Origin of Perovskite Multiferroicity and Magnetoelectric-Multiferroic Effects—The Role of Electronic Spin in Spontaneous Polarization of Crystals

In this semi-review paper, we show that the multiferroic properties of perovskite ABO3 crystals with B(dn), n > 0, centers are fully controlled by the influence of the electronic spin on the local dipolar instability that triggers the spontaneous polarization of the crystal. Contrary to the widespread statements, the multiferroicity of these crystals does not emerge due to the addition of unpaired electrons (carrying magnetic moments) to the spontaneously polarizing crystal; the spin states themselves are an important part of the local electronic structure that determines the very possibility of the spontaneous polarization. This conclusion emerges from vibronic theory, in which the ferroelectricity is due to the cooperative interaction of the local dipolar distortions induced by the pseudo-Jahn-Teller effect (PJTE). The latter requires sufficiently strong vibronic coupling between ground and excited electronic states with opposite parity but the same spin multiplicity. The detailed electronic structure of the octahedral [B(dn)O6] center in the molecular orbital presentation shows how this requirement plays into the dependence of the possible perovskite magnetic, ferroelectric, and multiferroic properties on the number of d electrons, provided the criterion of the PJTE is obeyed. Revealed in detail, the role of the electronic spin in all these properties and their combination opens novel possibilities for their manipulation by means of external perturbations and exploration. In particular, it is shown that by employing the well-known spin-crossover phenomenon, a series of novel effects become possible, including magnetic-ferroelectric (multiferroic) crossover with electric-multiferroic, magnetic-ferroelectric, and magneto-electric effects, some of which have already been observed experimentally.

New Semi-Prime Factorization and Application in Large RSA Key Attacks

Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.

Intrinsic and Extrinsic Exciton Recombination Pathways in AgInS2 Colloidal Nanocrystals

Ternary I-III-VI2 nanocrystals (NCs), such as AgInS2 and CuInS2, are garnering interest as heavy-metal-free materials for photovoltaics, luminescent solar concentrators, LEDs, and bioimaging. The origin of the emission and absorption properties in this class of NCs is still a subject of debate. Recent theoretical and experimental studies revealed that the characteristic Stokes-shifted and long-lived luminescence of stoichiometric CuInS2 NCs arises from the detailed structure of the valence band featuring two sublevels with different parity. The same valence band substructure is predicted to occur in AgInS2 NCs, yet no experimental confirmation is available to date. Here, we use complementary spectroscopic, spectro-electrochemical, and magneto-optical investigations as a function of temperature to investigate the band structure and the excitonic recombination mechanisms in stoichiometric AgInS2 NCs. Transient transmission measurements reveal the signatures of two subbands with opposite parity, and photoluminescence studies at cryogenic temperatures evidence a dark state emission due to enhanced exchange interaction, consistent with the behavior of stoichiometric CuInS2 NCs. Lowering the temperature as well as applying reducing electrochemical potentials further suppress electron trapping, which represents the main nonradiative channel for exciton decay, leading to nearly 100% emission efficiency.

Photonic bandgap engineering using second-order supersymmetry

First-order supersymmetry (SUSY) adapted from quantum physics to optics manipulates the transverse refractive index of guided-wave structures using a nodeless ground state to obtain intended modal content. Second-order SUSY can be implemented using excited states as a seed function, even with the presence of nodes. We apply second-order SUSY to the coupled-mode equations by recasting them as the Dirac equation. This enables the engineering of non-uniform surface corrugation of waveguide gratings and coupling potential, which encapsulates the Bragg interaction between counterpropagating modes. We show that the added bound states appear as transmission resonances inside the bandgap of the finite grating. The probability density of each state provides the longitudinal modal energy distribution in the waveguide grating. The smooth modal energy distribution of the states obtained by SUSY can mitigate longitudinal spatial hole burning in high power laser operation. We demonstrate that degenerate second-order SUSY allows the insertion of two states, which can coalesce into Friedrich-Wintgen type bound states in the continuum (BIC) for one-dimensional grating. We show that the eigenfunctions of BIC states are doubly degenerate with opposite parity, and the corresponding transmission resonances have phase changes of 2π across these states. One-dimensional BIC states can find application as robust high-speed all-optical temporal integrators by lifting restrictions on the length of various sections in the phase-shifted grating.

Perovskite Crystals: Unique Pseudo-Jahn–Teller Origin of Ferroelectricity, Multiferroicity, Permittivity, Flexoelectricity, and Polar Nanoregions

