Diabolical Points Research Articles

Photonic Berry curvature in double liquid crystal microcavities with broken inversion symmetry

We investigate a photonic device consisting of two coupled optical cavities possessing Rashba-Dresselhaus spin-orbit coupling, TE-TM splitting, and linear polarization splitting that opens a tunable energy gap at the diabolic points of the photon dispersion; giving rise to an actively addressable local Berry curvature. The proposed architecture stems from recent advancements in the design of artificial photonic gauge fields in liquid crystal cavities [K. Rechci\ifmmode \acute{n}\else \'{n}\fi{}ska et al., Science 366, 727 (2019)]. Our study opens perspectives for topological photonics, room-temperature spinoptronics, and studies on the quantum geometrical structure of photonic bands in extreme settings.

Berry phase in quantum field theory: Diabolical points and boundary phenomena

We study aspects of Berry phase in gapped many-body quantum systems by means of effective field theory. Once the parameters are promoted to spacetime-dependent background fields, such adiabatic phases are described by Wess-Zumino-Witten (WZW) and similar terms. In the presence of symmetries, there are also quantized invariants capturing generalized Thouless pumps. Consideration of these terms provides constraints on the phase diagram of many-body systems, implying the existence of gapless points in the phase diagram which are stable for topological reasons. We describe such diabolical points, realized by free fermions and gauge theories in various dimensions, which act as sources of "higher Berry curvature" and are protected by the quantization of the corresponding WZW terms or Thouless pump terms. These are analogous to Weyl nodes in a semimetal band structure. We argue that in the presence of a boundary, there are boundary diabolical points---parameter values where the boundary gap closes---which occupy arcs ending at the bulk diabolical points. Thus the boundary has an "anomaly in the space of couplings" in the sense of C\'ordova et al. Consideration of the topological effective action for the parameters also provides some new checks on conjectured infrared dualities and deconfined quantum criticality in 2+1d.

Impact of Diabolical Singular Points on Nonadiabatic Dynamics and a Remedy: Photodissociation of Ammonia in the First Band.

Several recent publications have pointed out a potentially severe drawback in some widely used diabatization methods based on the electronic properties of molecules. In a diabatic representation defined by a property-based method, artificial singularities may arise due to the defining equation of the adiabatic-to-diabatic (AtD) transformation. Such diabolical singular points (DSPs) may seriously affect nuclear dynamics if they lie in the relevant configuration space. Their impact is demonstrated here using the A-band photodissociation of ammonia as an example. To this end, quantum dynamics calculations are performed based on a diabatic potential energy matrix (DPEM) constructed using the generalized Mulliken-Hush method, which is based on dipoles. These property-based results are compared with the results obtained with a DPEM determined using derivative coupling explicitly. A DSP seam is found near the Franck-Condon region, which results in a complete failure to reproduce the absorption spectrum. A modification of the generalized Mulliken-Hush method is proposed to remove the DSPs while preserving the conical intersection, which leads to an accurate reproduction of the absorption spectrum and the NH2(Ã)/NH2(X̃) product branching ratio.

Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point

High-sensitivity gyroscope is widely used for rotation detection in several practical applications. Recently, exceptional points (EPs) have garnered considerable attention for enhancing the sensitivity of sensors based on optical cavities. Here, we propose an EP-enhanced optical gyroscope based on mechanical parity-time (PT) symmetry in a microcavity system. We demonstrate that by pumping the two optical modes with different colors, i.e., blue and red detuning, an effective mechanical PT-symmetric system can be obtained, and the system can be prepared at EP with appropriate parameters. The sensitivity of gyroscope at EP was enhanced by more than one order of magnitude in the weak perturbation regime as compared to that at diabolic point. This indicates that the sensitivity of gyroscope can be effectively enhanced by monitoring the mechanical modes rather than the optical modes. Overall, our work provides a promising approach to design high-sensitivity gyroscopes in optical microcavities and is potentially useful in a variety of research fields including fundamental physics and precision measurement.

Non-Hermitian Dirac Cones.

