wave method generalized to the noncollinear magnetic structures as an example. Particular attention is devoted to the symmetry of the problem. It is shown that a traditional approach of the space group theory fails in the case of a noncollinear magnetic state. A generalized approach based on the notion of the spin space groups is presented which allows a consequent treatment of the symmetry properties of both nonrelativistic and relativistic problems. This approach allows the development of an exact procedure for a first-principles calculation of an incommensurate spiral structure and gives a sound basis for the calculation and phys...
292 citations
TL;DR: In this article, the authors resolve an existing paradox regarding the causality and symmetry properties of response functions within time-dependent density-functional theory by defining a new action functional within the Keldysh formalism.
Abstract: We resolve an existing paradox regarding the causality and symmetry properties of response functions within time-dependent density-functional theory. We do this by defining a new action functional within the Keldysh formalism. By functional differentiation the new functional leads to response functions which are symmetric in the Keldysh time contour parameter, but which become causal when a transition to physical time is made. The new functional is further used to derive the equations of the timedependent optimized potential method. [S0031-9007(97)05233-2]
235 citations
28 Jun 1998
TL;DR: It is proved that the orbit problem is equivalent to an important problem in computational group theory which is at least as hard as the graph isomorphism but not known to be NP-complete.
Abstract: The use of symmetry to alleviate state-explosion problems during model-checking has become a important research topic. This paper investigates several problems which are important to techniques exploiting symmetry. The most important of these problems is the orbit problem. We prove that the orbit problem is equivalent to an important problem in computational group theory which is at least as hard as the graph isomorphism but not known to be NP-complete. This paper also shows classes of commonly occurring groups for which the orbit problem is easy. Some methods of deriving symmetry for a shared variable model of concurrent programs are also investigated. Experimental results providing evidence of reduction in state space by using symmetry are also provided.
158 citations
TL;DR: In this paper, the elastic field around a buried, strained quantum dot is solved with a scalar potential that obeys Poisson's equation and is analogous to the charge density and electric field.
Abstract: The elastic field around a buried, strained quantum dot is solved with a scalar potential that obeys Poisson’s equation. Standard methods from electrostatics can therefore be used. The lattice mismatch and displacement are analogous to the charge density and electric field. The dilation is proportional to the local lattice mismatch and therefore vanishes outside a dot. Expressions are also given for the piezoelectric potential. The results agree remarkably well with previous numerical calculations for a pyramidal dot. Thermoelasticity provides another analogy with many useful solutions available. These results are for an isotropic medium but cubic symmetry is considered briefly.
147 citations
TL;DR: In this article, the authors considered the symmetry properties of solutions and the existence of ground states of a system and found interesting results on asymptotic behavior of solutions, on symmetry properties, and on ground states.
Abstract: Our utmost aim was originally to establish the existence of solutions of system. However, in this effort we got sidetracked to consider other questions. In this way we come to interesting results on asymptotic behavior of solutions, on symmetry properties of such solutions, and the existence of ground states.
147 citations
TL;DR: In this article, a slave-boson theory for the finite doping model at finite doping was developed, which respects an SU(2) symmetry, a symmetry previously known to be important at half filling.
Abstract: We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping that respects an SU(2) symmetry: a symmetry previously known to be important at half filling. The mean-field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain $d$-wave superconductor, spin-gap, strange metal, and Fermi-liquid phases. The spin-gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The physical electron spectral function shows small Fermi segments at low doping that continuously evolve into the large Fermi surface at high-doping concentrations. The close relation between the SU(2) and the U(1) slave-boson theory is discussed. The low-energy effective theory for the low-lying fluctuations is derived and additional lying modes [which were overlooked in the U(1) theory] are identified.
145 citations
TL;DR: In this paper, conditions for oscillation of a high-aperture resonator, such as a microcavity, are derived for higher-order modes in systems with cylindrical symmetry.
