Diabolical Points Research Articles

Manifestation of bifurcations and diabolic points in molecular energy spectra

The general scheme of the qualitative analysis of finite-particle quantum problems is discussed. Theoretical methods for the qualitative analysis of energy spectra in quantum problems are applied to the study of the rovibrational energy levels of spherical top molecules. The existence of modifications in the rotational cluster structure and of redistributions of the energy levels between different branches under the rotational excitation is shown for the ν2 /ν4 dyad of the 12 CH4 and 28 SiH4 molecules. An interpretation of the qualitative features of energy spectra in terms of bifurcations and formations of conical intersection points (diabolic points) on the energy surfaces of the corresponding classical problem is proposed.

Berry's phase on a quantized path. An example from nuclear physics

It is shown that Berry's phase occurs not only along a continuous path in parameter space, but also in situations in which the parameters belong to a quantized space, where the closed path consists of a finite number of discrete points in this space, and where diabolical points do not exist. This necessitates that a new and more physical phase propagation rule be invoked. As an example we study diabolic pair transfer in rotating nuclei.

Critical phenomena and diabolic points in rovibrational energy spectra of spherical top molecules

The qualitative modifications of the rovibrational energy-level structure under rotational excitation are studied for spherical top molecules. The ν2, ν4 bands of 12CD4 and 120SnH4 and the ν1, ν3 bands of 120SnH4 are treated as examples. Two types of qualitative changes are shown to exist: the critical phenomena corresponding to the modification of the cluster structure and the so-called diabolic points associated with the redistribution of energy levels between different branches as the rotational quantum number increases. A simple model interpretation of energy-level redistribution is given to show that such a phenomenon is typical for tetrahedral molecules. Manifestation of critical phenomena and diabolic point formation in high-resolution infrared spectra are discussed.

Bandcrossings at high spin: Level interactions and diabolic points

Cranking calculations have been performed for many particles in a deformed single- j shell interacting via realistic two-body forces. Although the particle number is conserved in our problem, it can still be treated as a continuous variable. This allows us to consider the diabolic structure of our hamiltonian, which is important in understanding the two-particle transfer reaction. We also discuss the level interaction which determines whether a system will backbend or upbend. We find an analytic solution for the level interaction strength for small deformations and calculate numerical ones for large deformations. These permit a comparison with other models and a general discussion of the AB interaction strength.

The role of avoided crossings in the dynamics of strong laser field ? matter interactions

The structure of quasi-energy surfaces, i.e. of quasi-energies as functions of external parameters like the laser frequency, laser intensity, atomic distances etc., is discussed and avoided crossings are traced to the existence of diabolical points at the origin of doublecone structures. The notion of non-contractable closed loops produces non-trivial Berry-phases. We then consider the interaction of atomic or molecular systems with relatively short smooth pulses and find that avoided crossings and, hence, diabolical points essentially, dominate non-adiabatic effects; away from avoided crossings the system evolves adiabatically in Floquet states, phases picked up during this motion lead to a new interference phenomenon which we study numerically in an appropriately simplified model. A scenario for the measurement of Berry phases is proposed which is linked to this interference in an essential manner; its eventual feasibility stems from the fact that at least two external parameters can be freely tuned in laser-atom (molecule) systems.

Cosmiclike domain walls in superfluid 3He-B: Instantons and diabolical points in (k,r) space.