Non-Hermitian systems containing gain or loss commonly host exceptional point degeneracies, not the diabolic points that, in Hermitian systems, play a key role in topological transitions and related phenomena. Non-Hermitian Hamiltonians with parity-time symmetry can have real spectra but generally nonorthogonal eigenstates, impeding the emergence of diabolic points. We introduce a pair of symmetries that induce not only real eigenvalues but also pairwise eigenstate orthogonality. This allows non-Hermitian systems to host Dirac points and other diabolic points. We construct non-Hermitian models exhibiting three exemplary phenomena previously limited to the Hermitian regime: Haldane-type topological phase transition, Landau levels without magnetic fields, and Weyl points. This establishes a new connection between non-Hermitian physics and the rich phenomenology of diabolic points.

Magnetic degeneracy points in interacting two-spin systems: Geometrical patterns, topological charge distributions, and their stability

Spectral degeneracies of quantum magnets are often described as diabolical points or magnetic Weyl points, which carry topological charge. Here, we study a simple, yet experimentally relevant quantum magnet: two localized interacting electrons subject to spin-orbit coupling. In this setting, the degeneracies are not necessarily isolated points, but can also form a line or a surface. We identify ten different possible geometrical patterns formed by these degeneracy points, and study their stability under small perturbations of the Hamiltonian. Stable structures are found to depend on the relative sign of the determinants of the two $g$-tensors, $\cal S$. Both for ${\cal S}=+1$ and ${\cal S}=-1$, two stable configurations are found, and three out of these four configurations are formed by pairs of Weyl points. These stable regions are separated by a surface of almost stable configurations, with a structure akin to co-dimension one bifurcations.

Multiple exceptional points in bulk parity-time symmetric optical media

We study bulk optical modes in media with a parity-time symmetric permittivity tensor via the k-surface approach. We show that k-surfaces support multiple exceptional points (EPs), and the off-axial EPs can be manipulated by tuning the tensor elements. They merge into diabolic points if they have opposite handednesses, and annihilate each other otherwise. The underlying physical mechanisms and the potential applications are discussed.

Diabolical points in coupled active cavities with quantum emitters

In single microdisks, embedded active emitters intrinsically affect the cavity modes of the microdisks, resulting in trivial symmetric backscattering and low controllability. Here we demonstrate macroscopic control of the backscattering direction by optimizing the cavity size. The signature of the positive and negative backscattering directions in each single microdisk is confirmed with two strongly coupled microdisks. Furthermore, diabolical points are achieved at the resonance of the two microdisks, which agrees well with theoretical calculations considering the backscattering directions. Diabolical points in active optical structures pave the way for an implementation of quantum information processing with geometric phase in quantum photonic networks.

On the Impact of Singularities in the Two-State Adiabatic to Diabatic State Transformation: A Global Treatment.

Diabatizations based on molecular properties can remove the singularity in the derivative coupling at a conical intersection in the diabatic representation. Yet, they can also create new singularities in the derivative couplings because of the defining equations of the adiabatic to diabatic state transformation. In the iconic two-state case, these singularities occur at points termed diabolical singular points and form a seam of dimension Nint-2, where Nint is the number of internal degrees of freedom. This seam is of the same dimension as the conical intersection seam, but is distinct. Here, the global topography of the diabolical singularity seam of 1,21A states of ammonia is reported using a Boys localization (BL) dipole-based diabatization and juxtaposed with a previously reported global representation of the coupled electronic state potential energy surfaces. The principal finding is that the seam of BL diabatization-induced singularities passes very close to the key saddle point on the 21A potential energy surface which connects the 21A equilibrium structure with the NH2(X̃,Ã) + H channel. The construction and detailed study of the reported diabolical singularity seam is made possible by a recently constructed analytic representation of the dipole and transition dipole moment surfaces.