Abstract: Beam modes are considered by a high-aperture scalar theory based on the complex source-point method. A different form for the beam mode that avoids nonphysical singularities in intensity is introduced. The amplitude in the focal region is explored for the lowest-order mode. Conditions for oscillation of a high-aperture resonator, such as a microcavity, are derived. The amplitude in the focal region is explored for higher-order modes in systems with cylindrical symmetry. These modes are expressed in terms of Jacobi polynomials. @S1050-2947~98!01103-2#
142 citations
TL;DR: In this paper, a nonlinear electromagnetic gyrokinetic equation for plasmas with large flow velocities on the order of the ion thermal speed is derived for general magnetic geometries including the slab, cylindrical and toroidal configurations.
Abstract: A new nonlinear electromagnetic gyrokinetic equation is derived for plasmas with large flow velocities on the order of the ion thermal speed. The gyrokinetic equation derived here retains a collision term and is given in the form which is valid for general magnetic geometries including the slab, cylindrical and toroidal configurations. The source term for the anomalous viscosity arising through the Reynolds stress is identified in the gyrokinetic equation. For the toroidally rotating plasma, particle, energy and momentum balance equations as well as the detailed definitions of the anomalous transport fluxes and the anomalous entropy production are shown. The quasilinear anomalous transport matrix connecting the conjugate pairs of the anomalous fluxes and the forces satisfies the Onsager symmetry.
TL;DR: In this article, new identities relating the Euler-Lagrange, Lie-Backlund and Noether operators are obtained, and the symmetry based results deduced from the new identities are used to construct Lagrangians for partial differential equations.
Abstract: New identities relating the Euler–Lagrange, Lie–Backlund and Noether operators are obtained Some important results are shown to be consequences of these fundamental identities Furthermore, we generalise an interesting example presented by Noether in her celebrated paper and prove that any Noether symmetry is equivalent to a strict Noether symmetry, ie a Noether symmetry with zero divergence We then use the symmetry based results deduced from the new identities to construct Lagrangians for partial differential equations In particular, we show how the knowledge of a symmetry and its corresponding conservation law of a given partial differential equation can be utilised to construct a Lagrangian for the equation Several examples are given
TL;DR: In this paper, a perturbative renormalization-group analysis reveals that at half-filling the model scales onto an exactly soluble Gross-Neveu model for arbitrary finite-ranged interactions, provided they are sufficiently weak.
Abstract: We revisit the problem of interacting electrons hopping on a two-leg ladder. A perturbative renormalization-group analysis reveals that at half-filling the model scales onto an exactly soluble Gross-Neveu model for arbitrary finite-ranged interactions, provided they are sufficiently weak. The Gross-Neveu model has an enormous global SO(8) symmetry, manifest in terms of eight real Fermion fields that, however, are highly nonlocal in terms of the electron operators. For generic repulsive interactions, the two-leg ladder exhibits a Mott insulating phase at half-filling with $d$-wave pairing correlations. Integrability of the Gross-Neveu model is employed to extract the exact energies, degeneracies, and quantum numbers of all the low-energy excited states, which fall into degenerate SO(8) multiplets. One SO(8) vector includes two charged Cooper pair excitations, a neutral $s=1$ triplet of magnons, and three other neutral $s=0$ particle-hole excitations. A triality symmetry relates these eight two-particle excitations to two other degenerate octets, which are comprised of single-electron-like excitations. In addition to these 24 degenerate ``particle'' states costing an energy (mass) $m$ to create, there is a 28-dimensional antisymmetric tensor multiplet of ``bound'' states with energy $\sqrt{3}m.$ Doping away from half-filling liberates the Cooper pairs, leading to quasi-long-range $d$-wave pair field correlations, but maintaining a gap to spin and single-electron excitations. For very low doping levels, integrability allows one to extract exact values for these energy gaps. Enlarging the space of interactions to include attractive interactions reveals that there are four robust phases possible for the weak coupling two-leg ladder. While each of the four phases has a (different) SO(8) symmetry, they are shown to all share a common SO(5) symmetry---the one recently proposed by Zhang as a unifying feature of magnetism and superconductivity in the cuprates.
TL;DR: In this paper, the integrable systems in higher dimensions which can be written by the trilinear form instead of by the Hirota's bilinear forms were studied.
