Quantal Symmetries in the Nonlinear Sigma Model with Maxwell–Chern–Simons Term

The quantal symmetry property in the CP¹ nonlinear sigma model with Abelian–Maxwell–Chern–Simons (AMCS) term in 2 + 1 dimensions is studied. In the Coulomb gauge, the system is quantized in the Faddeev–Senjanovic (FS) path-integral formalism. The canonical Ward identities for proper vertices under local gauge transformation are derived. Based on the quantal symmetries of a constrained Hamiltonian system, the fractional spin at the quantum level of this system is also presented as those of the system without Maxwell term.     

Systematic Approach to the Analysis of Perturbed Spectra. Part II: Solvina the Problem—A Case Study

Abstract

Part I of this review [1] presented an overview of the study of perturbed spectra, the formal structure of vibration-rotation energy states, the systematic determination of symmetry allowed interactions, the representation and calculation of perturbed energies, and the inverse eigenvalue problem. In this concluding part we first consider perturbation effects on transition frequency and intensity in light of the formalism of Part I. Following this we develop a formalism for fitting perturbed spectra and carry out several example cases. Following these examples, we present a real case study of a perturbed system of vibration-rotation bands that illustrates a series of nearly insuperable problems in analysis. In the process of describing the approach to analysis, much of what was presented in Part I will be made concrete. In addition, the case study should give light to a number of areas only mentioned in passing in Part I.     

A fluctuation-dissipation theorem approach to mechanical relaxation in solid polymers

Abstract

Kubo theory formalism has been used to obtain expressions for shear and dilatational stress relaxation functions in terms of statistical mechanical time-dependent correlation functions. This is equivalent to obtaining expressions for the complex modulus or the complex viscosity for all frequencies. These results provide a basis for calculating the macroscopic consequences of molecular models presently used to provide qualitative understanding of relaxation peaks for solid polymers.

The shear and dilatational stress relaxation functions are quite different formally. For a particularly simple model it will be shown that the former is related to the frequency distribution of the kinetic energy and is also closely related to the dielectric relaxation function. The familiar results of the Rouse model are recovered in the results but no friction constant need be assumed in the present approach.     

Quantal Symmetries in the Non-Linear Sigma Model with Maxwell–Chern–Simons Term

The quantal symmetry property in the CP1 non-linear sigma model with Abelian–Maxwell–Chern–Simons (AMCS) term in 2 + 1 dimensions is studied. In the Coulomb gauge, the system is quantized in the Faddeev–Senjanovic (FS) path-integral formalism. The canonical Ward identities for proper vertices under local gauge transformation are derived. Based on the quantal symmetries of a constrained Hamiltonian system, the fractional spin at the quantum level of this system is also presented as those of the system without Maxwell term.     

Toward a canonical formalism of non-perturbative two-dimensional gravity

On the basis of the previously proposed action principle describing the theory space of 2D gravity in less than one-dimension, we develop a systematic canonical formalism for studying the properties of the string equation in the phase space of the cosmological constant and its canonical conjugate, the puncture operator. The string equation is written in a manifestly invariant form under the group of regular canonical transformations in the phase space of generalized coordinate and momentum. As a consequence, the geometrical origin of the generalized Virasoro condition on the partition function (or more precisely, the -function) is understood to be the symmetry under the regular area-preserving diffeomorphisms (w ₁₊ symmetry) in the deformed phase space. The deformed canonical formalism can be regarded as a quantization of a classical canonical formalism describing the sphere limit of the theory.     

Ward Identities in CP1 Nonlinear σ Model with Maxwell – Chern – Simons Term

In (2+1) space-time dimensions, CP¹ nonlinear σ model with Maxwell–Chern–Simons (MCS) term is studied by Ward identities. Firstly we revised, in the Coulomb gauge, the system is quantized in Faddeev–Senjanovic (FS) path integral quantization formalism. The canonical Ward identities are then given. Based on the Ward identities, the relations of the generating functional of proper vertex can be derived, and be expressed in Feynman rules with one-loop graphs.     

Symmetry ascent eigenvectors in finite point groups and an expanded matrix element selection rule

A formalism is proposed in which wave-functions quantized in a finite point group may be unambiguously labelled by their relative behaviour under the mapping of individual components from the generic point group R ₃. Such a mapping is only possible into highly symmetric finite groups including O _h and D _6h . Mapping to lower point groups by conventional symmetry descent creates ambiguities which can be removed by retaining the effect of discriminating virtual operators as parity labels for components. With such labelled wave functions, the formation of unambiguous direct products is possible with the introduction of Symmetry Ascent V Coefficients. By quantizing the wave functions about the desired n-fold axis in complex space, a commutative set of components is obtained. This allows component combination rules similar to those for 3-j symbols to be stated, modified to accommodate the possible mappings in finite groups and to retain the effect of parity. Hamiltonian operators in complex tensor form are treated similarly. The spin-orbit matrix elements for finite groups can thus be written in fully labelled form which when expanded as a scalar product of elements reflects all of the relevant selection rules pertaining to both the representations and components. With the violation of any one such rule, the matrix element vanishes. The electronic symmetry of these systems therefore is higher than that implied by the molecular geometry. The formalism further implies that during descent in symmetry, the number of selection rules for matrix elements can only increase.     