Borrmann effect in Laue diffraction in one-dimensional photonic crystals under a topological phase transition

We present experimental and theoretical studies of the anomalous high transmission of light (the Borrmann effect) under the Laue diffraction in a one-dimensional photonic crystal (PhC) characterized by spatial modulation of both refractive index and absorption. We show that a strong modulation of the refractive index along with the large PhC period provide new features of the Borrmann effect as compared to the well-known x-ray Borrmann effect in crystals appearing in PhC wavelength-angular transmission spectra. Namely, the maximal transmission is attained at the Bragg angles of incidence and corresponds alternatively to even or odd orders of the Bragg angles depending on the light wavelength. Second, a dramatic decrease of the angular width of the high transmission areas in the spectrum appear near the diabolic points. According to our description, this effect can be treated as a result of the topological phase transition accompanied by exchange of the parity of spatial distribution of the electromagnetic field of the two eigenmodes experiencing degeneracy. We demonstrate that these peculiarities are inherent to the PhC with the optical losses located in layers with higher refractive index, and disappear if the losses are specific for the PhC layers with lower refractive index. The suggested underlying mechanism involves the contribution of the waveguide PhC modes to the PhC transmission spectra.

Capture and chaotic scattering of a charged particle by a magnetic monopole under a uniform electric field

Motivated by the realization of magnetic monopole of Berry curvature by the energy crossing point, we theoretically study the effect of magnetic monopole under a uniform electric field in the semiclassical dynamics, which is relevant to many physical situations such as relaxation through the diabolic point. We found that the competition between the backward scattering by the monopole magnetic field and the acceleration by the electric field leads to the bound state, i.e., capture of a particle near the monopole. Furthermore, the nonlinearity induced by the magnetic monopole leads to the chaotic behavior in the transient dynamics, i.e., the transient chaos. We computed characteristic quantities of the strange saddle which gives rise to the transient chaos, and verified that the abrupt bifurcation occurs as we tune the system parameter toward the parameter region in which the system is solvable.

Determining whether diabolical singularities limit the accuracy of molecular property based diabatic representations: The 1,21A states of methylamine.

An efficient, easily implemented method for locating singularities attributable to the failure of the defining equations in a molecular property based diabatization, termed diabolical singular points, is reported. For two state diabatizations, the singular points form a seam of dimension N int - 2, where N int is the number of internal degrees of freedom. The dynamical outcomes of nuclear trajectories that reach the region of this seam are flawed. The algorithm easily identifies these otherwise hard to anticipate regions of fallaciously large derivative coupling. The fact that the algorithm is easily incorporated into a two state diabatization code based on molecular properties makes it a practical tool for determining whether the existence of diabolical singularities is relevant to the problem being considered. The algorithm is illustrated using a multireference single and double excitation configuration interaction description of the 1,21A states of CH3NH2.

Exact Mapping from Many-Spin Hamiltonians to Giant-Spin Hamiltonians.

Thermodynamic and spectroscopic data of exchange-coupled molecular spin clusters (e.g. single-molecule magnets) are routinely interpreted in terms of two different models: the many-spin Hamiltonian (MSH) explicitly considers couplings between individual spin centers, while the giant-spin Hamiltonian (GSH) treats the system as a single collective spin. When isotropic exchange coupling is weak, the physical compatibility between both spin Hamiltonian models becomes a serious concern, due to mixing of spin multiplets by local zero-field splitting (ZFS) interactions ('S-mixing'). Until now, this effect, which makes the mapping MSH→GSH ('spin projection') non-trivial, had only been treated perturbationally (up to third order), with obvious limitations. Here, based on exact diagonalization of the MSH, canonical effective Hamiltonian theory is applied to construct a GSH that exactly matches the energies of the relevant (2S+1) states comprising an effective spin multiplet. For comparison, a recently developed strategy for the unique derivation of effective ('pseudospin') Hamiltonians, now routinely employed in ab initio calculations of mononuclear systems, is adapted to the problem of spin projection. Expansion of the zero-field Hamiltonian and the magnetic moment in terms of irreducible tensor operators (or Stevens operators) yields terms of all ranks k (up to k=2S) in the effective spin. Calculations employing published MSH parameters illustrate exact spin projection for the well-investigated [Ni(hmp)(dmb)Cl]4 ('Ni4 ') single-molecule magnet, which displays weak isotropic exchange (dmb=3,3-dimethyl-1-butanol, hmp- is the anion of 2-hydroxymethylpyridine). The performance of the resulting GSH in finite field is assessed in terms of EPR resonances and diabolical points. The large tunnel splitting in the M=± 4 ground doublet of the S=4 multiplet, responsible for fast tunneling in Ni4 , is attributed to a Stevens operator with eightfold rotational symmetry, marking the first quantification of a k=8 term in a spin cluster. The unique and exact mapping MSH→GSH should be of general importance for weakly-coupled systems; it represents a mandatory ultimate step for comparing theoretical predictions (e.g. from quantum-chemical calculations) to ZFS, hyperfine or g-tensors from spectral fittings.