Abstract: We study the integrable systems in higher dimensions which can be written by the trilinear form instead of by the Hirota's bilinear form We explicitly discuss the Bogoyavlenskii-Schiff equation in (2 + 1) dimensions Its analytical proof of multisoliton solution and a new feature are given Being guided by the strong symmetry, we also propose a new equation in (3 + 1) dimensions
TL;DR: Elastic anisotropy factors were derived for each of the three modes of propagation from the special in-plane phonon-focusing considerations arising when wave vectors are constrained to the symmetry planes of orthorhombic, tetragonal, and hexagonal crystals as discussed by the authors.
Abstract: Elastic anisotropy factors are derived for each of the three modes of propagation from the special in-plane phonon-focusing considerations arising when wave vectors are constrained to the symmetry planes of orthorhombic, tetragonal, and hexagonal crystals. Elastic anisotropy factors for the pure transverse and quasi-transverse modes depend upon the symmetry plane, whereas anisotropy factors for the quasilongitudinal mode depend both upon a symmetry plane and a symmetry axis. These anisotropy factors provide a convenient measure of in-plane phonon focusing.
TL;DR: In this article, a detailed analysis of finite energy solitary waves in two-and three-dimensional nonlinear periodic structures exhibiting a complete photonic band gap is presented, where the important physical features such as the size, shape, peak intensity, and total energy of the solitary waves are derived using a variational method.
Abstract: We present a detailed analysis of finite energy solitary waves in two- and three-dimensional nonlinear periodic structures exhibiting a complete photonic band gap. Solitary waves in photonic crystals with a two-dimensional (2D) square and triangular symmetry group as well as a 3D fcc symmetry group are described in terms of an effective nonlinear Dirac equation derived using the slowly varying envelope approximation for the electromagnetic field. Unlike one-dimensional Bragg solitons, the multiple symmetry points of the 2D and 3D Brillouin zones give rise to two distinct classes of solitary wave solutions. Solutions associated with a higher order symmetry point of the crystal exist for both positive and negative Kerr nonlinearities, whereas solutions associated with a twofold symmetry point occur only for positive Kerr coefficient. Using a variational method we derive the important physical features such as the size, shape, peak intensity, and total energy of the solitary waves. This is then confirmed numerically using the finite element Ritz-Galerkin method. It is shown that the initial variational method and the finite element numerical method are in good agreement. We discuss the stability of these solitary waves with respect to small perturbations. It is suggested that an analytical stability criterion for spinor fields satisfying the nonlinear Dirac type of equation may exist, similar to the well known stability criterion for solitary waves in the nonlinear Schr\"odinger equation. Our stability criterion correctly reproduces the stability conditions of other nonlinear Dirac type of equations which have been studied numerically. Our study suggests that for an ideal Kerr medium, two-dimensional solitary waves in a band gap are stable, whereas three-dimensional ones are stable only in certain regions of the gap.
TL;DR: In this article, a slave-boson theory for the finite doping model at finite doping was developed, which respects an SU(2) symmetry, a symmetry previously known to be important at half filling.
Abstract: We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping that respects an SU(2) symmetry: a symmetry previously known to be important at half filling. The mean-field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain $d$-wave superconductor, spin-gap, strange metal, and Fermi-liquid phases. The spin-gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The physical electron spectral function shows small Fermi segments at low doping that continuously evolve into the large Fermi surface at high-doping concentrations. The close relation between the SU(2) and the U(1) slave-boson theory is discussed. The low-energy effective theory for the low-lying fluctuations is derived and additional lying modes [which were overlooked in the U(1) theory] are identified.
TL;DR: In this article, the low-energy supersymmetric quantum mechanics describing D-particles in the background of D8-branes and orientifold planes is analyzed in detail, including a careful discussion of Gauss' law and normal ordering of operators.
