The trapping of positive energy electrons on an infinite sheet

The one electron Schrödinger equation for an infinite sheet may have Bloch states that live on the sheet and have positive energy with respect to the vacuum. Such states occur for special values of k in the two dimensional Brillouin zone and form, in general, “line bands” and “point bands”. The decay into the continuum is prevented by the space group symmetry.     

HighT c superconductivity in the itinerant ising model: Spin singlet pairing

We study the spin singlet superconductivity exhibited in an itinerant Ising model Hamiltonian. This Hamiltonian models the Cu–O layers in highT _c oxide superconductors. Electrons are itinerant through nearest neighbor hopping. An Ising term is introduced to describe the antiferromagnetic superexchange interaction between electrons nominally on nearest neighbor Cu sites. We discuss various symmetry states allowed by the model, and give detailed predictions of the superconducting energy gap, specific heat, susceptibility, andT _c variation with carrier concentration. Results are compared to experimental data on highT _c superconductors and reasonable agreement is obtained.     

Energy bands: Chern numbers and symmetry

Energy bands formed by rotation–vibrational states of molecules in the presence of symmetry and their qualitative modifications under variation of some control parameters are studied within the semi-quantum model. Rotational variables are treated as classical whereas a finite set of vibrational states is considered as quantum. In the two-state approximation the system is described in terms of a fiber bundle with the base space being a two-dimensional sphere, the classical phase space for rotational variables. Generically this rank 2 complex vector bundle can be decomposed into two complex line bundles characterized by a topological invariant, the first Chern class. A general method of explicit calculation of Chern classes and of their possible modifications under variation of control parameters in the presence of symmetry is suggested. The construction of iso-Chern diagrams which split the space of control parameters into connected domains with fixed Chern numbers is suggested. A detailed analysis of the rovibrational model Hamiltonian for a D₃ invariant molecule possessing two vibrational states transforming according to the two-dimensional irreducible representation is done to illustrate non-trivial restrictions imposed by symmetry on possible values of Chern classes.     

On the magnetic space group and Raman tensors of antiferromagnetic RbCoF3

The magnetic unit cell of the antiferromagnetic RbCoF₃ is identified theoretically, and the magnetic space group is found to be of type IIIb Shubnikov. The study of the splitting of the fundamental energy level of the magnetic ion Co²⁺(⁴F) at the center of BZ is presented, using the theory of crystalline field adapted for magnetic crystals. The symmetries of the electronic Raman activities are determined by group theory, and the respective Raman tensors are presented. Comparison between the theory and the existing experimental data is discussed.     

Quantum competitions between the effects of the interlayer exchange couplings and the magnetically structural symmetry in a four-layer ferrimagnetic superlattice

The magnon energy spectra, the sublayer magnetization and the quantum fluctuations in a ferrimagnetic superlattice consisting of four different magnetic sublayers are studied by employing the linear spin-wave approach and Green's function technique. The effects of the interlayer exchange couplings and the spin quantum numbers on the sublayer magnetization and the quantum fluctuations of the systems are discussed for three different spin configurations. The roles of quantum competitions among the interlayer exchange couplings and the symmetry of the different spin configurations have been understood. The magnetizations of some sublayers increase monotonously, while those of others can exhibit their maximum, and the quantum fluctuations of the whole superlattice system can show a minimum when one of the antiferromagnetic interlayer exchange couplings increases. This is due to the quantum competition/transmission of effects of the interlayer exchange couplings. When the spin quantum number of sublayers varies, the system goes through from a quantum region of small spin numbers to a classical region of large spin numbers. The quantum fluctuations of the system exhibit a maximum as a function of the spin quantum number of a sublayer, which is related with higher symmetry of the system. It belongs to the type III Shubnikov group of magnetic groups. This magnetically structural symmetry consists of not only the symmetry of space group, but also the symmetry of the direction and strength of spins.     

Galaxies: A Very Short Introduction,by J. Gribbin

A brief outline of conventional symmetry studies of crystalline solids using Bravais lattices, point groups and space groups is presented, leading up to the recent development by Shubnikov of the idea of antisymmetry and the subsequent derivation of the black and white Bravais lattices and of the black and white point groups and space groups. These groups are also called magnetic groups or Shubnikov groups.

The results of neutron-diffraction experiments have made it clear that there are many crystals whose structures must be described by one of these Shubnikov groups rather than by one of the ordinary point groups or space groups. The relationship between a structure with the symmetry of one of the Shubnikov point groups and the magnetic properties, paramagnetism, diamagnetism, ferromagnetism, ferrimagnetism or anti-ferromagnetism, which it is able to exhibit is also considered.     

