Diabolical Points Research Articles

On Krein space related perturbation theory for MHD α2‐dynamos

AbstractThe spectrum of the spherically symmetric α2–dynamo is studied in the case of idealized boundary conditions. Starting from the exact analytical solutions of models with constant α –profiles a perturbation theory and a Galerkin technique are developed in a Krein‐space approach. With the help of these tools a very pronounced α –resonance pattern is found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points located at the nodes of this mesh. Non‐oscillatory as well as oscillatory dynamo regimes are obtained. An estimation technique is developed for obtaining the critical α –profiles at which the eigenvalues enter the right spectral half‐plane with non‐vanishing imaginary components (at which overcritical oscillatory dynamo regimes form). (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Conical diffraction complexified: dichroism and the transition to double refraction

With dichroism (anisotropic absorption), the dielectric tensor of a crystal becomes non-Hermitian, and each optic axis (diabolical point) splits into two singular axes (branch-point degeneracies). For a Gaussian beam incident on a dichroic crystal in a direction near the singular axes, the polarized light beyond the crystal is given by the same diffraction integrals as for a transparent crystal illuminated along the optic axis, but with the variables complexified. Unexpectedly, the effects of absorption and beam direction can be described with a single parameter. The theory predicts several new types of interference, visible in logarithmic intensity plots: between the nonorthogonally polarized geometrical rays, and between these rays and waves diffracted by the singular axes. The phenomena are analysed in terms of saddle-point, end-point and uniform asymptotics of the diffraction integrals.

A Krein space related perturbation theory for MHD α2-dynamos and resonant unfolding of diabolical points

The spectrum of the spherically symmetric alpha-2 dynamo is studied in the case of idealized boundary conditions. Starting from the exact analytical solutions of models with constant alpha-profiles a perturbation theory and a Galerkin technique are developed in a Krein-space approach. With the help of these tools a very pronounced alpha-resonance pattern is found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points located at the nodes of this mesh. Non-oscillatory as well as oscillatory dynamo regimes are obtained. A Fourier component based estimation technique is developed for obtaining the critical alpha-profiles at which the eigenvalues enter the right spectral half-plane with non-vanishing imaginary components (at which overcritical oscillatory dynamo regimes form). Finally, Frechet derivative (gradient) based methods are developed, suitable for further numerical investigations of Krein-space related setups like MHD alpha-2-dynamos or models of PT-symmetric quantum mechanics.

Berry Phase, Topology, and Degeneracies in Quantum Nanomagnets

A topological theory of the diabolical points (degeneracies) of quantum magnets is presented. Diabolical points are characterized by their diabolicity index, for which topological sum rules are derived. The paradox of the missing diabolical points for Fe8 molecular magnets is clarified. A new method is also developed to provide a simple interpretation, in terms of destructive interferences due to the Berry phase, of the complete set of diabolical points found in biaxial systems such as Fe8.

Singularities of energy surfaces under non-Hermitian perturbations

In this work, we study singularities of the energy surfaces formed by the eigenvalues of the real symmetric and Hermitian matrices depending on parameters under arbitrary complex perturbation. Using the theory of eigenvalue bifurcations, which was developed in [10], we derive general asymptotic formulas describing the deformation of the energy surface near the conic singularity for various complex perturbations. The deformation of the eigenvalue surfaces appears to be described by the eigenvalues, eigenvectors, and derivatives of the Hamiltonian with respect to the parameters at the diabolic point. As an application, the singularities of the refractive-index surfaces in crystal optics are studied. Explicit expressions are obtained for these surfaces as functions of the properties of a crystal. Singular axes are found for crystals with weak absorption and optical activity. In terms of the components of the inverse dielectric tensor, we obtain a new condition that distinguishes crystals with prevailing absorption and with prevailing optical activity. 1. We consider the eigenvalue problem

Avoided level crossings and singular points

Avoided crossings of discrete states as well as of resonance states are traced back to the existence of branch points (BPs) which are singular points in the complex energy plane. They cannot be identified with exceptional points. At a BP width bifurcation occurs, different Riemann sheets evolve and the levels do not cross anymore when the system is still further opened. The BPs are physically meaningful: they influence the dynamics of open as well as of closed quantum systems. The geometric phase that arises by encircling a BP, is different from the phase that appears by encircling a diabolic point. This is found to be true even for the two BPs into which the diabolic point is unfolded by opening the system.

Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation

The paper presents a new theory of unfolding of eigenvalue surfaces of real symmetric and Hermitian matrices due to an arbitrary complex perturbation near a diabolic point. General asymptotic formulae describing deformations of a conical surface for different kinds of perturbing matrices are derived. As a physical application, singularities of the surfaces of refractive indices in crystal optics are studied.