Abstract: The low-energy supersymmetric quantum mechanics describing D-particles in the background of D8-branes and orientifold planes is analyzed in detail, including a careful discussion of Gauss' law and normal ordering of operators. This elucidates the mechanism that binds D-particles to an orientifold plane, in accordance with the predictions of heterotic/type I duality. The ocurrence of enhanced symmetries associated with massless bound states of a D-particle with one orientifold plane is illustrated by the enhancement of SO(14) × U(1) to E8 and SO(12) × U(1) to E7 at strong type I' coupling. Enhancement to higher-rank groups involves both orientifold planes. For example, the enhanced E8 × E8 × SU(2) symmetry at the self-dual radius of the heterotic string is seen as the result of two D8-branes coinciding midway between the orientifold planes, while the enhanced SU(18) symmetry results from the coincidence of all sixteen D8-branes and SO(34) when they also coincide with an orientifold plane. As a separate by-product, the s-rule of brane-engineered gauge theories is derived by relating it through a chain of dualities to the Pauli exclusion principle.
TL;DR: In this article, the violation of the fluctuation-dissipation theorem in the three and four-dimensional Gaussian Ising spin glasses using on and off equilibrium simulations was studied.
Abstract: We study the violation of the fluctuation-dissipation theorem in the three- and four-dimensional Gaussian Ising spin glasses using on and off equilibrium simulations. We have characterized numerically the function X(C) that determine the violation and we have studied its scaling properties. Moreover we have computed the function x(C) which characterize the breaking of the replica symmetry directly from equilibrium simulations. The two functions are numerically equal and in this way we have established that the conjectured connection between the violation of fluctuation-dissipation theorem in the off-equilibrium dynamics and the replica symmetry breaking at equilibrium holds for finite-dimensional spin glasses. These results point to a spin-glass phase with spontaneously broken replica symmetry in finite-dimensional spin glasses.
TL;DR: In this paper, the H5O2+ system has been studied using a variety of coupled cluster methods based on a Brueckner reference determinant with levels of correlation up to double and perturbatively treated connected triple excitations.
Abstract: The H5O2+ system has been studied using a variety of coupled cluster methods based on a Brueckner reference determinant with levels of correlation up to double and perturbatively treated connected triple excitations [B–CCD(T)]. Basis sets as large as the triple-ζ plus double polarization basis augmented with f functions on oxygen and d functions on hydrogen [TZ2P(f,d)] were used. Harmonic vibrational frequencies were also predicted. In contrast with previous high-level ab initio studies, a stationary point of C1 symmetry was not found. An absence of imaginary vibrational frequencies at all levels of theory for the stationary point of C2 symmetry proves it to be the global minimum, lying only ∼0.4 kcal/mol lower in energy than the transition state of Cs symmetry.
TL;DR: In this article, the authors extend the Dirac-Brueckner approach with a Bonn one-boson-exchange nucleon-nucleon interaction to the general case of asymmetric nuclear matter.
Abstract: To study the nuclear symmetry energy, we extend the Dirac-Brueckner approach with a Bonn one-boson-exchange nucleon-nucleon interaction to the general case of asymmetric nuclear matter. We extract the symmetry energy coefficient at the saturation to be about 31 MeV, which is in good agreement with the empirical value of 30{plus_minus}4thinspMeV. The symmetry energy is found to increase almost linearly with the density, which differs considerably from the results of nonrelativistic approaches. This finding also supports the linear parametrization of Prakash, Ainsworth, and Lattimer. We find, furthermore, that the higher-order dependence of the nuclear equation of state on the asymmetry parameter is unimportant. {copyright} {ital 1998} {ital The American Physical Society}
TL;DR: The size and shape of this integration region (IR) is investigated by measuring human detection of spatially band–pass symmetrical patches embedded in noise by showing resistance to disruption of symmetry improves with increasing patch size, and then asymptotes when the embedded region fills the IR.
Abstract: Symmetry is a complex image property that is exploited by a sufficiently wide range of species to indicate that it is detected using simple visual mechanisms. These mechanisms rely on measurements made close to the axis of symmetry. We investigated the size and shape of this integration region (IR) by measuring human detection of spatially band–pass symmetrical patches embedded in noise. Resistance to disruption of symmetry (in the form of random phase noise) improves with increasing patch size, and then asymptotes when the embedded region fills the IR. The size of the IR is shown to vary in inverse proportion to spatial frequency; i.e. symmetry detection exhibits scale–invariance. The IR is shown to have rigid dimensions, elongated in the direction of the axis of symmetry, with an aspect ratio of ca. 2:1. These results are consistent with a central role for spatial filtering in symmetry detection.