Symmetry analysis of magnetic structures and spin-waves on the basis of a stabilizer concept

The common scheme for the group theory analysis of a crystal magnetic structure and a spin-wave spectrum is developed. This scheme is based on the representation theory of a crystal space group G (for magnetic structure analysis) and on the co-representation theory of the Shubnikov group M of a magnetically ordered crystal (for spin wave analysis).Unlike the standard calculation, the procedure proposed in this paper is based on the conception of the stabilizer-the symmetry group of single atoms, and this considerably symplified the calculation work, reducing it to the work with the elements of the stabilizer and with the representatives of the expansion of the G or M groups with respect to the group of the stabilizer. It is enough to calculate the components of the basic functions only at one atom, as for the other atoms they may be found by the action of the representatives.For the symmetrization of the spin wave Hamiltonian matrix a new mathematical procedure is used the expansion of group G over the double coset classes. It lets us easily find the symmetric relations between the different matrix elements. The method is illustrated in the example of the complicated magnetic structure of a garnet Dy₃Al₅O₁₂. The advantages of the new method are especially obvious in the case of structures with a great number of magnetic atoms in a primitive cell.     

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)].     

Magnetic and ferroelectric ordering in BiMn<Subscript>2</Subscript>O<Subscript>5</Subscript> oxide

A group-theoretical analysis of the magnetic phase of BiMn₂O₅ oxide is performed using the space symmetry group of the compound. Using the projection operator method, we determine the basis functions of the irreducible representation of the space group, which are expressed in terms of the magnetic vector components. This representation can govern two phase transitions from the paramagnetic state to the antiferromagnetic phase with close temperatures and ordering of the spins of manganese ions in two crystallographic positions. It is found from renorm group analysis of these transitions that when these transitions occur as second- order transitions, the electric polarization does not appear in the system because spin fluctuations in this case elevate the symmetry of the system. Polarization appears when at least one of these transitions becomes a first-order transition as a result of spin fluctuations.     

IBM: Discrete symmetry viewpoint

It is shown that the set of transformations of the s and d boson operators that maintain the IBM-like form of the Hamiltonian comprises a discrete point symmetry group D ₂′. The transformations manifest themselves as a parameter symmetry of the IBM-1 Hamiltonian. The transformations considered are also necessary for constructing the most general IBM-2 Hamiltonian. The properties of the potential energy surfaces arising in connection with these transformations are discussed.     

Dynamical supersymmetry of the magnetic monopole and the 1/r 2-potential

We examine the recently discovered dynamical OSp(1, 1) supersymmetry of the Pauli Hamiltonian for a spin 1/2 particle with gyromagnetic ratio 2, in the presence of a Dirac magnetic monopole. Using this symmetry and algebraic methods only, we construct the spectrum and obtain the wave functions. At all but the lowest angular momenta, the states transform under a single irreducible representation of OSp(1, 1). On the lowest angular momentum states, it is impossible to define self-adjoint supercharges, and the states transform under an irreducible representation of SO(2, 1) only. The Hamiltonian is not self-adjoint in thes-wave sector, but admits a one parameter family of self-adjoint extensions. The full SO(2, 1) algebra can be realized only for two specific values of the parameter.The Pauli Hamiltonian is generalized to accommodate a ²/r ² potential. The new Hamiltonian exhibits a dynamical OSp(2, 1) supersymmetry. The spectrum and the wave functions are obtained. The states at all but the lowest angular momenta transform under the sum of two irreducible representations of OSp(2, 1). These two representations are distinguished by the chirality of their ground state. On the lowest angular momentum states, the OSp(2, 1) group is still realized, since the supercharges can all be rendered self-adjoint simultaneously, but the states only transform according to a single irreducible representation of OSp(2, 1). The chirality of the ground state for this representation is related to the signs of andeg. The Hamiltonian is not self-adjoint in thes-wave sector when ||<3/2. Only one of its self-adjoint extensions supports the OSp(2, 1) supersymmetry, and yields the wave functions obtained from the group theoretic approach. The supersymmetry is always spontaneously broken as there exists no normalizable zero energy states.The massless Dirac Hamiltonian in the presence of a magnetic monopole and a/r potential is related to a generator of an OSp(2, 1) superalgebra which also contains the Pauli Hamiltonian. This symmetry is used to generate the complete spectrum of the Dirac Hamiltonian.This work is supported in part through funds provided by the U.S. Department of Energy (DOE) under contract DE-AC02-76ER03069, and by the Natural Science and Engineering Research Council (NSERC) of Canada     

Scalar-relativistic spline augmented plane-wave method using a Douglas Kroll transformation in coordinate representation