The possible planar superfluid B-B boundaries between inequivalent B-phase vacua are considered; such B-B interfaces provide an analogy with the cosmic domain walls that are believed to have precipitated in the phase transitions of the early Universe. Several of them display nontrivial structure in (k,r) space (i.e., the union of the momentum and real spaces). Such a wall represents an instanton connecting two B-phase vacua with different k-space topology. The transition between the vacua occurs through the formation of a pointlike defect either in the (k,r) space, or in the (k,t) space. These defects are so-called diabolical points of codimension 4, at which the fermionic energy tends to zero, thus providing the fermionic zero modes. Such points are new examples (within condensed-matter physics) of the peculiar diabolical points, which are characterized by the occurrence of a contact between the different branches of the quasiparticle spectra; in the present case, the branches of particles and holes, respectively. These points are here discussed for the case of the superfluid phases of liquid $^{3}\mathrm{He}$ in close analogy with the quantum field theory of fermions interacting with classical bosonic fields. The cosmiclike domain walls in superfluid $^{3}\mathit{B}$ are observable in principle; in particular, the motion of the superfluid A-B interface is governed at low temperatures by the periodical emission of these topological excitation planes. The edges of B-B interfaces serve to generate fractionally quantized pure and mixed mass and spin supercurrent vortices in $^{3}\mathit{B}$, while holes in these surfaces may give rise to the corresponding vortex rings and combined vortex and/or spin-disclination rings.

A nuclear squid: Diabolic pair transfer in rotating nuclei

A new unexpected behavior of pair transfer matrix elements in superfluid rotating nuclei is predicted. With increasing angular velocity they drop to zero, change their sign and in some cases even oscillate between positive and negative values. This effect is related to diabolical points in rotating quasiparticle spectra and is closely analogous to the DC-Josephson effect in superconductors in the presence of a magnetic field.

Diabolic pair transfer and oscillating behavior of backbending in rotating nuclei

Matrix elements for the transfer of a pair of nucleons coupled to angular momentum zero are investigated within the framework of the cranking model. An oscillating behavior of this matrix element as a function of the angular velocity ω is found, which can be understood in terms of diabolic points of the rotating shell model hamiltonian.

Energy-eigenvalue degeneracies in the quasiparticle spectra of rotating nuclei

New “exotic” diabolical points (i.e. degeneracies between the energy eigenvalues) are predicted to exist in the independent quasiparticle spectra of rotating nuclei. Distributions and properties of all the diabolical points are discussed.

Diabolical points in one-dimensional Hamiltonians quartic in the momentum

The authors study the family of quantal Hamiltonians defined on -1<x<+1 by H=p4=d4/dx4 which depend via the boundary conditions on four real parameters a, b, c, d. The spectra of states with even and odd parity have degeneracies with codimension three, that is, on lines in abcd space; near the degeneracies, the level splitting has a double-cone (diabolical) structure, and during a circuit of a diabolical point in the subspace of real H (i.e. d=0) the wavefunctions change sign. These are generic properties which in this model can be studied analytically. The degeneracies could be realised classically in vibrating beams with positive and negative linear feedback in the boundary conditions.

Diabolical points in the spectra of triangles

‘Accidental’ degeneracies between energy levelsEjandEj+1of a real Hamiltonian can occur generically in a family of Hamiltonians labelled by at least two parametersX,Y,... Energy-level surfaces inE,X,Yspace have (locally) a double-cone (diabolo) connection and we refer to the degeneracies themselves as ‘diabolical points’. We studied the family of systems in which a particle moves freely within hard-walled triangles (vibrations of triangular membranes), withXandYlabelling two of the angles. Using an efficient Green-function technique to compute the levels, we found several diabolical points for low-lying levels (as well as some symmetry degeneracies); the lowest diabolical point occurred for levels 5 and 6 of the triangle 130.57°, 30.73°, 18.70°. The conical structure was confirmed by noting that the normal derivativeuof the wavefunctionψat a boundary point changed sign during a small circuit of the diabolical point. The form of the variation ofuaround a circuit, and the changing pattern of nodal lines ofψ, agreed with theoretical expectations. An estimate of the total number of degeneraciesNd(j) involving levels 1 throughj, based on the energy-scaling of cone angles and the level spacing distribution, gaveNd(j) ~ (j+ ½)2.5, and our limited data support this prediction. Analytical theory confirmed that for thin triangles (where our computational method is slow) there are no degeneracies in the energy range studied.