TL;DR: In this article, a 2×2 nonorthogonal CI is used to recombine the two symmetry broken Hartree-Fock determinants, and the necessary matrix elements closely resemble those used in the spin projection calculations.
Abstract: Spatial symmetry breaking can occur in Hartree–Fock wave functions when there are two or more close lying configurations that can mix strongly, such as in HCO2, NO2, and allyl radical. Like spin contamination, spatial symmetry breaking can cause sizeable errors when perturbation theory is used to estimate the correlation energy. With conventional methodology, very large MCSCF and MRCI calculations are necessary to overcome the spatial symmetry breaking problem. This paper explores an alternative approach in which a 2×2 nonorthogonal CI is used to recombine the two symmetry broken Hartree–Fock determinants. The necessary matrix elements closely resemble those used in the spin projection calculations. Second order perturbation theory is used to include electron correlation energy in this approach. With perturbative corrections for correlation energy, this approach predicts that the 2B2 structure is a minimum, in agreement with the best available calculations.
TL;DR: In this paper, it was shown that any four-dimensional hyper-Hermitian manifold admitting a triholomorphic Killing vector field is locally determined by the solution of a monopole-like equation on a three-dimensional Einstein-Weyl space of a special type.
TL;DR: In this paper, a spin free massless fermionic fields corresponding to mixed symmetry representations of the SO (d −1) compact group and propagating in even d -dimensional anti-de Sitter spacetime are investigated.
TL;DR: In this article, the bound-state spectrum of the generalized Morse potential (GMP) was studied for diatomic molecules and the exact solvability of the problem was shown by symmetry algebra.
Abstract: We study in detail the bound-state spectrum of the generalized Morse potential (GMP), which was proposed by Deng and Fan as a potential function for diatomic molecules. By connecting the corresponding Schrodinger equation with the Laplace equation on the hyperboloid and the Schrodinger equation for the Poschl-Teller potential, we explain the exact solvability of the problem by an symmetry algebra, and obtain an explicit realization of the latter as . We prove that some of the generators connect among themselves wavefunctions belonging to different GMPs (called satellite potentials). The conserved quantity is some combination of the potential parameters instead of the level energy, as for potential algebras. Hence, belongs to a new class of symmetry algebras. We also stress the usefulness of our algebraic results for simplifying the calculation of Frank-Condon factors for electromagnetic transitions between rovibrational levels based on different electronic states.
TL;DR: In this article, full distributions of conductance through quantum dots with single-mode leads are reported for both broken and unbroken time-reversal symmetry, and they agree well with random matrix theory calculations that account for a finite dephasing time, once broadening due to finite temperature.
Abstract: Full distributions of conductance through quantum dots with single-mode leads are reported for both broken and unbroken time-reversal symmetry. Distributions are nongaussian and agree well with random matrix theory calculations that account for a finite dephasing time, $\tau_\phi$, once broadening due to finite temperature $T$ is also included. Full distributions of the derivatives of conductance with respect to gate voltage $P(dg/dV_g)$ are also investigated.
01 Jan 1998
TL;DR: In this paper, the cohomological approach to the problem of consistent interactions between fields with a gauge freedom is reviewed and the role played by the BRST symmetry is explained, applications to massless vector fields and 2-form gauge fields are surveyed.
Abstract: The cohomological approach to the problem of consistent interactions between fields with a gauge freedom is reviewed. The role played by the BRST symmetry is explained. Applications to massless vector fields and 2-form gauge fields are surveyed.
TL;DR: In this paper, it was shown that the S-wave surface is always an ellipsoid under the assumption of 3rd order nonlinear isotropic hyperelasticity (i.e., no hysteresis and existence of an elastic energy function developed to the 3rd-order in the strain components).