A scalar-relativistic Hamiltonian, which contains all relativistic corrections up to second order in the fine structure constant, is derived with coordinate representation of the first order Douglas Kroll transformation. In addition to the correction of second order in the fine structure constant, this Hamiltonian contains the exact relativistic kinetic energy as well as the exact relativistic potential correction up to terms linear in the external potential. Based on this Hamiltonian, we develop a scalar-relativistic extension of the Spline Augmented Plane-Wave method, and show that the matrix elements with the new operator can be evaluated elegantly when using an alternative basis of Spline functions. As a first test we investigate solid silver and gold. By comparing the energies of the core states with the solutions of the radial Dirac equation we find that the stabilization of the s levels are slightly overestimated. Even smaller deviations from the Dirac energies are found for higher angular momentum. By comparing the valence band structure with the results for other scalar-relativistic operators, which can be used in a variational context, we find the new operator superior in all aspects: s-type bands are reproduced quite well, and again bands which are dominated by higher angular momenta behave even better. On the contrary, the results obtained with simpler scalar-relativistic Hamiltonians are unsatisfactory. Received 21 November 1996     

The Generalized <Emphasis Type="Italic">PT</Emphasis>-Symmetric Sinh-Gordon Potential Solvable within Quantum Hamilton–Jacobi Formalism

The generalized Sinh-Gordon potential is solved within quantum Hamiltonian Jacobi approach in the framework of PT symmetry. The quasi exact solutions of energy eigenvalues and eigenfunctions of the generalized Sinh-Gordon potential are found for n=0,1 states.     

Electronic correlation effects on the properties of an half-filled octahedron cluster in a magnetic field

The present study focuses on electronic correlation effects on magnetic energy, the spin-spin correlation function of an octahedron cluster in the (3↑, 3 ↓) electronic configuration threaded by a magnetic field. Some other spin configurations are also discussed and various field directions are considered. An accurate diagonalisation technique has been used to solve the Hubbard Hamiltonian. A result is analysed on a linear energy stabilisation at low magnetic flux. Moreover, two types of antiferromagnetic transition versus the flux occurring for a correlation term larger than a critical one have been observed, i.e. the likelihood of a charge excitation before the antiferromagnetic transition. Finally, a comparison between the results obtained from the exact diagonalisation and the Gutzwiller method has been carried out, leading to a suggested modification of the Gutzwiller approach in order to improve it. Received 23 June 1999 and Received in final form 28 July 2000     

A study of the internal dynamics of inverting trimethylamine by means of the non-rigid group theory

The non-rigid molecule group (NRG), defined as the complete set of the symmetry physical operations which commute with some Hamiltonian operator, is applied to determine the character table for the three-fold methyl rotation and pyramidal inversion in trimethylamine. The restricted NRG of this molecule is seen to be a group of order 324, isomorphic to the G₃₂₄ of planar trimethylborane. For this purpose, the group structure of the r-NRG of inverting trimethylamine is first deduced, i.e. the character table, the number of classes and irreducible representations, as well as their dimensions. The symmetry eigenvectors for each representation were deduced by developing them on the basis of quadruple products of trigonometric functions. The dynamical properties of trimethylamine are analysed by inspecting the symmetries of the potential energy function. The r-NRG molecule group theory is seen to be a powerful tool for studying the internal dynamics of such highly symmetric molecules.     

Photoemission from Au single-crystals. Location of transitions in k space along lines of high symmetry

Using angle-resolved photoemission we have measured single transitions normal to the surface of Au(111) and identified the same transitions in a family of energy distribution curves obtained from Au(112) and Au(110) at different polar angles. The conservation of the electron momentum parallel to the surfaces makes it possible to locate the transitions in k space along lines of high symmetry without additional assumptions on the final state energy bands.     

Influence of the degenerate d level and of the Jahn-Teller effect on the manganite electronic structure calculated in the tight-binding approximation

E(k) dispersion curves for the charge carriers in the LaMnO₃-like perovskites were calculated for the basic types of canted antiferromagnetic ordering of the Mn sublattice in the framework of the tight-binding approximation. The E(k) spectrum of the antiferromagnetic structures was calculated for the first time taking into account the degeneracy of the Mn e _g level and the Jahn-Teller distortion of the cubic perovskite structure. This calculation involved diagonalization of the 8×8 Hamiltonian matrix. Analytical expressions for the E(k) function at separate points and symmetry lines of the Brillouin zone were derived. The calculations showed that the properties of the La_1?x Ca_xMnO₃ system do not have electron-hole symmetry.     

Shape Transition and Triaxial Interaction Effect in the Structure of 152-166Dy Isotopes

The positive parity states in even-even ^152-166Dy are studied systematically in the framework of the interacting boson model (IBM). A cubic term, L=3, has been added to the Hamiltonian in order to produce the effect of triaxiality on the energy spectrum. The potential energy surfaces as a function of β and γ deformation parameters, for all isotopes have been produced. Energy levels and reduced electric quadrupole transition probabilities are calculated in framework of IBM with Cubic term (IBMC). All results are compared with available experimental data. It is found that these isotopes can be described by a schematic Hamiltonian in transition from U(5) (vibration) to SU(3) (rotation) dynamic symmetry.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.