Avoided level crossings, diabolic points, and branch points in the complex plane in an open double quantum dot

We study the spectrum of an open double quantum dot as a function of different system parameters in order to receive information on the geometric phases of branch points in the complex plane (BPCP). We relate them to the geometrical phases of the diabolic points (DPs) of the corresponding closed system. The double dot consists of two single dots and a wire connecting them. The two dots and the wire are represented by only a single state each. The spectroscopic values follow from the eigenvalues and eigenfunctions of the Hamiltonian describing the double dot system. They are real when the system is closed, and complex when the system is opened by attaching leads to it. The discrete states as well as the narrow resonance states avoid crossing. The DPs are points within the avoided level crossing scenario of discrete states. At the BPCP, width bifurcation occurs. Here, different Riemann sheets evolve and the levels do not cross anymore. The BPCP are physically meaningful. The DPs are unfolded into two BPCP with different chirality when the system is opened. The geometric phase that arises by encircling the DP in the real plane, is different from the phase that appears by encircling the BPCP. This is found to be true even for a weakly opened system and the two BPCP into which the DP is unfolded.

Coupling of eigenvalues of complex matrices at diabolic and exceptional points

The paper presents a general theory of coupling of eigenvalues of complex matrices of arbitrary dimension depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two physical examples illustrate effectiveness and accuracy of the presented theory.

The MHD 2-Dynamo, Z2-Graded Pseudo-Hermiticity, Level Crossings and Exceptional Points of Branching Type

The spectral branching behavior of the 2×2 operator matrix of the magneto-hydrodynamic α2-dynamo is analyzed numerically. Some qualitative aspects of level crossings are briefly discussed with the help of a simple toy model which is based on a Z 2-graded-pseudo-Hermitian 2×2 matrix. The considered issues comprise: the underlying SU(1,1) symmetry and the Krein space structure of the system, exceptional points of branching type and diabolic points, as well as the algebraic and geometric multiplicity of corresponding degenerate eigenvalues.

Unfolding a diabolic point: a generalized crossing scenario

The typical avoided crossings for Hermitian quantum systems depending on parameters, the diabolic crossing scenario, are generalized to the non-Hermitian case, e.g. for resonances. Two types of crossings appear: for type I, the real parts show an avoided and the imaginary parts a true crossing of the eigenenergies, and for type II the opposite is found. A simple symmetric non-Hermitian two-state matrix Hamiltonian is analysed in detail. The diabolic point bifurcates into two exceptional ones on exceptional lines where the matrices are defective. The adiabatic transport of eigenvectors and eigenstates in parameter space is discussed in this generalized diabolic crossing scenario, in particular the geometric Berry phases for a cyclic variation of system parameters, depending on the topology of the closed curves with respect to the exceptional lines.

Branch points in the complex plane and geometric phases.

Laser-induced degenerate states (LIDS) are equivalent to double poles of the S matrix that are branch points in the complex plane (BPCP). These branch points cause geometric phase changes by encircling them adiabatically around a closed circuit by varying certain parameters. They cause also the well-known phase changes appearing by encircling a diabolic point (DP) being a singularity associated with level repulsion. In both cases, the wave functions are exchanged, Phi(i) --> +/- iPhi(j not equal i), at the critical value of the parameter where the states avoid crossing. Such a critical point is passed twice by encircling a DP but only once by surrounding a BPCP. As a consequence, the phase changes are different in both cases. A second surrounding restores the wave functions including their phases in both cases (when the BPCP is well isolated from others and the time of encircling is shorter than the lifetime of the two states). The different interference pictures appearing in surrounding LIDS adiabatically in opposite directions on a closed circuit represent a completion of the work by Berry.

Topological quenching of spin tunneling in magnetic molecules with a fourfold easy axis

Spin tunneling is investigated in magnetic molecules that have an easy axis with fourfold symmetry, such as ${\mathrm{Mn}}_{12}$-acetate, with emphasis on understanding the topological quenching of tunneling and the diabolical point locations in the magnetic field space. This is done using a model spin Hamiltonian that has a fourth-order term describing the tetragonal anisotropy. The problem is studied qualitatively using instantons and quantitatively using two methods: a discrete phase integral or Wentzel-Kramers-Brillouin method and perturbation theory in the fourth-order anisotropy and transverse magnetic field. The former method is used to find the splitting between various levels when the applied magnetic field is along the hard axis and is found to give good quantitative answers. The latter method is employed for fields which may have an easy component in addition to a hard one and is found to be effective in locating all the diabolical points. These points are found as the roots of a small number of polynomials in the hard component of the magnetic field and the basal plane anisotropy. These roots are used to obtain approximate formulas that apply to any system with total spin $Sl~10.$ The analytic results are found to compare reasonably well with exact numerical diagonalization for the case of ${\mathrm{Mn}}_{12}$-acetate. In addition, perturbation theory shows that the diabolical points may be indexed by the magnetic quantum numbers of the levels involved, even at large transverse fields. Certain points of degeneracy are found to be mergers (or near mergers) of two or three diabolical points because of the symmetry of the problem.