A maximally particle-hole asymmetric spectrum emanating from a semi-Dirac point

Tight binding models have proven an effective means of revealing Dirac (massless) dispersion, flat bands (infinite mass), and intermediate cases such as the semi-Dirac (sD) dispersion. This approach is extended to a three band model that yields, with chosen parameters in a two-band limit, a closed line with maximally asymmetric particle-hole dispersion: infinite mass holes, zero mass particles. The model retains the sD points for a general set of parameters. Adjacent to this limiting case, hole Fermi surfaces are tiny and needle-like. A pair of large electron Fermi surfaces at low doping merge and collapse at half filling to a flat (zero energy) closed contour with infinite mass along the contour and enclosing no carriers on either side, while the hole Fermi surface has shrunk to a point at zero energy, also containing no carriers. The tight binding model is used to study several characteristics of the dispersion and density of states. The model inspired generalization of sD dispersion to a general ± form, for which analysis reveals that both n and m must be odd to provide a diabolical point with topological character. Evolution of the Hofstadter spectrum of this three band system with interband coupling strength is presented and discussed.

Topological phase transition and interface states in hybrid plasmonic-photonic systems

The geometric phase and topological property for one-dimensional hybrid plasmonic-photonic crystals consisting of a simple lattice of graphene sheets are investigated systematically. For transverse magnetic waves, both plasmonic and photonic modes exist in the momentum space. The accidental degeneracy point of these two kinds of modes is identified to be a diabolic point accompanied with a topological phase transition. For a closed loop around this degeneracy point, the Berry phase is π as a consequence of the discontinuous jump of the geometric Zak phase. The wave impedance is calculated analytically for the semi-infinite system, and the corresponding topological interface states either start from or terminate at the degeneracy point. This type of localized interface state may find potential applications in manipulation of photon emission of quantum dots, optical sensing and enhancement of nonlinear effects, etc.

Visualising Berry phase and diabolical points in a quantum exciton-polariton billiard.

Diabolical points (spectral degeneracies) can naturally occur in spectra of two-dimensional quantum systems and classical wave resonators due to simple symmetries. Geometric Berry phase is associated with these spectral degeneracies. Here, we demonstrate a diabolical point and the corresponding Berry phase in the spectrum of hybrid light-matter quasiparticles—exciton-polaritons in semiconductor microcavities. It is well known that sufficiently strong optical pumping can drive exciton-polaritons to quantum degeneracy, whereby they form a macroscopically populated quantum coherent state similar to a Bose-Einstein condensate. By pumping a microcavity with a spatially structured light beam, we create a two-dimensional quantum billiard for the exciton-polariton condensate and demonstrate a diabolical point in the spectrum of the billiard eigenstates. The fully reconfigurable geometry of the potential walls controlled by the optical pump enables a striking experimental visualization of the Berry phase associated with the diabolical point. The Berry phase is observed and measured by direct imaging of the macroscopic exciton-polariton probability densities.

Construction of Giant-Spin Hamiltonians from Many-Spin Hamiltonians by Third-Order Perturbation Theory and Application to an Fe3 Cr Single-Molecule Magnet.