Abstract: This summary contains formulas (***) which can not be displayed on the screenA general principle outlined by P. Curie (1894) regarding the influence of symmetry in physical phenomena states, in modern language, that the symmetry group of the causes is a sub-group of the symmetry group of the effects. For instance, regarding stress-induced seismic anisotropy, the most complex symmetry exhibited by an initially isotropic medium when tri-axially stressed is orthorhombic, or orthotropic, symmetry characterized by three symmetry planes mutually perpendicular (Nur, 1971). In other respects, Schwartz et al. (1994) demonstrated that two very different rock models, namely a cracked model and a weakly consolidated granular model, always lead to elliptical anisotropy when uniaxially stressed. The addressed questions are : Is this result true for any rock model? and more generally : Do initially isotropic rock form a well-defined sub-set of orthorhombic media when triaxially stressed?Under the hypothesis of 3rd order nonlinear isotropic hyperelasticity (i. e. , no hysteresis and existence of an elastic energy function developed to the 3rd order in the strain components) it is demonstrated that the qP-wave stress-induced anisotropy is always ellipsoidal, for any strength of anisotropy. For instance point sources generate ellipsoidal qP-wave fronts. This result is general and absolutely independent of the rock model, that is to say independent of the causes of nonlinearity, as far as the initial assumptions are verified. This constitutes the main result of this paper. Thurston (1965) pointed out that an initially isotropic elastic medium, when non-isotropically pre-stressed, is never strictly equivalent to an unstressed anisotropic crystal. For instance the components of the stressed elastic tensor lack the familiar symmetry with respect to indices permutation. This would prohibit Voigt's notation of contracted indices. However if the magnitude of the components of the stress deviator is small compared to the wave moduli, which is always verified in practical situations of seismic exploration, the perfect equivalence is re-established. Under this condition, the 9 elastic stiffnesses C'ij (in contracted notation) of an initially isotropic solid, when triaxially stressed, are always linked by 3 ellipticity conditions in the coordinate planes associated with the eigen directions of the static pre-stress, namely :(***)Thus only 6 of the 9 elastic stiffnesses of the orthorhombic stressed solid are independent (Nikitin and Chesnokov, 1981), and are simple functions of the eigen stresses, and of the 2 linear (2nd order) and the 3 nonlinear (3rd order) elastic constants of the unstressed isotropic solid. Furthermore, given the state of pre-stress, the strength of the stress-induced P- or S-wave anisotropy and S-wave birefringence (but not the magnitude of the wave moduli themselves) are determined by only 2 intrinsic parameters of the medium, one for the P-wave and one for the S-waves. Isotropic elastic media, when triaxially stressed, constitute a special sub-set of orthorhombic media, here called ellipsoidal media , verifying the above conditions. Ellipsoidal anisotropy is the natural generalization of elliptical anisotropy. Ellipsoidal anisotropy is to orthorhombic symmetry what elliptical anisotropy is to transversely isotropic (TI) symmetry. Elliptical anisotropy is a special case of ellipsoidal anisotropy restricted to TI media. In other words, ellipsoidal anisotropy degenerates in elliptical anisotropy in TI media. In ellipsoidal media the qP-wave slowness surface is always an ellipsoid. The S-wave slowness surfaces are not ellipsoidal, except in the degenerate elliptical case, and have to be considered as a single double-valued self-intersecting sheet (Helbig, 1994). The intersections of these latter surfaces with the coordinate planes are either ellipses, for the S-vave polarized out of the coordinate planes, or circles, for the qS-wave polarized in the coordinate planes. The nearly exhaustive collection of experimental data on seismic anisotropy in rocks (considered as transverse isotropic) by Thomsen (1986) show that elliptical anisotropy is more an exception than a rule. Since stress-induced anisotropy is essentially elliptical when restricted to transversely isotropic media, as a consequence this work clearly shows that stress can be practically excluded as a unique direct cause of elastic anisotropy in rocks.
TL;DR: In this paper, the spontaneous appearance of an internal magnetic field below the transition temperature of the superconductor Sr2RuO4 has been observed, and the appearance of such a field indicates that the superconducting state in this material is characterized by the breaking of time-reversal symmetry.
Abstract: We report muon spin relaxation measurements on the superconductor Sr2RuO4 that reveal the spontaneous appearance of an internal magnetic field below the transition temperature: the appearance of such a field indicates that the superconducting state in this material is characterized by the breaking of time-reversal symmetry. These results, combined with other symmetry considerations, suggest that superconductivity in Sr2RuO4 is of p-wave (odd-parity) type, analogous to superfluid 3He.