New symmetries and constants of the motion from dynamical groups

It is shown that every dynamical group (sometimes also called spectrum generating group) gives rise to a proper Noether symmetry group of the action. For each generator there is a constant of the motion. Those which do not commute with the hamiltonian but connect states of different energy contain an explicit time dependence when expressed as a function of the Heisenberg variables p(t), q(t) which ensures their conservation. If the hamiltonian is in the Lie algebra, this time dependence is given by a simple “rotation” matrix in the adjoint representation. The statements are illustrated by exhibiting the conserved symmetry operators for the bound state problem with electric and magnetic charges.     

Perturbation solution of the spin hamiltonian of low symmetry using the projector technique

Second-order perturbation solutions of the spin hamiltonian

are presented for the case of most general symmetry conditions, when the principal axes of the different tensors may be non-collinear and the tensors g and A may be asymmetric. In order to obtain a concise formalism we introduce projectors on the quantization directions of the electron and nuclear spins, which make the determination of complete Euler transformation matrices unnecessary. The angular dependence of the intensities are discussed both for the allowed and different first-order for bidden transitions for randomly oriented samples.     

Construction of the effective Lagrangian of pionic processes for arbitrary SU(2)×SU(2) breaking

By assuming that SU(2)×SU(2) symmetry is broken by the isoscalar element of the representation (l, l), effective Lagrangians reproducing the results of current algebra and the PCAC assumption can be constructed by a direct method suggested byR. Dashen andM. Weinstein. The symmetry-breaking parts of these Lagrangians are the solutions (in closed form) of the differential equation for the breaking parts in Weinberg’s formalism.

     

Relativistic version of the imaginary-time formalism

A relativistic version of the quasiclassical imaginary-time formalism is developed. It permits calculation of the tunneling probability of relativistic particles through potential barriers, including barriers lacking spherical symmetry. Application of the imaginary-time formalism to concrete problems calls for finding subbarrier trajectories which are solutions of the classical equations of motion, but with an imaginary time (and thus cannot be realized in classical mechanics). The ionization probability of an s level, whose binding energy can be of the order of the rest energy, under the action of electric and magnetic fields of different configuration is calculated using the imaginary-time formalism. Besides the exponential factor, the Coulomb and pre-exponential factors in the ionization probability are calculated. The Hamiltonian approach to the tunneling of relativistic particles is described briefly. Scrutiny of the ionization of heavy atoms by an electric field provides an additional argument against the existence of the “Unruh effect.” Zh. éksp. Teor. Fiz. 114, 798–820 (September 1998)     

Discrete Symmetries in Translation Invariant Cosmological Models

In this paper we investigate a class of (d + 1) dimensional cosmological models with a cosmological constant possessing an R ^d simply transitive symmetry group and show that it can be written in a form that manifests the effect of a permutation symmetry. We investigate the solution orbifold and calculate the probability of a certain number of dimensions that will expand or contract. We use this to calculate the probabilities up to dimension d = 5.     

Ab initio ro-vibrational Hamiltonian in irreducible tensor formalism: a method for computing energy levels from potential energy surfaces for symmetric-top molecules

A theoretical approach to study ro-vibrational molecular states from a full nuclear Hamiltonian expressed in terms of normal-mode irreducible tensor operators is presented for the first time. Each term of the Hamiltonian expansion can thus be cast in the tensor form in a systematic way using the formalism of ladder operators. Pyramidal XY₃ molecules appear to be good candidates to validate this approach which allows taking advantage of the symmetry properties when doubly degenerate vibrational modes are considered. Examples of applications will be given for PH₃ where variational calculations have been carried out from our recent potential energy surface [Nikitin et al., J. Chem. Phys. 130, 244312 (2009)].     

Group theory in solid-state physics is not dead yet alias some recent developments in the use of group theory in solid-state physics

In this article a review is given of the principal applications of group theory in solid-state physics.

Some of these applications are well established, such as the simplification of the forms of tensors representing physical properties of crystals, the labelling of electronic energy band structures, and the study of the splitting of atomic or ionic energy levels in crystals. The general principles involved in these applications are discussed. However, no attempt is made to give a comprehensive review of all the work which has been done in these areas; for further details references are given to the existing literature.

The main intention of the article is to show that apart from the well-established applications, which are adequately described in the existing literature, there have been many new developments in recent years. Group theory has come to be applied to many other types of problems in solid-state physics and these applications have not been discussed extensively in the existing review and textbook literature on the subject. These applications include: the study of the symmetry, in k space, of constant energy surfaces and in particular the symmetry of the Fermi surface; the labelling and the degeneracies of dispersion relations for phonons, magnons, and other kinds of quasiparticles; selection rules for processes involving various particle or quasiparticle states in crystals; structure determination and phase transitions; the use of two-dimensional space groups for surfaces and thin films; and the problem of the symmetry of a (non-magnetic) crystal situated in a uniform external magnetic field. The treatment given in the article is not restricted to the use of the classical point groups and space groups but, where magnetic ordering is important, the appropriate generalized symmetry groups are considered.     