A general giant-spin Hamiltonian (GSH) describing an effective spin multiplet of an exchange-coupled metal cluster with dominant Heisenberg interactions was derived from a many-spin Hamiltonian (MSH) by treating anisotropic interactions at the third order of perturbation theory. Going beyond the existing second-order perturbation treatment allows irreducible tensor operators of rank six (or corresponding Stevens operator equivalents) in the GSH to be obtained. Such terms were found to be of crucial importance for the fitting of high-field EPR spectra of a number of single-molecule magnets (SMMs). Also, recent magnetization measurements on trigonal and tetragonal SMMs have found the inclusion of such high-rank axial and transverse terms to be necessary to account for experimental data in terms of giant-spin models. While mixing of spin multiplets by local zero-field splitting interactions was identified as the major origin of these contributions to the GSH, a direct and efficient microscopic explanation had been lacking. The third-order approach developed in this work is used to illustrate the mapping of an MSH onto a GSH for an S=6 trigonal Fe3 Cr complex that was recently investigated by high-field EPR spectroscopy. Comparisons between MSH and GSH consider the simulation of EPR data with both Hamiltonians, as well as locations of diabolical points (conical intersections) in magnetic-field space. The results question the ability of present high-field EPR techniques to determine high-rank zero-field splitting terms uniquely, and lead to a revision of the experimental GSH parameters of the Fe3 Cr SMM. Indeed, a bidirectional mapping between MSH and GSH effectively constrains the number of free parameters in the GSH. This notion may in the future facilitate spectral fitting for highly symmetric SMMs.

Chiral surface waves supported by biaxial hyperbolic metamaterials

AbstractWe discover a new kind of surface wave on biaxial hyperbolic metamaterial, which, in the k-space, connects the two diabolical points (or conical singularities) of the equifrequency surface. Interestingly, the propagation of such surface wave is found to be sensitive to the refractive index of the surrounding dielectric medium, showing a convex, concave or flat phase front when the refractive index is varied. Furthermore, the surface wave shows an elliptically polarized state, in which helicity is dependent on the propagation direction. This feature can be utilized for the spin-controllable excitation of surface waves, opening a gateway towards integrated photonic circuits with reconfigurable functionalities.

Osculations of spectral lines in a layered medium

When the dense web of spectral curves for the normal modes of wave propagation in a layered medium are examined carefully, it is often observed that pairs of curves approach each other to the point that they appear to intersect, yet a closer examination reveals that they do not. In the technical literature these are referred to as avoided crossings, curve veering, mode-splitting, double cusps, diabolical points, or osculation points. We consider herein their presence–or absence–in a homogeneous elastic layer underlain by an elastic half-space, and include the two extreme cases of a homogeneous stratum where the half-space is infinitely rigid, and a free plate when it is infinitely flexible. First we show rigorously that the spectral curves for generalized Love waves (SH waves) in a layered medium will not ever intersect, even if in some cases they may come close and osculate. We then demonstrate that the spectral lines for generalized Rayleigh waves (SV-P waves) in a homogeneous stratum will only intersect when Poisson’s ratio is either exactly 14, or is in some other fractional (but not continuous) ratio below 13, in which case we provide the intersection points in closed form. We also provide the general conditions that must be satisfied by the SVP spectral lines for double roots to exist in a layered medium, whether they are plates, strata, half-spaces of full spaces. We then posit that intersections of SVP spectral lines (i.e. double roots) can only occur when the layered medium exhibits material symmetry with respect to some horizontal plane. Although at first sight the stratum case might seem to contradict this principle we show that ultimately it does not. Thus, neither a non-symmetric free plate nor a half-space will ever have intersecting lines. By contrast, a symmetric full space does have such intersections, as do also symmetrically layered plates, among which the homogeneous free plate and the homogeneous doubly-clamped plate exhibit an abundance of double roots.

Diabolical points in multi-scatterer optomechanical systems.

Diabolical points, which originate from parameter-dependent accidental degeneracies of a system's energy levels, have played a fundamental role in the discovery of the Berry phase as well as in photonics (conical refraction), in chemical dynamics, and more recently in novel materials such as graphene, whose electronic band structure possess Dirac points. Here we discuss diabolical points in an optomechanical system formed by multiple scatterers in an optical cavity with periodic boundary conditions. Such configuration is close to experimental setups using micro-toroidal rings with indentations or near-field scatterers. We find that the optomechanical coupling is no longer an analytic function near the diabolical point and demonstrate the topological phase arising through the mechanical motion. Similar to a Fabry-Perot resonator, the optomechanical coupling can grow with the number of scatterers. We also introduce a minimal quantum model of a diabolical point, which establishes a connection to the motion of an arbitrary-spin particle in a 2D parabolic quantum dot with spin-orbit coupling.