In a semi-review paper, we show that the local pseudo-Jahn–Teller effect (PJTE) in transition metal B ion center of ABO3 perovskite crystals, notably BaTiO3, is the basis of all their main properties. The vibronic coupling between the ground and excited electronic states of the local BO6 center results in dipolar distortions, leading to an eight-well adiabatic potential energy surface with local tunneling or over-the-barrier transitions between them. The intercenter interaction between these dipolar dynamic units results in the formation of the temperature-dependent three ferroelectric and one paraelectric phases with order–disorder phase transitions. The local PJTE dipolar distortion is subject to the presence of sufficiently close in energy local electronic states with opposite parity but the same spin multiplicity, thus limiting the electronic structure and spin of the B(dn) ions that can trigger ferroelectricity. This allowed us to formulate the necessary conditions for the transition metal perovskites to possess both ferroelectric and magnetic (multiferroic) properties simultaneously. It clarifies the role of spin in the spontaneous polarization. We also show that the interaction between the independently rotating dipoles in the paraelectric phase may lead to a self-assembly process resulting in polar nanoregions and relaxor properties. Exploring interactions of PJTE ferroelectrics with external perturbations, we revealed a completely novel property—orientational polarization in solids—a phenomenon first noticed by P. Debye in 1912 as a possibility, which was never found till now. The hindered rotation of the local dipole moments and their ordering along an external field is qualitatively similar to the behavior of polar molecules in liquids, thus adding a new dimension to the properties of solids—notably, the perovskite ferroelectrics. We estimated the contribution of the orientational polarization to the permittivity and flexoelectricity of perovskite crystals in different limiting conditions.

Bootstrapping conformal four-point correlators with slightly broken higher spin symmetry and 3D bosonization

Three-dimensional conformal field theories (CFTs) with slightly broken higher spin symmetry provide an interesting laboratory to study general properties of CFTs and their roles in the AdS/CFT correspondence. In this work we compute the planar four-point functions at arbitrary ’t Hooft coupling λ in the CFTs with slightly broken higher spin symmetry. We use a bootstrap approach based on the approximate higher spin Ward identity. We show that the bootstrap equation is separated into two parts with opposite parity charges, and it leads to a recursion relation for the λ expansions of the correlation functions. The λ expansions terminate at order λ2 and the solutions are exact in λ. Our work generalizes the approach proposed by Maldacena and Zhiboedov to four-point correlators, and it amounts to an on-shell study for the 3D Chern-Simons vector models and their holographic duals in AdS4. Besides, we show that the same results can also be obtained rather simply from bosonization duality of 3D Chern-Simons vector models. The odd term at order O(λ) in the spinning four-point function relates to the free boson correlator through a Legendre transformation. This provides new evidence on the 3D bosonization duality at the spinning four-point function level. We expect this work can be generalized to a complete classification of general four-point functions of single trace currents.

Quasi-Exactly Soluble Potentials and Deformed Oscillators

It is proved that quasi-exactly soluble potentials corresponding to an oscillator with harmonic, quartic and sextic terms, for which the n+1 lowest levels of a given parity can be determined exactly, may be approximated by WKB equivalent poten- tials corresponding to deformed anharmonic oscillators of SUq(1,1) symmetry, which have been used for the description of vibrational spectra of diatomic molecules. This connection allows for the immediate approximate determination of the levels of the same parity lying above the fewest n+1 known levels, as well as of all levels of the opposite parity. Such connections are not possible in the cases of the q-deformed oscillator, the Q-deformed oscillator, and the modified Pöschl-Teller potential with SU(1,1) symmetry.

Bridging the Physics and Chemistry of Graphene(s): From Hückel’s Aromaticity to Dirac’s Cones and Topological Insulators

By bridging graphene and benzene through a well-defined sequence of polycyclic aromatic hydrocarbons and their inherent shell structure, it is shown that graphene is actually a coherent arrangement of interwoven benzene molecules, coordinated by aromaticity, shell structure, and topology, all interrelated and microscopically realized through dynamical flipping of the atomic pz-orbitals, playing the role of pseudospin or “qubits”. This renders graphene resonance structure, “resonating” between two complementary aromaticity patterns, involving 2k, k→∞ kekulé type of resonances resulting in “robust electronic coherence”, with dual “molecular-crystalline” nature, and two valence-conduction bands of opposite parity, driven by inversion symmetry competition, which is essentially a “molecule-versus-crystal” competition, in accord with topological-insulator and many-body theory. The “average picture” converges to the usual band structure with two aromatic π-electrons per ring, and the fingerprints of inversion-competition at the D3h-symmetric Dirac points, which for rectangular nanographene(s) appear as gapless topological edge states without real spin-polarization, contrary to opposite claims. <br>

Bridging the Physics and Chemistry of Graphene(s): From Hückel’s Aromaticity to Dirac’s Cones and Topological Insulators

By bridging graphene and benzene through a well-defined sequence of polycyclic aromatic hydrocarbons and their inherent shell structure, it is shown that graphene is actually a coherent arrangement of interwoven benzene molecules, coordinated by aromaticity, shell structure, and topology, all interrelated and microscopically realized through dynamical flipping of the atomic pz-orbitals, playing the role of pseudospin or “qubits”. This renders graphene resonance structure, “resonating” between two complementary aromaticity patterns, involving 2k, k→∞ kekulé type of resonances resulting in “robust electronic coherence”, with dual “molecular-crystalline” nature, and two valence-conduction bands of opposite parity, driven by inversion symmetry competition, which is essentially a “molecule-versus-crystal” competition, in accord with topological-insulator and many-body theory. The “average picture” converges to the usual band structure with two aromatic π-electrons per ring, and the fingerprints of inversion-competition at the D3h-symmetric Dirac points, which for rectangular nanographene(s) appear as gapless topological edge states without real spin-polarization, contrary to opposite claims. <br>