The quasiclassical limit of the symmetry constraint of the KP hierarchy and the dispersionless KP hierarchy with self-consistent sources

Abstract

For the first time we show that the quasiclassical limit of the symmetry constraint of the Sato operator for the KP hierarchy leads to the generalized Zakharov reduction of the Sato function for the dispersionless KP (dKP) hierarchy which has been proved to be result of symmetry constraint of the dKP hierarchy recently. By either regarding the symmetry constrained dKP hierarchy as its stationary case or taking the dispersionless limit of the KP hierarchy with self-consistent sources directly, we construct a new integrable dispersionless hierarchy, i.e., the dKP hierarchy with self-consistent sources and find its associated conservation equations (or equations of Hamilton-Jacobi type). Some solutions of the dKP equation with self-consistent sources are also obtained by hodograph transformations.     

Dielectric and anelastic relaxation of crystals containing point defects

A theory is developed which describes the linear, reversible, time-dependent response of a crystal containing point defects to stress or electric fields, respectively known as anelastic and dielectric relaxation. Such relaxation occurs because of the redistribution of the defects among sites which are initially equivalent, but which becomes inequivalent in the presence of the external field. The macroscopic behaviour of such a crystal is found to be describable in terms of the symmetry which can be assigned to the defect. This defect symmetry determines whether or not the crystal will undergo dielectric or anelastic relaxation and, if relaxation can occur, which specific coefficients of elastic compliance or electric susceptibility show the relaxation effect. The latter information, called the ‘selection rules’ tells, in effect, which combination of stress or electric field components is capable of redistributing the defects. Tables are given for these selection rules for all possible defect symmetries in each of the 32 crystal classes. It is also shown that a hitherto unobserved phenomenon of piezoelectric relaxation may occur; the selection rules for this effect are also given.

Aside from its symmetry, the defect can be described as an electric dipole in terms of a suitable dipole moment vector μ, and as an ‘elastic dipole’ in terms of a tensor λ. It is shown that the defect symmetry determines the number of independent components of μ and λ. Finally, a thermodynamic theory is developed which permits calculation of the relaxation strengths for those compliance, susceptibility, and piezoelectric coefficients which undergo relaxation, in terms of the independent components of μ and λ. Applications of the theory to specific cases are then reviewed.     

Exact Solutions of the Klein–Gordon Equation with Position-Dependent Mass for Mixed Vector and Scalar Kink-Like Potentials

The relativistic problem of spinless particles with position-dependent mass subject to kink-like potentials (~tanh αx) is investigated. By using the basic concepts of the supersymmetric quantum mechanics formalism and the functional analysis method, we solve exactly the position-dependent effective mass Klein–Gordon equation with the vector and scalar kink-like potential coupling, and obtain the bound state solutions in the closed form. It is found that in the presence of position-dependent mass there exists the symmetry that the discrete positive energy spectra and negative energy spectra are symmetric about zero energy for the case of a mixed vector and scalar kink-like potential coupling, and in the presence of constant mass this symmetry only appears for the cases of a pure scalar kink-like potential coupling or massless particles.     

Normal Coordinate Analysis of the Tricyanomethanide ion C(CN)− 3 (D3h)

Abstract

A complete normal coordinate analysis has been performed for the tricyanomethanide ion C(CN)^? ₃ for which a planar structure of symmetry D_3h was assumed. The symmetric Fand internal f valence force constant matrices were derived in the general case by using a GVFF and the results are applied to C(CN)^? ₃.     

Asymmetric d-wave superconducting topological insulator in proximity with a magnetic order

In the framework of the Dirac–Bogoliubov–de Gennes formalism, we investigate the transport properties in the surface of a 3-dimensional topological insulator-based hybrid structure, where the ferromagnetic and superconducting orders are simultaneously induced to the surface states via the proximity effect. The superconductor gap is taken to be spin-singlet d-wave symmetry. The asymmetric role of this gap respect to the electron–hole exchange, in one hand, affects the topological insulator superconducting binding excitations and, on the other hand, gives rise to forming distinct Majorana bound states at the ferromagnet/superconductor interface. We propose a topological insulator N/F/FS junction and proceed to clarify the role of d-wave asymmetry pairing in the resulting subgap and overgap tunneling conductance. The perpendicular component of magnetizations in F and FS regions can be at the parallel and antiparallel configurations leading to capture the experimentally important magnetoresistance (MR) of junction. It is found that the zero-bias conductance is strongly sensitive to the magnitude of magnetization in FS region $m_{zfs}$ and orbital rotated angle α of superconductor gap. The negative MR only occurs in zero orbital rotated angle. This result can pave the way to distinguish the unconventional superconducting state in the relating topological insulator hybrid structures.