Quenched spin tunneling and diabolical points in magnetic molecules. II. Asymmetric configurations

The perfect quenching of spin tunneling first predicted for a model with biaxial symmetry, and recently observed in the magnetic molecule Fe_8, is further studied using the discrete phase integral (or Wentzel-Kramers-Brillouin) method. The analysis of the previous paper is extended to the case where the magnetic field has both hard and easy components, so that the Hamiltonian has no obvious symmetry. Herring's formula is now inapplicable, so the problem is solved by finding the wavefunction and using connection formulas at every turning point. A general formula for the energy surface in the vicinity of the diabolo is obtained in this way. This formula gives the tunneling apmplitude between two wells unrelated by symmetry in terms of a small number of action integrals, and appears to be generally valid, even for problems where the recursion contains more than five terms. Explicit results are obtained for the diabolical points in the model for Fe_8. These results exactly parallel the experimental observations. It is found that the leading semiclassical results for the diabolical points appear to be exact, and the points themselves lie on a perfect centered rectangular lattice in the magnetic field space. A variety of evidence in favor of this perfect lattice hypothesis is presented.

Quenched spin tunneling and diabolical points in magnetic molecules. I. Symmetric configurations

The perfect quenching of spin tunneling that has previously been discussed in terms of interfering instantons, and has recently been observed in the magnetic molecule \Fe8, is treated using a discrete phase integral (or Wentzel-Kramers-Brillouin) method. The simplest model Hamiltonian for the phenomenon leads to a Schr\"odinger equation that is a five-term recursion relation. This recursion relation is reflection-symmetric when the magnetic field applied to the molecule is along the hard magnetic axis. A completely general Herring formula for the tunnel splittings for all reflection-symmetric five-term recursion relations is obtained. Using connection formulas for a new type of turning point that may be described as lying "under the barrier", and which underlies the oscillations in the splitting as a function of magnetic field, this Herring formula is transformed into two other formulas that express the splittings in terms of a small number of action and action-like integrals. These latter formulas appear to be generally valid, even for problems where the recursion contains more than five terms. The results for the model Hamiltonian are compared with experiment, numerics, previous instanton based approaches, and the limiting case of no magnetic field.

Spin tunnelling in mesoscopic systems

We study spin tunnelling in molecular magnets as an instance of a mesoscopic phenomenon, with special emphasis on the molecule Fe8. We show that the tunnel splitting between various pairs of Zeeman levels in this molecule oscillates as a function of applied magnetic field, vanishing completely at special points in the space of magnetic fields, known as diabolical points. This phenomena is explained in terms of two approaches, one based on spin-coherent-state path integrals, and the other on a generalization of the phase integral (or WKB) method to difference equations. Explicit formulas for the diabolical points are obtained for a model Hamiltonian.

Diabolical points in magnetic molecules: An exactly solvable model

The magnetic molecule Fe_8 has been observed to have a rich pattern of degeneracies in its magnetic spectrum as the static magnetic field applied to the molecule is varied. The points of degeneracy, or diabolical points in the magnetic field space, are found exactly in the simplest model Hamiltonian for this molecule. The points are shown to form a perfect centered rectangular lattice, and are shown to be multiply diabolical in general. The multiplicity is found. An earlier semiclassical solution to this problem is thereby shown to be exact in leading order in 1/J where J is the spin.

Repulsion of resonance states and exceptional points

Level repulsion is associated with exceptional points which are square root singularities of the energies as functions of a (complex) interaction parameter. This is also valid for resonance state energies. Using this concept it is argued that level anticrossing (crossing) must imply crossing (anticrossing) of the corresponding widths of the resonance states. Further, it is shown that an encircling of an exceptional point induces a phase change of one wave function but not of the other. An experimental setup is discussed where this phase behavior, which differs from the one encountered at a diabolic point, can be observed.

Monodromy, diabolic points, and angular momentum coupling

Monodromy, or the most basic obstruction to global action-angle coordinates is shown to be present in the well known problem of two coupled angular momenta. It is also shown that in the corresponding quantum problem monodromy manifests itself as the redistribution of energy levels between different multiplets of the quantum spectrum.

Diabolical points in the resonance spectra of vibrating smectic films

Recent theoretical studies reveal the existence of so-called diabolical points in the energetic spectra of rectangular quantum billiards with a pointlike scatterer. The wave equation that rules the drum-head oscillations of free-standing smectic films is similar to the two-dimensional Schr\odinger equation, which makes vibrating smectic films the analogues of appropriate quantum billiards. In this Rapid Communication we study experimentally the diabolical points in a family of rectangular quantum billiards with an infinite scatterer via the resonance spectra of appropriate smectic films.